Handbook of Exact Solutions for Ordinary Differential Equations

HANDBOOK OF

EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION

HANDBOOK OF

EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION

Andrei D. Polyanin Valentin F. Zaitsev

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

C2972_frame_DISC Page 1 Thursday, August 29, 2002 1:45 PM

Library of Congress Cataloging-in-Publication Data Polyanin, A. D. (Andrei Dmitrievich) Handbook of exact solutions for ordinary differential equations / Andrei D. Polyanin, Valentin F. Zaitsev.--2nd ed. p. cm. Includes bibliographical references and index. ISBN 1-58488-297-2 (alk. paper) 1. Differential equations--Numerical solutions. I. Zaitsev, V. F. (Valentin F.) II. Title. QA372 .P725 2002 515′.352--dc21

2002073735

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CONTENTS Authors Foreword Notation and Some Remarks Introduction Some Definitions, Formulas, Methods, and Transformations 0.1. First-Order Differential Equations 0.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 0.1.1-1. Equations solved for the derivative. General solution 0.1.1-2. The Cauchy problem. The uniqueness and existence theorems 0.1.1-3. Equations not solved for the derivative. The existence theorem 0.1.1-4. Singular solutions 0.1.1-5. Point transformations 0.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration 0.1.2-1. Equations with separated or separable variables 0.1.2-2. Equation of the form 
 =  (  +  ) 0.1.2-3. Homogeneous equations and equations reducible to them 0.1.2-4. Generalized homogeneous equations and equations reducible to them 0.1.2-5. Linear equation 0.1.2-6. Bernoulli equation 0.1.2-7. Equation of the form  
 = +  ( ) (  ) 0.1.2-8. Darboux equation 0.1.3. Exact Differential Equations. Integrating Factor 0.1.3-1. Exact differential equations 0.1.3-2. Integrating factor 0.1.4. Riccati Equation 0.1.4-1. General Riccati equation. Simplest integrable cases 0.1.4-2. Polynomial solutions of the Riccati equation 0.1.4-3. Use of particular solutions to construct the general solution 0.1.4-4. Some transformations 0.1.4-5. Reduction of the Riccati equation to a second-order linear equation 0.1.4-6. Reduction of the Riccati equation to the canonical form 0.1.5. Abel Equations of the First Kind 0.1.5-1. General form of Abel equations of the first kind. Simplest integrable cases 0.1.5-2. Reduction to the canonical form. Reduction to an Abel equation of the second kind 0.1.6. Abel Equations of the Second Kind 0.1.6-1. General form of Abel equations of the second kind. Simplest integrable cases 0.1.6-2. Reduction to the canonical form. Reduction to an Abel equation of the first kind 0.1.6-3. Use of particular solutions to construct self-transformations 0.1.6-4. Use of particular solutions to construct the general solution 0.1.7. Equations Not Solved for the Derivative 0.1.7-1. The method of “integration by differentiation.” 0.1.7-2. Equations of the form =  (
 )

Page v

0.1.7-3. Equations of the form  =  (
 ) 0.1.7-4. Clairaut’s equation =  
 +  (
 ) 0.1.7-5. Lagrange’s equation =   (
 ) + (
 ) 0.1.8. Contact Transformations 0.1.8-1. General form of contact transformations 0.1.8-2. A method for the construction of contact transformations 0.1.8-3. Examples of contact transformations linear in the derivative 0.1.8-4. Examples of contact transformations nonlinear in the derivative 0.1.9. Approximate Analytic Methods for Solution of Equations 0.1.9-1. The method of successive approximations (Picard method) 0.1.9-2. The method of Taylor series expansion in the independent variable 0.1.9-3. The method of regular expansion in the small parameter 0.1.10. Numerical Integration of Differential Equations 0.1.10-1. The method of Euler polygonal lines 0.1.10-2. Single-step methods of the second-order approximation 0.1.10-3. Runge–Kutta method of the fourth-order approximation 0.2. Second-Order Linear Differential Equations 0.2.1. Formulas for the General Solution. Some Transformations 0.2.1-1. Homogeneous linear equations. Various representations of the general solution 0.2.1-2. Wronskian determinant and Liouville’s formula 0.2.1-3. Reduction to the canonical form 0.2.1-4. Reduction to the Riccati equation 0.2.1-5. Nonhomogeneous linear equations. The existence theorem 0.2.1-6. Nonhomogeneous linear equations. Various representations of the general solution 0.2.1-7. Reduction to a constant coefficient equation (a special case) 0.2.1-8. Kummer–Liouville transformation 0.2.2. Representation of Solutions as a Series in the Independent Variable 0.2.2-1. Equation coefficients are representable in the ordinary power series form 0.2.2-2. Equation coefficients have poles at some point 0.2.3. Asymptotic Solutions 0.2.3-1. Equations not containing 
 . Leading asymptotic terms 0.2.3-2. Equations not containing 
 . Two-term asymptotic expansions 0.2.3-3. Equations of special form not containing 
0.2.3-4. Equations not containing 
 . Equation coefficients are dependent on 0.2.3-5. Equations containing 
 0.2.3-6. Equations of the general form 0.2.4. Boundary Value Problems 0.2.4-1. The first, second, third, and mixed boundary value problems 0.2.4-2. Simplification of boundary conditions. Reduction of equation to the self-adjoint form 0.2.4-3. The Green’s function. Boundary value problems for nonhomogeneous equations 0.2.4-4. Representation of the Green’s function in terms of particular solutions 0.2.5. Eigenvalue Problems 0.2.5-1. The Sturm–Liouville problem 0.2.5-2. General properties of the Sturm–Liouville problem (1), (2) 0.2.5-3. Problems with boundary conditions of the first kind 0.2.5-4. Problems with boundary conditions of the second kind

Page vi

0.2.5-5. Problems with boundary conditions of the third kind 0.2.5-6. Problems with mixed boundary conditions 0.3. Second-Order Nonlinear Differential Equations 0.3.1. Form of the General Solution. Cauchy Problem 0.3.1-1. Equations solved for the derivative. General solution 0.3.1-2. Cauchy problem. The existence and uniqueness theorem 0.3.2. Equations Admitting Reduction of Order 0.3.2-1. Equations not containing explicitly 0.3.2-2. Equations not containing  explicitly (autonomous equations) 0.3.2-3. Equations of the form  (  +  , 
 , 

 ) = 0 0.3.2-4. Equations of the form  ( ,  
− , 

 ) = 0 0.3.2-5. Homogeneous equations 0.3.2-6. Generalized homogeneous equations 0.3.2-7. Equations invariant under scaling–translation transformations 0.3.2-8. Exact second-order equations 0.3.2-9. Reduction of quasilinear equations to the normal form 0.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable or Small Parameter 0.3.3-1. Method of expansion in powers of the independent variable 0.3.3-2. Method of regular (direct) expansion in powers of the small parameter 0.3.3-3. Pad´e approximants 0.3.4. Perturbation Methods of Mechanics and Physics 0.3.4-1. Preliminary remarks. A summary table of basic methods 0.3.4-2. The method of scaled parameters (Lindstedt–Poincar e´ method) 0.3.4-3. Averaging method (Van der Pol–Krylov–Bogolyubov scheme) 0.3.4-4. Method of two-scale expansions (Cole–Kevorkian scheme) 0.3.4-5. Method of matched asymptotic expansions 0.3.5. Galerkin Method and Its Modifications (Projection Methods) 0.3.5-1. General form of an approximate solution 0.3.5-2. Galerkin method 0.3.5-3. The Bubnov–Galerkin method, the moment method, and the least squares method 0.3.5-4. Collocation method 0.3.5-5. The method of partitioning the domain 0.3.5-6. The least squared error method 0.3.6. Iteration and Numerical Methods 0.3.6-1. The method of successive approximations (Cauchy problem) 0.3.6-2. The Runge–Kutta method (Cauchy problem) 0.3.6-3. Shooting method (boundary value problems) 0.3.6-4. Method of accelerated convergence in eigenvalue problems 0.4. Linear Equations of Arbitrary Order 0.4.1. Linear Equations with Constant Coefficients 0.4.1-1. Homogeneous linear equations 0.4.1-2. Nonhomogeneous linear equations 0.4.2. Linear Equations with Variable Coefficients 0.4.2-1. Homogeneous linear equations. Structure of the general solution 0.4.2-2. Utilization of particular solutions for reducing the order of the original equation 0.4.2-3. Wronskian determinant and Liouville formula

Page vii

0.4.2-4. Nonhomogeneous linear equations. Construction of the general solution 0.4.3. Asymptotic Solutions of Linear Equations 0.4.3-1. Fourth-order linear equations 0.4.3-2. Higher-order linear equations 0.5. Nonlinear Equations of Arbitrary Order 0.5.1. Structure of the General Solution. Cauchy Problem 0.5.1-1. Equations solved for the highest derivative. General solution 0.5.1-2. The Cauchy problem. The existence and uniqueness theorem 0.5.2. Equations Admitting Reduction of Order 0.5.2-1. Equations not containing , 
 ,  , (  ) explicitly 0.5.2-2. Equations not containing  explicitly (autonomous equations) ( )  0.5.2-3. Equations of the form  +  , 
 ,  ,  =0 


 )  = 0 and its 0.5.2-4. Equations of the form  ,   − ,  ,  , ( generalizations 0.5.2-5. Homogeneous equations 0.5.2-6. Generalized homogeneous equations  0.5.2-7. Equations of the form    , 
  , 

  ,  , (  )   = 0  ) = 0 0.5.2-8. Equations of the form      ,  
 ,  2 

 ,  ,   ( 0.5.2-9. Other equations 0.5.3. A Method for Construction of Solvable Equations of General Form 0.5.3-1. Description of the method 0.5.3-2. Examples 0.6. Lie Group and Discrete-Group Methods 0.6.1. Lie Group Method. Point Transformations 0.6.1-1. Local one-parameter Lie group of transformations. Invariance condition 0.6.1-2. Group analysis of second-order equations. Structure of an admissible operator 0.6.1-3. Utilization of local groups for reducing the order of equations and their integration 0.6.2. Contact Transformations. B¨acklund Transformations. Formal Operators. Factorization Principle 0.6.2-1. Contact transformations 0.6.2-2. B¨acklund transformations. Formal operators and nonlocal variables 0.6.2-3. Factorization principle 0.6.3. First Integrals (Conservation Laws) 0.6.4. Discrete-Group Method. Point Transformations 0.6.5. Discrete-Group Method. The Method of RF-Pairs 1. First-Order Differential Equations 1.1. Simplest Equations with Arbitrary Functions Integrable in Closed Form 1.1.1. Equations of the Form 
 =  ( ) 1.1.2. Equations of the Form 
 =  ( ) 1.1.3. Separable Equations 
 =  ( ) ( ) 1.1.4. Linear Equation ( )
 =  1 ( ) +  0 ( ) 1.1.5. Bernoulli Equation ( )
 =  1 ( ) +   ( )  1.1.6. Homogeneous Equation 
 =  (  )

Page viii

1.2. Riccati Equation ( ) 
=  2 ( ) 2 +  1 ( ) +  0 ( ) 1.2.1. Preliminary Remarks 1.2.2. Equations Containing Power Functions 1.2.2-1. Equations of the form ( )
 =  2 ( ) 2 +  0 ( ) 1.2.2-2. Other equations 1.2.3. Equations Containing Exponential Functions 1.2.3-1. Equations with exponential functions 1.2.3-2. Equations with power and exponential functions 1.2.4. Equations Containing Hyperbolic Functions 1.2.4-1. Equations with hyperbolic sine and cosine 1.2.4-2. Equations with hyperbolic tangent and cotangent 1.2.5. Equations Containing Logarithmic Functions 1.2.5-1. Equations of the form ( )
 =  2 ( ) 2 +  0 ( ) 1.2.5-2. Equations of the form ( )
 =  2 ( ) 2 +  1 ( )
 +  0 ( ) 1.2.6. Equations Containing Trigonometric Functions 1.2.6-1. Equations with sine 1.2.6-2. Equations with cosine 1.2.6-3. Equations with tangent 1.2.6-4. Equations with cotangent 1.2.6-5. Equations containing combinations of trigonometric functions 1.2.7. Equations Containing Inverse Trigonometric Functions 1.2.7-1. Equations containing arcsine 1.2.7-2. Equations containing arccosine 1.2.7-3. Equations containing arctangent 1.2.7-4. Equations containing arccotangent 1.2.8. Equations with Arbitrary Functions 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives) 1.2.8-2. Equations containing arbitrary functions and their derivatives 1.2.9. Some Transformations 1.3. Abel Equations of the Second Kind 1.3.1. Equations of the Form !
 − =  ( ) 1.3.1-1. Preliminary remarks. Classification tables 1.3.1-2. Solvable equations and their solutions 1.3.2. Equations of the Form !
 =  ( ) + 1 1.3.3. Equations of the Form !
 =  1 ( ) +  0 ( ) 1.3.3-1. Preliminary remarks 1.3.3-2. Solvable equations and their solutions 1.3.4. Equations of the Form [ 1 ( ) + 0 ( )]
 =  2 ( ) 2 +  1 ( ) +  0 ( ) 1.3.4-1. Preliminary remarks 1.3.4-2. Solvable equations and their solutions 1.3.5. Some Types of First- and Second-Order Equations Reducible to Abel Equations of the Second Kind 1.3.5-1. Quasi-homogeneous equations 1.3.5-2. Equations of the theory of chemical reactors and the combustion theory 1.3.5-3. Equations of the theory of nonlinear oscillations 1.3.5-4. Second-order homogeneous equations of various types 1.3.5-5. Second-order equations invariant under some transformations

Page ix

1.4. Equations Containing Polynomial Functions of 1.4.1. Abel Equations of the First Kind "
 =  3 ( ) 3 +  2 ( ) 2 +  1 ( ) +  0 ( ) 1.4.1-1. Preliminary remarks 1.4.1-2. Solvable equations and their solutions 1.4.2. Equations of the Form ( # 22 2 + # 12  + # 11  2 + # 0 )
 = $ 22 2 + $ 12  + $ 11  2 + $ 1.4.2-1. Preliminary remarks. Some transformations 1.4.2-2. Solvable equations and their solutions 1.4.3. Equations of the Form ( # 22 2 + # 12  + # 11  2 + # 2 + # 1  )
 = $ 22 2 + $ 12  + $ 11  2 + $ 2 + $ 1  1.4.3-1. Preliminary remarks 1.4.3-2. Solvable equations and their solutions 1.4.4. Equations of the Form ( # 22 2 + # 12  + # 11  2 + # 2 + # 1  + # 0 )
 = $ 22 2 + $ 12  + $ 11  2 + $ 2 + $ 1  + $ 0 1.4.4-1. Preliminary remarks. Some transformations 1.4.4-2. Solvable equations and their solutions 1.4.5. Equations of the Form ( # 3 3 + # 2  2 + # 1  2 + # 0  3 +  1 +  0  )
 = $ 3 3 + $ 2 2 + $ 1 2 + $ 0 3 +  1 +  0

0

142

1.5. Equations of the Form  ( , )
 = ( , ) Containing Arbitrary Parameters 1.5.1. Equations Containing Power Functions 1.5.1-1. Equations of the form 
 =  ( , ) 1.5.1-2. Other equations 1.5.2. Equations Containing Exponential Functions 1.5.2-1. Equations with exponential functions 1.5.2-2. Equations with power and exponential functions 1.5.3. Equations Containing Hyperbolic Functions 1.5.4. Equations Containing Logarithmic Functions 1.5.5. Equations Containing Trigonometric Functions 1.5.6. Equations Containing Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions

1.6. Equations of the Form  ( , , 
 ) = 0 Containing Arbitrary Parameters 1.6.1. Equations of the Second Degree in 
1.6.1-1. Equations of the form  ( , )(
 )2 = ( , ) 1.6.1-2. Equations of the form  ( , )(
 )2 = ( , )
 + % ( , ) 1.6.2. Equations of the Third Degree in 
 1.6.2-1. Equations of the form  ( , )(
 )3 = ( , )
 + % ( , ) 1.6.2-2. Equations of the form  ( , )(
 )3 = ( , )(
 )2 + % ( , )
 + & ( , ) 180 1.6.3. Equations of the Form (
 )  =  ( ) + ( ) 1.6.3-1. Some transformations 1.6.3-2. Classification tables and exact solutions 1.6.4. Other Equations 1.6.4-1. Equations containing algebraic and power functions with respect to
 1.6.4-2. Equations containing exponential, logarithmic, and other functions with respect to 
1.7. Equations of the Form  ( , )
 = ( , ) Containing Arbitrary Functions 1.7.1. Equations Containing Power Functions 1.7.2. Equations Containing Exponential and Hyperbolic Functions 1.7.3. Equations Containing Logarithmic Functions 1.7.4. Equations Containing Trigonometric Functions 1.7.5. Equations Containing Combinations of Exponential, Logarithmic, and Trigonometric Functions

Page x

1.8. Equations of the Form  ( , , 
 ) = 0 Containing Arbitrary Functions 1.8.1. Some Equations 1.8.1-1. Arguments of arbitrary functions depend on  and 1.8.1-2. Argument of arbitrary functions is "
 1.8.1-3. Arguments of arbitrary functions are linear with respect to
 1.8.1-4. Arguments of arbitrary functions are nonlinear with respect to
 1.8.2. Some Transformations 2. Second-Order Differential Equations 2.1. Linear Equations 2.1.1. Representation of the General Solution Through a Particular Solution 2.1.2. Equations Containing Power Functions 2.1.2-1. Equations of the form 

 +  ( ) = 0 2.1.2-2. Equations of the form 

 +  ( )
 + ( ) = 0 2.1.2-3. Equations of the form (  +  )

 +  ( )
 + ( ) = 0 2.1.2-4. Equations of the form  2 

 +  ( )
 + ( ) = 0 2.1.2-5. Equations of the form (  2 + ' + ( )

 +  ( )
 + ( ) = 0 2.1.2-6. Equations of the form (  3  3 +  2  2 +  1  +  0 )

 +  ( )
 + ( ) = 0 2.1.2-7. Equations of the form (  4  4 + ))) +  1  +  0 )

 +  ( )
 + ( ) = 0 2.1.2-8. Other equations 2.1.3. Equations Containing Exponential Functions 2.1.3-1. Equations with exponential functions 2.1.3-2. Equations with power and exponential functions 2.1.4. Equations Containing Hyperbolic Functions 2.1.4-1. Equations with hyperbolic sine 2.1.4-2. Equations with hyperbolic cosine 2.1.4-3. Equations with hyperbolic tangent 2.1.4-4. Equations with hyperbolic cotangent 2.1.4-5. Equations containing combinations of hyperbolic functions 2.1.5. Equations Containing Logarithmic Functions 2.1.5-1. Equations of the form  ( )

 + ( ) = 0 2.1.5-2. Equations of the form  ( )

 + ( )
 + % ( ) = 0 2.1.6. Equations Containing Trigonometric Functions 2.1.6-1. Equations with sine 2.1.6-2. Equations with cosine 2.1.6-3. Equations with tangent 2.1.6-4. Equations with cotangent 2.1.6-5. Equations containing combinations of trigonometric functions 2.1.7. Equations Containing Inverse Trigonometric Functions 2.1.7-1. Equations with arcsine 2.1.7-2. Equations with arccosine 2.1.7-3. Equations with arctangent 2.1.7-4. Equations with arccotangent 2.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions 2.1.9. Equations with Arbitrary Functions 2.1.9-1. Equations containing arbitrary functions (but not containing their derivatives) 2.1.9-2. Equations containing arbitrary functions and their derivatives 2.1.10. Some Transformations

Page xi

2.2. Autonomous Equations 

 =  ( , 
 ) 2.2.1. Equations of the Form 

 − 
 =  ( ) 2.2.2. Equations of the Form 

 +  ( ) 
+ = 0 2.2.2-1. Preliminary remarks 2.2.2-2. Solvable equations and their solutions 2.2.3. Lienard Equations 

 +  ( )
 + ( ) = 0 2.2.3-1. Preliminary remarks 2.2.3-2. Solvable equations and their solutions 2.2.4. Rayleigh Equations 

 +  (
 ) + ( ) = 0 2.2.4-1. Preliminary remarks. Some transformations 2.2.4-2. Solvable equations and their solutions

2.3. Emden–Fowler Equation 

 = #*  + 2.3.1. Exact Solutions 2.3.1-1. Preliminary remarks. Classification table 2.3.1-2. Solvable equations and their solutions 2.3.2. First Integrals (Conservation Laws) 2.3.2-1. First integrals with , = 2 2.3.2-2. First integrals with , = 3 2.3.2-3. First integrals with , = 4 2.3.2-4. First integrals with , = 5 2.3.3. Some Formulas and Transformations

2.4. Equations of the Form 

 = # 2.4.1. Classification Table 2.4.2. Exact Solutions

1

  1 +

1

+#

2

  2 +

2

2.5. Generalized Emden–Fowler Equation 

 = #*  + ( 
)2.5.1. Classification Table 2.5.2. Exact Solutions 2.5.3. Some Formulas and Transformations 2.5.3-1. A particular solution 2.5.3-2. Discrete transformations of the generalized Emden–Fowler equation 2.5.3-3. Reduction of the generalized Emden–Fowler equation to an Abel equation 2.6. Equations of the Form 

 = # 1   1 + 1 (
 )- 1 + # 2   2 + 2 (
 )- 2 2.6.1. Modified Emden–Fowler Equation 

 = # 1  −1 
 + # 2   + 2.6.1-1. Preliminary remarks. Classification table 2.6.1-2. Solvable equations and their solutions 2.6.2. Equations of the Form 

 = ( # 1   1 + 1 + # 2   2 + 2 )(
 )2.6.2-1. Classification table 2.6.2-2. Solvable equations and their solutions 2.6.3. Equations of the Form 

 = ./#*  + (
 )- + #*  −1 + +1 (
 )- −1 2.6.3-1. Classification table 2.6.3-2. Solvable equations and their solutions 2.6.4. Other Equations ( 0 1 ≠ 0 2 ) 2.6.4-1. Classification table 2.6.4-2. Solvable equations and their solutions

2.7. Equations of the Form 

 =  ( ) ( ) % ( 
) 2.7.1. Equations of the Form 

 =  ( ) ( ) 2.7.2. Equations Containing Power Functions ( %21 const) 2.7.3. Equations Containing Exponential Functions ( %31 const)

Page xii

2.7.3-1. Preliminary remarks 2.7.3-2. Solvable equations and their solutions 2.7.4. Equations Containing Hyperbolic Functions ( %31 const) 2.7.5. Equations Containing Trigonometric Functions ( %31 const) 2.7.6. Some Transformations 2.8. Some Nonlinear Equations with Arbitrary Parameters 2.8.1. Equations Containing Power Functions 2.8.1-1. Equations of the form  ( , )

 + ( , ) = 0 2.8.1-2. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0 2.8.1-3. Equations of the form  ( , )

 + ( , )(
 )2 + % ( , )
 + & ( , ) = 0 2.8.1-4. Other equations 2.8.2. Painlev´e Transcendents 2.8.2-1. Preliminary remarks. Singular points of solutions 2.8.2-2. First Painlev´e transcendent 2.8.2-3. Second Painlev´e transcendent 2.8.2-4. Third Painlev´e transcendent 2.8.2-5. Fourth Painlev´e transcendent 2.8.2-6. Fifth Painlev´e transcendent 2.8.2-7. Sixth Painlev´e transcendent 2.8.3. Equations Containing Exponential Functions 2.8.3-1. Equations of the form  ( , )

 + ( , ) = 0 2.8.3-2. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0 2.8.3-3. Equations of the form  ( , )

 + ( , )(
 )2 + % ( , )
 + & ( , ) = 0 2.8.3-4. Other equations 2.8.4. Equations Containing Hyperbolic Functions 2.8.4-1. Equations with hyperbolic sine 2.8.4-2. Equations with hyperbolic cosine 2.8.4-3. Equations with hyperbolic tangent 2.8.4-4. Equations with hyperbolic cotangent 2.8.4-5. Equations containing combinations of hyperbolic functions 2.8.5. Equations Containing Logarithmic Functions 2.8.5-1. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0 2.8.5-2. Other equations 2.8.6. Equations Containing Trigonometric Functions 2.8.6-1. Equations with sine 2.8.6-2. Equations with cosine 2.8.6-3. Equations with tangent 2.8.6-4. Equations with cotangent 2.8.6-5. Equations containing combinations of trigonometric functions 2.8.7. Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions 2.9. Equations Containing Arbitrary Functions 2.9.1. Equations of the Form  ( , )

 + 4 ( , ) = 0 2.9.1-1. Arguments of arbitrary functions are algebraic and power functions of  and 2.9.1-2. Arguments of the arbitrary functions are other functions 2.9.2. Equations of the Form  ( , )

 + 4 ( , )
 + 5 ( , ) = 0 2.9.2-1. Argument of the arbitrary functions is  2.9.2-2. Argument of the arbitrary functions is 2.9.2-3. Other arguments of the arbitrary functions

Page xiii

7

2.9.3. Equations of the Form  ( , )

 + 6

+

2.9.4.

2.9.5. 2.9.6.

2.9.7.

=0

4 + ( , )(
 ) + = 0 ( 8

= 2, 3, 4)

2.9.3-1. Argument of the arbitrary functions is  . 2.9.3-2. Argument of the arbitrary functions is 2.9.3-3. Other arguments of arbitrary functions Equations of the Form  ( , , 
 )

 + 4 ( , , 
 ) = 0 2.9.4-1. Arguments of the arbitrary functions depend on  or 2.9.4-2. Arguments of the arbitrary functions depend on  and 2.9.4-3. Arguments of the arbitrary functions depend on  , , and
 Equations Not Solved for Second Derivative Equations of General Form 2.9.6-1. Equations containing arbitrary functions of two variables 2.9.6-2. Equations containing arbitrary functions of three variables Some Transformations

3. Third-Order Differential Equations 3.1. Linear Equations 3.1.1. Preliminary Remarks 3.1.2. Equations Containing Power Functions 3.1.2-1. Equations of the form  3 ( )


 +  0 ( ) = ( ) 3.1.2-2. Equations of the form  3 ( )


 +  1 ( )
 +  0 ( ) = ( ) 3.1.2-3. Equations of the form  3 ( )


 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ) 3.1.3. Equations Containing Exponential Functions 3.1.3-1. Equations with exponential functions 3.1.3-2. Equations with power and exponential functions 3.1.4. Equations Containing Hyperbolic Functions 3.1.4-1. Equations with hyperbolic sine 3.1.4-2. Equations with hyperbolic cosine 3.1.4-3. Equations with hyperbolic sine and cosine 3.1.4-4. Equations with hyperbolic tangent 3.1.4-5. Equations with hyperbolic cotangent 3.1.5. Equations Containing Logarithmic Functions 3.1.5-1. Equations with logarithmic functions 3.1.5-2. Equations with power and logarithmic functions 3.1.6. Equations Containing Trigonometric Functions 3.1.6-1. Equations with sine 3.1.6-2. Equations with cosine 3.1.6-3. Equations with sine and cosine 3.1.6-4. Equations with tangent 3.1.6-5. Equations with cotangent 3.1.7. Equations Containing Inverse Trigonometric Functions 3.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions 3.1.9. Equations Containing Arbitrary Functions 3.1.9-1. Equations of the form  3 ( )


 +  1 ( )
 +  0 ( ) = ( ) 3.1.9-2. Equations of the form  3 ( )


 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( )

3.2. Equations of the Form 


 = #* 9 : (
 ); (

 ) < 3.2.1. Classification Table 3.2.2. Equations of the Form 


 = # :

Page xiv

3.2.3. Equations of the Form 


 = #* 9 3.2.4. Equations with |= | + | > | ≠ 0 3.2.5. Some Transformations

:

3.3. Equations of the Form 


 =  ( ) (
 ) % (

 ) 3.3.1. Equations Containing Power Functions 3.3.2. Equations Containing Exponential Functions 3.3.3. Other Equations 3.4. Nonlinear Equations with Arbitrary Parameters 3.4.1. Equations Containing Power Functions 3.4.1-1. Equations of the form  ( , ) 


 = ( , ) 3.4.1-2. Equations of the form 


 =  ( , , 
) 3.4.1-3. Equations of the form  ( , , 
 )


 + ( , , 
 )

 + % ( , , 
 ) = 0 3.4.1-4. Other equations 3.4.2. Equations Containing Exponential Functions 3.4.2-1. Equations of the form 


 =  ( , , 
 ) 3.4.2-2. Other equations 3.4.3. Equations Containing Hyperbolic Functions 3.4.3-1. Equations with hyperbolic sine 3.4.3-2. Equations with hyperbolic cosine 3.4.3-3. Equations with hyperbolic tangent 3.4.3-4. Equations with hyperbolic cotangent 3.4.4. Equations Containing Logarithmic Functions 3.4.4-1. Equations of the form 


 =  ( , , 
 ) 3.4.4-2. Other equations 3.4.5. Equations Containing Trigonometric Functions 3.4.5-1. Equations with sine 3.4.5-2. Equations with cosine 3.4.5-3. Equations with tangent 3.4.5-4. Equations with cotangent 3.5. Nonlinear Equations Containing Arbitrary Functions 3.5.1. Equations of the Form  ( , )


 + 4 ( , ) = 0 3.5.1-1. Arguments of the arbitrary functions are  or 3.5.1-2. Arguments of the arbitrary functions depend on  and 3.5.2. Equations of the Form  ( , , 
 )


 + 4 ( , , 
 ) = 0 3.5.2-1. Arguments of the arbitrary functions depend on  and 3.5.2-2. Arguments of the arbitrary functions depend on  , , and
 3.5.3. Equations of the Form  ( , , 
) 


 + 4 ( , , 
) 

 + 5 ( , , 
) = 0 3.5.3-1. The arbitrary functions depend on  or 3.5.3-2. Arguments of arbitrary functions depend on  and 3.5.3-3. Arguments of arbitrary functions depend on  , , and
 7 3.5.4. Equations of the Form  ( , , 
 )


 + 4 ( , , 
 )(

 ) 9 = 0

9

9

3.5.4-1. Arbitrary functions depend on  or 3.5.4-2. Arguments of arbitrary functions depend on  , , and
 3.5.5. Other Equations 3.5.5-1. Equations of the form  ( , , 
 , 

 )


 + 4 ( , , 
 , 

 ) = 0 3.5.5-2. Equations of the form  ( , , 
 , 

 , 


 ) = 0

Page xv

xvi

CONTENTS

4. Fourth-Order Differential Equations 4.1. Linear Equations 4.1.1. Preliminary Remarks 4.1.2. Equations Containing Power Functions 4.1.2-1. Equations of the form  4 ( ) 



 +  0 ( ) = ( ) 4.1.2-2. Equations of the form  4 ( ) 



 +  1 ( ) 
+  0 ( ) = ( ) 4.1.2-3. Equations of the form  4 ( )



 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ) 4.1.2-4. Other equations 4.1.3. Equations Containing Exponential and Hyperbolic Functions 4.1.3-1. Equations with exponential functions 4.1.3-2. Equations with hyperbolic functions 4.1.4. Equations Containing Logarithmic Functions 4.1.5. Equations Containing Trigonometric Functions 4.1.5-1. Equations with sine and cosine 4.1.5-2. Equations with tangent and cotangent 4.1.6. Equations Containing Arbitrary Functions 4.1.6-1. Equations of the form  4 ( )



 +  1 ( )
 +  0 ( ) = ( ) 4.1.6-2. Equations of the form  4 ( )



 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ) 4.1.6-3. Other equations 4.2. Nonlinear Equations 4.2.1. Equations Containing Power Functions 4.2.1-1. Equations of the form 



 =  ( , ) 4.2.1-2. Equations of the form 



 =  ( , , 
 4.2.1-3. Equations of the form 



 =  ( , , 
 4.2.1-4. Equations of the form 



 =  ( , , 
 4.2.2. Equations Containing Exponential Functions 4.2.2-1. Equations of the form 



 =  ( , ) 4.2.2-2. Other equations 4.2.3. Equations Containing Hyperbolic Functions 4.2.3-1. Equations with hyperbolic sine 4.2.3-2. Equations with hyperbolic cosine 4.2.3-3. Equations with hyperbolic tangent 4.2.3-4. Equations with hyperbolic cotangent 4.2.4. Equations Containing Logarithmic Functions 4.2.4-1. Equations of the form 



 =  ( , ) 4.2.4-2. Other equations 4.2.5. Equations Containing Trigonometric Functions 4.2.5-1. Equations with sine 4.2.5-2. Equations with cosine 4.2.5-3. Equations with tangent 4.2.5-4. Equations with cotangent 4.2.6. Equations Containing Arbitrary Functions 4.2.6-1. Equations of the form 



 =  ( , ) 4.2.6-2. Equations of the form 



 =  ( , , 
 4.2.6-3. Equations of the form 



 =  ( , , 
 4.2.6-4. Equations of the form 



 =  ( , , 
 4.2.6-5. Other equations

) , 

 ) , 

 , 


 )

) , 

 ) , 

 , 


 )

Page xvi

5. Higher-Order Differential Equations 5.1. Linear Equations 5.1.1. Preliminary Remarks 5.1.2. Equations Containing Power Functions 5.1.2-1. Equations of the form   ( ) (  ) +  0 ( ) = ( ) 5.1.2-2. Equations of the form   ( ) (  ) +  1 ( ) 
+  0 ( ) = ( ) 5.1.2-3. Other equations 5.1.3. Equations Containing Exponential and Hyperbolic Functions 5.1.3-1. Equations with exponential functions 5.1.3-2. Equations with hyperbolic functions 5.1.4. Equations Containing Logarithmic Functions 5.1.5. Equations Containing Trigonometric Functions 5.1.5-1. Equations with sine and cosine 5.1.5-2. Equations with tangent and cotangent 5.1.6. Equations Containing Arbitrary Functions 5.1.6-1. Equations of the form   ( ) (  ) +  1 ( )
 +  0 ( ) = ( ) 5.1.6-2. Other equations 5.2. Nonlinear Equations 5.2.1. Equations Containing Power Functions 5.2.1-1. Fifth- and sixth-order equations 5.2.1-2. Equations of the form (  ) =  ( , ) 5.2.1-3. Equations of the form (  ) =  ( , , 
 , 

 ) 5.2.1-4. Other equations 5.2.2. Equations Containing Exponential Functions 5.2.2-1. Fifth- and sixth-order equations 5.2.2-2. Equations of the form (  ) =  ( , ) 5.2.2-3. Other equations 5.2.3. Equations Containing Hyperbolic Functions 5.2.3-1. Equations with hyperbolic sine 5.2.3-2. Equations with hyperbolic cosine 5.2.3-3. Equations with hyperbolic tangent 5.2.3-4. Equations with hyperbolic cotangent 5.2.4. Equations Containing Logarithmic Functions 5.2.4-1. Equations of the form (  ) =  ( , ) 5.2.4-2. Other equations 5.2.5. Equations Containing Trigonometric Functions 5.2.5-1. Equations with sine 5.2.5-2. Equations with cosine 5.2.5-3. Equations with tangent 5.2.5-4. Equations with cotangent 5.2.6. Equations Containing Arbitrary Functions 5.2.6-1. Fifth- and sixth-order equations 5.2.6-2. Equations of the form (  ) =  ( , ) 5.2.6-3. Equations of the form (  ) =  ( , , 
 ) 5.2.6-4. Equations of the form (  ) =  ( , , 
 , 

 ) 5.2.6-5. Equations of the form  ( , ) (  ) + ( , , 
 ) (  −1) =  −2)  %   , , 
 ,  , (?  −1)  5.2.6-6. Equations of the form (  ) =    , , 
 ,  , (? 
( )  =0 5.2.6-7. Equations of the general form    , ,  ,  , 

Page xvii

Supplements S.1. Elementary Functions and Their Properties S.1.1. Trigonometric Functions S.1.1-1. Simplest relations S.1.1-2. Relations between trigonometric functions of single argument S.1.1-3. Reduction formulas S.1.1-4. Addition and subtraction of trigonometric functions S.1.1-5. Products of trigonometric functions S.1.1-6. Powers of trigonometric functions S.1.1-7. Addition formulas S.1.1-8. Trigonometric functions of multiple arguments S.1.1-9. Trigonometric functions of half argument S.1.1-10. Euler and de Moivre formulas. Relationship with hyperbolic functions S.1.1-11. Differentiation formulas S.1.1-12. Expansion into power series S.1.2. Hyperbolic Functions S.1.2-1. Definitions S.1.2-2. Simplest relations S.1.2-3. Relations between hyperbolic functions of single argument ( ≥ 0) S.1.2-4. Addition formulas S.1.2-5. Addition and subtraction of hyperbolic functions S.1.2-6. Products of hyperbolic functions S.1.2-7. Powers of hyperbolic functions S.1.2-8. Hyperbolic functions of multiple arguments S.1.2-9. Relationship with trigonometric functions S.1.2-10. Differentiation formulas S.1.2-11. Expansion into power series S.1.3. Inverse Trigonometric Functions S.1.3-1. Definitions and some properties S.1.3-2. Simplest formulas S.1.3-3. Relations between inverse trigonometric functions S.1.3-4. Addition and subtraction of inverse trigonometric functions S.1.3-5. Differentiation formulas S.1.3-6. Expansion into power series S.1.4. Inverse Hyperbolic Functions S.1.4-1. Relationships with logarithmic functions S.1.4-2. Relations between inverse hyperbolic functions S.1.4-3. Addition and subtraction of inverse hyperbolic functions S.1.4-4. Differentiation formulas S.1.4-5. Expansion into power series S.2. Special Functions and Their Properties S.2.1. Some Symbols and Coefficients S.2.1-1. Factorials S.2.1-2. Binomial coefficients S.2.1-3. Pochhammer symbol S.2.2. Error Functions and Exponential Integral S.2.2-1. Error function and complementary error function S.2.2-2. Exponential integral S.2.2-3. Logarithmic integral

Page xviii

S.2.3. Gamma and Beta Functions S.2.3-1. Gamma function S.2.3-2. Logarithmic derivative of the gamma function S.2.3-3. Beta function S.2.4. Incomplete Gamma and Beta Functions S.2.4-1. Incomplete gamma function S.2.4-2. Incomplete beta function S.2.5. Bessel Functions S.2.5-1. Definitions and basic formulas S.2.5-2. Bessel functions for @ = A"BCA 12 , where B = 0, 1, 2,  S.2.5-3. Bessel functions for @ = A"B , where B = 0, 1, 2,  S.2.5-4. Wronskians and similar formulas S.2.5-5. Integral representations S.2.5-6. Asymptotic expansions S.2.5-7. Zeros and orthogonality properties of the Bessel functions S.2.5-8. Hankel functions (Bessel functions of the third kind) S.2.6. Modified Bessel Functions S.2.6-1. Definitions. Basic formulas S.2.6-2. Modified Bessel functions for @ = A"BCA 12 , where B = 0, 1, 2,  S.2.6-3. Modified Bessel functions for @ = B , where B = 0, 1, 2,  S.2.6-4. Wronskians and similar formulas S.2.6-5. Integral representations S.2.6-6. Asymptotic expansions as EDGF S.2.7. Degenerate Hypergeometric Functions S.2.7-1. Definitions. The Kummer’s series S.2.7-2. Some transformations and linear relations S.2.7-3. Differentiation formulas and Wronskian S.2.7-4. Degenerate hypergeometric functions for B = 0, 1, 2,  S.2.7-5. Integral representations S.2.7-6. Asymptotic expansion as | | DGF S.2.7-7. Whittaker functions S.2.8. Hypergeometric Functions S.2.8-1. Definition. The hypergeometric series S.2.8-2. Basic properties S.2.8-3. Integral representations S.2.9. Legendre Functions and Legendre Polynomials S.2.9-1. Definitions. Basic formulas S.2.9-2. Trigonometric expansions S.2.9-3. Some relations S.2.9-4. Integral representations S.2.9-5. Legendre polynomials S.2.9-6. Zeros of the Legendre polynomials and the generating function S.2.9-7. Associated Legendre functions S.2.10. Parabolic Cylinder Functions S.2.10-1. Definitions. Basic formulas S.2.10-2. Integral representations S.2.10-3. Asymptotic expansion as | H | DGF S.2.11. Orthogonal Polynomials S.2.11-1. Laguerre polynomials and generalized Laguerre polynomials S.2.11-2. Chebyshev polynomials S.2.11-3. Hermite polynomials

Page xix

S.2.11-4. Gegenbauer polynomials S.2.11-5. Jacobi polynomials S.2.12. The Weierstrass Function S.2.12-1. Definitions S.2.12-2. Some properties S.3. Tables of Indefinite Integrals S.3.1. Integrals Containing Rational Functions S.3.1-1. Integrals containing  + ' S.3.1-2. Integrals containing  +  and  +  S.3.1-3. Integrals containing  2 +  2 S.3.1-4. Integrals containing  2 −  2 S.3.1-5. Integrals containing  3 +  3 S.3.1-6. Integrals containing  3 −  3 S.3.1-7. Integrals containing  4 AI 4 S.3.2. Integrals Containing Irrational Functions S.3.2-1. Integrals containing  1 J 2 S.3.2-2. Integrals containing (  + ' )K J 2 S.3.2-3. Integrals containing ( 2 +  2 )1 J 2 S.3.2-4. Integrals containing ( 2 −  2 )1 J 2 S.3.2-5. Integrals containing (  2 −  2 )1 J 2 S.3.2-6. Reduction formulas S.3.3. Integrals Containing Exponential Functions S.3.4. Integrals Containing Hyperbolic Functions S.3.4-1. Integrals containing cosh  S.3.4-2. Integrals containing sinh  S.3.4-3. Integrals containing tanh  or coth  S.3.5. Integrals Containing Logarithmic Functions S.3.6. Integrals Containing Trigonometric Functions S.3.6-1. Integrals containing cos  S.3.6-2. Integrals containing sin  S.3.6-3. Integrals containing sin  and cos  S.3.6-4. Reduction formulas S.3.6-5. Integrals containing tan  and cot  S.3.7. Integrals Containing Inverse Trigonometric Functions References

Page xx

AUTHORS Andrei D. Polyanin, Ph.D., D.Sc., is a noted scientist of broad interests who works in various areas of mathematics, mechanics, and chemical engineering sciences. A. D. Polyanin graduated from the Department of Mechanics and Mathematics of the Moscow State University in 1974. He earned his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, A. D. Polyanin has been a member of the staff of the Institute for Problems in Mechanics of the Russian Academy of Sciences. He is a member of the Russian National Committee on Theoretical and Applied Mechanics. Professor Polyanin has made important contributions to developing new exact and approximate analytical methods of the theory of differential equations, mathematical physics, integral equations, engineering mathematics, nonlinear mechanics, theory of heat and mass transfer, and chemical hydrodynamics. He obtained exact solutions for several thousand ordinary differential, partial differential, mathematical physics, and integral equations. Professor Polyanin is an author of 30 books in English, Russian, German, and Bulgarian, as well as over 120 research papers and three patents. He has written a number of fundamental handbooks, including A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (first edition), CRC Press, 1995; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002; A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; and A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Mathematical Physics Equations, Fizmatlit, 2002. Professor Polyanin is Editor of the book series Differential and Integral Equations and Their Applications, Taylor & Francis Inc., London. In 1991, A. D. Polyanin was awarded a Chaplygin Prize of the Russian Academy of Sciences for his research in mechanics. Address: Institute for Problems in Mechanics, RAS, 101 Vernadsky Avenue, Building 1, 119526 Moscow, Russia E-mail: [email protected]

Valentin F. Zaitsev, Ph.D., D.Sc., is a noted scientist in the fields of ordinary differential equations, mathematical physics, and nonlinear mechanics. V. F. Zaitsev graduated from the Radio Electronics Faculty of the Leningrad Polytechnical Institute (now St. Petersburg Technical University) in 1969 and received his Ph.D. degree in 1983 at the Leningrad State University. His Ph.D. thesis was devoted to the group approach to the study of some classes of ordinary differential equations. In 1992, Professor Zaitsev earned his Doctor of Sciences degree; his D.Sc. thesis was dedicated to the discretegroup analysis of ordinary differential equations. In 1971–1996, V. F. Zaitsev worked in the Research Institute for Computational Mathematics and Control Processes of the St. Petersburg State University. Since 1996, Professor Zaitsev has been a member of the staff of the Russian State Pedagogical University (St. Petersburg). Professor Zaitsev has made important contributions to new methods in the theory of ordinary and partial differential equations. He is an author of more than 130 scientific publications, including 18 books and one patent. Address: Russian State Pedagogical University, 48 Naberezhnaya reki Moiki, 191186 St. Petersburg, Russia E-mail: [email protected]

Page xxi

FOREWORD Exact solutions of differential equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. These solutions can be used to verify the consistencies and estimate errors of various numerical, asymptotic, and approximate analytical methods. This book contains nearly 6200 ordinary differential equations and their solutions. A number of new solutions to nonlinear equations are described. In some sections of this book, asymptotic solutions to some classes of equations are also given. When selecting the material, the authors gave preference to the following two types of equations: • Equations that have traditionally attracted the attention of many researchers: those of the simplest appearance but involving the most difficulties for integration (Abel equations, Emden– Fowler equations, Painlev´e equations, etc.) • Equations that are encountered in various applications (in the theory of heat and mass transfer, nonlinear mechanics, elasticity, hydrodynamics, theory of nonlinear oscillations, combustion theory, chemical engineering science, etc.) Special attention is paid to equations that depend on arbitrary functions. All other equations contain one or more arbitrary parameters (in fact, this book deals with whole families of ordinary differential equations), which can be fixed by a reader at will. In total, the handbook contains many more equations than any other book currently available (for example, the number of nonlinear equations of the second and higher order is ten times more than in the well-known E. Kamke’s Handbook on Ordinary Differential Equations). For the reader’s convenience, the introductory chapter of the book outlines basic definitions, useful formulas, and some transformations. In a concise form, it also presents exact, asymptotic, and approximate analytical methods for solving linear and nonlinear differential equations. Specific examples of utilization of these methods are considered. Formulations of existence and uniqueness theorems are also given. Boundary-value problems and eigenvalue problems are described. The handbook consists of chapters, sections, subsections, and paragraphs. Equations and formulas are numbered separately in each subsection. The equations within subsections and paragraphs are arranged in increasing order of complexity. The extensive table of contents provides rapid access to the desired equations. The main material is followed by some supplements, where basic properties of elementary and special functions (Bessel, modified Bessel, hypergeometric, Legendre, etc.) are described. Here are three main distinguishing features of the second edition vs. the first edition: • 1200 nonlinear equations with solutions have been added. • An introductory chapter that outlines exact, asymptotic, and approximate analytical methods for solving ordinary differential equations has been included. • The overwhelming majority of subsections are organized into paragraphs. As a result, the table of contents has been increased threefold to help the readers get faster access to desired equations. We would like to express our deep gratitude to Alexei Zhurov for fruitful discussions and valuable remarks. We are very grateful to Alain Moussiaux who tested a number of solutions, which allowed us to remove some inaccuracies and misprints. The authors hope that this book will be helpful for a wide range of scientists, university teachers, engineers, and students engaged in the fields of mathematics, physics, mechanics, control, chemistry, and engineering sciences. Andrei D. Polyanin Valentin F. Zaitsev

Page xxiii

NOTATIONS AND SOME REMARKS 1. Throughout this book, in the original equations the independent variable is denoted by  , and the dependent one is denoted by . In the given solutions, the symbols L , L 0 , L 1 , L 2 ,  stand for arbitrary integration constants. Solutions are often represented in parametric form (e.g., see Subsections 1.3.1 and 2.3.1). 2. Notation for derivatives:


= M ,  M

2

3



 = M 2 ,  M




 = M 3 ,  M

3. Brief notation for partial derivatives:

   = N ,  N



4





 = M 4 ;  M

 (  ) = M   M

with

B ≥ 5.

2     = N , where  =  ( , ).  N N N  4. In some cases, we use the operator notation O M , which is defined by the recurrence 3P M relation   −1 O ( ) M ( ) =  ( ) M O ( ) M (  )S . 2P RQ 2P M M M 5. Brief operator notation corresponding to partial derivatives:  = N , = N . N N   N N

  = N

,

2    = N 2 ,  N

6. In some sections of the book (e.g., see 1.3, 2.3–2.6, 3.2–3.3), for the sake of brevity, solutions are represented as several formulas containing the terms with the signs “A ” and “T .” Two formulas are meant—one corresponds to the upper sign and the other to the lower sign. For example, the solution of equation 1.3.1.6 can be written in the parametric form:

 = 

−1

exp(T"U 2 ),

= 

−1

V exp(T"U 2 ) A 2UW ,

where

 = X exp(T U 2 ) U − L , M

# = T 2 2.

This is equivalent so that the solutions of equation 1.3.1.6 are given by the two formulas:

 = 

−1

exp(−U 2 ),

= 

−1

V exp(−U 2 ) + 2UW ,

where

 = X exp(−U 2 ) U − L , M

V exp(U 2 ) − 2UW ,

where

 = X exp(U 2 ) U − L , M

# = −2

2

(for the upper signs) and

 = 

−1

exp(U 2 ),

= 

−1

# = 2

2

(for the lower signs).

 ( ) 7. If a relation contains an expression like , it is often not stated that the assumption  ≠ 2  −2 is adopted. 1 8. In solutions, expressions like Y  ( ) =   +1 can usually be defined so as to cover the B +1 case B = −1 in accordance with the rule Y −1 ( ) = ln | |. This is accounted for by the fact that such expressions arise from the integration of the power-law function Y  ( ) = XZ   . M

9. The order symbol [ is used to compare two functions  =  ( ) and = ( ), where is a small parameter. So  = [ ( ) means that |  | is bounded as \D 0, or  and are of the same order of magnitude as ID 0.

Page xxv

10. When referencing a particular equation, a notation like “4.1.2.5” stands for “equation 5 in Subsection 4.1.2.” 11. The handbooks by Kamke (1977), Murphy (1960), Zaitsev and Polyanin (1993, 2001), Polyanin and Zaitsev (1995) were extensively used in compiling this book; references to these sources are frequently omitted.

]

^_

12. To highlight portions of the text, the following symbols are used throughout the book: indicates important information pertaining to a group of equations (Chapters 1–5); indicates the literature used in the preparation of the text in subsections, paragraphs, and specific equations.

Page xxvi

Introduction

Some Definitions, Formulas, Methods, and Transformations 0.1. First-Order Differential Equations 0.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 0.1.1-1. Equations solved for the derivative. General solution. A first-order ordinary differential equation* solved for the derivative has the form


=  ( , ).

(1)

=  ( , )  . Sometimes it is represented in terms of differentials as M M A solution of a differential equation is a function ( ) that, when substituted into the equation, turns it into an identity. The general solution of a differential equation is the set of all its solutions. In some cases, the general solution can be represented as a function = Y ( , L ) that depends on one arbitrary constant L ; specific values of L define specific solutions of the equation (particular solutions). In practice, the general solution more frequently appears in implicit form, ` ( , , L ) = 0, or parametric form,  =  (a , L ), = (a , L ). Geometrically, the general solution (also called the general integral) of an equation is a family of curves in the  -plane depending on a single parameter L ; these curves are called integral curves of the equation. To each particular solution (particular integral) there corresponds a single curve that passes through a given point in the plane. For each point ( , ), the equation 
 =  ( , ) defines a value of 
, i.e., the slope of the integral curve that passes through this point. In other words, the equation generates a field of directions in the  -plane. From the geometrical point of view, the problem of solving a first-order differential equation involves finding the curves, the slopes of which at each point coincide with the direction of the field at this point. 0.1.1-2. The Cauchy problem. The uniqueness and existence theorems. 1 b . The Cauchy problem: find a solution of equation (1) that satisfies the initial condition =

0

at

 =  0,

(2)

where 0 and  0 are some numbers. Geometrical meaning of the Cauchy problem: find an integral curve of equation (1) that passes through the point ( 0 , 0 ). Condition (2) is alternatively written ( 0 ) = 0 or |  =  0 = 0 . * In what follows, we often call an ordinary differential equation a “differential equation” or even shorter an “equation.”

Page 1

2 b . Existence theorem (Peano). Let the function  ( , ) be continuous in an open domain c of the  -plane. Then there is at least one integral curve of equation (1) that passes through each point ( 0 , 0 ) dec ; each of these curves can be extended at both ends up to the boundary of any closed domain c 0 f c such that ( 0 , 0 ) belongs to the interior of c 0 . 3 b . Uniqueness theorem. Let the function  ( , ) be continuous in an open domain c and have in c a bounded partial derivative with respect to (or the Lipschitz condition holds: |  ( , ) −  ( , H )| ≤ 8 | − H |, where 8 is some positive number). Then there is a unique solution of equation (1) satisfying condition (2). 0.1.1-3. Equations not solved for the derivative. The existence theorem. A first-order differential equation not solved for the derivative can generally be written as

 ( , , 
) = 0.

(3)

Existence and uniqueness theorem. There exists a unique solution = ( ) of equation (3) satisfying the conditions |  =  0 = 0 and 
 |  =  0 = a 0 , where a 0 is one of the real roots of the equation  ( 0 , 0 , a 0 ) = 0, if the following conditions hold in a neighborhood of the point ( 0 , 0 , a 0 ): 1. The function  ( , , a ) is continuous in each of the three arguments. 2. The partial derivative hg exists and is nonzero. 3. There is a bounded partial derivative with respect to , |  | ≤ 8 .



The solution exists for | −  0 | ≤  , where  is a (sufficiently small) positive number. 0.1.1-4. Singular solutions. 1 b . A point ( , ) at which the uniqueness of the solution to equation (3) is violated is called a singular point. If conditions 1 and 3 of the existence and uniqueness theorem hold, then

 ( , , a ) = 0,

hg ( , , a ) = 0

(4)

simultaneously at each singular point. Relations (4) define a a -discriminant curve in parametric form. In some cases, the parameter a can be eliminated from (4) to give an equation of this curve in implicit form, i ( , ) = 0. If a branch = j ( ) of the curve i ( , ) = 0 consists of singular points and, at the same time, is an integral curve, then this branch is called a singular integral curve and the function = j ( ) is a singular solution of equation (3). 2 b . The singular solutions can be found by identifying the envelope of the family of integral curves, ` ( , , L ) = 0, of equation (3). The envelope is part of the L -discriminant curve, which is defined by the equations ` ( , , L ) = 0, `lk ( , , L ) = 0. The branch of the L -discriminant curve at which (a) there exist bounded partial derivatives, | `  | < 8 (b) | `  | + | ` | ≠ 0 is the envelope.

1

and | `



 |<8

2,

and

0.1.1-5. Point transformations. In the general case, a point transformation is defined by

 =  (m , n ),

= 4 (m , n ),

(5)

Page 2

where m is the new independent variable, n = n (m ) is the new dependent variable, and  are some (prescribed or unknown) functions. The derivative 
under the point transformation (5) is calculated by


=

4po + 4Iqrn ho + s  qrn o

and 4


o ,


where the subscripts m and n denote the corresponding partial derivatives. Transformation (5) is invertible if tou4Iq − sqr4po ≠ 0. Point transformations are used to simplify equations and reduce them to known equations. Sometimes a point transformation allows the reduction of a nonlinear equation to a linear one. Examples of point transformations can be found in Subsections 0.1.2 and 0.1.4–0.1.6.

^_

References for Subsection 0.1.1: G. M. Murphy (1960), G. A. Korn and T. M. Korn (1968), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980), D. Zwillinger (1998).

0.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration 0.1.2-1. Equations with separated or separable variables. 1 b . An equation with separated variables (a separated equation) has the form

 ( ) 
= ( ).

Equivalently, the equation can be rewritten as  ( ) = ( )  (the right-hand side depends M M on  alone and the left-hand side on alone). The general solution can be obtained by termwise integration: Xv ( ) = Xw ( )  + L ,

M

where L is an arbitrary constant.

M

2 b . An equation with separable variables (a separable equation) is generally represented by

 1 ( ) 1( ) 
=  2 ( ) 2 ( ).

Dividing the equation by  2 ( ) 1 ( ), one obtains a separated equation. Integrating yields:

X x3y'z|{}~/€

 1( )  2( ) M

=X

2 ( )  +L . 1 ( ) M

In termwise division of the equation by  2 ( ) 1 ( ), solutions corresponding to

 2 ( ) = 0 can be lost.

0.1.2-2. Equation of the form 
 =  (  +  ). The substitution H =  +  Paragraph 0.1.2-1.

brings the equation to a separable equation, H 
=  ( H ) +  ; see

0.1.2-3. Homogeneous equations and equations reducible to them. 1 b . A homogeneous equation remains the same under simultaneous scaling (dilatation) of the independent and dependent variables in accordance with the rule ‚D„ƒ… , D†ƒ , where ƒ is an arbitrary constant ( ƒ ≠ 0). Such equations can be represented in the form


= *O

3P

.

The substitution ‡ =   brings a homogeneous equation to a separable one, ‡
 =  (‡ ) − ‡ ; see Paragraph 0.1.2-1.

Page 3

2 b . The equations of the form

+( 1 2 +( 2P can be reduced to a homogeneous equation. To this end, for  1  +  1 ≠ , (  2  +  2 ), one should use the change of variables ˆ =  −  0 , ‰ = − 0 , where the constants  0 and 0 are determined by solving the linear algebraic system


= *O

 1

 1 +   2 + 

+  2 0 +  0

1

+( 0+(

1 0

1

2

2

= 0, = 0.

As a result, one arrives at the following equation for ‰ = ‰ (ˆ ):  ˆ +  1‰ ‰ Š
= *O 1 .  2 ˆ +  2 ‰‹P Dividing the argument of  by ˆ , one obtains a homogeneous equation whose right-hand side depends on the ratio ‰ ˆ only. For  1  +  1 = , (  2  +  2 ), see the equation of Paragraph 0.1.2-2. 0.1.2-4. Generalized homogeneous equations and equations reducible to them. 1 b . A generalized homogeneous equation (a homogeneous equation in the generalized sense) remains the same under simultaneous scaling of the independent and dependent variables in accordance with the rule |D†ƒ… , D†ƒ  , where ƒ ≠ 0 is an arbitrary constant and , is some number. Such equations can be represented in the form


=  

−1

 (  −  ).

The substitution ‡ =  −  brings a generalized homogeneous equation to a separable equation, ‡
 =  (‡ ) − ,!‡ ; see Paragraph 0.1.2-1. 2 b . The equations of the form


=  ( 



) can be reduced to a generalized homogeneous equation. To this end, one should use the change of  variable H =  and set Œ = − , . 0.1.2-5. Linear equation. A first-order linear equation is written as


+  ( ) = ( ). The solution is sought in the product form = ‡" , where  =  ( ) is any function that satisfies the “truncated” equation  
+  ( ) = 0 [as  ( ) one takes the particular solution  =  − Ž , where  = e ( )  ]. As a result, one obtains the following separable equation for ‡ = ‡ ( ):  ( )‡
 = ( ). M Integrating it yields the general solution: ( ) =  − Ž O X  Ž ( )  + L

M

P

,

where

 = Xv ( )  . M

0.1.2-6. Bernoulli equation. A Bernoulli equation has the form


+  ( ) = ( ) ,

1−

 ≠ 0, 1.  brings it to a linear equation, H 
+ (1 −  )  ( )H = (1 −  ) ( ), which is

The substitution H = discussed in Paragraph 0.1.2-5. With this in view, one can obtain the general integral: 1−

 = L  − Ž + (1 −  )  − Ž X  Ž ( )  , M

where

 = (1 −  ) Xv ( )  . M

Page 4

0.1.2-7. Equation of the form  
 =

The substitution ‡ =   Paragraph 0.1.2-1.

+  ( ) (  ).

brings the equation to a separable equation, 

2

‡
 =  ( ) (‡ ); see

0.1.2-8. Darboux equation. A Darboux equation can be represented as

Q

*O

2P

+   %lO

S 
= rO

3P

+  

3P

−1

%lO

.

3P

Using the substitution = "H ( ) and taking H to be the independent variable, one obtains a Bernoulli equation, which is considered in Paragraph 0.1.2-6:

V ( H ) − H! ( H )W  ‘
=   ( H ) +  

^_

+1

% ( H ).

References for Subsection 0.1.2: G. M. Murphy (1960), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980), A. Moussiaux (1996), D. Zwillinger (1998).

0.1.3. Exact Differential Equations. Integrating Factor 0.1.3-1. Exact differential equations. An exact differential equation has the form

 ( , )  + ( , ) M M

= 0,

where

N

 N

= N

N





.

The left-hand side of the equation is the total differential of a function of two variables ’ ( , ). The general integral, ’ ( , ) = L , where the function ’ is determined from the system:

N

’ N

= ,



’ N

= .

N

Integrating the first equation yields ’ = e ( , )  + i ( ) (while integrating, the variable is M treated as a parameter). On substituting this expression into the second equation, one identifies the function i (and hence, ’ ). As a result, the general integral of an exact differential equation can be represented in the form:  where 

0

and

0

X   (ˆ , ) ˆ + X  ( 0 , ‰ ) ‰ = L , M M 0 0

are arbitrary numbers.

0.1.3-2. Integrating factor. An integrating factor for the equation

 ( , )  + ( , ) M M

=0

“

“ , *•

is a function “ ( , ) 1 0 such that the left-hand side of the equation, when multiplied by “ ( , ), becomes a total differential, and the equation itself becomes an exact differential equation. An integrating factor satisfies the first-order partial differential equation:

lN N



− 3N

N

“

= ”‹N

N



− N

N

which is not generally easier to solve than the original equation. Table 1 lists some special cases where an integrating factor can be found in explicit form.

^_

References for Subsection 0.1.3: G. M. Murphy (1960), N. M. Matveev (1967), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980), A. Moussiaux (1996), D. Zwillinger (1998).

Page 5

TABLE 1 An integrating factor “ = “ ( , ) for some types of ordinary differential equations   + = 0, where  =  ( , ) and = ( , )

M

No

Conditions for  and

1  = Y ( 2 3 4 5

M

), = j (

  =  , –›š − —'œ — –›š − – —'œ –›š —

− −

Integrating factor

“ = –

)

  = − 

“ = –

1 − 1

Remarks

  − *1 0; Y ( H ) and j ( H ) are any functions

˜—

 + ™ is an analytic function of the complex variable  + ™

2+ 2

—

“ = exp V  Y ( )  W M V  “ = exp − Y ( ) W M V  “ = exp Y ( H ) HžW , H =  + M “ = exp V  Y ( H ) HžW , H =  M “ = exp V −  Y ( H ) H W , H =  M “ = exp V 12  Y ( H ) HžW , H =  2 + M “ = exp V  Y ( )  +  j ( ) W M M “ = exp V  Y (  )   W M

= Y ( ) =Y ( )

—'– œ = Y ( + )

›– š − '— œ – = Y ( ) ˜— −  2 ( –›š − ) 7 —'– œ = Yu   ˜— + –›š  −−—'œ – = Y ( 2 + 2 ) 8 —  9   −  = Y (  ) − j ( )  –›š − —'– œ š = Y (  ) 10 —'Ÿ!œ − Ÿ 6

Y ( ) is any function Y ( ) is any function Y ( H ) is any function Y ( H ) is any function Y ( H ) is any function 2

Y ( H ) is any function Y ( ) and j ( ) are any functions   =   ( , ) is any function of two variables

0.1.4. Riccati Equation 0.1.4-1. General Riccati equation. Simplest integrable cases. A Riccati equation has the general form


=  2 ( )

2

+  1 ( ) +  0 ( ).

(1)

If  2 ≡ 0, we have a linear equation (see Paragraph 0.1.2-5), and if  0 ≡ 0, we have a Bernoulli equation (see Paragraph 0.1.2-6 for  = 2), whose solutions were given previously. For arbitrary  2 ,  1 , and  0 , the Riccati equation is not integrable by quadrature. Listed below are some special cases where the Riccati equation (1) is integrable by quadrature. 1 b . The functions  2 ,  1 , and 

0

are proportional, i.e.,


= Y ( )( 

2

+

+ ( ),

where  ,  , and ( are constants. This equation is a separable equation; see Paragraph 0.1.2-1. 2 b . The Riccati equation is homogeneous:


=  

2 2

+



+( .

See Paragraph 0.1.2-3.

Page 6

3 b . The Riccati equation is generalized homogeneous:


=  

2

+





+ (˜ − 

−2

.

See Paragraph 0.1.2-4 (with , = −B − 1). The substitution H =   equation: "H 
= !H 2 + (  + B + 1) H + ( .

+1

brings it to a separable

4 b . The Riccati equation has the form


=  2  By the substitution

= +

−

2

+ ¡



−B

+ (˜ 2 + .

 H , the equation is reduced to a separable equation:  −  − + H 
= !H 2 + ( .

For other Riccati equations integrable by quadrature, see Section 1.2. 0.1.4-2. Polynomial solutions of the Riccati equation. Let  2 = 1,  1 ( ), and  0 ( ) be polynomials. If the degree of the polynomial ¢ =  12 − 2(  1)
 − 4  0 is odd, the Riccati equation cannot possess a polynomial solution. If the degree of ¢ is even, the equation involved may possess only the following polynomial solutions: = − 12  

1

A ¤V £ ¥ ¢ W ,

where V £ ¢¥W denotes an integer rational part of the expansion of £ ¢ (for example, V £  2 − 2 + 3 W =  − 1).

in decreasing powers of 

0.1.4-3. Use of particular solutions to construct the general solution. 1 b . Given a particular solution written as: =

0

=

0(

0(

 ) of the Riccati equation (1), the general solution can be

 ) + ` ( ) L − X ` ( )  2 ( )  S Q M

−1

,

where

` ( ) = exp ¦/X V 2  2 ( ) 0 ( ) +  1 ( )W t  § . M To the particular solution 0 ( ) there corresponds L = F . 2 b . Let 1 = 1 ( ) and 2 = 2 ( ) be two different particular solutions of equation (1). Then the general solution can be calculated by: =

L

To the particular solution

+ ’ ( ) L + ’ ( ) 1

1(

2

,

X  2( 1 − 2)  S . ’ ( ) = exp v Q M

where

 ), there corresponds L = F ; and to

2(

 ), there corresponds L = 0.

3 b . Let 1 = 1 ( ), 2 = 2 ( ), and 3 = 3 ( ) be three distinct particular solutions of equation (1). Then the general solution can be found without quadrature: − −

2 1

− 3−

3

1 2

=L .

This means that the Riccati equation has a fundamental system of solutions.

Page 7

0.1.4-4. Some transformations. 1 b . The transformation ( Y , j

1,

j 2 , j 3 , and j

are arbitrary functions) j 4 (ˆ )‡ + j 3 (ˆ )  = Y (ˆ ), = j 2 (ˆ )‡ + j 1 (ˆ ) reduces the Riccati equation (1) to a Riccati equation for ‡ = ‡ (ˆ ). 4

2 b . Let 0 = 0 ( ) be a particular solution of equation (1). Then the substitution leads to a linear equation for ¨ = ¨ ( ):

=

0

+ 1 ¨

¨ 
+ V 2  ( ) ( ) +  ( )W ¨ +  ( ) = 0. 2 0 1 2

For solution of linear equations, see Paragraph 0.1.2-5. 0.1.4-5. Reduction of the Riccati equation to a second-order linear equation. The substitution

‡ ( ) = exp O − Xv

2

 M P

reduces the general Riccati equation (1) to a second-order linear equation:

 2 ‡

 − V (  2 )
 +  1  2 W‡
 +  0  22 ‡ = 0,

which often may be easier to solve than the original Riccati equation. 0.1.4-6. Reduction of the Riccati equation to the canonical form. 1 b . The general Riccati equation (1) can be reduced with the aid of the transformation

 = Y (ˆ ),

1 ¨ 1 = −  2 2

to the canonical form: Here, the function i

1 2




1 1 + ” , 2  2• Š

¨ Š
= ¨

2

where

h© (ˆ ) = © ( Y ) Y Š
,

+ i (ˆ ).

(2) (3)

is defined by the formula:

1 2 1
1  2
3 
1  2

 1 +  1−  1 − ” 2 + ; 4 2 2  2 4  2• 2  2 the prime denotes differentiation with respect to ˆ . Transformation (2) depends on a function Y = Y (ˆ ) that can be arbitrary. For a specific original Riccati equation, different functions Y in (2) will generate different functions i in equation (3). [In practice, transformation (2) is most frequently used with Y (ˆ ) = ˆ ].

i (ˆ ) =  0 

2

2

−

2 b . In the special case where the original equation has the canonical form


=

2

transformation (2) is written as

 = Y (ˆ ),

=

+  ( ),



1 ¨ 1 Y ŠªŠ − , Y Š
2 ( Y Š
)2

and the transformed equation (3) is determined by the function



3 Y ŠªŠ i (ˆ ) =  ( Y )(Y Š
)2 − ” Š
4 Y •

2

+




1 Y ŠªŠªŠ . 2 Y Š


If the original Riccati equation is integrable by quadrature, one may obtain, specifying different functions Y , a variety of different integrable equations of the form (3). In Subsection 1.2.9, some useful transformations are given for specific functions Y .

^_

References for Subsection 0.1.4: G. M. Murphy (1960), E. L. Ince (1964), W. T. Reid (1972), E. Kamke (1977), W. E. Boyce and R. C. DiPrima (1986), A. D. Polyanin and V. F. Zaitsev (1995).

Page 8

0.1.5. Abel Equations of the First Kind 0.1.5-1. General form of Abel equations of the first kind. Simplest integrable cases. An Abel equation of the first kind has the general form


=  3 ( )

3

2

+  2 ( )

+  1 ( ) +  0 ( ),

 3 ( ) 1 0.

(1)

In the degenerate case  2 ( ) =  0 ( ) = 0, we have a Bernoulli equation (see Paragraph 0.1.2-6 with  = 3). The Abel equation (1) is not integrable for arbitrary   ( ). Listed below are some special cases where the Abel equation of the first kind is integrable by quadrature. 1 b . If the functions   ( ) (B = 0, 1, 2, 3) are proportional, i.e.,   ( ) =   ( ), then (1) is a separable equation (see Paragraph 0.1.2-1). 2 b . The Abel equation is homogeneous: 3


=  

3

+

2



+(

2

+ .



M

See Paragraph 0.1.2-3. 3 b . The Abel equation is generalized homogeneous:


=  2 

+ ' 

+1 3

2

+



(

+  −

M

See Paragraph 0.1.2-4 for , = −B − 1. The substitution ¨  ¨2
=  ¨ 3 +  ¨ 2 + ( ( + B + 1)¨ + .

= 

−2

+1

. leads to a separable equation:

M

4 b . The Abel equation


=  3  − + can be reduced with the substitution

5 b . Let  0 ≡ 0,  1 ≡ 0, and (  3  2 )
 leads to a separable equation: ‡
 =  6 b . If

 0=

+ ' 2 

3

2

+ ¡



−B

+  2+

M + − = H to a separable equation:  −  − + H 
= !H 3 + ˜H 2 + ( . =  2 for some constant  . Then the substitution =  2  3−1 ‡ 2 −1 3 + ‡ 2 + ‡ ). 2  3 (‡

 1  2 2  23 1  − − M 2, 3 3 27  32 3   3 M

  =   ( ),

then the solution of equation (1) is given by: ( ) = « O L − 2 Xv 3 «

2



J

−1 2

M P



−

3

2 3

,

where

« = exp ¬Xv” 1 −



3

2 2 3

•­M

 ® .

For other solvable Abel equations of the first kind, see Subsection 1.4.1. 0.1.5-2. Reduction to the canonical form. Reduction to an Abel equation of the second kind. 1 b . The transformation = ’ ( )‰ (ˆ ) −



3

2 3

,

ˆ = X  3’

2

M

 ,

where

’ ( ) = exp ¬ X ” 1 −



3

2 2 3

•­M

 ® ,

brings equation (1) to the canonical (normal) form:

‰ Š
= ‰ 3 + ` (ˆ ).

Page 9

Here, the function ` (ˆ ) is defined parametrically ( is the parameter) by the relations: 1

` = 2 b . Let

0

=

0(

 3’

3

” 0 −

 1  2 2  23 1  + + M 2 , 3 3 27  32 3   3 • M

ˆ = X¯ 3 ’

2

M

 .

 ) be a particular solution of equation (1). Then the substitution =

0

+

« ( ) , H ( )

« ( ) = exp X (3  Q

where

2 3 0

+ 2

2 0

+  1)  S ,

M

leads to an Abel equation of the second kind:

HH 
= −(3 

3 0

2

+  2 ) «IH −  3 «

.

For equations of this type, see Subsection 0.1.6.

^_

References for Subsection 0.1.5: E. Kamke (1977), G. M. Murphy (1960).

0.1.6. Abel Equations of the Second Kind 0.1.6-1. General form of Abel equations of the second kind. Simplest integrable cases. An Abel equation of the second kind has the general form


[ + ( )]  =  2 ( )

2

+  1 ( ) +  0 ( ),

( ) 1 0.

(1)

The Abel equation (1) is not integrable for arbitrary   ( ). Given below are some special cases where the Abel equation of the second kind is integrable by quadrature. 1 b . If ( ) = const and the functions   ( ) (B = 0, 1, 2) are proportional, i.e.,   ( ) =   ( ), then (1) is a separable equation (see Paragraph 0.1.2-1). 2 b . The Abel equation is homogeneous:


( + °˜ )  = 



2

+

+ (˜ .

See Paragraph 0.1.2-3. 3 b . The Abel equation is generalized homogeneous:

 ( + °˜  ) 
=

2



+ ' 

+ (˜ 2 

−1

See Paragraph 0.1.2-4 for , = B . The substitution ¨  (¨ + ° )¨2
= (  − B )¨ 2 + (  − B±° )¨ + ( .

=

−1

.

 −  leads to a separable equation:

4 b . The general solution of the Abel equation ( + ) 
= 

2

2

+

1

+  1 −  2 2,

  =   ( ), = ( ),

is given by: = − + Lu« + « X (  5 b . If 

1

+ 
− 2  2 )«

−1

M

 ,

where

« = exp O X 

= 2  2 − 
, the general solution of the Abel equation (1) has the form: = − *AC«

^_

1

Q

2 X (

0

+  
−  2 2 ) «

−2

M

 + L2S

J

1 2

,

where

2

 . M P

« = exp O X 

2

 . M P

For other solvable Abel equations of the second kind, see Section 1.3. References for Paragraph 0.1.6-1: G. M. Murphy (1960), E. Kamke (1977).

Page 10

0.1.6-2. Reduction to the canonical form. Reduction to an Abel equation of the first kind. 1 b . The substitution

¨ = ( + )« ,

« = exp O − X 

where

 , M P

2

(2)

brings equation (1) to the simpler form:

¨*¨ 
=  ( )¨ +  ( ), 1 0 where



1

= (

1


− 2  2 +  )« , 

0

= (

0

(3)

−  1 +  2 2)«

2

.

2 b . In turn, equation (3) can be reduced, by the introduction of the new independent variable

H = X  1 ( )  , M

(4)

to the canonical form:

¨*¨ ‘
− ¨ = ² ( H ). Here, the function ² ( H ) is defined parametrically ( is the parameter) by the relations  0 ( ) ,  1 ( )

² =

(5)

H = X  1 ( )  . M

Substitutions (2) and (4), which take the Abel equation to the canonical form, are called canonical. x3y'z|{}~­³´€ The transformation ¨ =  ¨ ˆ , H =  H ˆ +  brings (5) to a similar equation, ¨ ˆ ¨I ˆ ‘
ˆ − ¨ ˆ = −1  ² (  H ˆ +  ). Therefore the function ² (H ) in the right-hand side of the Abel equation (5) can be identified with the two-parameter family of functions  −1 ² ( !H +  ).

x3y'z|{}~‚µ € Any Abel equations of the second kind related by linear (in ) transformations ¶  = Y 1 ( ), = Y 2 ( ) + Y 3 ( ) have identical canonical forms (up to the two-parameter family of ¶

functions specified in Remark 1). 3 b . The substitution

+ = 1 ‡ leads to an Abel equation of the first kind:

‡
 + (  0 −  1 +  2 2 )‡

3

+ (

1

− 2  2 + 
)‡

2

+  2 ‡ = 0.

For equations of this type, see Subsection 0.1.5.

^_

References for Paragraph 0.1.6-2: G. M. Murphy (1960), E. Kamke (1977), V. F. Zaitsev and A. D. Polyanin (1994).

0.1.6-3. Use of particular solutions to construct self-transformations. 1 b . Suppose a particular solution Then the substitution ’ = 1 ( −

0

0)

= 0 ( ) of an Abel equation of the second kind (1) is known. leads to a similar Abel equation:


−  1 ’ 
= 0 0 + ·•

”"’ +

1 0

− 2 +

2 0

’

2

−

 0

2

+

’ .

(6)

If  0 ≡ 0, equation (1) has the trivial particular solution 0 = 0. In this case, the change of variable ’ = 1  leads to an Abel equation of the form (6) with 0 = 0. 2 b . Given a particular solution

the substitution

¨ =

0

5 ( )

2 0( 0

=

− )

 ) of the Abel equation of the second kind ! 
=  ( ) +  ( ), 1 0

0(

,

where

5 ( ) = exp ” X



1 0

 , M •

(7) (8)

Page 11

brings (7) to another, similar Abel equation:

¨*¨ 
= ¸ ( )¨ + ¸ ( ). 1 0 Here, the functions ¸

=¸

1

1(

 ) and ¸ ¸

1

=

0

(

=¸

1 0

0(

 ) are defined by

+ 3  0)5 4 0

(9)

,

¸

0

=

 05

2 6 0

.

It is not difficult to verify by direct substitution that equation (9) has a particular solution:

¨ ( ) = − 5 ( ) . 0 2 0 ( )

(10)

The transformation based on the particular solution (10) brings the Abel equation (9) to the original equation (7) with  1 having the opposite sign. x3y'z|{}~E¹€ In general, the canonical forms of equations (1) and (6) and also those of equations (7) and (9) are different. See Paragraph 0.1.6-2.

x3y'z|{}~Eº€

Given , distinct particular solutions  of equation (7), , distinct Abel equations of the second kind related to (7) by known substitutions of the form (8) can be constructed.

^_

References for Paragraph 0.1.6-3: T. A. Alexeeva, V. F. Zaitsev, and T. B. Shvets (1992), V. F. Zaitsev and A. D. Polyanin (1994).

0.1.6-4. Use of particular solutions to construct the general solution. For some Abel equations of the second kind, the general solution can be found if B its distinct particular solutions  =  ( ), , = 1,  , B , are known. Below we consider Abel equations of the canonical form

! 
− = ² ( ),

(11)

whose general solutions can be represented in the special form:

» 

=1

| −  ( )| +‹¼ = L .

(12)

Here, the particular solutions  =  ( ) correspond to L = 0 (if  > 0) and L = F (if  < 0). ¡ ¡ The logarithmization of (12), followed by the differentiation of the resulting expression and rearrangement, leads to the equation

¾

½¾  ¬ =1

¡

¾

» ½  À −1 ( 
−
) ( −  )® ≡ 
` ¼ =1 =1 ¼ ≠¿

À À

+

½À

−1

i

À À

= 0,

(13)

=1

¾
where = ( )
 . We require that equation (13) be equivalent to the Abel equation (11). To this ¾

end, we set:

iuÁ = − `lÁ ,

iuÁ

−1

= − ² ( ) `lÁ

and equate the other `l© and i3© with zero.

¾

Selecting different values @ = 1, 2,  , B − 1, we obtain B − 1 systems of differential-algebraic equations; only one of the systems, corresponding to  ≠ 0 for all , = 1,  , B and © ≠ ¡ for ™ ≠  , leads to a nondegenerate solution of the form (12). Consider the Abel equations (11) corresponding to the simplest solutions of the form (12) in more detail.

Page 12

1 b . Case B = 2. The system of differential-algebraic equations has the form: 1

¡

1

¡

1

¡ where 8

+

2

¡




1 1 2

¡

=8

2+
¡ 1+

¡

2 2

+

¡

, = 0, 1
=8 , 2
2 1 2 = − 8ò (  ),

(14)

is an arbitrary constant. It follows from the second and third equations that 1

where Ä

=

¡

¡

2 1

1

−

2 2

¡

( 8 + Ä ),

2

2

¡

=−

2 1

¡

−

¡

2 2

( 8 + Ä ),

is an arbitrary constant. Introducing the new constants ( + ) ( + ) # = ¡ 1¡ 2 ¡ 1 ¡ 2 2 8 , $ = ¡ 1¡ 2 ¡ 1 ¡ 2 2 Ä , ( 1 − 2) ( 1 − 2) ¡ ¡ ¡ ¡ we find from the last relation in (14) that

² ( ) = #* + $ ,

(15)

which means that for B = 2 the right-hand side of the Abel equation is a linear function of  (see equation 1.3.1.2). The particular solutions 1 , 2 , and the corresponding exponents 1 , 2 in the general ¡ ¡ integral (12), are expressed in terms of the coefficients # , $ on the right-hand side (15) of the Abel equation (11) as follows: 1 + £ 4# + 1 ( #* + $ ), = 2 # + 1 + £ 4 # + 1, ¡ 1 2# 1 + £ 4# + 1 ( #* + $ ), = 2# . 2 =− ¡ 2 2# + 1 + £ 4# + 1 2 b . Case B = 3. Equation (13) with B = 3 leads to the Abel equation (11) with the right-hand side 1

=

² ( ) = − 29  + # + $I

J

−1 2

(16)

(see equation 1.3.1.3). The particular solutions in the general À and the exponents À À integral (12) are expressed as: À À $ 2 2 3 2# , =  + Œ  1J 2 + , = À 3 3 Œ ¡ 3(2 Œ 2 − 3 # ) where the Œ are roots of the cubic equation

Œ

3

− 92 3 # Œ − 92 $

° = 1, 2, 3.

= 0,

3 b . Case B = 4. Equation (13) with B = 4 leads to the Abel equation (11) with the right-hand side

² ( ) = − 163  + #*

J + $I

−1 3

J

−5 3

(see equation 1.3.3.61). The particular solutions and the exponents in (12) are expressed as: 1,2

= 34 u  AEÅ 3 # + 32 £ −3 $p 1 J 3 + £ −3 $I

−1 3

J ,

3,4

= 34 u  AEÅ 3 # − 32 £ −3À $p 1 J 3 − £ −3 $I

−1 3

J ,

¡

1,2

= T  2 # − £ −3 $  ,

¡

3,4

= A £ 4#

2

+ 3$ .

4 b . Case B > 4. The equations for are algebraic equations of degree B and, in the general case, are not soluble in radicals. The right-hand of equation (11) is expressed as B −1 ² ( ) = − 2  + Æ ( ), where the function Æ ( ) is bounded as EDGF

^_

B

(Æ

can be specified in parametric form).

References for Paragraph 0.1.6-4: B. M. Koyalovich (1894), T. A. Alexeeva, V. F. Zaitsev, and T. B. Shvets (1992).

Page 13

0.1.7. Equations Not Solved for the Derivative 0.1.7-1. The method of “integration by differentiation.” In the general case, a first-order equation not solved for the derivative,

 ( , , 
) = 0,

(1)

can be rewritten in the equivalent form

a = 
.

 ( , , a ) = 0, We look for a solution in parametric form:  =  (a ), in (2), the differential of  is given by:

  M

 +  M

(2)

= (a ). In accordance with the first relation

+ …g

M

a = 0.

(3)

Using the relation = a  , we eliminate successively and  from (3). As a result, we obtain M M M M the system of two first-order ordinary differential equations:

M M

 a

=−

…g ,   + a›  M

M

a

=−

a›…g .   + a› 

(4)

By finding a solution of this system, one thereby obtains a solution of the original equation (1) in parametric form,  =  (a ), = (a ). x3y'z|{}~­³´€ The application of this method may lead to loss of individual solutions; this issue should be additionally investigated. 0.1.7-2. Equations of the form

=  ( 
).

This equation is a special case of equation (1), with  ( , , a ) = in Paragraph 0.1.7-1 yields 
(a )  , =  (a ). M =

M

a

a

−  (a ). The procedure described (5)

Here, the original equation is used instead of the second equation in system (4); this is valid because the first equation in (4) does not depend on explicitly. Integrating the first equation in (5) yields the solution in parametric form:

 =X


(a ) a +L , aÇM

=  (a ).

0.1.7-3. Equations of the form  =  ( 
). This equation is a special case of equation (1), with  ( , , a ) =  −  (a ). The procedure described in Paragraph 0.1.7-1 yields

 =  (a ), M

M

a

= aª
(a ).

(6)

Here, the original equation is used instead of the first equation in system (4); this is valid because the second equation in (4) does not depend on  explicitly. Integrating the second equation in (5) yields the solution in parametric form:

 =  (a ),

= XÈaª
(a ) a + L .

M

Page 14

=  
 +  (
 ).

0.1.7-4. Clairaut’s equation

Clairaut’s equation is a special case of equation (1), with  ( , , a ) = − a −  (a ). It can be rewritten as = a +  (a ), a = 
. (7) and 
in

= 


 by their values in accordance with (7), we obtain [ + 
(a )] a = 0. M
This equation splits into a = 0 and  +  (a ) = 0. The solution of the first equation is obvious: a = L ; it gives the generalM solution of Clairaut’s equation, Substituting

M

M

M

= L3 +  ( L ),

(8)

which is a family of straight lines. The second equation generates a solution in parametric form,

 = − 
(a ),

= −aª
(a ) +  (a ),

which is a singular solution and is the envelope of the family of lines (8). 0.1.7-5. Lagrange’s equation

=   (
 ) + (
 ).

Lagrange’s equation is a special case of equation (1), with  ( , , a ) = −   (a ) − (a ). In the special case  (a ) ≡ a , it coincides with Clairaut’s equation; see Paragraph 0.1.7-4. The procedure described in Paragraph 0.1.7-1 yields

M M

 a

+


(a )
(a )  = ,  (a ) − a a −  (a )

=   (a ) + (a ).

(9)

Here, the original equation is used instead of the second equation in system (4); this is valid because the first equation in (4) does not depend on explicitly. The first equation of system (9) is linear. Its general solution has the form  = Y (a ) L + j (a ); the functions Y and j are defined in Paragraph 0.1.2-5. Substituting this solution into the second equation in (9), we obtain the general solution of Lagrange’s equation in parametric form:

 = Y (a ) L + j (a ),

= V Y (a ) L + j (a )W  (a ) + (a ).

x3y'z|{}~‚µ €

With the above method, solutions of the form = a   + (a  ), where the a  are roots of the equation  (a ) − a = 0, may be lost. These solutions can be particular or singular solutions of Lagrange’s equation.

^_

References for Subsection 0.1.7: G. M. Murphy (1960), E. Kamke (1977).

0.1.8. Contact Transformations 0.1.8-1. General form of contact transformations. A contact transformation has the form

 =  (m , n , n o
),
= 4 (m , n , n o ),

(1)

where the functions  (m , n , ’ ) and 4 (m , n , ’ ) are chosen so that the derivative
 does not depend on n ol

o :



= o
 o

=

4po + 4Iqrn ho + s  qrn




o + 4CÉsn olo


o + tÉsn olo

= 5 (m , n , n o
).

(2)

Page 15

The subscripts m , n , and ’ after  and 4 that tÉ\1 0 and 4CÉ|1 0). It follows from (2) that the relation

N

4

OrN ’ m N N





+ ’ÊN

N

holds; the derivative is calculated by

denote the respective partial derivatives (it is assumed



− N

OrN ’ m N N

n¥P

4

+ ’ÊN

N

4 =0 n P ¥

(3)

4CÉ , tÉ


=

(4)

where 4CÉ tÉ\1 const. The application of contact transformations preserves the order of differential equations. The inverse of a contact transformation can be obtained by solving system (1) and (4) for m , n , n o
.

^_

References for Paragraph 0.1.8-1: D. Zwillinger (1989).

0.1.8-2. A method for the construction of contact transformations. Suppose the function  =  (m , n , ’ ) in the contact transformation (1) is specified. Then relation (3) can be viewed as a linear partial differential equation for the second function 4 . The corresponding characteristic system of ordinary differential equations (see A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, 2002),

M

m

1

admits the obvious first integral:

= M

’

n

=−

tÉ

’  o + ’usq h M

 (m , n , ’ ) = L 1 ,

(5)

where L 1 is an arbitrary constant. It follows that, to obtain the general representation of the function 4 = 4 (m , n , ’ ), one has to deal with the ordinary differential equation

n o
= ’ ,

(6)

whose right-hand side is defined in implicit form by (5). Let the first integral of equation (6) has the form ` (m , n , L 1 ) = L 2 . Then the general representation of 4

= 4 (m , n , ’ ) in transformation (1) is given by:

¶

¶ 4 = i (  , ` ),

where i (  , ` ) is an arbitrary function of two variables, 

¶

=  (m , n , ’ ), and ` = ` (m , n ,  ).

0.1.8-3. Examples of contact transformations linear in the derivative. Example 1. Legendre transformation (E. Kamke, 1977):

Ë

ÌtÎ Í Ð = ÏÒÍ =

Ï

, ,

ÐuÌÑÎ Í − Ì Ì = Ë ÏÒÍ − Ï , =

,

ÏÒÍ = Ð ÌtÎ Í = Ë

(direct transformation); (inverse transformation).

Example 2. Contact transformation ( Ó ≠ 0):

Ë Ð

څÛÝÜ2Þßà´á

ÌtÎ Í

= =

1

Ó

+ ÓÌ , ln

Ï ÒÍ Ô

,

Ï

Î Î ÔÖÕª× t Ì Í , 1 Ë Ï Ì = − ÒÍ Ù Ó2Ø Ï…

ÏÒÍ

=

,

ÔÖÕª× Ì ÎÍ = ÏÒ ÏÍ =

Î

(direct transformation); (inverse transformation).

It is apparent from this example that a contact transformation that is linear in the derivative can have a nonlinear inverse, which is also a contact transformation.

Page 16

Example 3. Contact transformation ( Ó ≠ −1):

Ë

ÌtÎ Í

+

= (Ï

ÒÍ ) ×

=

Ð

Ð

Ó Ì

1 +1

Ï

,

Ì

,

Ð3× +1 Ñ Ì ÎÍ

= =

1 Ë ( +1

Ó

ÏÒÍ

− Ð3×Ì ,

Ï ÒÍ

)− ×

ÒÍ

− Ï )( Ï

Ð3× +1 Ë Ò Ì Î Í = Ï Í + ӁÏÒ ( Ó + 1) Ï Í

× +1 ,

=

(direct transformation); (inverse transformation).

Example 4. Contact transformation:

Ë â

= â ( Ð ) and

where

ã

=

âÌÑÎ Í

+ ãÌ ,

= ( âÌÑÎ Í + ãÌ ) Xåä

Ï

Ì

,

ØX

ã Ð â æ Ù

.

+ ( âè − Õ'é ) Ì

+ ãè −

= ã ( Ð ) are arbitrary functions,

= exp

ä

Ð

âçæ

−

ä

ÏÒÍ

=

Xåäâeæ Ð

,

Example 5. Contact transformation:

Ë where

â

= â ( Ð ) and

Ì ÎÍ

=

ã

+ âÌ

+ã ,

Ï

èÌ Î Í

=

= ã ( Ð ) are arbitrary functions,

ê

X â æÐ

=

è

,

=

X ãÕ'é æ Ð X Õé æÐ

,

Ï ÒÍ

=è ,

.

0.1.8-4. Examples of contact transformations nonlinear in the derivative. Example 1. Contact transformation:

Ë

=

Ð

Ì ÎÍ

+ ӁР,

=

Ë

1

Ӌë

−Ï

Ï

ÒÍ ì

Ì ÎÍ

Ï ÒÍ

)2 + ÓÌ ,

1 = 2Ï − (Ï 2 Ó2í

Ì

,

1 ( 2

=

Ò Í )2 î

,

Ì ÎÍ

,

Ï ÒÍ

= 2Ð

,

Ì ÎÍ

=

=

Ì ÎÍ

(direct transformation);

Ï ÒÍ

(inverse transformation).

=

Example 2. Contact transformation:

Ë

Ì ÎÍ

=

Ð

=

+

Ð

1

Ì

Ï

,

1 Ë ï l Ë ð Ï ÒÍ

−Ï ,

=

Ì

=

Ð

ÎÍ

(Ì

2

)2 − Ì

2

Ë Ï ÒÍ − 2 Ï Ë Ï ÒÍ − Ï

ï

2ð

2

Ë

Õ

=

Î

(Ì

ÎÍ

− Ì ),

Ë Ð = ln ñ Ë ï Ò ð ÏÍ

Ï

− ϱò

,

ÎÍ

= (Ì

)2 − Ì

2

Ë Ò Ì = ï 2Ï Ë − Ò Ï Í 2ð Ï Í −Ï

,

= 2Õ

−

(direct transformation); (inverse transformation).

)

Î

Ì ÎÍ Ë Ï ÒÍ Ì ÎÍ = ï t 2 ð Ë Ï ÒÍ

Ï ÒÍ

,

Ë 2 Ï ÒÍ Ï ÒÍ − Ï

2( Ë

Example 3. Contact transformation

Ì ÎÍ

(direct transformation); (inverse transformation).

−Ï

Example 4. Contact transformation:

Ë

= ( ÌtÎ Í )2 − Ì

2

Ï

,

=

ÌtÎ Í

cosh Ð

− Ì sinh Ð ,

ÏÒÍ

=

cosh Ð 2Ì Î Í

cos Ð

+ Ì sin Ð ,

ÏÒÍ

=

cos Ð 2Ì Î Í

.

Example 5. Contact transformation:

Ë

= ( ÌÑÎ Í )2 + Ì

2

,

Ï

=

ÌtÎ Í

.

Example 5. Contact transformation ( ÓÔ ≠ 0):

Ð

Ë

= Ó ( ÌÑÎ Í )2 − ÔÝÐ , Ó (ÏÒÍ )2 − 1 Ë , = 9Ô Ô

Ï Ì

= 2 Ó ( ÌrÎ Í )3 − 3 ÔÖÌ , 2Ó 1 = ( ÏÒÍ )3 − Ï, 81 Ô 3Ô

ÏÒÍ Ì ÎÍ Ñ

= 3 ÌÑÎ Í 1 = ÏÒÍ 3

(direct transformation); (inverse transformation).

Page 17

Example 7. Contact transformation (D. Zwillinger, 1989, p. 169):

Ë Ð

− ÌrÌtÎ Í ,

Ð

=

Ë

=

Ï

− ÏÏÒÍ ,

ÎÍ

Ì

= − Ì3ó ( Ì =

Ï ð ( Ï ÒÍ

Ì ÎÍ ó ( Ì Î Í )2 − 1 Ò ÌÑÎ Í = − Ï Í 2 Ò ð (Ï Í ) − 1

ÏÒÍ

)2 − 1,

)2 − 1,

=

(direct transformation); (inverse transformation).

Example 8. Contact transformation (D. Zwillinger, 1989, p. 169):

Ë Ð

=

Ð

=

Ë

− +

ÓÌ Î Í , Ï ó ( Ì Î Í )2 + 1 ÓÏ ÒÍ , Ì ð ( Ï Ò Í )2 + 1

= =

Ì

Ó

Ì ÎÍ , ÏÒÍ = t ó ( Ì Î Í )2 + 1 Ó , Ì Î Í = Ï ÒÍ ð ( Ï Ò Í )2 + 1

+

Ï

−

Example 9. Contact transformation ( ÓÔ ≠ 0,

Ë

= Ó ( ÌÑÎ Í ) õ − ÔÝÐ , Ë Ò Ð = Ó (Ï Í ) õ − Ô ( ô + 1) õ Ô

,

ô ≠ −1): +1 Î Ï = Óô ( ÌÑÍ ) õ − Ô ( ô + 1) Ì , Ï ÒÍ Ò +1 Ì = Óô (Ï Í ) õ +2 − Ï , Ì Î Í Ô ( ô + 1) õ Ô ( ô + 1)

(direct transformation); (inverse transformation).

= ( ô + 1) ÌÑÎ Í =

ô

Ï ÒÍ

+1

(direct transformation); (inverse transformation).

0.1.9. Approximate Analytic Methods for Solution of Equations 0.1.9-1. The method of successive approximations (Picard method). The method of successive approximations consists of two stages. At the first stage, the Cauchy problem


=  ( , ) ( 0 ) = 0 is reduced to the equivalent integral equation: ( ) =

0

+X 

 0

(equation), (initial condition)

(1) (2)

 (a , (a )) a . M

(3)

Then a solution of equation (3) is sought using the formula of successive approximations:

 +1 ( ) =

0+X 



0

 (a ,  (a )) a ; M

B = 0, 1, 2, 

The initial approximation 0 ( ) can be chosen arbitrarily; the simplest way is to take 0 to be a number. The iterative process converges as BwDöF , provided the conditions of the theorems in Paragraph 0.1.1-2 are satisfied. 0.1.9-2. The method of Taylor series expansion in the independent variable. A solution of the Cauchy problem (1)–(2) can be sought in the form of the Taylor series in powers of ( −  0 ): 

 ( ) 0 ( ) = ( 0 ) + 
( 0 )( −  0 ) + ( −  0 )2 + ))) . (4) 2! The first coefficient ( 0 ) in solution (4) is prescribed by the initial condition (2). The values of the derivatives of ( ) at  =  0 are determined from equation (1) and its derivative equations (obtained by successive differentiation), taking into account the initial condition (2). In particular, setting  =  0 in (1) and substituting (2), one obtains the value of the first derivative:


( 0 ) =  ( 0 , 0 ).

(5)

Page 18

Further, differentiating equation (1) yields



 =   ( , ) +  ( , ) 
. 

(6)

On substituting  =  0 , as well as the initial condition (2) and the first derivative (5), into the right-hand side of this equation, one calculates the value of the second derivative:



 ( 0 ) =   ( 0 , 0 ) +  ( 0 , 0 )  ( 0 , 0 ).  Likewise, one can determine the subsequent derivatives of at  =  0 .

Solution (4) obtained by this method can normally be used in only some sufficiently small neighborhood of the point  =  0 . 0.1.9-3. The method of regular expansion in the small parameter. Consider a general first-order ordinary differential equation with a small parameter :


=  ( , , ).

(7)

Suppose the function  is representable as a series in powers of :

½  ( , , ) = ÷    ( , ).  =0

ID

(8)

One looks for a solution of the Cauchy problem for equation (7) with the initial condition (2) as 0 in the form of a regular expansion in powers of the small parameter:

½

= ÷



=0

 n  ( ).

(9)

Relation (9) is substituted in equation (7) taking into account (8). Then one expands the functions   into a power series in and matches the coefficients of like powers of to obtain a system of equations for n  ( ):

n 0
=  0 ( , n 0 ), n 1
= ( , n 0 ) n 1 +  1 ( , n 0 ),

( , ) = N



(10) 0

.

(11)

N

Only the first two equations are written out here. The prime denotes differentiation with respect to  . The initial conditions for n  can be obtained from (2) taking into account (9):

n 0 ( 0 ) =

0,

n 1 ( 0 ) = 0.

Success in the application of this method is primarily determined by the possibility of constructing a solution of equation (10) for the leading term in the expansion of n 0 . It is significant that the remaining terms of the expansion, n  with B ≥ 1, are governed by linear equations with homogeneous initial conditions. x3y'z|{}~­³´€ Paragraph 0.3.3-2 gives an example of solving a Cauchy problem by the method of regular expansion for a second-order equation and also discusses characteristic features of the method.

x3y'z|{}~‚µ €

The methods of scaled coordinates, two-scale expansions, and matched asymptotic expansions are also used to solve problems defined by first-order differential equations with a small parameter. The basic ideas of these methods are given in Subsection 0.3.4.

^_

References for Subsection 0.1.9: G. M. Murphy (1960), G. A. Korn and T. M. Korn (1968), E. Kamke (1977), D. Zwillinger (1998).

Page 19

0.1.10. Numerical Integration of Differential Equations 0.1.10-1. The method of Euler polygonal lines. Consider the Cauchy problem for the first-order differential equation


=  ( , ) with the initial condition ( 0 ) = 0 . Our aim is to construct an approximate solution = ( ) of this equation on an interval [ 0 , /ø ]. /ø −  0 . We seek Let us split the interval [ 0 , /ø ] into B equal segments of length ¢C = B approximate values 1 , 2 ,  ,  of the function ( ) at the partitioning points  1 ,  2 ,  ,   = /ø . For a given initial value 0 = ( 0 ) and a sufficiently small ¢C , the values of the unknown function  = (  ) at the other points   =  0 + , ¢C are calculated successively by the formula:



+1

=  +  (  ,  ) ¢C

(Euler polygonal line),

where , = 0, 1,  , B −1. The Euler method is a single-step method of the first-order approximation (with respect to the step ¢C ). 0.1.10-2. Single-step methods of the second-order approximation. Two single-step methods for solving the Cauchy problem in the second-order approximation are specified by the recurrence formulas:



+1



+1

=  +     + 12 ¢C ,  + 12   ¢C ) ¢C , =  + 12 V   +  (  +1 ,  +   C ¢  )WC ¢  ,

where   =  (  ,  ); , = 0, 1,  , B − 1. 0.1.10-3. Runge–Kutta method of the fourth-order approximation. This is one of the widely used methods. The unknown values  are successively found by the formulas: ¢  ,  +1 =  + 16 (  1 + 2  2 + 2  3 +  4 ) C where

x3y'z|{}~­³´€

 1 =  (  ,  ),  2 =  (  + 12 ¢C ,  + 12  1 ¢C ),  3 =  (  + 12 C ¢  ,  + 12  2 ¢C ),  4 =  (  + ¢C ,  +  3 ¢C ).

All methods described in Subsection 0.1.10 are special cases of the Runge—Kutta method (a detailed description of this method can be found in the monographs listed below).

x3y'z|{}~‚µ €

In practice, calculations are performed on the basis of any of the above recurrence formulas with two different steps ¢C , 12 ¢C and an arbitrarily chosen small ¢C . Then one compares the results obtained at common points. If these results coincide within the given order of accuracy, one assumes that the chosen step ¢C ensures the desired accuracy of calculations. Otherwise, the step is halved and the calculations are performed with the steps 12 ¢C and 14 ¢C , after which the results are compared again, etc. (Quite often, one compares the results of calculations with steps varying by ten or more times.)

^_

References for Subsection 0.1.10: G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), E. Kamke (1977), D. Zwillinger (1998).

Page 20

0.2. Second-Order Linear Differential Equations 0.2.1. Formulas for the General Solution. Some Transformations 0.2.1-1. Homogeneous linear equations. Various representations of the general solution. 1 b . Consider a second-order homogeneous linear equation in the general form:

 2 ( ) 

 +  1 ( ) 
+  0 ( ) = 0.

(1)

The trivial solution, = 0, is a particular solution of the homogeneous linear equation. Let 1 ( ), 2 ( ) be a fundamental system of solutions (nontrivial linearly independent particular solutions) of equation (1). Then the general solution is given by: = L 1 1 ( ) + L 2 2 ( ), (2) where L 1 and L 2 are arbitrary constants. 2 b . Let 1 = 1 ( ) be any nontrivial particular solution of equation (1). Then its general solution can be represented as: =

1

OL

1

+L

3 b . Consider the equation

2

X

 −Ž

2 1

 , M P

where

 =X 



1 2

M

 .

(3)



 +  ( ) = 0,

which is written in the canonical form; see Paragraph 0.2.1-3 for the reduction of equations to this form. Let 1 ( ) be any nontrivial partial solution of this equation. The general solution can be constructed by formula (3) with  = 0 or formula (2) in which 
[  ( ) − 1][ 12 − (
1 )2 ] 1  + 2 ( ) = 1 ù
2. 2 M [ 12 + ( 1
)2 ]2 1 + ( 1) Here, 1 = 1 ( ) and the prime denotes differentiation with respect to  . The last formula is suitable where 1 vanishes at some points. 0.2.1-2. Wronskian determinant and Liouville’s formula. The Wronskian determinant (or Wronskian) is defined by: ú ( ) 2 ( ) ( ) = û 
1 û ≡ 1 ( 2 )
 − 2 ( 1 )
 , û 1 ( ) 
2 ( ) û û û û where 1 ( ), 2 ( ) is a fundamentalû system of solutions of equation (1). Liouville’s formula:   (a ) ú ú 1 ( ) = ( 0 ) exp ¬ − X  aÝ® . 0  2 (a ) M 0.2.1-3. Reduction to the canonical form. 1 b . The substitution



1 X 2 brings equation (1) to the canonical (or normal) form: = ‡ ( ) exp ” −





1 2

 M •

(4) 2




1  1 1  1 ” − ” .  2 4  2• 2  2•  2 b . The substitution (4) is a special case of the more general transformation ( Y function) 1  1 (Y )  = Y (ˆ ), = ‡ (ˆ ) Å | Y Š
(ˆ )| exp ” − X Y , 2  2 (Y ) M • which also brings the original equation to the canonical form.

‡

 +  ( )‡ = 0,

where

 =

0

−

(5) is an arbitrary

Page 21

0.2.1-4. Reduction to the Riccati equation.

The substitution ‡ = 
 brings the second-order homogeneous linear equation (1) to the Riccati equation:  2 ( )‡
 +  2 ( )‡ 2 +  1 ( )‡ +  0 ( ) = 0, which is discussed in Subsection 0.1.4. 0.2.1-5. Nonhomogeneous linear equations. The existence theorem. A second-order nonhomogeneous linear equation has the form

 2 ( ) 

 +  1 ( ) 
+  0 ( ) = ( ). (6) Existence and uniqueness theorem. On an open interval  <  <  , let the functions  2 ,  1 ,  0 , and be continuous and  2 ≠ 0. Also let 
( 0 ) = $ ( 0 ) = # , be arbitrary initial conditions, where  0 is any point such that  <  0 <  , and # and $ are arbitrary prescribed numbers. Then a solution of equation (6) exists and is unique. This solutions is defined for all Ed ( ,  ). 0.2.1-6. Nonhomogeneous linear equations. Various representations of the general solution. 1 b . The general solution of the nonhomogeneous linear equation (6) is the sum of the general solution of the corresponding homogeneous equation (1) and any particular solution of the nonhomogeneous equation (6). 2 b . Let 1 = 1 ( ), 2 = 2 ( ) be a fundamental system of solutions of the corresponding homogeneous equation, with ≡ 0. Then the general solution of equation (6) can be represented as:   = L 1 1 + L 2 2 + 2 X 1 Mú − 1 X 2 (7) Mú , where

ú

=


1 ( 2) −





2


2 ( 1 )  is the Wronskian determinant.

2

3 b . Given a nontrivial particular solution 1 = 1 ( ) of the homogeneous equation (with ≡ 0), a second particular solution 2 = 2 ( ) can be calculated from the formula: 2

=

1

X

 −Ž

2 1

M

 ,

where

 =X 



1 2

M

 ,

ú

=  −Ž .

(8)

Then the general solution of equation (6) can be constructed by (7). Subsections 2.1.2–2.1.8 present only homogeneous equations; the solutions of the corresponding nonhomogeneous equations can be obtained using (7) and (8). 4 b . Let ¯1 and ¯2 be respective solutions of the nonhomogeneous differential equations L [ ¯ 1 ] = 1 ( ) and L [ ¯2 ] = 2 ( ), which have the same left-hand side but different right-hand sides; where L [ ] is the left-hand side of equation (6). Then the function ¯ = ¯ 1 + ¯2 is a solution of the equation L [ ¯] = 1 ( ) + 2 ( ). 0.2.1-7. Reduction to a constant coefficient equation (a special case). Let 

2

= 1, 

0

≠ 0, and the condition 1 1 | 0| + £ =  = const M |  0 | Cü |  0|

M

be satisfied. Then the substitution ˆ = X

|  |  leads to a constant coefficient linear equation, ü 0M
Šª
Š +  Š
+ sign  = 0. 0

Page 22

0.2.1-8. Kummer–Liouville transformation. The transformation

 = ƒ (a ),

= ý (a ) H + = (a ),

(9)

where ƒ (a ), ý (a ), and = (a ) are arbitrary sufficiently smooth functions (ýþ1 0), takes any linear differential equation for ( ) to a linear equation for H = H (a ). In the special case = ≡ 0, a homogeneous equation is transformed to a homogeneous one. Special cases of transformation (9) are widely used to simplify second- and higher-order linear differential equations.

^_

References for Subsection 0.2.1: G. A. Korn and T. M. Korn (1968), E. Kamke (1977), A. D. Polyanin and V. F. Zaitsev (1995), S. Yu. Dobrokhotov (1998), D. Zwillinger (1998).

0.2.2. Representation of Solutions as a Series in the Independent Variable 0.2.2-1. Equation coefficients are representable in the ordinary power series form. Let us consider a homogeneous linear differential equation of the general form



 +  ( ) 
+ ( ) = 0.

(1)

Assume that the functions  ( ) and ( ) are representable, in the vicinity of a point  =  0 , in the power series form,

½  ( ) = ÷ #  ( −  0 )  ,  =0

½ ( ) = ÷ $  ( −  0 )  ,  =0

(2)

on the interval | −  0 | < ² , where ² stands for the minimum radius of convergence of the two series in (2). In this case, the point  =  0 is referred to as an ordinary point, and equation (1) possesses two linearly independent solutions of the form: 1(

½  ) = ÷   ( −  0 )  ,  =0

2(

½  ) = ÷   ( −  0 )  .  =0

(3)

The coefficients   and   are determined by substituting the series (2) into equation (1) followed by extracting the coefficients of like powers of ( −  0 ).* 0.2.2-2. Equation coefficients have poles at some point. Assume that the functions  ( ) and ( ) are representable, in the vicinity of a point  =  0 , in the form

½ ½  ( ) = ÷ #  ( −  0 )  , ( ) = ÷ $  ( −  0 )  , (4)  =−1  =−2 on the interval | −  0 | < ² . In this case, the point  =  0 is referred to as a regular singular point. Let Œ 1 and Œ 2 be roots of the quadratic equation Œ

2 1

+ (#

−1

− 1) Œ + $

−2

= 0.

There are three cases, depending on the values of the exponents of the singularity. * Prior to that, the terms containing the same powers ( Ë − Ë

0)

õ,ô

= 0, 1, ÿ›ÿ›ÿ , should be collected.

Page 23

1. If Œ 1 ≠ Œ of the form:

2

and Œ

1

−Œ

2

is not an integer, equation (1) has two linearly independent solutions 1(

 ) = | −  0 | 

2 ( ) = | −  0 | 

2. If Œ

1

=Œ

2

½

1+ ÷

1

2

Q



=1

½

Q

1+ ÷



=1

  ( −  0 )  S , (5)

  ( −  0 )  S .

= Π, equation (1) possesses two linearly independent solutions:

½  ) = | −  0 |  1 + ÷   ( −  0 )  S , Q  =1 ½   ( −  0 )  . 2 (  ) = 1 (  ) ln |  −  0 | + |  −  0 |  ÷  =0 1(

3. If Œ 1 = Œ 2 + Ä , where Ä solutions of the form: 1(

2(

is a positive integer, equation (1) has two linearly independent

 ) = | −  0 |   )=,

1(

½

1

Q

1+ ÷



=1

  ( −  0 )  S ,

 ) ln | −  0 | + | −  0 | 

2

½ 

÷ =0

  ( −  0 )  ,

where , may be equal to zero. To construct the solution in each of the three cases, the following procedure should be performed: substitute the above expressions of 1 and 2 into the original equation (1) and equate the coefficients of ( −  0 )  and ( −  0 )  ln | −  0 | for different values of B to obtain recurrence relations for the unknown coefficients. From these recurrence relations the solution sought can be found.

^_

References for Subsection 0.2.2: G. M. Murphy (1960), G. A. Korn and T. M. Korn (1968), E. Kamke (1977), D. Zwillinger (1989).

0.2.3. Asymptotic Solutions This subsection presents asymptotic solutions,as hD 0 ( > 0), of some second-order linear ordinary differential equations containing arbitrary functions (sufficiently smooth), with the independent variable being real. 0.2.3-1. Equations not containing 
 . Leading asymptotic terms. 1 b . Consider the equation

2



 −  ( ) = 0

(1)

on a closed interval  ≤  ≤  . Case 1. With the condition  ≠ 0, the leading terms of the asymptotic expansions of the fundamental system of solutions, as ID 0, are given by the formulas:

J exp O − 1 X   ,

ü M P 1 −1 J 4 cos O X −  , 1 = (−  )

ü M P 1

=

−1 4

J exp O 1 X  

ü M P 1 −1 J 4 sin O X −  2 = (−  )

ü M P 2

=

−1 4

if  > 0, if  < 0.

Page 24

Case 2. Discuss the asymptotic solution of equation (1) in the vicinity of the point  =  0 , where function  ( ) vanishes,  ( 0 ) = 0 (such a point is referred to as a transition point). We assume that the function  can be presented in the form

 ( ) = (

0

−  )j ( ),

In this case, the fundamental solutions, as ID

1

2








£






=

=

1 |  ( )|1 J





j ( ) > 0.

0, are described by three different formulas:

 X  |  ( )|  +  S Q M 4 0 ü 1

 Ai( H ) [ j ( 0 )]1 J 6 0 1 1 exp −  ( )  S X Q  ü M 2[  ( )]1 J 4  1 1 cos |  ( )|  +  S X Q 0ü M 4 |  ( )|1 J 4





4

sin

where



















£

 Bi( H ) [ j ( 0 )]1 J 6  1 1 exp X Q  [  ( )]1 J 4

if  − 

0

≥> ,

if | −  0 | ≤ > , if 

0

− ≥ > ,

if  − 

0

≥> ,

if | −  0 | ≤ > , 0

ü

 ( )  S M

if 

0

− ≥ > ,

where Ai( H ) and Bi( H ) are the Airy functions of the first and second kind, respectively (see equation 2.1.2.2), H = −2 J 3 [j ( 0 )]1 J 3 ( 0 −  ), and > = [ ( 2 J 3 ). 0.2.3-2. Equations not containing 
 . Two-term asymptotic expansions. The two-term asymptotic expansions of the solution of equation (1) with  > 0, as \D closed interval  ≤  ≤  , has the form:

 1 J exp O −   ¦ 1− X 1 M P

0ü  1 −1 J 4 exp O X    ¦ 1 + ÑX  2 = 

M P 0 ü =

−1 4

 1  

 5 (  
)2 − S  + [ ( 2 ) § , X  3 J 2 32  5 J 2 M 0 Q 8   1  

 5 (  
)2 − S  + [ ( 2 ) § , 3J 2 32  5 J 2 M 0 Q 8 

0, on a

(2)

where  0 is an arbitrary number satisfying the inequality  ≤  0 ≤  . The asymptotic expansions of the fundamental system of solutions of equation (1) with  < 0 are derived by separating the real and imaginary parts in either formula (2). 0.2.3-3. Equations of special form not containing "
 . Consider the equation

2



 −  +

−2

 ( ) = 0

(3)

on a closed interval  ≤  ≤  , where  < 0 and  > 0, under the conditions that is a positive ¡ integer and  ( ) ≠ 0. In this case, the leading term of the asymptotic solution, as D 0, in the vicinity of the point  = 0 is expressed in terms of a simpler model equation, which results from substituting the function  ( ) in equation (3) by the constant  (0) (the solution of the model equation is expressed in terms of the Bessel functions of order 1 , see equation 2.1.2.7). ¡ We specify below formulas by which the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (3) with  <  < 0 and 0 <  <  are related (excluding a small vicinity of the point  = 0). Three different cases can be extracted.

Page 25

1 b . Let

be an even integer, and  ( ) > 0. Then,

¡

=

1

=

2






[  ( )]−1 J 4 exp



 ( )  S ü  M 1 −1 −1 J 4 , [  ( )] exp  ( )  S X Q 0 ü M  1 [  ( )]−1 J 4 exp − X  ( )  S Q 0 ü M 1 −1 J 4 , [  ( )] exp − X  ( )  S Q 0 ü M


 

1





where  =  ( ), , = sin O

Q

X

¡

. where  =  ( ), , = tan O 2 P ¡ 3 b . Let be an odd integer. Then,

¡

1

2





=

=

|  ( )|−1 J 4 cos −

 


1

if  < 0, if  > 0,



( )|  +

M

4 S

|  ( )|  −

M

( )|  −

M

( )|  +

M

if  < 0,

4 S

4 S

4 S

if  > 0, if  < 0, if  > 0,

4 S

if  < 0,

 ( )  S X Q 0 ü M  1 |  ( )|−1 J 4 cos − X |  ( )|  −  S Q 0 ü M 4 1 −1 J 4 , [  ( )] exp − X  ( )  S Q 0 ü M

if  > 0,

Q

1 , 2




if  > 0,

.

P ¡ be an even integer, and  ( ) < 0. Then,
  |  ( )|−1 J 4 cos Q − 1 X 0 ü | 
  1 = 1 , −1 |  ( )|−1 J 4 cos X Q 0 ü
1 −1 J 4  |  ( )| cos Q − X 0 ü | 
 =  2 1 , |  ( )|−1 J 4 cos X | Q 0 ü

2 b . Let

if  < 0,

0

where  =  ( ), , = sin O 2

¡

P

−1

X

0

[  ( )]−1 J 4 exp

ü

1

|  ( )|  +



M

if  < 0, if  > 0,

.

0.2.3-4. Equations not containing 
 . Equation coefficients are dependent on . Consider an equation of the form



 −  ( , ) = 0 (4) on a closed interval  ≤  ≤  under the condition that  ≠ 0. Assume that the following asymptotic relation holds:

2

½  (  , ) = ÷   (  )  ,  =0

ID

0.

Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (4) are given by the formulas: 1 J  ) exp − 1 X  ( )  + X Q ü 0 M 2

1

=

−1 4 ( 0

2

=

−1 4 ( 0

1 J  ) exp 1 X  0 ( )  + X Q ü M 2

 1 ( ) V £  0 (  ) M  S 1 + [ ( )W ,  1 ( ) V £  0 (  ) M  S 1 + [ ( )W .

Page 26

0.2.3-5. Equations containing 
 . 1 b . Consider an equation of the form



 + ( ) 
+  ( ) = 0 on a closed interval 0 ≤  ≤ 1. With ( ) > 0, the asymptotic solution of this equation, satisfying the boundary conditions (0) = L 1 and (1) = L 2 , can be represented in the form: = (L where , = exp X

Q

1 0

1

−, L

2 ) exp

V −

−1

(0) W + L

2

exp X  Q

 ( )  S + [ ( ), ( ) M

1

 ( )  S . ( ) M

2 b . Now let us take a look at an equation of the form 2



 +  ( ) 
+  ( ) = 0

(5)

on a closed interval  ≤  ≤  . Assume

c ( ) ≡ [ ( )]2 − 4  ( ) ≠ 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (5), as pD 0, are expressed by 1

2

1 1 X c ( )  − X £ Q 2 ü M 2 1 1 
= | c ( )|−1 J 4 exp X c ( )  − X £ Q 2 ü M 2 c = | c ( )|−1 J 4 exp −


( )  S V 1 + [ ( )W , c ( ) M ( )  S V 1 + [ ( )W . ( ) M

0.2.3-6. Equations of the general form. The more general equation

2



 +  ( , ) 
+  ( , ) = 0

is reducible, with the aid of the substitution

= ¨ exp O −

2 ¨ 

 + (  − 14

2

1 XÈ  , to an equation of the form (4), 2 M P

− 12  
)¨ = 0,

to which the asymptotic formulas given above in Paragraph 0.2.3-4 are applicable.

^_

References for Subsection 0.2.3: W. Wasov (1965), F. W. J. Olver (1974), A. H. Nayfeh (1973, 1981), M. V. Fedoryuk (1993).

0.2.4. Boundary Value Problems 0.2.4-1. The first, second, third, and mixed boundary value problems (

1

≤  ≤  2 ).

We consider the second-order nonhomogeneous linear differential equation



 +  ( ) 
+ ( ) = % ( ).

(1)

1 b . The first boundary value problem: Find a solution of equation (1) satisfying the boundary conditions =  1 at  =  1 , =  2 at  =  2 . (2) (The values of the unknown are prescribed at two distinct points 

1

and  2 ).

Page 27

2 b . The second boundary value problem: Find a solution of equation (1) satisfying the boundary conditions 
=  1 at  =  1 , 
=  2 at  =  2 . (3) (The values of the derivative of the unknown are prescribed at two distinct points 

1

and  2 ).

3 b . The third boundary value problem: Find a solution of equation (1) satisfying the boundary conditions 
− , 1 =  1 at  =  1 , (4) 
+ , 2 =  2 at  =  2 . 4 b . The third boundary value problem: Find a solution of equation (1) satisfying the boundary conditions 
=  2 at  =  2 . =  1 at  =  1 , (5) (The unknown itself is prescribed at one point, and its derivative at another point.) Conditions (1), (2), (3), and (4) are called homogeneous if  1 =  2 = 0. 0.2.4-2. Simplification of boundary conditions. Reduction of equation to the self-adjoint form. 1 b . Nonhomogeneous boundary conditions can be reduced to homogeneous ones by the change of variable H = # 2  2 + # 1  + # 0 + (the constants # 2 , # 1 , and # 0 are selected using the method of undetermined coefficients). In particular, the nonhomogeneous boundary conditions of the first kind (1) can be reduced to homogeneous boundary conditions by the linear change of variable  − H = − 2 1 ( −  1 ) −  1 .  2− 1 2 b . On multiplying by  ( ) = exp X

Q

 ( )  S , one reduces equation (1) to the self-adjoint form: M  (6) [ ( )
]
 +  ( ) = & ( ).

Without loss of generality, we further consider equation (6) instead of (1). We assume that the functions  , 
 ,  , and & are continuous on the interval  1 ≤  ≤  2 , and  is positive. 0.2.4-3. The Green’s function. Boundary value problems for nonhomogeneous equations. The Green’s function of the first boundary value problem for equation (6) with homogeneous boundary conditions (2) is a function of two variables 4 ( , ° ) that satisfies the following conditions: 1 b . 4 ( , ° ) is continuous in  for fixed ° , with 

1

≤ ≤

2

and 

1

≤ ° ≤  2.

2 b . 4 ( , ° ) is a solution of the homogeneous equation (6), with & = 0, for all  of the point  = ° .

1

< <

2

exclusive

3 b . 4 ( , ° ) satisfies the homogeneous boundary conditions 4 ( 1 , ° ) = 4 ( 2 , ° ) = 0.

4 b . The derivative 4
 ( , ° ) has a jumpÀ of À 1  ( ° ) at the point À À  = ° , that is, 1 4
 ( , ° )  ,  > − 4
 ( , ° )  ,  < = . û û  (° ) û û For the second, third, and mixed boundary value problems, the Green’s function is defined likewise except that in 3 b the homogeneous boundary conditions (3), (4), and (5), with  1 =  2 = 0, are adopted, respectively. The solution of the nonhomogeneous equation (6) subject to appropriate homogeneous boundary conditions is expressed in terms of the Green’s function as follows:* ( ) = X  * The homogeneous boundary value problem—with solution.



2 1

 (Ë

4 ( , ° )& ( ° ) ° . M ) = 0 and

Ó

1

=

Ó

2

= 0—is assumed to have only the trivial

Page 28

0.2.4-4. Representation of the Green’s function in terms of particular solutions. We consider the first boundary value problem. Let 1 = 1 ( ) and 2 = 2 ( ) be linearly independent particular solutions of the homogeneous equation (6), with & = 0, that satisfy the conditions



1( 1)



= 0,

2( 2 )

= 0.

(Each of the solutions satisfies one of the homogeneous boundary conditions.) The Green’s function is expressed in terms of solutions of the homogeneous equation as follows:





4 ( ú ( ) = x3y'z|{}~/€

where

1(

( ) 2 ( ° ) (° ) 2 ( ) (° )

  1(° ) ú , ° ) =
 1(° )  (° ) ú

for 

1

≤ ≤° ,

for ° ≤  ≤  2 ,

(7)

 )
2 ( ) − 
1 ( ) 2 ( ) is the Wronskian determinant.

Formula (7) can also be used to construct the Green’s functions for the second, third, and mixed boundary value problems. To this end, one should find two linearly independent solutions, 1 = 1 (  ) and 2 = 2 (  ), of the homogeneous equation; the former satisfies the corresponding homogeneous boundary condition at  =  1 and the latter satisfies the one at  =  2 .

^_

References for Subsection 0.2.4: L. E. El’sgol’ts (1961), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980).

0.2.5. Eigenvalue Problems 0.2.5-1. The Sturm–Liouville problem. Consider the second-order homogeneous linear differential equation



[  ( )  ]  + [ Œ ( ) − % ( )] = 0

(1)

subject to linear boundary conditions of the general form



1



2


+  
+ 

1 2

= 0 at = 0 at

 =  1,  =  2.

(2)

It is assumed that the functions  ,  
, , and % are continuous, and  and are positive on an interval  1 ≤  ≤  2 . It is also assumed that |  1 | + |  1 | > 0 and |  2 | + |  2 | > 0. The Sturm–Liouville problem: Find the values Œ  of the parameter Œ at which problem (1), (2) has a nontrivial solution. Such Œ  are called eigenvalues and the corresponding solutions  =  ( ) are called eigenfunctions of the Sturm–Liouville problem (1), (2). 0.2.5-2. General properties of the Sturm–Liouville problem (1), (2). 1 b . There are infinitely (countably) many eigenvalues. All eigenvalues can be ordered so that Œ 1 < Œ 2 < Œ 3 < ))) . Moreover, Œ  D F as BD F ; hence, there can only be a finite number of negative eigenvalues. Each eigenvalue has multiplicity 1. 2 b . The eigenfunctions are defined up to a constant factor. Each eigenfunction  ( ) has precisely B − 1 zeros on the open interval ( 1 ,  2 ). 3 b . Any two eigenfunctions  ( ) and + ( ), B ≠ , are orthogonal with weight ( ) on the ¡ interval  1 ≤  ≤  2 : 2 X  ( )  ( ) + ( )  = 0 if B ≠ . 1

M

¡

Page 29

4 b . An arbitrary function  ( ) that has a continuous derivative and satisfies the boundary conditions of the Sturm–Liouville problem can be decomposed into an absolutely and uniformly convergent series in the eigenfunctions:

½  ( ) = ÷    ( ),  =1 where the Fourier coefficients   of  ( ) are calculated by 2 2 2 1   = 2 X  ( )  ( )  ( )  ,  = X  ( ) 2 ( ) M  . M 1 1  5 b . If the conditions

% ( ) ≥ 0,

 1  1 ≤ 0,

 2 2 ≥ 0

(3)

hold true, there are no negative eigenvalues. If % ≡ 0 and  1 =  2 = 0, the least eigenvalue is Œ 1 = 0, to which there corresponds an eigenfunction 1 = const. In the other cases where conditions (3) are satisfied, all eigenvalues are positive. 6 b . The following asymptotic formula is valid for eigenvalues as B­DGF :

Œ  =  ¢

2 2

B

2

+ [ (1),

¢ =X 



2

1

( )  .  ( ) M

(4)

Paragraphs 0.2.5-3 through 0.2.5-6 will describe special properties of the Sturm–Liouville problem which depend on the specific form of the boundary conditions. x3y'z|{}~­³´€ Equation (1) can be reduced to the case where  ( ) ≡ 1 and ( ) ≡ 1 by the change of variables  

( ) 1J 4  , ‡ ( ) = V  (  ) (  )W =X ( ).  ( ) M In this case, the boundary conditions are transformed to boundary conditions of similar form. x3y'z|{}~‚µ € The second-order linear equation

Y 2 ( ) 

 + Y 1 ( ) 
+ [ Π+ Y 0 ( )] = 0 can be represented in the form of equation (1) where  ( ), ( ), and % ( ) are given by:

 ( ) = exp X Q

Y 1(  ) 1  S , ( ) = exp X Y 2(  ) M Y 2(  ) Q

Y 1(  ) Y ( )  S , % ( ) = − 0 exp X Y 2(  ) M Y 2( ) Q

0.2.5-3. Problems with boundary conditions of the first kind (case 

1

=

2

Y 1(  )  S . Y 2(  ) M

= 0 and 

1

=

2

= 1).

Let us note some special properties of the Sturm–Liouville problem that is the first boundary value problem for equation (1) with the boundary conditions

 =  1,

= 0 at

= 0 at

 =  2.

(5)

1 b . For B¥DöF , the asymptotic relation (4) can be used to estimate the eigenvalues Œ  . In this case, the asymptotic formula

 ( ) = 

4 ¬ 2 ® ¢  (  ) (  )

J

1 4

sin ¬

¢

B

X 

 1

( ) 1  ® +[ O ,  ( ) M B P

¢ =X 



2 1

( )   ( ) M

holds true for the eigenfunctions  ( ).

Page 30

TABLE 2 Example estimates of the first eigenvalue Œ 1 in Sturm–Liouville problems with boundary conditions of the first kind (0) = (1) = 0 obtained using the Rayleigh–Ritz principle [the right-hand side of relation (6)] Equation

Test function

Π1, approximate

Π1, exact



 + Œ (1+  2)−2 = 0

H = sin 

15.337

15.0



 + Œ (4−  2)−2 = 0



H = sin 

135.317

134.837

[(1+  )−1
 ]
 + Œ



=0

H = sin 

7.003

6.772


)
 + Œ



=0

H = sin 



11.9956

11.8985

( £ 1+ 

H = sin   ) H =  (1−



 + Π(1+sin  ) = 0



2

0.54105  0.55204



2

0.54032  0.54032

2



2

2 b . If % ≥ 0, the following upper estimate holds for the least eigenvalue (Rayleigh–Ritz principle):

Œ

1

≤

X 



V  ( )( H 
)2 + % ( ) H 2 W  M , 2 2 X  ( ) H  M 1

2 1

(6)

where H = H ( ) is any twice differentiable function that satisfies the conditions H ( 1 ) = H ( 2 ) = 0. The equality in (6) is attained if H = 1 ( ), where 1 ( ) is the eigenfunction corresponding to the ( −  1 ) S in (6) to obtain specific eigenvalue Œ 1 . One can take H = ( −  1 )( 2 −  ) or H = sin  Q  2− 1 estimates. It is significant to note that the left-hand side of (6) usually gives a fairly precise estimate of the first eigenvalue (see Table 2). 3 b . The extension of the interval [ 1 ,  2 ] leads to decreasing in eigenvalues. 4 b . Let the inequalities 0<

min

≤  ( ) ≤ 

max ,

0<

min

≤ ( ) ≤

0<%

max ,

≤ % ( ) ≤ %

min

max

be satisfied. Then the following bilateral estimates hold:





min max

(



2

B

2 2

−

1

)2

+



%

min max

≤Œ  ≤



max min

(



2

B

2 2

+

−  1 )2

%

max min

.

5 b . In engineering calculations for eigenvalues, the approximate formula

Œ  =  ¢

2 2

B

2

1 +  2−

1

X 



2 1

% ( )  , ( ) M

¢ =X 



2 1

( )   ( ) M

(7)

may be quite useful. This formula provides an exact result if  ( ) ( ) = const and % ( ) ( ) = const (in particular, for constant equation coefficients,  =  0 , % = % 0 , and = 0 ) and gives a correct asymptotic behavior of (4) for any  ( ), % ( ), and ( ). In addition, relation (7) gives two correct leading asymptotic terms as BCD F if  ( ) = const and ( ) = const [and also if  ( ) ( ) = const].

Page 31

6 b . Suppose  ( ) = ( ) = 1 and the function % = % ( ) has a continuous derivative. The following asymptotic relations hold for eigenvalues Œ  and eigenfunctions  ( ) as B­DGF :

Œ = ü  

 −B 

+

1

B

5 ( 1 ,  2 ) + [­O

B ( −   1 ) 1 − (  ( ) = sin   −  B Q w 2 1 2

1



1

1

B

2

,

P

−  ) 5 ( ,  2 ) + (

where

5 (‡ ,  ) =

2

−  ) 5 ( 1 ,  )S cos



1 B ( −  1 ) + [­O 2 ,  2− 1 B P

1 X % ( )  . 2  M

(8)

7 b . Let us consider the eigenvalue problem for the equation with a small parameter



 + [ Œ + ´% ( )] = 0

( pD

0)

subject to the boundary conditions (5) with  1 = 0 and  2 = 1. We assume that % ( ) = % (− ). This problem has the following eigenvalues and eigenfunctions:

Π = 2B 

2

− ž# ž +

 ( ) = £ 2 sin( B/ ) −



2

½

 ≠ B £ 2½ 2



2

# 2´ 

2

 ≠ B

−,

2

# ´

2

−,

+ [ ( 3 ),

2

# ´ = 2 X

1 0

% ( ) sin( B/ ) sin( ,! )  ;   M

sin( ,! ) + [ ( 2 ).



Here, the summation is carried out over , from 1 to F . The next term in the expansion of be found in Nayfeh (1973). 0.2.5-4. Problems with boundary conditions of the second kind (case 



1 = 2 =1

 can

and  1 =  2 = 0).

Let us note some special properties of the Sturm–Liouville problem that is the second boundary value problem for equation (1) with the boundary conditions


= 0 at  =  1 ,


= 0 at  =  2 .

1 b . If % > 0, the upper estimate (6) is valid for the least eigenvalue, with H = H ( ) being any twice-differentiable function that satisfies the conditions H 
( 1 ) = H 
( 2 ) = 0. The equality in (6) is attained if H = 1 ( ), where 1 ( ) is the eigenfunction corresponding to the eigenvalue Œ 1 . 2 b . Suppose  ( ) = ( ) = 1 and the function % = % ( ) has a continuous derivative. The following asymptotic relations hold for eigenvalues Œ  and eigenfunctions  ( ) as B­DGF : (B − 1) 1 1 + 5 ( 1 ,  2 ) + [ O 2 , B P 2− 1  (B − 1) (B − 1)( −  1 ) 1 + ( 1 −  ) 5 ( ,  2 )  ( ) = cos   2− 1  ((B B −−1)1)(Q  −  1 ) 1 + [­O 2 , + ( 2 −  ) 5 ( 1 ,  )S sin   2− 1 B P

Œ =  ü  

where 5 (‡ ,  ) is given by (8).

Page 32

0.2.5-5. Problems with boundary conditions of the third kind (case 

1

=

2

= 1 and  1 

2

≠ 0).

We consider the third boundary value problem for equation (1) subject to condition (2) with  1 =  2 = 1. We assume that  ( ) = ( ) = 1 and the function % = % ( ) has a continuous derivative. The following asymptotic formulas hold for eigenvalues Œ  eigenfunctions  ( ) as B­DGF :

ü

1 (B − 1) V 5 ( 1 ,  2 ) −  1 +  2 W + [­O 1 , + (B − 1) B 2P 2− 1  (B − 1)( −  1 ) 1 + ¦ ( 1 −  ) V 5 ( ,  2 ) +  2 W  ( ) = cos   2− 1 (B − 1)  (B − 1)( −  1 ) 1 + ( 2 −  ) V 5 ( 1 ,  ) −  1 W!§ sin  + [­O 2 ,  2− 1 B P

Π =  

where 5 (‡ ,  ) is defined by (8). 0.2.5-6. Problems with mixed boundary conditions (case 

1

=

2

= 1 and 

2

=

1

= 0).

Let us note some special properties of the Sturm–Liouville problem that is the a mixed boundary value problem for equation (1) with the boundary conditions


= 0 at  =  1 ,

= 0 at

 =  2.

1 b . If % ≥ 0, the upper estimate (6) is valid for the least eigenvalue, with H = H ( ) being any twice-differentiable function that satisfies the conditions H 
( 1 ) = 0 and H ( 2 ) = 0. The equality in (6) is attained if H = 1 ( ), where 1 ( ) is the eigenfunction corresponding to the eigenvalue Œ 1 . 2 b . Suppose  ( ) = ( ) = 1 and the function % = % ( ) has a continuous derivative. The following asymptotic relations hold for eigenvalues Œ  and eigenfunctions  ( ) as B­DGF :

ü

(2B − 1) 2 1 5 ( 1 ,  2 ) + [­O 2 , + −  ) (2 B − 1) B P 2 1  (2B − 1)( −  1 ) 2 ( 1 −  ) 5 ( ,  2 ) +  ( ) = cos  2( −  ) (2B − 1) Q 2 1  (2B − 1)( −  1 ) 1 + ( 2 −  ) 5 ( 1 ,  )S sin  + [­O 2 , 2( 2 −  1 ) B P

Π =  2(

where 5 (‡ ,  ) is defined by (8).

^_

References for Subsection 0.2.5: L. Collatz (1963), E. Kamke (1977), A. G. Kostyuchenko and I. S. Sargsyan (1979), V. A. Marchenko (1986), B. M. Levitan and I. S. Sargsyan (1988), V. A. Vinokurov and V. A. Sadovnichii (2000), A. D. Polyanin (2002).

0.3. Second-Order Nonlinear Differential Equations 0.3.1. Form of the General Solution. Cauchy Problem 0.3.1-1. Equations solved for the derivative. General solution. A second-order ordinary differential equation solved for the highest derivative has the form



 =  ( , , 
).

(1)

The general solution of this equation depends on two arbitrary constants, L 1 and L 2 . In some cases, the general solution can be written in explicit form, = Y ( , L 1 , L 2 ), but more often implicit or parametric forms of the general solution are encountered.

Page 33

0.3.1-2. Cauchy problem. The existence and uniqueness theorem. 1 b . Cauchy problem: Find a solution of equation (1) satisfying the initial conditions ( 0 ) =

0,


( 0 ) =

(At a point  =  0 , the value of the unknown function,

0,

1.

and its derivative,

(2) 1,

are prescribed.)

2 b . Existence and uniqueness theorem. Let  ( , , H ) be a continuous function in all its arguments ‘ in a neighborhood of a point ( 0 , 0 , 1 ) and let  have bounded partial derivatives  and  in  this neighborhood, or the Lipschitz condition is satisfied: |  ( , , H )−  ( , ¯ , H ¯)| ≤ #  | − ¯ |+| H − H ¯|  , where # is some positive number. Then a solution of equation (1) satisfying the initial conditions (2) exists and is unique.

^_

References for Subsection 0.3.1: G. A. Korn and T. M. Korn (1968), I. G. Petrovskii (1970), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980).

0.3.2. Equations Admitting Reduction of Order 0.3.2-1. Equations not containing

explicitly.

In the general case, an equation that does not contain

implicitly has the form

 ( , 
, 

 ) = 0.

(1)

Such equations remain unchanged under an arbitrary translation of the dependent variable: D + const . The substitution 
 = H ( ), 

 = H 
( ), brings (1) to a first-order equation:  ( , H , H 
) = 0. 0.3.2-2. Equations not containing  explicitly (autonomous equations). In the general case, an equation that does not contain  implicitly has the form

 ( , 
, 

 ) = 0.

(2)

Such equations remain unchanged under an arbitrary translation of the independent variable: vD  + const. Using the substitution 
 = ¨ ( ), where plays the role of the independent variable, and taking into account the relations "

 = ¨2
= ¨2

 = ¨2
¨ , one can reduce (2) to a   first-order equation:  ( , ¨ , ¨*¨u
) = 0. 

x3y'z|{}~­³´€ The equation  =  ( +  2 + ' + ( ) is reduced by the change of variable 2 ‡ = +  + ' + ( to an autonomous equation, ‡

 =  (‡ ) + 2  . 0.3.2-3. Equations of the form  (  +  , 
 , 

 ) = 0. Such equations are invariant under simultaneous translations of the independent and dependent variables in accordance with the rule ED  + ˜( , D − !( , where ( is an arbitrary constant. For  = 0, see equation (1). For  ≠ 0, the substitution  ¨ =  +  leads to equation (2):  (  ¨ , ¨2
−   , ¨2

 ) = 0. 0.3.2-4. Equations of the form  ( ,  
 − , 

 ) = 0. The substitution ¨ ( ) = 


−

leads to a first-order equation:  ( , ¨ , ¨u
 ) = 0.

Page 34

0.3.2-5. Homogeneous equations. 1 b . The equations homogeneous in the independent variable remain unchanged under scaling of the independent variable, ED ƒ… , where ƒ is an arbitrary nonzero number. In the general case, such equations can be written in the form

 ( ,  
, 

The substitution H ( ) = 

2



 ) = 0.

(3)


leads to a first-order equation:  ( , H , HH
− H ) = 0. 

2 b . The equations homogeneous in the dependent variable remain unchanged under scaling of the variable sought, D ƒ , where ƒ is an arbitrary nonzero number. In the general case, such equations can be written in the form

 ( , 
 , 

  ) = 0.

(4)

The substitution H ( ) = 
 leads to a first-order equation:  ( , H , H 
+ H 2 ) = 0. 3 b . The equations homogeneous in both variables are invariant under simultaneous scaling (dilatation) of the independent and dependent variables,  DGƒ… and DGƒ , where ƒ is an arbitrary nonzero number. In the general case, such equations can be written in the form The transformation a = ln | |, ¨  (¨ , ¨2g
+ ¨ , ¨2g

g + ¨2g
) = 0.

=  

 (  , 
,  

 ) = 0.

(5)

leads to an autonomous equation (see Paragraph 0.3.2-2):

0.3.2-6. Generalized homogeneous equations. 1 b . The generalized homogeneous equations remain unchanged under simultaneous scaling of the independent and dependent variables in accordance with the rule ‚D„ƒ… and D†ƒ  , where ƒ is an arbitrary nonzero number and , is some number. Such equations can be written in the form

 ( −  ,  1−  
,  2−  

 ) = 0. (6) The transformation a = ln  , ¨ =  −  leads to an autonomous equation (see Paragraph 0.3.2-2):   ¨ , ¨ g
+ , ¨ , ¨ g

g + (2 , − 1)¨ g
+ , ( , − 1)¨  = 0. 2 b . The most general form of representation of generalized homogeneous equations is as follows:

¸ (  + ,  
 ,  2 

  ) = 0. (7) + 


The transformation H =   , ‡ =  brings this equation to the first-order equation ¸ÊH , ‡ , H ( ‡ + B )‡
‘ − ‡ + ‡ 2  = 0. ¡ x3y'z|{}~‚µ € For ≠ 0, equation (7) is equivalent to equation (6) in which , = −B . To ¡ ¡ the particular values B = 0 and = 0 there correspond equations (3) and (4) homogeneous in ¡ the independent and dependent variables, respectively. For B = − ≠ 0, we have an equation ¡ homogeneous in both variables (5). 0.3.2-7. Equations invariant under scaling–translation transformations. 1 b . The equations of the form

 ( 



, 


  ,   

 ) = 0

(8)

remain unchanged under simultaneous translation and scaling of variables, çD  + ƒ and D ý ,  where ý =  − 9  and ƒ is an arbitrary number. The substitution ¨ =  brings (8) to an ¨ p ¨
¨ 2 ¨

2 ¨
autonomous equation (see Paragraph 0.3.2-2):  ( ,  − Œ ,  − 2 Œ  + Œ 2 ¨ ) = 0.

Page 35

2 b . The equation

 


,  , 

  ) = 0

 ( 

(9)

is invariant under the simultaneous translation and scaling of variables, ­D  + ƒ and  where ý = exp(− ƒsŒ B ) and ƒ is an arbitrary number. The transformation H = !  , ¨ ‘ ¨ ¨ 2 ¨
¨ 2 brings (9) to a first-order equation:   H , , H (B +Œ ) + = 0. 3 b . The equation

 (    ,  
, 

2

DGý , = 
 



 ) = 0

(10) +ý ,

is invariant under the simultaneous scaling and translation of variables, ­DGƒ… and D where ƒ = exp(−ýhŒ B ) and ý is an arbitrary number. The transformation H =    , ¨ brings (10) to a first-order equation:   H , ¨ , H ( Œ ¨ + B )¨2‘
− ¨  = 0.

=  


0.3.2-8. Exact second-order equations. The second-order equation

 ( , , 
, 

 ) = 0

(11)

is said to be exact if it is the total differential of some function,  = Y
 , where Y = Y ( , , 
). If equation (11) is exact, then we have a first-order equation for Y : Y ( , , 
) = L , (12) where L is an arbitrary constant. If equation (11) is exact, then  ( , , 
 , 

 ) must have the form:  ( , , 
, 

 ) =  ( , , 
) 

 + ( , , 
). (13) Here,  and are expressed in terms of Y by the formulas: Y Y Y
 .  ( , , 
) = N
, ( , , 
) = N + N (14)   N N N By differentiating (14) with respect to  , , and  = 
, we eliminate the variable Y from the two formulas in (14). As a result, we have the following test relations for  and :   + 2    +  2   =  K +   K −  , (15)   K +    K + 2   = K'K . Here, the subscripts denote the corresponding partial derivatives. If conditions (15) hold, then equation (11) with  of (13) is exact. In this case, we can integrate the first equation in (14) with respect to  = 
 to determine Y = Y ( , , 
):

Y = X  ( , ,  )

M



+ j ( , ),

(16)

where j ( , ) is an arbitrary function of integration. This function is determined by substituting (16) into the second equation in (14). Example. The left-hand side of the equation

ÏÏ Ò˜Í Í Ò

+ (Ï

ÒÍ

Ë ÏÏ ÒÍ

)2 + 2 Ó



can be represented in the form (13), where â = Ï and ã = 2 + 2 Ó satisfied. Hence, equation (17) is exact. Using (16), we obtain

ä

2

=0

(17)

2.

It is easy to verify that conditions (15) are

ӁÏ

+

+ ( Ë , Ï ).

Ï

=

+ ӁÏ

Ë Ï



(18)

Substituting this expression into the second equation in (14) and taking into account the relation ã = 2 + 2 Ó Ë Ï + ÓÏ 2 , we find that 2 Ó Ë Ï + ÓÏ 2 = Ò + . Since = ( Ë , Ï ), we have 2 Ó Ë Ï = and ÓÏ 2 = Ò . Integrating yields Ë 2 = Ó Ï + const. Substituting this expression into (18) and taking into account relation (12), we find a first integral of equation (17): Ï + Ó Ë Ï 2 = 1 , where = Ï ÒÍ . (19) = Ï 2 , we arrive at the first-order linear equation ÒÍ + 2 Ó Ë = 2 1 , which is easy to integrate. Thus, we find the Setting solution of the original equation in the form:

















Ï

2

=2



1





exp − Ó

ë

Ë 2ì X



exp

ë



Ó Ë 2ì Ë æ



+











2

exp − Ó

ë

Ë 2ì

.

Page 36

0.3.2-9. Reduction of quasilinear equations to the normal form. We consider the equation



 +  ( ) 
+ ( ) = ` ( , ) (20) with linear left-hand side and nonlinear right-hand side. Let 1 ( ) and 2 ( ) form a fundamental system of solutions of the truncated linear equation corresponding to ` ≡ 0. The transformation ˆ =

 ) , ( 1  ) 2(

‡ =

1(

(21)

 )

brings equation (20) to the normal form:

‡ Šª

Š = i (ˆ , ‡ ),

where

i (ˆ , ‡ ) = ú

ú

 ) `ç , 1 ( )‡  .  )

3 1( 2(

( ) = 1 
2 − 2 
1 is the Wronskian of the truncated equation; and the variable  must be Here, expressed in terms of ˆ using the first relation in (21). Transformation (21) is convenient for the simplification and classification of equations having the form (20) with ` ( , ) = % ( )  , thus reducing the number of functions from three to one: {  , , % }  {0, 0, % 1}.

^_

References for Subsection 0.3.2: G. M. Murphy (1960), E. Kamke (1977), V. F. Zaitsev and A. D. Polyanin (1993, 2001), A. D. Polyanin and V. F. Zaitsev (1995), D. Zwillinger (1998).

0.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable or Small Parameter 0.3.3-1. Method of expansion in powers of the independent variable. A solution of the Cauchy problem



 =  ( , , 
), 
( 0 ) = ( 0 ) = 0 ,

1

(1) (2)

can be sought in the form of a Taylor series in powers of the difference ( −  0 ), specifically:



 ( ) 


 ( ) 0 0 ( ) = ( 0 ) + 
( 0 )( −  0 ) + (  −  0 )2 + ( −  0 )3 + ))) . 2! 3!

(3)



 ( 0 ) =  ( 0 , 0 , 1 ).

(4)

The first two coefficients ( 0 ) and 
 ( 0 ) in solution (3) are defined by the initial conditions (2). The values of the subsequent derivatives of at the point  =  0 are determined from equation (1) and its derivative equations (obtained by successive differentiation of the equation) taking into account the initial conditions (2). In particular, setting  =  0 in (1) and substituting (2), we obtain the value of the second derivative:

Further, differentiating (1) yields




 =   ( , , 
) +  ( , , 
) 
+  ( , , 
) 

 . (5)   œ "

On substituting  =  0 , the initial conditions (2), and the expression of  ( 0 ) of (4) into the right-hand side of equation (5), we calculate the value of the third derivative:




 ( 0 ) =   ( 0 , 0 , 1 ) +  ( 0 , 0 , 1 ) 1 +  ( 0 , 0 , 1 )  ( 0 , 0 , 1 ).   œ The subsequent derivatives of the unknown are determined likewise. The thus obtained solution (3) can only be used in a small neighborhood of the point  = 

0.

Page 37

0.3.3-2. Method of regular (direct) expansion in powers of the small parameter. We consider an equation of general form with a parameter :



 +  ( , , 
, ) = 0.

(6)

We assume that the function  can be represented as a series in powers of :

½  ( , , 
, ) = ÷    ( , , 
).  =0

ID

(7)

Solutions of the Cauchy problem and various boundary value problems for equation (6) with 0 are sought in the form of a power series expansion:

½

  ( ).

= ÷



=0

(8)

One substitutes expression (8) into equation (6) taking into account (7). Then the functions   are expanded into power series with respect to the small parameter, and the coefficient of like powers of are collected. Equating the resulting expressions (the coefficient of like powers of ) to zero, one arrives at a system of equations for  :



+  ( , ,
) = 0, 0 0 0

+  ( , ,
)
+ 4 ( , ,
) +  ( , ,
) = 0, 0 0 1 0 0 1 1 0 0 1 0

 = N N



0


,

4 = N



(9) 0

.

(10)

N

Here, only the first two equations are written out. The prime denotes differentiation with respect to  . To obtain the initial (or boundary) conditions for  , the expansion (8) is taken into account. The success in the application of this method is primarily determined by the possibility of constructing a solution of equation (9), for the leading term 0 . It is significant to note that the other terms  with B ≥ 1 are governed by linear equations with homogeneous initial conditions. Example. The Duffing equation with initial conditions

ÏÒ˜Í Í Ò



+ Ï + Ï

3

Ï

= 0,

(0) = Ó ,

ÏÒÍ

(0) = 0,

describes the motion of a cubic oscillator, i.e., oscillations of a point mass on a nonlinear spring. Here, Ï is the deviation of the point mass from the equilibrium and Ë is dimensionless time. For 0, an approximate solution of the problem is sought in the form of the asymptotic expansion (8). We substitute (8) into the equation and initial conditions and expand in powers of . On equating the coefficients of like powers of the small parameter to zero, we obtain the following problems for Ï 0 and Ï 1 :

!



The solution of the problem for

Ï

Ï 0Í Í + Ï Ï 1Í Í + Ï

0

0

= 0,

1

= − Ï 03 ,

is given by:

Ï

0

Ï Ï

=

Ó

0

=Ó ,

1

= 0,

Ï 0Í = 0; Ï 1Í = 0.

cos Ë .

Substituting this expression into the equation for Ï 1 and taking into account the identity cos3 Ë = obtain Ï 1Í Í + Ï 1 = − 14 Ó 3 (cos 3Ë + 3 cos Ë ), Ï 1 = 0, Ï 1Í = 0. Integrating yields Ï 1 = − 38 Ó 3 Ë sin Ë + 321 Ó 3 (cos 3Ë − 3 cos Ë ). Thus the two-term solution of the original problem is given by:

څÛÝÜ2Þßà$#˜á

Ï

=

Ó

cos Ë + 'Ó



3

í

− 38 Ë sin Ë +

1 (cos 3 32

Ë

− 3 cos Ë )î +

1 4

cos 3 Ë +

3 4

cos Ë , we

" ( 2 ).

The term Ë sin Ë causes Ï 1 Ï 0 as Ë . For this reason, the solution obtained is unsuitable at large times. It can only be used for Ë 1; this results from the condition of applicability of the expansion, Ï 0 Ï 1 . This circumstance is typical of the method of regular series expansions with respect to the small parameter; in other words, the expansion becomes unsuitable at large values of the independent variable. This method is also inapplicable if the expansion (8) begins with negative powers of . Methods that allow avoiding the above difficulties are discussed in Subsection 0.3.4. څÛÝÜ2Þßà á Growing terms as Ë , like Ë sin Ë , that narrow down the domain of applicability of asymptotic expansions are called secular.

,+

^_

 )(

% '&

'&

* 



-&

References for Paragraph 0.3.3-2: A. H. Nayfeh (1973, 1981).

Page 38

0.3.3-3. Pad´e approximants. Suppose the , + 1 leading coefficients in the Taylor series expansion of a differential equation about the point  = 0 are obtained by the method presented in Paragraph 0.3.3-1, so that



 ) =  0 +  1  + ))) +     .

+1 (

(11)

The partial sum (11) pretty well approximates the solution at small  but is poor for intermediate and large values of  , since the series can be slowly convergent or even divergent. This is also related to the fact that  DGF as EDGF , while the exact solution can well be bounded. cases, instead of the expansion (11), it is reasonable to consider a Pad e´ approximant .0/ (In ),many which is the ratio of two polynomials of degree Ä and 8 , specifically,

6

. / 6

( ) =

#

+ # 1  + ))) + # 1 + $ 1  + )) ) + $

/ /

0



6

,

6

Ä +8

where

=, .

(12)

are selected so that the , + 1 leading terms in the The coefficients # 1 ,  , # / and $ 1 ,  , $ Taylor series expansion of (12) coincide with 6 the respective terms of the expansion (11). In other words, the expansions (11) and (12) must be asymptotically equivalent as þD 0. In practice, one usually takes Ä = 8 (the diagonal sequence). It often turns out that formula (12) pretty well approximates the exact solution on the entire range of  (for sufficiently large Ä ). Example 1. Consider the following Cauchy problem for a second-order nonlinear equation:

ÏÒ˜Í Í Ò

=

ÏÏÒÍ

+ Ï 3;

The Taylor series expansion of the solution about |Ë

Ë

Ï =1+Ë

Ï

ÏÒÍ

(0) =

(0) = 1.

(13)

= 0 has the form (see Paragraph 0.3.3-1): +Ë

2

+Ë

3

+Ë

4

12131 .

+

(14)

This geometric series is convergent only for | < 1. The diagonal sequence of Pad´e approximants corresponding to series (14) is:

4

1 1(

Ë

)=

4

1 , 1−Ë

Ï (Ë

2 2(

Ë

)=

4

1 , 1−Ë

3 3(

Ë

1 . 1−Ë

)=

(15)

1 is the exact solution of the Cauchy problem (13). Hence, in 1−Ë this case, the diagonal sequence of Pad´e approximants recovers the exact solution from only a few terms in the Taylor series. It is not difficult to verify that the function

)=

Example 2. Consider the Cauchy problem for a second-order nonlinear equation:

ÏÒ˜Í Í Ò

= 2ÏÏÒÍ ;

Ï

ÏÒÍ

(0) = 0,

(0) = 1.

(16)

Following the method presented in Paragraph 0.3.3-1, we obtain the Taylor series expansion of the solution to problem (16) in the form: 17 Ë 7 Ï (Ë ) = Ë + 13 Ë 3 + 152 Ë 5 + 315 + . (17)

Ï (Ë

13121

) = tan Ë

The exact solution of problem (16) is given by: . Hence it has singularities at any finite segment of the Taylor series (17) does not have any singularities. With series (17), we construct the diagonal sequence of Pad´e approximants:

4

2 2(

Ë

)=

3Ë 3−Ë

, 2

4

3 3(

Ë

)=

Ë (Ë

3(2 Ë

2

− 15) , 2 − 5)

4

4 4(

Ë

)=

Ë

=ï

1 (2 2

5

6

+ 1) . However,

5Ë (21 − 2 Ë 2 ) . 4 − 45 Ë 2 + 105

Ë

(18)

These Pad´e approximants have singularities (at the points where the denominators vanish):

Ë)7 ï Ë)7 ï Ë)7 ï

1.732 1.581 1.571 and

Ë7 ï

for

4

for

4

4 for 6.522

2 2( 3 3( 4 4(

Ë

),

Ë

Ë

), ).

2 and It is apparent that the Pad´e approximants are attempting to recover the singularities of the exact solution at Ë = ï Ë = ï 3 2. The Pad´e approximant 4 (Ë ) gives an accurate numerical approximation of the exact solution on the interval 4 | Ë | ≤ 2 (the error is everywhere less than 1%, except for a very small neighborhood of the point Ë = ï 2).

^_

6%

4

6 %

6 %

References for Paragraph 0.3.3-3: G. A. Baker (Jr.) and P. Graves–Morris (1981), D. Zwillinger (1989, pp. 450–453).

Page 39

0.3.4. Perturbation Methods of Mechanics and Physics 0.3.4-1. Preliminary remarks. A summary table of basic methods. Perturbation methods are widely used in nonlinear mechanics and theoretical physics for solving problems that are described by differential equations with a small parameter . The primary purpose of these methods is to obtain an approximate solution that would be equally suitable at all (small, intermediate, and large) values of the independent variable as ID 0. Equations with a small parameter can be classified according to the following: (i) the order of the equation remains the same at = 0; (ii) the order of the equation reduces at = 0. For the first type of equations, solutions of related problems* are sufficiently smooth (little varying as decreases). The second type of equation is said to be degenerate at = 0, or singularly perturbed. In related problems, thin boundary layers usually arise whose thickness is significantly dependent on ; such boundary layers are characterized by high gradients of the unknown. All perturbation methods have a limited domain of applicability; the possibility of using one or another method depends on the type of equations or problems involved. The most commonly used methods are summarized in Table 3 (the method of regular series expansions is set out in Paragraph 0.3.3-2). In subsequent paragraphs, additional remarks and specific examples are given for some of the methods. In practice, one usually confines oneself to few leading terms of the asymptotic expansion. In many problems of nonlinear mechanics and theoretical physics, the independent variable is dimensionless time a . Therefore, in this subsection we use the conventional a , 0 ≤ a < F instead of  . 0.3.4-2. The method of scaled parameters (Lindstedt–Poincar e´ method). We illustrate the characteristic features of the method of scaled parameters with a specific example (the transformation of the independent variable we use here as well as the form of the expansion are specified in the first row of Table 3). Example 1. Consider the Duffing equation:

Ï89Í Í 8 + Ï

 3 = 0. (1) On performing the change of variable : = ; (1 + 2< 1 + 12131 ), we have Ï =2Í Í = + (1 + 2< 1 + 12131 )2 (Ï +  Ï 3 ) = 0. (2) The solution is sought in the series form Ï = Ï 0 ( ; ) + Ï 1 ( ; ) + 13131 . Substituting it into equation (2) and matching the coefficients of like powers of  , we arrive at the following system of equation for two leading terms of the series: Ï 0Í Í + Ï Ï 1Í Í + Ï

= 0,

0

= −Ï

1

;

where the prime denotes differentiation with respect to . The general solution of equation (3) is given by:

Ï where

<

Ó

and

Ô

=

0

+ Ï

Ó

3 0

−2

<

(3)

Ï

1 0,

(4)

;

cos( + Ô ),

(5)

are constants of integration. Taking into account (5) and rearranging terms, we reduce equation (4) to

Ï 1Í Í + Ï

1

= − 14 Ó

3

;

cos 3( + Ô )î − 2 Ó

í

ë

3 8

Ó

2

+

<

1

ì

;

cos( + Ô ).

;

(6)

;

For 1 ≠ Ó the particular solution of equation (6) contains a secular term proportional to cos( + Ô ). In this case, the condition of applicability of the expansion Ï 1 Ï 0 = (1) (see the first row and the last column of Table 3) cannot be satisfied at sufficiently large . For this condition to be met, one should set − 38

2,

;

"

%

<

1

= − 38 Ó 2 .

(7)



* Further on, we assume that the initial and/or boundary conditions are independent of the parameter .

Page 40

TABLE 3 Perturbation methods of nonlinear mechanics and theoretical physics (the third column gives B leading asymptotic terms with respect to the small parameter ) Method name

Examples of problems solved by the method

Method of scaled parameters (0 ≤ < )

One looks for periodic solutions of the equation Í Í + 02 = ( , Í ); see also Paragraph 0.3.4-2

Method of strained coordinates (0 ≤ < )

Cauchy problem: Í = ( , , ); ( 0) = 0 ( is of a special form); see also the problem in the method of scaled parameters

> ? > ?

Averaging method (0 ≤ < )

> ?

@ 98 8 A @ BDC @ @ 8

@8 C > @ B @ > C

@

Form of the solution sought

@ (> ) = E F −1 B õ @ (G ), õ õ =0F −1 > = GIH 1+ E B õ A õ2J õ =1 @ (> ) = E F −1 B õ @ (G ), õ

õ F =0 −1

Additional conditions and remarks

@

Unknowns:

A

õ and õ ; = (1); õ õ secular terms are eliminated through selection of the constants

@

+1

K@

L

A

@ M @ = õL and K õM õM = L (1), (1) õ +1 K õ

Unknowns:

@

;

õ

+1

õ

> = G + E B õNM õ (G ) õ =1 @ = P (> ) cos M (> ), problem: Unknowns: P and M ; the amplitude P and phase M @ Cauchy 2X 89Í Í 8 + A 02@ = BDC (@ , @ 8Í ), 1 C s = 2 X!Y 0 sin M[Z]\^M , are governed @ (0) = @ 0, @ 8Í (0) = @ 1; Q Q 8 by the equations × = − RS CUT (P ), C c = 21X Y 02X cos M[Z]\^M , for more general problems, N Q V Z = C (P cos M ,−P A 0 sin M ) see Paragraph 0.3.4-3, Item 2 O Q 8 = A 0 − RS CUW (P ) × @ = P cos M + E F −1 B õ @ (P , M ), õ One looks for periodic õ =1 Unknowns: @ õ , _ õ , ` õ ; P and M are determined @ are 2a -periodic solutions of the equation õ functions of M ; by the equations @ 89Í Í 8 + A 02@ = BDC (@ , @ 8Í ); Q Q 8 F õ2_ × = E B Cauchy problem for this the @ õ are assumed õ (P ), =1 and other equations not to contain cos M õ QNQ V 8 = A + F E B õ` (P ) 0

0

Krylov– Bogolyubov– Mitropolskii method (0 ≤ < )

> ?

0

Cauchy problem: Method Í Í + 02 = ( , Í ), of two-scale (0) = 0, Í (0) = 1; expansions for boundary value problems, (0 ≤ < ) see Paragraph 0.3.4-4, Item 2

> ?

Multiple scales method (0 ≤ < )

> ?

Method of matched asymptotic expansions (0 ≤ ≤ )

p q

Method of composite expansions (0 ≤ ≤ )

p q

@ 98 8 A @ DB C @ @ 8 @ @ @8 @

õ

=1

@ = E F −1 B õ @ (b , c õ õ

=0

õ

), where

−1 b = UB > , c = > H 1+ E F B õ A õ J , O Q Q 8 = Bd + f 1+ B 2õA =22 + gNgNg ) d de dih One looks for periodic F= E −1 B õ @ , where @ solutions of the equation õ @ 89Í Í 8 + A 02@ = BDC (@ , @ 8Í ); @ = @ (j 0õ, =0j 1, kNkNk , j ), j = B õ > Cauchy problem for this õQ õ Q 8 = d + Bmd + gUgNE g + B E õ d and other equations dil dil diln Outer expansion: Boundary value problem: B @ Ò˜Í Í @Ò + C (p @ , @ )@@ ÒÍ = r (@ p s , @ ), @ =E F −1 õ (B )@ õ (p ), L (B ) ≤ p ≤ q . (0) = 0, (q ) = õ =0 t Inner expansion ( G = p K B ): ( C assumed positive); for other problems, see u@ =E F −1u (B )u@ (G ), 0 ≤ p ≤ L (B ) Paragraph 0.3.4-5, Item 2 O õ õ õ =0 t @ = y (p , B )+ y u (G , B ), Boundary value problem: F −1 (B )y (p ), B @ Ò˜Í Í @Ò + C (p @ , @ )@@ ÒÍ = r ( @ p s , @ ), y =E õ õ (0) = 0, (q ) = u F −1 u õ =0 t u ( C assumed positive); y = E õ (B )y õ (G ), G = p B . boundary value problems õ =0 t u for other equations Here, y õ v 0 as G[vx? 0

1

Unknowns:

@

A

õ and õ; = (1); õ õ secular terms are eliminated through selection of õ

@

@ +1 K

L

@

A

Unknowns: õ ; õ +1 õ = (1); for = 1, this method is equivalent to the averaging method

@

o

K@

Unknowns:

u @@

@ õ +1 K u @ +1 K

@

L

u@

õ =,L (1), õ ,t õ

õ = L (1); õ õ the procedure of

,

u @ G[vx? u u Unknowns: y õ , y õ , õ , y (q ,u B ) = @ s , t t y (0, B )+ y (0, B ) = @ 0;

u

õ

t

;

matching expansions is used: ( 0) = ( )

@ pwv

õ

;

two forms of representation of the equation (in terms of and ) are used to obtain solutions

p

G

Page 41

In this case, the solution of equation (6) is given by:

Ï

1

=

1 32

Ó

3

;

cos 3( + Ô )î .

í

(8)

Subsequent terms of the expansion can be found likewise. With (5), (7), and (8), we obtain a solution of the Duffing equation in the form:

<: + Ô ) + 321 'Ó 3 cos í 3(<: + Ô )î + " ( 2), < = í 1 − 38 ' Ó 2 + " ( 2)î −1 = 1 + 38 'Ó 2 + " ( 2). Ï

Ó

=

cos(

0.3.4-3. Averaging method (Van der Pol–Krylov–Bogolyubov scheme). 1 b . The averaging method involved two stages. First, the second-order nonlinear equation



g g + 

2 0

= ´ ( , g
)

(9)

is reduced with the transformation


g = −  0  sin Y ,

=  cos Y ,

 =  (a ),

where

Y = Y (a ),

to an equivalent system of two first-order differential equations:

 g
= −  (  cos Y , −  0  sin Y ) sin Y ,   0

Y g
=  

0

−

 (  cos Y , −  0  sin Y ) cos Y .   0

(10)

The right-hand sides of equations (10) are periodic in Y , with the amplitude  being a slow function of time a . The amplitude and the oscillation character are changing little during the time the phase Y changes by 2 .  At the second stage, the right-hand sides of equations (10) are being averaged with respect to Y . This procedure results in an approximate system of equations:

 g
= −  s (  ),   0 where

 s ( ) =

1 X 2



2 0

z

sin Ye (  cos Y , − 

0

 sin Y ) Y , M

Y g
=  

0

−

 c ( ) =

 (  ),   0 c 1 X 2



2 0

z

cos Ye (  cos Y , − 

(11)

0

 sin Y ) Y . M

System (11) is substantially simpler than the original system (10)—the first equation in (11), for the oscillation amplitude  , is a separable equation and, hence, can readily be integrated; then the second equation in (11), which is linear in Y , can also be integrated. Note that the Krylov–Bogolyubov–Mitropolskii method (see the fourth row in Table 3) generalizes the above approach and allows obtaining subsequent asymptotic terms as ID 0. 2 b . Below we outline the general scheme of the averaging method. We consider the second-order nonlinear equation with a small parameter:





g g + (a , , g ) = ´ (a , , g ).

(12)

Equation (12) should first be transformed to the equivalent system of equations


g =‡ ,

‡ g
+ (a , , ‡ ) = ´ (a , , ‡ ).

(13)

Suppose the general solution of the “truncated” system (13), with = 0, is known: 0

= Y (a , L

1,

L 2 ),

‡

0

= j (a , L

1,

L 2 ),

(14)

Page 42

where L 1 and L 2 are constants of integration. Taking advantage of the method of variation of constants, we pass from the variables , ‡ in (13) to Lagrange’s variables  1 ,  2 according to the formulas = Y (a ,  1 ,  2 ), ‡ = j (a ,  1 ,  2 ), (15) where Y and j are the same functions that define the general solution of the “truncated” system (14). Transformation (15) allows the reduction of system (13) to the standard form:

 1
= ž 1 (a ,  1 ,  2 ),

 2
= ž 2 (a ,  1 ,  2 ).

(16)

Here, the prime denotes differentiation with respect to a and



1

=

Y 2  (a , Y , j ) Y 1  (a , Y , j ) Y ,  2=− ; Y  = N ,  Y 2j 1 − Y 1j 2 Y 2j 1 − Y 1 j 2 N  Y = Y (a ,  1 ,  2 ), j = j (a ,  1 ,  2 ).

j j  = N ,  N 

It is significant to note that system (16) is equivalent to the original equation (12). The unknowns and  2 are slow functions of time. As a result of averaging, system (16) is replaced by a simpler, approximate autonomous system of equations:  1
= ¸ 1 ( 1 ,  2 ),  2
= ¸ 2 ( 1 ,  2 ), (17)



1

where

{ 1 ù}|

¸  ( 1 ,  2 ) =

0

¸  ( 1 ,  2 ) = lim x3y'z|{}~­³´€

|

  (a ,  1 ,  2 ) a , M

{ 1 ù}|

{

-periodic function of a ;

  (a ,  1 ,  2 ) a , if   is not periodic in a . M

0

÷

if   is a

The averaging method is applicable to equations (9) and (12) with nonsmooth right-hand sides.

x3y'z|{}~‚µ €

The averaging method has rigorous mathematical substantiation. There is also a procedure that allows finding subsequent asymptotic terms. For this procedure, e.g., see the books by Bogolyubov and Mitropolskii (1974), Zhuravlev and Klimov (1988), and Arnold, Kozlov, and Neishtadt (1993). 0.3.4-4. Method of two-scale expansions (Cole–Kevorkian scheme). 1 b . We illustrate the characteristic features of the method of two-scale expansions with a specific example. Thereafter we outline possible generalizations and modifications of the method. Example 2. Consider the Van der Pol equation:

Ï89Í Í 8

8



+ Ï = (1 − Ï 2 ) ÏÍ .

(18)

The solution is sought in the form (see the fifth row in Table 3):

~ ,  ) + Ï 1(~ ,  ) +  2 Ï 2 (~ ,  ) + 13131 , (19) ~ = : ,  = ë 1 +  2 < 2 + 13131 ì : . On substituting (19) into (18) and on matching the coefficients of like powers of  , we obtain the following system for two leading terms: € €2 (20) €  Ï 20 + Ï 0 = 0, € € € € € € Ï

=

Ï

0(

2



Ï

1 2

+Ï

1

= −2

2

Ï

0

~ 

+ (1 − Ï 02 )

Ï



0

.

(21)

Page 43

The general solution of equation (20) is given by:



Ï

‚

0

=

~

 (~ ) cos  + ‚ (~ ) sin  .

(22)

The dependence of and on the slow variable is not being established at this stage. We substitute (22) into the right-hand side of equation (21) and perform elementary manipulations to obtain

€

€ 2Ï 1 2 2 î 1  2 + Ï 1 = í −2 ‚seÍ + 4 ‚ (4 −  − ‚ ) + 14 ( ‚ 3 − 3  2 ‚ ) cos 3  + 14 ( 



 e

cos + 2 …Í −

í

1 4



(4 −



2

−

‚

2

)î sin



ƒ‚ 2 ) sin 3 . (23) The solution of this equation must not contain unbounded terms as „& ; otherwise the necessary condition Ï 1 % Ï 0 = " (1) is not satisfied. Therefore the coefficients of cos  and sin  must be set equal to zero: −2 ‚ Í + 14 ‚ (4 −  2 − ‚ 2 ) = 0, e (24) 2…  Íe − 14  (4 −  2 − ‚ 2 ) = 0. Equations (24) serve to determine  =  ( ~ ) and ‚ = ‚ ( ~ ). We multiply the first equation in (24) by − ‚ and the second by  and add them together to obtain ' Í − 18  (4 −  2 ) = 0, where  2 =  2 + ‚ 2 . (25) e 3

−3

The integration by separation of variables yields:

2  2 =  2 + (44− 0 2 )Õ −e 0



,

(26)

0

where 0 is the initial oscillation amplitude. On expressing and in terms of the amplitude and phase , we have = cos and = − sin . Substituting ä we find that Í =ä 0 or = 0 = const. ä these expressions into either of the two equations in (24) and using (25), Therefore ä ä ä the leading asymptotic term can be represented as:

where



~ =  :

‚



Ï

and

0

 

 ~



= ( ) cos( +

ä

e

‚



0 ),

 = : , and the function  (~ ) is determined by (26).

2 b . The method of two-scale expansions can also be used for solving boundary value problems where the small parameter appears together with the highest derivative as a factor (such problems for 0 ≤  ≤  are indicated in the seventh row of Table 3 and in Paragraph 0.3.4-5). In the case where a boundary layer arises near the point  = 0 (and its thickness has an order of magnitude of ), the solution is sought in the form:

ˆ , ‰ ) + 1 (ˆ , ‰ ) + 2 2 (ˆ , ‰ ) + ))) , ˆ =  , ‰ = −1 V 0 ( ) +  1 ( ) + 2 2 ( ) + ))) W , =

0(

where the functions  =  (ˆ , ‰ ) and  =  ( ) are to be determined. The derivative with respect to  is calculated in accordance with the rule:

M

M 

= N

N

ˆ

+ ‰ 
N

N

‰

= N

N

ˆ

+

1

 0
+  1
+ 2 2
+ )))  N . ‰ N

Additional conditions are imposed on the asymptotic terms in the domain under consideration; namely,  +1   = [ (1) and  +1  = [ (1) for , = 0, 1,  , and 0 ( ) D„ as þD 0. x3y'z|{}~/€ The two-scale method is also used to solve problems that arise in mechanics and physics and are described by partial differential equations. 0.3.4-5. Method of matched asymptotic expansions. 1 b . We illustrate the characteristic features of the method of matched asymptotic expansions with a specific example (the form of the expansions is specified in the seventh row of Table 3). Thereafter we outline possible generalizations and modifications of the method.

Page 44

Example 3. Consider the linear boundary value problem

+ Ï ÒÍ + â ( Ë ) Ï = 0, (0) = Ó , Ï (1) = Ô ,

Ï Ò˜Í Í Ò Ï

&

(27) (28)

where 0 < â (0) < . At = 0 equation (27) degenerates; the solution of the resulting first-order equation



Ï ÒÍ

+ â (Ë )Ï = 0

Ë

(29)

cannot meet the two boundary conditions (28) simultaneously. It can be shown that the condition at = 0 has to be omitted in this case (a boundary layer arises near this point). The leading asymptotic term of the outer expansion, Ï = Ï 0 ( Ë ) + ( ), is determined by equation (29). The solution of (29) that satisfies the second boundary condition in (28) is given by:

" 

Ï 0 (Ë

) = Ô exp

…X

1

~ æ ~† .

Ò â

( )

(30)

We seek the leading term of the inner expansion, in the boundary layer adjacent to the left boundary, in the following form (see the seventh row and third column in Table 3):

Ë

=

%,

(31)

 −1 , we obtain ‡ÏÍ Í + ‡ÏÍ = 0, (32) 0 0 where the prime denotes differentiation with respect to ; . The solution of equation (32) that satisfies the first boundary condition in (28) is given by: ‡Ï 0 = Ó −  + s Õ −= . (33) The constant of integration  is determined from the condition of matching the leading terms of the outer and inner expansions: ‡ Ï 0 (Ë 0) = Ï 0 ( ;ˆ'& ). (34)

where

;

‡Ï = ‡Ï 0 (; ) + " ( ), ;

is the extended variable. Substituting (31) into (27) and extracting the coefficient of

Substituting (30) and (33) into condition (34) yields



=

Ó

− ÔÖÕ

‰

〈 〉

,

〈â 〉 =

where

X

1 0

â (Ë

)

Ë æ

.

(35)

Taking into account relations (30), (31), (33), and (35), we represent the approximate solution in the form:

Ï

‹Œ Š

=

Ó − ÔÖÕ 〈 ‰ 〉 ì Õ − ҍ ë1 Ô exp … X Ò â (~ ) ~† æ

‰

ÔÖÕ

〈 〉

+

R

for 0 ≤ for

Ë

≤

" ( ) ≤ Ë

" ( ), ≤ 1.

(36)



It is apparent that inside the thin boundary layer, whose thickness is proportional to , the solution rapidly changes by a finite value, = ÔÖÕ 〈 〉 − Ó . To determine the function Ï on the entire interval Ë [0, 1] using formula (36), one has to “switch” at some intermediate point Ë = Ë 0 from one part of the solution to the other. Such switching is not convenient and, in practice, one often resorts to a composite solution instead of using the double formula (36). In similar cases, a composite solution is defined as:

Ž

‰

Ï



=

Ï 0 (Ë

‡ ;

) + Ï 0( ) −

For  (¯Ë

=

ë

Ó



,

‘

= Ò lim

0

Ï 0 (Ë

=‘)’

) = lim

‡Ï 0 (; ).

‰ and hence the composite solution becomes: 1 Ò  − ÔÖÕ 〈 ‰ 〉 ì Õ − R + Ô exp … X â (~ ) ~ † .

In the problem under consideration, we have

Ï





=

ÔÖÕ

〈 〉

Ò

Ï 0 (Ë

≤ 1, this solution transforms to the outer solution ) and for 0 ≤ an approximate representation of the unknown over the entire domain.

æ

˓(

 , to the inner solution, thus providing

2 b . We now consider an equation of the general form



 =  ( , , 
) subject to boundary conditions (28). For the leading term of the outer expansion

=

0(

(37)

 ) + ))) , we have the equation:

 ( , 0 , 0
) = 0.

Page 45

In the general case, when using the method of matched asymptotic expansions, the position of the boundary layer and the form of the inner (extended) variable have to be determined in the course of the solution of the problem. First we assume that the boundary layer is located near the left boundary. In (37), we make a change of variable H =  > ( ) and rewrite the equation as 1 ‘'

‘ = > 2   O>H , , ‘
.

> P

(38)

The function > = > ( ) is selected so that the right-hand side of equation (38) has a nonzero limit ‘ value as ID 0, provided that H , , and 
are of the order of 1. Example 4. For

ê ( Ë , Ï , Ï ÒÍ

) = −ô

Ë^” Ï ÒÍ

+ Ï , where 0 ≤

Ï =2Í Í =

=−

•

< 1, the substitution

– 1+ ” ô ; ” Ï =Í + – 2 Ï .  

In order that the right-hand side of this equation has a nonzero limit value as

!

;

=

Ë

%N– ( ) brings equation (37) to

0, one has to set

– 1+ ” % 

= 1 or

– ” %  = const, where const is any positive number. It follows that – =  ” . ‡ leading asymptotic term of the inner expansion in the boundary layer, Ï = Ï 0 ( ; ) + 1131 , is determined by the equation ‡Ï Í Í + The ‡ ô ; ” Ï 0Í = 0, where the prime denotes differentiation with respect to ; . 0 1 1+

1+

If the position of the boundary layer is selected incorrectly, the outer amd inner expansions cannot be matched. In this situation, one should consider the case where an arbitrary boundary layer is located on the right (this case is reduced to the previous one with the change of variable  = 1 − H ). In example 4 above, the boundary layer is on the left if , > 0 and on the right if , < 0. There is a procedure for matching subsequent asymptotic terms of the expansion (see the seventh row and last column in Table 3). In its general form, this procedure can be represented as: inner expansion of the outer expansion ( -expansion for þD 0) ¶ = outer expansion of the inner expansion ( -expansion for HCDGF ).

x3y'z|{}~­³´€

The method of matched asymptotic expansions can also be applied to construct periodic solutions of singularly perturbed equations (e.g., in the problem of relaxation oscillations of the Van der Pol oscillator).

x3y'z|{}~‚µ €

Two boundary layers can arise in some problems (e.g., in cases where the right-hand side of equation (37) does not explicitly depend on "
 ).

x3y'z|{}~E¹€

The method of matched asymptotic expansions is also used for solving equations (in semi-infinite domains) that do not degenerate at = 0. In such cases, there are no boundary layers; the original variable is used in the inner domain, and an extended coordinate is introduced in the outer domain.

x3y'z|{}~Eº€

The method of matched asymptotic expansions is successfully applied for the solution of various problems in mathematical physics that are described by partial differential equations; in particular, it plays an important role in the theory of heat and mass transfer and in hydrodynamics.

^_

References for Subsection 0.3.4: M. Van Dyke (1964), A. Blaquiere (1966), G. D. Cole (1968), G. E. O. Giacaglia (1972), A. H. Nayfeh (1973, 1981), N. N. Bogolyubov and Yu. A. Mitropolskii (1974) J. Kevorkian and J. D. Cole (1981, 1996), P. A. Lagerstrom (1988), V. Ph. Zhuravlev and D. M. Klimov (1988), V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt (1993).

0.3.5. Galerkin Method and Its Modifications (Projection Methods) 0.3.5-1. General form of an approximate solution. Consider a boundary value problem for the equation

—

[ ] −  ( ) = 0

(1)

Page 46

with linear homogeneous boundary conditions at the points  =  1 and  =  2 ( 1 ≤  ≤  2 ). Here, — is a linear or nonlinear differential operator of the second order (or a higher order operator); — = ( ) is the unknown function and  =  ( ) is a given function. It is assumed that [0] = 0. Let us choose a sequence of linearly independent functions

Y = Y  ( )

(B = 1, 2,  , Ä )

(2)

satisfying the same boundary conditions as = ( ). According to all methods that will be considered below, an approximate solution of equation (1) is sought as a linear combination

/

=

/

½ 

=1

#  Y  ( ),

(3)

with the unknown coefficients #  to be found in the process of solving the problem. The finite sum (3) is called an approximation function. The remainder term ² / obtained after the finite sum has been substituted into the equation (1) has the form

²

/

—

= [

/

]− .

If the remainder ² / is identically equal to zero, then the function equation (1). In general, ² / 1 0.

/

(4) is the exact solution of

0.3.5-2. Galerkin method. In order to find the coefficients #  in (3), consider another sequence of linearly independent functions j = j  ( ) ( , = 1, 2,  , Ä ). (5)

Let us multiply both sides of (4) by j  and integrate the resulting relation over the region ˜ = { ≤  ≤  }, in which we seek the solution of equation (1). Next, we equate the cor1 2 responding integrals to zero (for the exact solutions, these integrals are equal to zero). Thus, we obtain the following system of linear algebraic equations for the unknown coefficients #  :

X 



2 1

j  ²

/ M

 =0

( , = 1, 2,  , Ä ).

(6)

Relations (6) mean that the approximation function (3) satisfies equation (1) “on the  2 average” (i.e., in the integral sense) with weights j  . Introducing the scalar product 〈 , % 〉 = X  %  of 1

M

arbitrary functions and % , we can consider equations (6) as the condition of orthogonality of the remainder ² / to all weight functions j  . The Galerkin method can be applied not only to boundary value problems but also to eigenvalue problems (in the latter case, one takes  = Œ and seeks eigenfunctions  , together with eigenvalues Œ  ). Mathematical justification of the Galerkin method for specific boundary value problems can be found in the literature listed at the end of Subsection 0.3.5. Below we describe some other methods, which are in fact special cases of the Galerkin method. x3y'z|{}~/€ As a rule, one takes suitable sequences of polynomials or trigonometric functions as Y  ( ) in the approximation function (3). 0.3.5-3. The Bubnov–Galerkin method, the moment method, and the least squares method.

1 b . The sequences of functions (2) and (5) in the Galerkin method can be chosen arbitrarily. In the case of equal functions, Y  ( ) = j  ( ) ( , = 1, 2,  , Ä ), (7) the method is often called the Bubnov–Galerkin method.

Page 47

2 b . The moment method is the Galerkin method with the weight functions (5) being powers of  ,

j  =  .

(8)

3 b . Sometimes, the functions j  are expressed in terms of Y  by the relations

—

j  = [Y  ]

—

( , = 1, 2,  ),

where is the differential operator of equation (1). This version of the Galerkin method is called the least squares method. 0.3.5-4. Collocation method. In the collocation method, one chooses a sequence of points   , , = 1,  , Ä , and imposes the condition that the remainder (4) be zero at these points,

²

/

 =  ( , = 1,  , Ä ). (9) When solving a specific problem, the points   , at which the remainder ² / is set equal to zero, are regarded as most significant. The number of collocation points Ä is taken equal to the number = 0 at

of the terms of the series (3). This allows one to obtain a complete system of algebraic equations for the unknown coefficients #  (for linear boundary value problems, this algebraic system is linear). Note that the collocation method is a special case of the Galerkin method with the sequence (5) consisting of the Dirac delta functions:

j  = > ( −   ). In the collocation method, there is no need to calculate integrals, and this essentially simplifies the procedure of solving nonlinear problems (although usually this method yields less accurate results than other modifications of the Galerkin method). 0.3.5-5. The method of partitioning the domain.

˜

The domain = { 1 ≤  ≤  2 } is split into Ä subdomains: In this method, the weight functions are chosen as follows:

˜

 = {  1 ≤  ≤   2 }, , = 1,  , Ä .

˜

1 for ‚d  , ˜ 0 for  ∉  .

™

j  ( ) =

˜

The subdomains  are chosen according to the specific properties of the problem under consider˜ ation and can generally be arbitrary (the union of all subdomains  may differ from the domain ˜ , and some ˜ and ˜ + may overlap).  0.3.5-6. The least squared error method. Sometimes, in order to find the coefficients #  of the approximation function (3), one uses the least squared error method based on the minimization of the functional:

` =X 



2 1

²

/

2

M

þD

min .

(10)

For given functions Y  in (3), the integral ` is a quadratic polynomial with respect to the coefficients #  . In this case, the necessary conditions of minimum in (10) have the form:

N

N

`

# 

=0

(B = 1,  , Ä ).

is a system of linear algebraic equations for the coefficients ^This _

#  .

References for Subsection 0.3.5: L. V. Kantorovich and V. I. Krylov (1962), M. A. Krasnoselskii, G. M. Vainikko, P. P. Zabreiko et al. (1969), S. G. Mikhlin (1970), B. A. Finlayson (1972), D. Zwillinger (1989).

Page 48

0.3.6. Iteration and Numerical Methods 0.3.6-1. The method of successive approximations (Cauchy problem). The method of successive approximations is implemented in two steps. First, the Cauchy problem



 =  ( , , 
) 
( 0 ) = 0
( 0 ) = 0 ,

(equation), (boundary conditions)

(1) (2)

is reduced to an equivalent system of integral equations by the introduction of the new variable ‡ ( ) = 
 . These integral equations have the form

 ‡ ( ) = 0
+ X    a , (a ), ‡ (a )  a , M 0

( ) =

0

+X 



0

‡ (a ) a . M

(3)

Then the solution of system (3) is sought by means of successive approximations defined by the following recurrence formulas:


+1 (  ) = 0 + X 

‡ 



0

  a ,  (a ), ‡  (a )  a , M

As the initial approximation, one can take

^_

 +1 ( ) = 0(

 )=

0,

0+X 



0

‡ 0 ( ) = 
0 .

‡  (a ) a ; M

B = 0, 1, 2, 

References for Paragraph 0.3.6-1: G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), E. Kamke (1977).

0.3.6-2. The Runge–Kutta method (Cauchy problem). For the numerical integration of the Cauchy problem (1)–(2), one often uses the Runge–Kutta method. Let ¢C be sufficiently small. We introduce the following notation:




 = (  ),  =  (  ),   =  (  ,  ,  ); The desired values  and 
 are successively found by the formulas:
 +1 =  +  ¢C + 16 (  1 +  2 +  3 )( ¢C )2 ,

1  +1 =  + 6 (  1 + 2  2 + 2  3 +  4 ) ¢C ,   =

0

+ , ¢C ,

, = 0, 1, 2, 

where

 1 =     ,  , 
 ,  2 =     + 12 ¢C ,  + 12 
¢C , 
+ 12  1 ¢C  ,  3 = l  + 12 ¢C ,  + 12 
¢C + 14  1 ( ¢C )2 , 
+ 12  2 ¢C  ,  4 =     + ¢C ,  + 
¢C + 12  2 ( ¢C )2 , 
+  3 ¢C  . In practice, the step ¢C is determined in the same way as for first-order equations (see Remark 2 in Paragraph 0.1.10-3).

^_

References for Paragraph 0.3.6-2: G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), E. Kamke (1977), D. Zwillinger (1989).

0.3.6-3. Shooting method (boundary value problems). In order to solve the boundary value problem for equation (1) with the boundary conditions ( 1 ) =

1,

( 2 ) =

2,

(4)

one considers an auxiliary Cauchy problem for equation (1) with the initial conditions ( 1 ) =

1,


( 1 ) =  .

(5)

Page 49

(The solution of this Cauchy problem can be obtained by the Runge–Kutta method or some other numerical method.) The parameter  is chosen so that the value of the solution = ( ,  ) at the point  =  2 coincides with the value required by the second boundary condition in (4): ( 2 ,  ) =

2.

In a similar way one constructs the solution of the boundary value problem with mixed boundary conditions 
( 2 ) + , ( 2 ) = 2 . ( 1 ) = 1 , (6) In this case, one also considers the auxiliary Cauchy problem (1), (5). The parameter  is chosen so that the solution = ( ,  ) satisfies the second boundary condition in (6) at the point  =  2 .

^_

References for Paragraph 0.3.6-3: S. K. Godunov and V. S. Ryaben’kii (1973), N. N. Kalitkin (1978).

0.3.6-4. Method of accelerated convergence in eigenvalue problems. Consider the Sturm–Liouville problem for the second-order nonhomogeneous linear equation [  ( ) 
]
 + [ Œ ( ) − % ( )] = 0

(7)

with linear homogeneous boundary conditions of the first kind (0) = (1) = 0.

It is assumed that the functions  ,  
, , % are continuous and  > 0, > 0.

(8)

First, using the Rayleigh–Ritz principle, one finds an upper estimate for the first eigenvalue Œ 01 [this value is determined by the right-hand side of relation (6) from Paragraph 0.2.5-3]. Then, one solves numerically the Cauchy problem for the auxiliary equation [  ( ) 
]
 + [ Œ 01 ( ) − % ( )] = 0

with the boundary conditions

(0) = 0,


(0) = 1.

(9) (10)

satisfies the condition ( 0 , Œ where  0 < 1. The criterion of closeness The function ( , Œ of the exact and approximate solutions, Œ 1 and Œ the form of the inequality |1 −  0 | ≤ > , where > is a sufficiently small given constant. If this inequality does not hold, one constructs a refinement for the approximate eigenvalue on the basis of the formula: 0 1)

0 1 ) = 0, 0 1 , has

1 1

Œ



=Œ

0 1

− 0  (1)

[
 (1)]2

2

,

0

= 1 −  0,

(11)

1

2 where = X ( ) 2 ( )  . Then the value Œ 11 is substituted for Œ 01 in the Cauchy problem 0 M (9)–(10). As a result, a new solution and a new point  1 are found; and one has to check whether the criterion |1 −  1 | ≤ > holds. If this inequality is violated, one refines the approximate eigenvalue by means of the formula:

Œ

2 1

=Œ

1 1

− 1  (1)

[
 (1)]2

2

,

1

= 1 −  1,

(12)

and repeat the above procedure. x3y'z|{}~­³´€ Formulas of the type (11) are obtained by a perturbation method based on a transformation of the independent variable  (see Paragraph 0.3.4-1). If   > 1, the functions  , , and % are smoothly extended to the interval (1, ˆ ], where ˆ ≥   .

x3y'z|{}~‚µ €

The algorithm described above has the property of accelerated convergence

2 −4 to 10−8 +1 š  , which ensures that the relative error of the approximate solution becomes 10 after two or three iterations for 0 š 0.1. This method is quite effective for high-precision calcula-



tions, is fail-safe, and guarantees against accumulation of roundoff errors.

Page 50

x3y'z|{}~E¹€

In a similar way, one can compute subsequent eigenvalues Œ + , ¡ that end, a suitable initial approximation Œ 0+ should be chosen).

= 2, 3,  (to

x3y'z|{}~Eº€

A similar computation scheme can also be used in the case of boundary conditions of the second and the third kinds, periodic boundary conditions, etc. (see the references below). Example 1. The eigenvalue problem for the equation

ÏÒ˜Í Í Ò

+ (1 + Ë

•

2 −2

)

Ï

=0

•

•

•

5

= 16 2 − 1. with the boundary conditions (8) admits an exact analytic solution and has eigenvalues 1 = 15, 2 = 63, ÿªÿ›ÿ , According the Rayleigh–Ritz principle, formula (6) of Paragraph 0.2.5-3 for = sin( Ë ) yields the approximate value 01 = 15.33728. The solution of the Cauchy problem (9)–(10) with â (Ë ) = 1, ã (Ë ) = (1 + Ë 2 )−2 , ( Ë ) = 0 yields Ë 0 = 0.983848, 1 − Ë 0 = 0.016152, ÖÏ 2 = 0.024585, Ï ÒÍ (Ë 0 ) = −0.70622822. The first iteration for the first eigenvalue is determined by (11) and results in the value 11 = 14.99245 with the relative 1 = 5 × 10−4 . error 1 2 < 10−6 . The second iteration results in 21 = 14.999986 with the relative error 1

•

;

œ ^œ

Ž• % •

•

6

›

E

•

•

Ž• % •

Example 2. Consider the eigenvalue problem for the equation ( 1+Ë

ž

Ï ÒÍ ) ÍÒ

•

+ Ï = 0

with the boundary conditions (8). The Rayleigh–Ritz principle yields 01 = 11.995576. The next two iterations result in the values 2 = 11.898458. For the relative error we have 2 < 10−5 . 1 1

^• _

•

Ž• % •

•

1 1

= 11.898578 and

References for Paragraph 0.3.6-4: L. D. Akulenko and S. V. Nesterov (1996, 1997).

]

For more details about finite-difference methods and other numerical methods, see, for instance, the books by Lambert (1973), Keller (1976), and Zwillinger (1998).

0.4. Linear Equations of Arbitrary Order 0.4.1. Linear Equations with Constant Coefficients 0.4.1-1. Homogeneous linear equations. An B th-order homogeneous linear equation with constant coefficients has the general form

 +  ( )

−1



( −1)

+ ))) + 

1


+ 

0

= 0.

(1)

The general solution of this equation is determined by the roots of the characteristic equation:

.

( Π) = 0,

where

.

(Π) = Π +  

−1

Π

−1

+ ))) +  1 Π+  0 .

(2)

The following cases are possible: 1 b . All roots Π1 , Π2 ,  , Π of the characteristic equation (2) are real and distinct. Then the general solution of the homogeneous linear differential equation (1) has the form: =L

1

exp( Π1  ) + L

2

exp( Π2  ) + ))) + L  exp( Π  ).

2 b . There are equal real roots Œ 1 = Œ 2 = ))) = Œ + ¡ distinct. In this case, the general solution is given by:

(

¡

≤ B ), and the other roots are real and

= exp( Œ 1  )( L 1 + L 2  + ))) + L +  + −1 ) + L + +1 exp( Œ + +1  ) + L + +2 exp( Œ + +2  ) + ))) + L  exp( Œ   ). 3 b . There are equal complex conjugate roots Œ = ƒ\AC™Ýý (2 ≤ B ), and the other roots are real ¡ ¡ and distinct. In this case, the general solution is:

 + ))) + # +  + −1 ) + exp( ƒ… ) sin(ý± )( $ 1 + $ 2  + ))) + $ +  + −1 ) + L 2 + +1 exp( Œ 2 + +1  ) + L 2 + +2 exp( Œ 2 + +2  ) + ))) + L  exp( Œ   ), where # 1 ,  , # + , $ 1 ,   , $ + , L 2 + +1 ,  , L  are arbitrary constants. = exp( ƒ… ) cos(ý± )( #

1

+#

2

Page 51

4 b . In the general case, where there are & different roots Œ 1 , Œ 2 ,  , ŒŸ of multiplicities , ,  , Ÿ , respectively, the right-hand side of the characteristic equation (2) can be rep¡ 1 ¡ 2 ¡ resented as the product

.

where

¡

1+

¡

2+

))) + ¡

Ÿ =

(Œ ) = (Œ − Œ 1)

+ 1 ( Œ − Œ )+ 2

2

+  ,  ( Œ − ŒŸ ) 

= B . The general solution of the original equation is given by the formula:

Ÿ

½ 

=1

exp( Π  )( L 

,0

+ L  ,1  + ))) + L  , +‹¼

−1

 +‹¼

−1

),

where L  , are arbitrary constants. If the characteristic equation (2) has complex conjugate roots, then in the above solution, one should extract the real part on the basis of the relation exp( ƒeAI™Ýý ) =  9 (cos ýeAI™ sin ý ). 0.4.1-2. Nonhomogeneous linear equations. An B th-order nonhomogeneous linear equation with constant coefficients has the general form

(  ) +  

( 

−1

−1)

+ ))) + 

1


+ 

0

=  ( ).

(3)

The general solution of this equation is the sum of the general solution of the corresponding homogeneous equation (see Paragraph 0.4.1-1) and any particular solution of the nonhomogeneous equation. If all the roots Π1 , Π2 ,  , Π of the characteristic equation (2) are different, equation (3) has the general solution: =

½  Á

=1

  ½ LlÁ   ¡ + . 
 (¡ Œ"Á ) ù Á =1 

 ( )  −  ¡

 M



(for complex roots, the real part should be taken). Table 4 lists the forms of particular solutions corresponding to some special forms of functions on the right-hand side of the linear nonhomogeneous equation. In the general case, a particular solution can be constructed on the basis of the formulas from Paragraph 0.4.2-3.

^_

References for Subsection 0.4.1: G. A. Korn and T. M. Korn (1968), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980), D. Zwillinger (1989).

0.4.2. Linear Equations with Variable Coefficients 0.4.2-1. Homogeneous linear equations. Structure of the general solution. The general solution of the B th-order homogeneous linear differential equation

  ( ) (  ) +  

−1 (

 ) ( 

−1)

+ ))) +  1 ( ) 
+  0 ( ) = 0

(1)

has the form: =L

1 1(

 )+L

2 2(

 ) + ))) + L   ( ).

(2)

Here, 1 ( ), 2 ( ),  ,  ( ) is a fundamental system of solutions (the  are linearly independent particular solutions,  1 0); L 1 , L 2 ,  , L  are arbitrary constants.

Page 52

TABLE 4 Forms of particular solutions of the constant coefficient nonhomogeneous linear equation (  ) +   −1 (  −1) + ))) +  1 
 +  0 =  ( ) that correspond to some special forms of the function  ( ) Form of the function  ( )

. .

.¶

Zero is not a root of the characteristic equation (i.e.,  0 ≠ 0)

+ ( )

+ ( )  9

Form of a particular ¶ solution = ( )

Roots of the characteristic equation Œ  +   −1 Œ  −1 + ))) +  1 Œ +  0 = 0

Zero is a root of the characteristic equation (multiplicity & )

.¶

characteristic equation

( ƒ is a real constant)

ƒ is a root of the



characteristic equation (multiplicity & )

.¶

™Ýý is not a root of the

.

+ ( ) 

+ ( )  9

Ÿ .¶

+ ( )  9



Á¶ ( ) cos ý± + ÆIÁ ( ) sin ý±

characteristic equation

+ ( ) cos ý± + Æ  ( ) sin ý±

Ÿ .¶ 

ƒ is not a root of the



+ ( )

.¶  Ÿ [¶ Á ( ) cos ý± characteristic equation (multiplicity & ) + ÆIÁ ( ) sin ý± ] .¶ ƒ + ™Ýý is not a root of the ¶ [ Á ( ) cos ý±  . characteristic equation + ÆIÁ ( ) sin ý± ]  9 [ + ( ) cos ý±  .¶ 9 + Æ  ( ) sin ý± ]  ƒ + ™Ýý is a root of the  ¶ Ÿ [ Á ( ) cos ý±  characteristic equation (multiplicity & ) + ÆIÁ ( ) sin ý± ]  9 . .¶ .¶ Notation: + and Æ  are polynomials of degrees and B with given coefficients; + , Á , ¶ ¡ and ÆIÁ are polynomials of degrees and @ whose coefficients are determined by substituting ¡ the particular solution into the basic equation; @ = max( , B ); and ƒ and ý are real numbers, ¡ ™ 2 = −1. ™Ýý is a root of the

0.4.2-2. Utilization of particular solutions for reducing the order of the original equation. 1 b . Let

1

=

1(

 ) be a nontrivial particular solution of equation (1). The substitution

 ) XZH ( )  M results in a linear equation of order B − 1 for the function H ( ). 2 b . Let 1 = 1 ( ) and 2 = 2 ( ) be two nontrivial linearly independent solutions of equation (1). =

The substitution

=

1

X

1(

2

¨

 −

M

results in a linear equation of order B − 2 for ¨ ( ).

2

X

1

¨ M



3 b . Suppose that linearly independent solutions 1 ( ), 2 ( ),  , + ( ) of equation (1) are ¡ known. Then one can reduce the order of the equation to B − by successive application of the ¡ following procedure. The substitution = + ( ) X H ( )  leads to an equation of order B − 1 for M the function H ( ) with known linearly independent solutions:

H 1= O

1




, + P 

H 2= O

2




, + P 

 ,

H +

−1

= O

+ +

−1




. P 

Page 53

The substitution H = H + −1 ( ) X ¨ ( )  yields an equation of order B − 2. Repeating this procedure M times, we arrive at a homogeneous linear equation of order B − .

¡

¡

0.4.2-3. Wronskian determinant and Liouville formula. The Wronskian determinant (or simply, Wronskian) is the function defined as: 1 ( ) û 
( ) 1 ( ) = ûû ))) û (  −1) û û 1 ( ) û

ú

))) ))) ))) )))

 ( ) 
( ) û  û, ))) ûû (  −1) ( ) ûû  û

(3)

1 (  ),  ,  (  ) is a fundamental system of solutions of the homogeneous equation (1); +3 + ) ( ) = M  , = 1,  , B − 1; , = 1,  , B .  ¡  + M The following Liouville formula holds:

where (

ú

( ) =

ú

( 0 ) exp ¬ − X 

   −1 (a ) aÝ® .  0  (a ) M

0.4.2-4. Nonhomogeneous linear equations. Construction of the general solution. 1 b . The general nonhomogeneous B th-order linear differential equation has the form

  ( ) (  ) +  



( −1)

−1

+ ))) +  1 ( ) 
+  0 ( ) = ( ).

(4)

The general solution of the nonhomogeneous equation (4) can be represented as the sum of its particular solution and the general solution of the corresponding homogeneous equation (1). 2 b . Let 1 ( ),  ,  ( ) be a fundamental system of solutions of the homogeneous equation (1), ú and let ( ) be the Wronskian determinant (3). Then the general solution of the nonhomogeneous linear equation (4) can be represented as: =

ú

½ Á

LlÁ Á ( ) + =1

½  Á

Á ( ) X =1

ú

Á ( )  ú M ,   ( ) ( )

where Á ( ) is the determinant obtained by replacing the @ th column of the matrix (3) by the column vector with the elements 0, 0,  , 0, .

^_

References for Subsection 0.4.2: G. A. Korn and T. M. Korn (1968), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980), D. Zwillinger (1989).

0.4.3. Asymptotic Solutions of Linear Equations This subsection presents asymptotic solutions, as ÑD 0 ( > 0), of some higher-order linear ordinary differential equations containing arbitrary functions (sufficiently smooth), with the independent variable being real. 0.4.3-1. Fourth-order linear equations. 1 b . Consider the equation

4





 −  ( ) = 0

Page 54

on a closed interval  ≤  ≤  . With the condition  > 0, the leading terms of the asymptotic expansions of the fundamental system of solutions, as pD 0, are given by the formulas: 1 1 −3 J 8 exp ™ − ù [  ( )]1 J 4 ˆ¢ , 2 = [  ( )]−3 J 8 exp ™ 1 = [  (  )] ù [  ( )]1 J 4 ˆ¢ ,

3

= [  ( )]−3 J 8 cos ™

M

1

[  ( )]1 J 4 ˆ¢ ,

ù M

4

= [  ( )]−3 J 8 sin ™

M

1

[  ( )]1 J 4 ˆ¢ .

ù M

2 b . Now consider the “biquadratic” equation:

4 



 − 2 2 ( ) 

 −  ( ) = 0. (1) Introduce the notation: c ( ) = [ ( )]2 +  ( ). In the range where the conditions  ( ) ≠ 0 and c ( ) ≠ 0 are satisfied, the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (1) are described by the formulas: 1 [ Œ  ( )]
 1 ˆ¢ ; , = 1, 2, 3, 4,  = [ Œ  ( )]−1 J 2 [ c ( )]−1 J 4 exp ™ ù Œ  ( ) M  − 2 ù £ c ( ) M where Œ 1 ( ) = ( ) + £ c ( ), Œ 2 ( ) = − ( ) + £ c ( ),

Œ 3 ( ) = ü

ü

ü Œ 4 ( ) = − ( ) − £ c ( ). ü

( ) − £ c ( ),

0.4.3-2. Higher-order linear equations. 1 b . Consider an equation of the form

 (  ) −  ( ) = 0 on a closed interval  ≤  ≤  . Assume that  ≠ 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions, as pD 0, are given by: 1   + −1 + 1 V  ( )W  ˆ¢ V 1 + [ ( )W , + = V  ( )W 2 2  exp ™ ù

M  where   1 ,   2 ,  ,    are roots of the equation   = 1: 2

2

  + = cos O /¡ + ™ sin O /¡ , B P B P

¡

= 1, 2,  , B .

2 b . Now consider an equation of the form

 (  ) +  −1   −1 ( ) (  −1) + ))) + ´ 1 ( ) 
+  0 ( ) = 0 (2) on a closed interval  ≤  ≤  . Let Œ + = Œ + ( ) ( = 1, 2,  , B ) be the roots of the characteristic ¡ equation: . ( , Œ ) ≡ Œ  +  ( ) Œ  −1 + ))) +  ( ) Œ +  ( ) = 0.  −1 1 0 Let all the roots of the characteristic equation be different on the interval  ≤  ≤  , i.e., the conditions Œ + ( ) ≠ Œ  ( ), ≠ , , are satisfied, which is equivalent to the fulfillment of the . ( , Œ + ) ≠ 0. Then ¡ the leading terms of the asymptotic expansions of the fundamental conditions  system of solutions of equation (2), as ID 0, are given by: .  , Œ + ( )  1 1
 Œ + ( )  − ù [ Œ + ( )] .  ˆ¢ , + = exp ™  M

ù M 2   , Œ + ( ) where .

.

 −1 + (B − 1)   −1 ( ) Œ  −2 + ))) + 2 Œ" 2 ( ) +  1 ( ),  ( , Œ ) ≡ N Œ = B…Œ . ( , Œ ) ≡ N N 2 . = B (B − 1) Œ  −2 + (B − 1)(B − 2)  ( ) Œ  −3 + ))) + 6 Œ" ( ) + 2  ( ).  −1 3 2  Œ 2 ^_ N References for Subsection 0.4.3: W. Wasov (1965), M. V. Fedoryuk (1993).

Page 55

0.5. Nonlinear Equations of Arbitrary Order 0.5.1. Structure of the General Solution. Cauchy Problem 0.5.1-1. Equations solved for the highest derivative. General solution. An B th-order differential equation solved for the highest derivative has the form

  =  ( , , 
,  , (  ( )

−1)

).

(1)

The general solution of this equation depends on B arbitrary constants L 1 ,  , L  . In some cases, the general solution can be written in explicit form as = Y ( , L 1 ,  , L  ). 0.5.1-2. The Cauchy problem. The existence and uniqueness theorem. 1 b . The Cauchy problem: find a solution of equation (1) with the initial conditions ( 0 ) =

0,


( 0 ) =

(1) 0 ,

 ,

( 

−1)

( 0 ) =



( −1) . 0

(2)

(At a point  0 , the values of the unknown function ( ) and all its derivatives of orders ≤ B − 1 are prescribed.) 2 b . The existence and uniqueness theorem. Suppose the function  ( , , H 1,  , H  −1 ) is continuous in all its arguments in a neighborhood of the point ( 0 , 0 , 0(1) ,  , 0(  −1) ) and has bounded derivatives with respect to , H 1 ,  , H  −1 in this neighborhood. Then a solution of equation (1) satisfying the initial conditions (2) exists and is unique.

^_

References for Subsection 0.5.1: G. A. Korn and T. M. Korn (1968), I. G. Petrovskii (1970), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980).

0.5.2. Equations Admitting Reduction of Order 0.5.2-1. Equations not containing , 
 ,  , (  ) explicitly. An equation that does not explicitly contain the unknown function and its derivatives up to order , inclusive can generally be written as

   , ( 

+1)

 ) = 0 ,  , (

(1 ≤ , + 1 < B ).

(1)

Such equations are invariant under arbitrary translations of the unknown function, D +const (the form of such equations is also preserved under the transformation ‡ ( ) = +     + ))) +  1  +  0 , where the  + are arbitrary constants). The substitution H ( ) = (  +1) reduces (1) to an equation whose order is by , + 1 smaller than that of the original equation, Ê , H , H 
,  , H (  −  −1)  = 0. 0.5.2-2. Equations not containing  explicitly (autonomous equations). An equation that does not explicitly contain  has in the general form

 )  = 0.   , 
,  , (

(2)

Such equations are invariant under arbitrary translations of the independent variable, CD  + const . The substitution 
= ¨ ( ) (where plays the role of the independent variable) reduces by one the order of an autonomous equation. Higher derivatives can be expressed in terms of ¨ and its derivatives with respect to the new independent variable, "

 = ¨*¨2
, 


 = ¨ 2 ¨2

+ ¨ (¨2
)2 , 







Page 56

( )  0.5.2-3. Equations of the form    +  , 
 ,  ,  = 0.

Such equations are invariant under simultaneous translations of the independent variable and the unknown function, ED„ + ˜( and D − !( , where ( is an arbitrary constant. For  = 0, see equation (1). For  ≠ 0, the substitution ¨ ( ) = +(  ) leads to an autonomous equation of the form (2).

 )  = 0 and its generalizations. 0.5.2-4. Equations of the form  ,  
 − , 

 ,  , (

The substitution ¨ ( ) =  
− reduces the order of this equation by one. This equation is a special case of the equation

   ,  
− , ( + ¡ The substitution ¨ ( ) =  
− ¡

+1)

,  , (  )  = 0,

where

¡

= 1, 2,  , B − 1.

(3)

reduces by one the order of equation (3).

0.5.2-5. Homogeneous equations. 1 b . Equations homogeneous in the independent variable are invariant under scaling of the independent variable, |D†ƒ… , where ƒ is an arbitrary constant (ƒ ≠ 0). In general, such equations can be written in the form  )  = 0.   ,  
,  2 

 ,  ,   ( The substitution H ( ) = 


reduces by one the order of this equation.

2 b . Equations homogeneous in the unknown function are invariant under scaling of the unknown function, D ƒ , where ƒ is an arbitrary constant ( ƒ ≠ 0). Such equations can be written in the general form   , 
 , 

  ,  , (  )   = 0. The substitution H ( ) = 
 reduces by one the order of this equation.

3 b . Equations homogeneous in both variables are invariant under simultaneous scaling (dilatation) of the independent and dependent variables, CD ƒ… and D ƒ , where ƒ is an arbitrary constant ( ƒ ≠ 0). Such equations can be written in the general form

 )  = 0.     , 
,  

 ,  ,   −1 ( The transformation a = ln | |, ¨ =   leads to an autonomous equation considered in Para-

graph 0.5.2-2.

0.5.2-6. Generalized homogeneous equations. 1 b . Generalized homogeneous equations (equations homogeneous in the generalized sense) are invariant under simultaneous scaling of the independent variable and the unknown function, uD ƒ… and D ƒ  , where ƒ ≠ 0 is an arbitrary constant and , is a given number. Such equations can be written in the general form ( )     −  ,  1−  
,  ,   −   = 0. ¨ The transformation a = ln  , =  − leads to an autonomous equation considered in Para-

graph 0.5.2-2.

2 b . The most general form of generalized homogeneous equations is

¸    + ,  
 ,  ,   (  )   = 0. The transformation H =   + , ‡ =  
  reduces the order of this equation by one.

Page 57

  
 



,  ,  ,  , (  )   = 0.

0.5.2-7. Equations of the form  

Such equations are invariant under simultaneous translation and scaling of variables, eD  + ƒ and  DGý , where ý = exp(−ƒsŒ B ), and ƒ is an arbitrary constant. The transformation H = !  ,
¨ =   leads to an equation of order B − 1. 0.5.2-8. Equations of the form       ,  
 ,  2 

 ,  ,  

   = 0.  ( )

Such equations are invariant under simultaneous scaling and translation of variables,  D ƒ… and D + ý , where ƒ = exp(−ýhŒ B ), and ý is an arbitrary constant. The transformation H =    , ¨ =  
 leads to an equation of order B − 1. 0.5.2-9. Other equations. Consider the nonlinear differential equation

   , L1 [ ],  , L  [ ]  = 0, À

(4)

where the L [ ] are linear homogeneous differential forms, À

À

L [ ]= Let

0

=

0(

½  £

+

=0

Y (+ ) ( ) ( + ) ,

° = 1,  , , .

 ) be a common particular À solution of the linear equations: ( ° = 1,  , , ).

L [ 0] = 0 Then the substitution

¨ = Y ( ) V ( ) 
−
( ) W 0 0

(5)

with an arbitrary function Y ( ) reduces by one the order of equation (4). Example 1. Consider the third-order equation

Ë ÏÒ˜Í Í Ò˜Í Ò

= â (Ë

ÏÒÍ

− 2Ï ).

It can be represented in the form (4) with

ô

= 2,

ê (Ë

¤ 

, , )=

The linear equations L [ Ï ] = 0 are

õ

Ë

¤



− â ( ),

ÏÒ˜Í Í Ò˜Í Ò

= 0,

Ë ÏÒÍ

ÏÒ˜Í Í Ò˜Í Ò

L2 [ Ï ] =

,

Ë ÏÒÍ

− 2 Ï = 0.

− 2Ï .

2 . Therefore, the substitution These equations have a common particular solution = Ë Ï ÒÍ − 2 Ï Ë Ë ˜ Ò Ò with ( ) = 1 ) leads to an autonomous second-order equation of the form 2.9.1.1: Í Í = â ( ).

ä

%

5

Example 2. The th-order equation

Ï (Ò E

can be represented in the form (4) with

ô

= 2,

ê (Ë

The linear equations have a common particular solution, leads to an ( − 1)st-order equation.

^_

5

¤ 

, , )=

¤

L1 [ Ï ] ≡

Ï0=ÕÒ

Ï 0=Ë

L1 [ Ï ] =

)

= â (Ë , Ï

− â ( Ë , ),



Ï (Ò E

)

− Ï = 0,





Ò˜Í Í Ò



(see formula (5)

−Ï )+Ï

L1 [ Ï ] =

Ï (Ò E

L2 [ Ï ] ≡

Ï Ò˜Í Í Ò

. Therefore, the substitution

−Ï ,

)



L2 [ Ï ] =

−Ï =0 =

Ï ÒÍ

−

Ï

ÏÒ˜Í Í Ò

−Ï .

(see formula (5) with

ä

(Ë ) =

Õ −Ò

)

References for Subsection 0.5.2: G. M. Murphy (1960), G. A. Korn and T. M. Korn (1968), E. Kamke (1977), V. F. Zaitsev and A. D. Polyanin (1993, 2001), A. D. Polyanin and V. F. Zaitsev (1995), D. Zwillinger (1998).

Page 58

0.5.3. A Method for Construction of Solvable Equations of General Form 0.5.3-1. Description of the method. Consider a function

=  ( , L

L 2 ,  , L 

1,

+1 )

(1)

depending on B + 1 free parameters L  . Differentiating relation (1) B times, we obtain the following sequence of equations:

  =   (  ) ( , L 1 , L 2 ,  , L  ( )

, = 1, 2,  , B .

+1 ),

(2)

Treating relations (1), (2) as an algebraic (transcendental) system of equations for the parameters L 1 , L 2 ,  , L  +1 and solving this system, we obtain

L  = Y    , , 
,  , (  )  ,

, = 1, 2,  , B + 1.

(3)

Consider a general B th-order equation of the form

 ( Y 1 , Y 2 ,  , Y 

+1 )

= 0,

(4)

 )  are the functions where  is an arbitrary function of (B +1) variables and Y  = Y    , , 
 ,  , ( from (3). Equation (4) is satisfied by the function (1), where the (B + 1) arbitrary parameters L 1 , L 2 ,  , L  +1 are related by a single constraint:  ( L 1 , L 2 ,  , L 

+1 )

= 0.

x3y'z|{}~­³´€

Equation (4) may also have singular solutions depending on a smaller number of arbitrary constants. In order to examine these solutions, one should differentiate equation (4); see Example 1.

x3y'z|{}~‚µ €

Instead of (4), one can consider a more general equation

 (j 1 , j 2 ,  , j 

+1 )

= 0,

j  = j  ( Y 1 , Y 2 ,  , Y 

where

+1 ).

x3y'z|{}~E¹€ The original expression (1) can be specified in an implicit form. x3y'z|{}~Eº€ The original expression (1) can be written as an th-order differential equation ¡ ( < B ) with B − + 1 free parameters L  . The solution of the B th-order differential equation ¡obtained in this way¡ can be expressed in terms of the solution of an th-order differential equation ¡ (see Example 4). 0.5.3-2. Examples. Example 1. Consider the function

Ï

By differentiation we obtain Let us solve equations (5)–(6) for the parameters



ÏÒÍ



1

1

=



=−

and

Ò Õ Ï ÒÍ



Õ

1

=

−



Ò

1



+

Õ

−

Ò

2.

(5)

.

(6)

2 . We have

,



2

=

ÏÒÍ

+Ï .

Using the above method, we construct an equation in accordance with (4):

Ò ê Õ Ï ÒÍ , Ï ÒÍ ë

This equation admits a solution of the form (5) with constants



+Ï 1

ì

= 0.

and



2

(7) related by the constraint

ê

(

 1, 

2)

= 0.

Page 59

Ë

, we get

Singular solution. Differentiating equation (7) with respect to

¤

Ò

( ÏҘ Í Í Ò + ÏÒÍ )( Õ

¨

ꦥ

¦§ ) = 0,

+ê

(8)

¤ ¨

where the subscripts and indicate the respective partial derivatives of the function ê = ê ( , ). Equating the first factor in (8) to zero, we obtain solution (5). Equating the second factor to zero, we obtain an expression which, combined with equation (7), yields a singular solution in parametric form:

One should eliminate

:

ê =

Ï ÒÍ

Ò Õ ¦ê ¥

¤ ¨

( , ) = 0,

¦§

+ê

= 0,

¤

where

Ò

Õ

=

:, ¨ =:

+Ï .

from these expressions.

Example 2. Consider the function

Ï

Differentiating this function twice, we get

Solving (9)–(10) for the parameters



1

=

 1 2

õ

=



1

ÏÒÍ Ï Ò˜Í Í Ò

Ë

+





1

2

=2 =2



2

Ë

Ë

+



+



2,

3.

(9) (10)

1.

, we find that

ÏÒ˜Í Í Ò



,

2

=

ÏÒÍ

−Ë

ÏÒ˜Í Í Ò



,

3

−Ë

Ï

=

ÏÒÍ

+

1 2

Ë 2 Ï Ò˜Í Í Ò

.

These relations lead to a second-order equation of general form:

ê

1 2

ë

−Ë

ÏÒ˜Í Í Ò , Ï ÒÍ

−Ë

ÏÒ˜Í Í Ò , Ï

ÏÒÍ

+

1 2

Ë 2 Ï Ò˜Í Í Òì

= 0,

 1 ,  2 , and  3 related by the constraint ê (  1 ,  2 ,  Example 3. In Example 2, one can choose the functions  of the form (see Remark 2) õ  1 = 2 ä 1 ,  2 = − ä 2 ,  3 = 4 ä 1 ä 3 − ä 22 ,

which has a solution of the type (9) with the three constants

where

ä

1

=

1 2

Ï Ò˜Í Í Ò

,

ä

2

=

Ï ÒÍ

−Ë

Ï Ò˜Í Í Ò

,

ä

3

©

=

ë

Ï

−Ë

Ï ÒÍ

+

Ï Ò˜Í Í Ò , Ë Ï Ò˜Í Í Ò

Its solution is given by (9) with three constants

 1, 

1 2

Ë 2 Ï Ò˜Í Í Ò ÒÍ

−Ï

2 , and

Example 4. Consider the autonomous equation

Ï Ò˜Í Í Ò

=





1

= 0.

. As a result, we obtain the differential equation:

, 2 ÏÏ 3

3)

Ò˜Í Í Ò

Ò Í )2 ì

− (Ï

= 0.

related by a single constraint

Ï −×

+



© (2 

1, −



 

2, 4 1

3−



2 3)

2.

= 0.

(11)

Its solution can be represented in implicit form (see 2.4.2.1 and 2.9.1.1). Differentiating (11), we obtain

D 1 Ï −× −1 ÏÒÍ . Let us solve equations (11)–(12) for the parameters  1 and  2 :  1 = −Ï × +1 Ï ÓÒ˜Í ÏÍ Ò˜Í ÒÍ Ò ,  2 = Ï Ò˜Í Í Ò + Ï Taking  1 = − Ó 1 and  2 = Ó 2 (see Remark 2), we obtain the equation: ä ä ÏÒ˜Í Í Ò˜Í Ò

êCñÏ ×

+1

= −Ó

Ï ˜ÒÍ Í Ò˜Í Ò , Ï Ï Ò˜Í Í Ò˜Í Ò Ï ÒÍ Ï ÒÍ

+

ÓÏ Ò˜Í Í Ò ò

(12)

Ï ˜ÒÍ Í Ò˜Í Ò ÓÏ ÒÍ

.

= 0.

This equation is satisfied by the solutions of a second-order autonomous equation of the form (11), where the constants and 2 are related by the constraint ê (− Ó 1 , Ó 2 ) = 0.



D D



1

0.6. Lie Group and Discrete-Group Methods 0.6.1. Lie Group Method. Point Transformations 0.6.1-1. Local one-parameter Lie group of transformations. Invariance condition. Here, we examine transformations of the ordinary differential equation

  =    , ,  , (  ( )

−1)

.

(1)

Page 60

Consider the set of transformations Tª =

 ¯ |ª

 ¯ = Y ( , , ), ¯ = j ( , , ),

™

=0

¯|ª

=0

= , = ,

(2)

where Y , j are smooth functions of their arguments and is a real parameter. The set T ª is called a continuous one-parameter Lie group of point transformations if, for any 1 and 2 , the following relation holds: T ª 1 « Tª 1 = Tª 1 +ª 2 , i.e., consecutive application of two transformations of the form (1) with parameters 1 and 2 is equivalent to a single transformation of the same form with parameter

1 + 2. In what follows, we consider local continuous one-parameter Lie groups of point transformations (briefly called point groups) corresponding to an infinitesimal transformation (2) for ·D 0. Taylor’s expansion of  ¯ and ¯ in (2) with respect to the parameter about = 0 yields:

­ ¯ ¬v + ˆ ( , ) ,

¯

¬

+ ‰ (  , ) ,

(3)

where

Y ( , , ) j ( , , ) , ‰ ( , ) = N û û ª =0 . ª

û =0 û N N û At each point ( , ), the vector (ˆ , ‰ ) is tangent to the curve described û by the transformed points ( ¯ , ¯ ). ˆ ( , ) = N

The first-order linear differential operator

X = ˆ ( , ) N

+ ‰ ( , ) N



N

(4)

N

corresponding to the infinitesimal transformation (3), is called the infinitesimal operator (or infinitesimal generator) of the group. By definition, the universal invariant (briefly, invariant) of the group (2) and the operator (4) is a function ® 0 ( , ), satisfying the condition ® 0 ( ¯ , ¯ ) = ® 0 ( , ). Taylor’s expansion with respect to the small parameter yields the following linear partial differential equation for ® 0 : X®

0

= ˆ ( , ) N

N

®0

+ ‰ ( , ) N



®0

N

= 0.

Equation (1) will be treated as a relation for B +2 variables  , , 
,  , (  constraints



( +1)

= M



( )

M



(5) )

with the differential

.

(6)

The space of these B + 2 variables is called the space of B th prolongation; and in order to work with differential equations, one has to define the action of operator (4) on the “new” variables 
 ,  , (  ) , taking into account the differential constraints (6). For example, let us calculate the infinitesimal transformation of the first derivative. We have

M

M

¯ c  ( + ‰ ) =  ¯ c  ( + ˆ )

¬


 + (‰  + ‰ 
 )  1 + ( ˆ  + ˆ 
) 

( c  =  + 
 + ))) is the operator of total derivative). Expanding the right-hand side into a power N N  series with respect to the parameter and preserving  the first-order terms, we obtain where




¯ ¯

 1

= ‰  + (‰

¬


+ 1 (  , , 
) ,

 
 
2 
  − ˆ ) − ˆ  ( ) = c (‰ ) − c (ˆ ).

The action of the group on higher-order  derivatives  is determined by the recurrence formula:



+1

= c  (  ) − ( 

+1)

c  (ˆ ).

Page 61

To a prolonged group there corresponds a prolonged operator: X = ˆ ( , ) N



N

+ ‰ ( , ) N



+

N



½ 


( )     , , ,  , È

=1

N

N ( ) . 

(7)

The ordinary differential equation (1) admits the group (2) if X V (  ) −    , , 
,  , ( 



−1)

W

= 0. û  (œ n ) = Ž û

(8)

Relation (8) is called the invariance condition. x3y'z|{}~/€ The invariant ® 0 , which is a solution of equation (5), also satisfies the equation X ® 0 = 0.



0.6.1-2. Group analysis of second-order equations. Structure of an admissible operator. For second-order nonlinear equations



 =  ( , , 
),

(9)

the invariance condition (8) is written in the form

‰  + (2‰   − ˆ   ) 
+ (‰   − 2ˆ   )( 
)2 − ˆ  ( 
)3


= (2ˆ  − ‰ + 3ˆ  )  + ˆž  + ‰ + [‰  + (‰ − ˆ  )  − ˆ (  )2 ]       œ , where  =  ( , , 
). This condition is in fact a second-order partial differential equation for two unknown functions ˆ ( , ) and ‰ ( , ). Since the unknown functions do not depend on the derivative 
, this equation can be represented (after  has been expanded in a power series with respect to 
, unless it is already a polynomial) in the form ½ 

÷ =0

`  ( 
)  = 0,

(10)

with the `  independent of 
. In order to ensure that condition (10) holds identically, one should set `  = 0, , = 0, 1,  Thus, the invariance condition for a second-order equation can be “split” and represented as a system of equations (whose number can generally be infinite).

Example 1. If ê = ê ( Ë , Ï ), i.e., the right-hand side of equation (9) does not depend on equation can be “split” and represented as the system:

~2N = 0, N − 2~ Ò  = 0, 2  Ò  − ~ Ò˜Ò − 3 ê ( Ë , Ï ) ~2 = 0,  Ò˜Ò + ( − 2~ Ò )ê (Ë , Ï ) − ê Ò (Ë

~ ¦ (Ë

Ï ÒÍ

, then the determining



,Ï ) − ê

, Ï ) = 0.

From the first two equations we find that where Ó ( Ë ), Ô ( Ë ), equations, we get

¯ (Ë

~

), and 3 ÓÍ Í

Ó ÍÍÍÏ

Ï

æ 2

= Ó ( Ë ) Ï + Ô ( Ë ),



=

ÓÍ (Ë )Ï

+ ( Ë ) Ï + ( Ë ),

¯

2

æ

(Ë ) are arbitrary functions. Substituting these expressions into the third and the fourth + 2 ÖÍ − ÔÝÍ Í − 3 ê ( Ë , Ï ) Ó = 0,

¯

+

¯ ÍÍÏ

+

æ

ÍÍ

¯

+ ( − 2Ô Í )ê

− ( ÓÏ + Ô ) ê

Ò

− (Ó

ÍÏ

2

¯

¦

+ ÖÏ + ) ê

æ

= 0.

(11)

In what follows, it is assumed that the function ê ( Ë , Ï ) is nonlinear with respect to the second argument. Then from the first equation in (11), we find that Ó = 0 and = 12 Ô Í + , where is an arbitrary constant. The second equation in (11) becomes 1 ÔÝÍ Í Í Ï + Í Í + − 32 ÔÝÍ ì ê − ÔÖê Ò − 12 ÔÝÍ + ì Ï + î ê = 0. (12) 2

æ

ë

°

¯

°

Equation (12) allows us to solve two different problems.

°

íë

°

æ

¦

Page 62

1 . If the function ê ( Ë , Ï ) is given, then, splitting equation (12) with respect to powers of Ï (the unknown functions Ô and are independent of Ï ), we obtain a new system, from which Ô , , and can be found; i.e., we ultimately obtain an æ æ admissible operator.

O

°

°

O

2 . Assuming that the functions Ô , and the constant are known but arbitrary, one can regard relation (12) as an equation æ for the unknown function ê ( Ë , Ï ). Solving this equation, we obtain a class of equations admitting a point operator.

ê (Ë , Ï

Example 2. Let 1 2

ÔÝÍ Í Í Ï

 Ë E ÏD± , i.e., we are dealing with the Emden–Fowler equation. Then equation (12) becomes + Í Í + ° − 32 ÔÝÍ ì  Ë Ï ± − Ô²5³ Ë −1 Ï ± − 12 ÔÝÍ + ° ì Ï + îµ´  Ë Ï ± −1 = 0. E E E æ ë íë æ )=

This relation must be satisfied identically by any function Ï = Ï ( Ë ), and therefore, the coefficients of different powers of must be equal to zero. As a result, we obtain a new system whose structure essentially depends on the value of .

ê (Ë , Ï

O

1 . It was assumed above that Then the system has the form:

´

) is nonlinear in its second argument, and therefore,

≠ 0 and

´

´

≠ 1. Let

´

Ï

≠ 2.

Í Í = 0, Ô Í Í Í = 0, æ æ

It follows that

æ

= 0 and Ô ( Ë ) = (

Ô 2Ë

´

+

+Ô

Ë

= 0,

° í

´

(1 −

)−

1 (3 2

´

−

îË

) ÔÝÍ

5

− !Ô = 0.

Ô 0 , and the last equation of the system can be written in the form + 3) Ô 2 Ë 2 + 12 ( ´ + 2 5 + 3) Ô 1 + ° ( ´ − 1)î Ë + 5!Ô 0 = 0. í

2

5

1

+

(13)

To ensure relation (13), we equate all coefficients of this quadratic trinomial to zero to obtain (

´

+

5

+ 3) Ô

2

1 ( 2

= 0,

´

€

5

+ 2 + 3) Ô

€

° ´

+ (

1

5!Ô 0 = 0.

− 1) = 0,

(14)

Analysis of system (14) yields solutions of the determining system corresponding to three different operators:

€ X1 = ( ´ X2 =

O

2 . Let

´

X3 =

Ë

€ Ò 2

− 1) Ë

Ò

Ò

+Ë

5



− ( + 2) Ï

€

if if



Ï

5 5

if

´

°

ìË

and = 0, +

´

are arbitrary,

5

+ 3 = 0.

= 2. Then equation (12) becomes

æ Equating the term

æ

æ ÍÍ

Ô

(Ë ) =

ÍÍ

+

ë

1 2

Ô ÍÍÍ

−2

Ï

æ

and the coefficient of 1

Ë

+

æ

 æ

Ë ìÏ E

−

5 2

íë

ÔÍ

+

in parentheses to zero, we get

0,

4 Ó 0 Ë +3 4 Ó 1 Ë +4 + +Ô (Ë ) = æ æ ( + 2)( + 3)( + 4) ( + 1)( + 2)( + 3)

5

5 î  Ë E −1 Ï 2 = 0.

+ !Ô

5 E 5

5 E 5

5

2

Ë

2

+Ô

1

Ë

+ Ô 0,

5

The expression in square brackets (the coefficient of Ï 2 ) can be split with respect to powers of system which, to within nonzero coefficients, has the form: (7 (7 ( (2

5

5!Ô

5

5

5

≠ −1, −2, −3, −4.

Ë

and we obtain an algebraic

+ 20) 1 = 0, æ + 15) 0 = 0, æ + 5) Ô 2 = 0, + 5) Ô 1 + 2 = 0, 0 = 0.

°

The last three equations coincide with the corresponding equations of system (14), whose solutions are already known. The first two equations yield two cases of prolongation of the admissible group:

€  X1 = 343  8 7 € Ò  X2 = 343  Ë 6 7 Ò Ë

€  Ë + 4 49  1 7 Ï − 3 ì €  ë  + 3 49  Ë −1 7 Ï + 4 ì  ë Ë

if if

5

5

= − 20 , 7 = − 15 . 7

Page 63

0.6.1-3. Utilization of local groups for reducing the order of equations and their integration. Suppose that an ordinary differential equation (1) admits an infinitesimal operator X of the form (4). Then the order of the equation can be reduced by one. Below we describe two methods for reducing the order of an equation. 1 b . The first method. The transformation

a =  ( , ),

‡ = ( , ),

(15)

with  and ( *1 0) being arbitrary particular solutions of the first-order linear partial differential equations

ˆ ( , ) N



N



+ ‰ ( , ) N





N

=, , (16)



ˆ ( , ) N + ‰ ( , ) N  N N

= 0,

reduces equation (1) to an autonomous equation (the constant , ≠ 0 can be chosen arbitrarily). The function = ( , ) is a universal invariant of the operator X. Suppose that the general solution of the characteristic equation



M

has the form

M ‰ ( , )

=

ˆ ( , )

’ ( , ) = L ,

where L is an arbitrary constant. Then the general solutions of equations (16) are given by (see A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, 2002):

 = ,‹X



+ i 1 ( ’ ), ˆ ( , ’ ) = i 2 ( ’ ), ’ = ’ ( , ),

M

where i 1 ( ’ ) and i 2 ( ’ ) are arbitrary functions, ˆ   , ’ ( , )  ≡ ˆ ( , ), and ’ regarded as a parameter.

€

Ï ˜ÒÍ Í Ò

€

Example 3. The Emden–Fowler equation Example 2): X = (Ë , Ï )

~

Equations (16) for

Solving (15) for

Ë

ô

= 49



€

Ë

+ (Ë , Ï )



€

,

Ï

Ï

~ (Ë

where

=



Ë

1 7

,

ã

Ï

−15 7 2

=

Ë



−3 7

Ï

+

 Ë 6  7 ,  (Ë

6 49



, we obtain the transformation

Ë

: 7,

=

Ï

=

: 3¤

= 49

¤

2

Ë



−2 7

, Ï ) = 147



Ë



−1 7

Ï

+ 12.

.

 :,

6 49

−

which reduces the original equation to the autonomous equation

¤ Í89Í 8

O

admits the operator (cf. the operator X2 in Item 2 of

, Ï ) = 343

admit the particular solutions

â and



Ë



=

in the integral is

,

which can easily be integrated by quadrature.

2 b . The second method. Suppose that we know two invariants of the admissible operator X:

® 0 = ® 0 ( ® 1 = ® 1 (

´

, ) , , 
)

(universal invariant), (first differential invariant*).

(17) (18)

* By definition, an th-order differential invariant of the operator X is a function è = = 0 with the operator X defined by (7). satisfying the linear partial differential equation X è

± ±

±

±

è

± ë

Ë , Ï , Ï ÒÍ , ªÿ ÿ›ÿ , Ï (Ò ± ) ì

,

Page 64

Then the second differential invariant can be found by differentiation,

® 1,

, , 
, 

 ) = M

® 2 (

(19)

®0

M

where ® + = ( c  ® + )  . Using (18)–(19), let us eliminate the derivatives 
and 

 from the M M original equation and take into account relation (17). Thus we obtain the first-order equation:

®1 M M

®0

Example 4. The Emden–Fowler equation whose first prolongation has the form: X= 1

This operator admits the invariants:

Ï Ò˜Í Í €Ò Ë

2

 Ë −6€ Ï 3 (see 2.3.1.3, € the special case ´ + Ë Ï  + ( Ï − Ë ÏÍ )  . 

Ò

=

è0=Ï % Ë

€

= 4 ( ® 0 , ® 1 ).

€

è 1 = Ë ÏÒÍ

,

= 3) admits an operator

−Ï ,

(20)

€

which form an integral basis of the first-order linear partial differential equation

Ë

€

2

Ë

è

+Ë

€

Ï è Ï

+ (Ï − Ë

€

Ï Í) è ÏÍ

= 0.

Using (19) and (20), we find the second invariant:

è2= æ è1 è æ 0

=

Ë 3 Ï ˜ÒÍ Í Ò Ë Ï ÒÍ − Ï

.

(21)

Let us express the unknown function and its derivatives from (20)–(21) to obtain

Ï

=

¤

Ë

,

ÏÒÍ

=

Ë

¤

Ë

+



,

ÏÒ˜Í Í Ò

=

„ ¥Í Ë 3

,

where

¤

Substituting these expressions into the original equation, we see that the variable form hÍ = 3 ,

è 0,

=

Ë



=

è 1.

is canceled and the equation takes the

„ ¥ ¤

i.e., it becomes a first-order separable equation.

^_

References for Subsection 0.6.1: G. W. Bluman and J. D. Cole (1974), L. V. Ovsiannikov (1982), J. M. Hill (1982), P. J. Olver (1986), G. W. Bluman and S. Kumei (1989), H. Stephani (1989), N. H. Ibragimov (1994).

¨ 0.6.2. Contact Transformations. Backlund Transformations. Formal Operators. Factorization Principle 0.6.2-1. Contact transformations. The set of transformations Tª =



 ¯ = Y ( , , 
 , ), ¯ = j ( , , 
 , ), 
¯  = ¶ ( , , 
 , ), ¯

 ¯ |ª

=0 =  ¯ |ª =0 = 
¯  |ª = ¯ =0

, ,




(1)

(here, Y , j , ¶ are smooth functions of their arguments and is a real parameter) is called a continuous one-parameter Lie group of tangential transformations (or simply, a tangential or contact group) if T ª 1 « Tª 2 = Tª 1 +ª 2 , i.e., if successive application of transformations (1) with parameters 1 and 2 is equivalent to the same transformation with parameter 1 + 2 . The transformed derivative ¯ 
¯ depends only on the first derivative 
and does not depend on the second derivative. Thus, the functions Y and j in (1) cannot be arbitrary but are related by (see Paragraph 0.1.8-1):

Y j j Y Y j N
O…N + 
N − N
… O N + 
N = 0,    P  P N N N N N N

Page 65

where the function

¶

is defined by

¶

j

= N
¸·

N

N

Y N
. 

Proceeding as in Paragraph 0.6.1-1, we consider the Taylor expansions of  ¯ , ¯ , and /
¯  ¯ in (1) with respect to the parameter about = 0, preserving only the first-order terms. We have 

¹ ¯ ¬¯ + ˆ ( , , 
) ,

where

Y ( , , 
, ˆ ( , , 
) = N

N

+ ‰ (  , , 
) ,

¬

¯

¯ 
¯



) = N j ( , ,  ,  , ‰ (  , , û ª =0

û N û

)

On the other hand,

¯ c  ( + ‰ ) =  ¯ c  ( + ˆ )

¯
¯ ≡ M

M


+ (  , , 
) ,

¬



)

ûª û û

=0

,


( , ,  ) = N


 + (‰  + ‰ 
 + ‰  œ 1+  (ˆ  + ˆ  
+ ˆ  œ

¬

¶

( , , 
, )

N



 ) . 

 )

û

û

ûª

=0

.

(2)

 Expanding (2) with respect to and requiring that be independent of

 , we find that three the ú functions ˆ , ‰ , and are expressed in terms of a single function ( , ,
 ) as follows: ˆ =−N N

ú




‰ =

,

ú

− 
N



N

ú




= N

,

N

ú



ú

+ 
N

.

(3)

N

To an infinitesimal tangential transformation (1) there corresponds the infinitesimal operator: X = ˆ ( , , 
) N + ‰ ( , , 
) N  N N



+ ( , , 
) N


(4)

N

whose coordinates satisfy relations (3). The action of the group on higher  derivatives  is determined by the recurrence formula:



 

+1

= c  (  ) − ( 

+1)

c  (ˆ ),

where 1 = . The invariance condition and the algorithm of finding tangential operators (4) admitted by ordinary differential equations are similar as those for point operators. The only difference is that the coordinates of the tangential operator depend on the first derivative; therefore, the determining equation can be split and reduced to a system only in the case of equations whose order is greater or equal to three. x3y'z|{}~/€ There are no tangential transformations of finite order , > 1 other than prolonged point transformations and contact transformations [these transformations are described by formulas similar to (1) and, in addition to 
 , 
¯  ¯ , contain higher derivatives of up to order , inclusive]. 0.6.2-2. B¨acklund transformations. Formal operators and nonlocal variables. 1 b . If the coordinates of the infinitesimal operator are allowed to depend on the derivatives of arbitrary (up to infinity) orders, we obtain Lie–B¨acklund groups (of tangential transformations of infinite order). However, on the manifold determined by an ordinary differential equation, all higher derivatives are expressed through finitely many lower derivatives, as dictated by the structure of the equation itself and the differential relations obtained from the equation. The substitution of the right-hand side of equation (1) into an infinite series with derivatives usually results in very cumbersome formulas hardly suitable for practical calculations. For this reason, the Lie–B a¨ cklund groups are widely used only for the investigation of partial differential equations, whereas in the case of ordinary differential equations, a more effective approach is that based on the canonical form of an operator and the notion of a formal operator.

Page 66

¶

2 b . The canonical form X is defined by the relation

¶
X = X − ˆ (  , ) c  = [ ‰ (  , ) − ˆ ( , )  ] N ,

where X = ˆ ( , ) º  + ‰ ( , )

º

ºº

N

is the infinitesimal operator of the group [see formula (4) in

¶

Paragraph 0.6.1-1], and c  is the operator of total derivative. The operators X and X are equivalent in the sense that if one of them is admissible for the equation, then the other is also admissible (the operator of total derivative is admissible for any ordinary differential equation). The function ® 0 ( , ) ≡  is an invariant of any operator in canonical form.  order derivatives  The action of the group on higher for an operator in canonical form is determined ¶ ¶ by the simple recurrence formula  +1 = c  (  ). The order of an equation that admits an operator in canonical form can be reduced on the basis of the algorithm described in Paragraph 0.6.1-3 (see Item 2 b , the second method). 3 b . By definition, a formal operator is an infinitesimal operator of the form

m

=`

, N 

(5)

where the function ` depends on  , , 
 ,  , (  ) (with , smaller than the order of the equation under investigation) and auxiliary variables whose definition involves the symbol of indefinite  integral, for instance, X ( , , 
) 

M

(the integration is with respect to the variable  which is involved both explicitly and implicitly, through the dependence of on  ). Such auxiliary variables are called nonlocal, in contrast to the coordinates of the prolonged space defined pointwise. The nonlocal variables depend on derivatives of arbitrarily high order, for instance,

X

½  + +1 ( + )  .  = ÷ (−1) + M ( + 1)! + =0 ¡

This formula is obtained by successive integration by parts of its left-hand side. Thus, a nonlocal variable can be represented as an infinite formal series; and this allows us to express the coordinates of the Lie–B¨acklund operator in concise form. A formal operator is a far-reaching generalization of an operator in canonical form. The function ® 0 ( , ) ≡  is an invariant of the formal operator (5) for any ` . When solving the direct problem, one usually prescribes the nonlocal operator in the general   form X = ‰ 1 exp O X  + ‰ 2S or X = O ‰ 1 X  +‰ 2 , (6)

Q

N 

M P

P/N 

M

and then, in order to find an admissible operator, one uses a search algorithm similar to that    described in Paragraph 0.6.1-2. The coordinates of the prolonged operator are found by the formulas   = c  (  −1 ), where 0 = ` . In contrast to the method of finding a point operator, in the present case, there are three unknown functions (‰ 1 , ‰ 2 , ); and the splitting procedure to obtain a system can be realized with respect to all “independent” variables, in particular, the nonlocal variables. Suppose that the differential equation

  =   , , 
,  , can be written in new variables  = ® 0 , H = ® 1 ( , , 
 ), H ( )

  ? 
, H 

 ,  , H (  ( −1)

invariants of an admissible operator of the form (5). Then the coordinate ` the equation

N N

® 1`

+ N

N

®1c 


(7) , where ® 0 and ® 1 are of this operator satisfies

−1)

 [ ` ] = 0,

Page 67

which is an analogue of a linear ordinary differential equation for a function of several variables, since it involves the total derivative of the unknown function (exact differential equation). Its solution has the form:

` = exp O − X N

N

® 1

® 1

N

N
 ,  M P

(8)

where the integral is taken with respect to  involved explicitly and implicitly (through the dependence of , 
 ,  on  ), which means that this representation of an operator through a nonlocal variable is most universal. The function (8) generates a nonlocal exponential operator of the form (5) [the class of nonlocal exponential operators is specified by the first expression in (6) with ‰ 2 ≡ 0]. Example 1. The equation

ÏÒ˜Í Í Ò

admits two Lie–B¨acklund operators: X1 = ( Ë , Ï , ÏÒÍ )

~

~

» Ò

,

X2 =

›

±

¼’ =0

=0

» Ò± … Ïã (Ï

−Ë

ÏÒÍ , Ï ÒÍ

) + (Ï − Ë

›

ÏÒÍ , Ï ÒÍ

)

†

€

€ Ï (Ò ±

)

,

where , ã , are arbitrary functions of their variables. The first operator is trivial (the operator of total derivative is admissible for any differential equation), while the second operator determines the maximal group of contact transformations admitted by the equation under consideration.

A Lie–B¨acklund operator admitted by an ordinary differential equation can by found by three methods: (i) in the form of an infinite formal series; (ii) by passing to an equivalent system of ordinary first-order differential equations:


=

1

2,


=

2

3,

 ,


 =  ( , 1 , 2 ,  ,  ),

and finding an admissible point group; (iii) by its representation as a formal operator whose coordinates depend on nonlocal variables (the general form of the operator is chosen by the investigator). In all cases, the search algorithm amounts to solving the determining system which is constructed by a procedure similar to that of Subsection 0.6.1. From the standpoint of simplicity and the possibility of integrating equations, the third method seems to be the most effective if one takes into account that an equation admitting an operator can be written in terms of new variables—invariants of the admissible operator—as a new ordinary differential equation whose order is by one less than that of the original equation. 0.6.2-3. Factorization principle. The use of formal operators allows us to formulate universal principles for reducing the order of an equation, independently of the specific structure of the operator (it can be a point operator, a tangential or nonlocal operator, or a Lie–B¨acklund operator). Theorem 1. An arbitrary B th-order differential equation (7) can be factorized to a system of special form H (  −1) = 4  , H , H 
,  , H (  −2)  , (9) H = 5 ( , , 
), if and only if equation (7) admits the nonlocal exponential operator: X = exp O − X

5   N . 5   M P œ N

(10)

Page 68

The function 5 ( , , 
) is the first differential invariant of the operator (10). Therefore, having found an admissible operator (10) of the form X=`

N

` ≡ exp X Æ ( , , 
)  S , Q M

N ,

(11)

we can calculate 5 by solving the first-order linear partial differential equation 5  + Æp5  œ = 0. The function Æ ( , , 
) is found as a solution of the determining system obtained by “splitting” the invariance condition for operator (11):

 −1)  W X V (  ) −    , , 
,  , (?

= 0, û  (œ n ) = Ž û



where X=



½ 

=0

` 

`  =c  ` 

N

N ( ) , 

−1 ,

`

c  = N + 
N  N N

=` ,

0



+   N

N
+ ))) . 

Theorem 1 generalizes the classical Lie algorithm, which is restricted to the case of unconditional solvability of the second equation of system (9). On the other hand, the introduction of the factor system (9) allows for two more cases, since the first equation is independent of . These cases are the following: 1. The first equation of system (9) allows for the reduction of the order or is solvable. 2. The first equation of system (9) has some special properties, for instance, admits a fundamental system of solutions. Example 2. The equation

â (Ë

= â ( Ë ) Ï + ãÒÍ ( Ë ) Ï

ÏÒ˜Í Í Ò

ã (Ë

−1

− [ ã ( Ë )]2 Ï

−3

(12)

for arbitrary functions ) and ) is the only equation of the form (its uniqueness is to within a Kummer–Liouville equivalence transformation; see Paragraph 0.2.1-8)

Ï Ò˜Í Í Ò

€

=

ê (Ë , Ï

)

admitting the nonlocal exponential operator:

€

€

€

 Ò +  Ï ÒÍ Ë † ,  =  (Ë , Ï ), ½ = ½ (Ë , Ï ). … + X ½ Ø  Ù æ Ï Ï The second prolongation€ of the operator X has€ the form: X = exp X ½ Ë   + ( Ò +  Ï ÒÍ +  ½ )  œ € 2 æ Ù¿¾ Ø +  Ò˜Ò + 2 ½  Ò +  ½ Ò + ½ 2  + (2  Ò  + 2 ½  +  ½  ) Ï ÒÍ + N ( Ï ÒÍ )2 + Ï Ò˜ ÍÍÒ î   . í œœ¦À X = exp

Ë Ø X}½ æ Ù



Applying this operator to the equation invariance condition in the form:

 ҘÒ

+2

= exp

Ï Ò˜Í Í Ò

=

ê (Ë , Ï

) and replacing all instances of

½  Ò +  ½ Ò + ½ 2  +  ê −  ê¦

+ (2

Ò

+2

by

ê

=

ê (Ë , Ï

), we obtain the

½  +  ½  )Ï ÒÍ +  N (Ï ÒÍ )2 = 0.

Splitting this relation with respect to powers of the “independent” variable equations for the functions , , and ê :

 ½

Ï Ò˜Í Í Ò

N = 0,  Ò  + 2½  +  ½  = 0,  Ò˜Ò + 2½  Ò +  ½ Ò + ½ 2  + ê −  ê¦

Ï ÒÍ

, we obtain the following system of three

2

= 0.

From the first two equations it follows that = Ó ( Ë ) Ï + Ô ( Ë ),

 ½

=−

ÓÓ Í Ï

2

+ 2 Ó Í ÔÝÏ + ( Ë ) , ( ÓÏ + Ô )2

¯

Page 69

where Ó = Ó ( Ë ), Ô = Ô ( Ë ), and = ( Ë ) are arbitrary functions. The third equation can be treated as a first-order linear differential equation for the unknown function ê = ê (Ë , Ï ):

¯ ¯



Substituting the above expressions of

ê (Ë , Ï

) = ( ÓÏ + Ô ) â ( Ë ) +

[ ÓÓ

ê æ Ï æ

 

−

½

and

Ó

Ò˜Ò  ( 1

=

+2

½  Ò +  ½ Ò +  ½ 2 ).

into this relation and integrating the result, we obtain

− 2( Ó Í )2 ] Ô − ( ÓÔ

ÍÍ

ê

3

− 2Ó

ÍÍ

Í Ô Í )Ó

−

[ ÓÓ

− 3( Ó Í )2 ] Ô

ÍÍ

i¯ Í − 2Ó Í ¯ )Ó

+ 2 ÓÓ Í ÔÖÔ Í − ( Ó 2 Ó 3 ( ÓÏ + Ô )

2

where â ( Ë ) is an arbitrary function. The differential invariant of the operator X satisfies the linear partial differential equation

€

;

(obtained after the division by exp(  equation

where



=

Ï ÒÍ

½

Ë



æ

æ Ï æ

€  Ï;

+(

 Ò +  Ï ÒÍ

€ € ; + ½ ) Ï ÒÍ

)). Substituting the above =

ӁÏ

Ó

+Ô

+

2 ÓÓ

ÍÏ

2

+ (3 Ó



−

i¯

( Ó Í Ô 2 − Ó )2 , 4 Ó 3 ( ÓÏ + Ô )3

=0

½

and

into this equation, we pass to the characteristic

¯

Í Ô + ÓÔ Í )Ï + ÔÖÔ Í + , ( ÓÏ + Ô )

. Integrating this equation, we find the differential invariant:

;

Ó Í Ô − ÓÔ Í + Ó Í Ô 2 − Ói¯ . Ó 2 ( Ó Ï + Ô ) 2 Ó 2 ( ÓÏ + Ô )2 ÒÍ , one can find Ï Ò˜Í Í Ò and, taking into account the known structure of the function =

ӁÏ

Ï ÒÍ

−

+Ô

Having calculated the derivative ; ê (Ë , Ï ), one obtains the factorization of the original equation: ;˜ÒÍ + Ói; 2 + (ÓÍ % Ó ); = â , ( ÓÏ + Ô ) ÏÒÍ = ( ÓÏ + Ô )2 ; + Ó −2 ( ÓÍ¤Ô − ÓÔÝÍ )( ÓÏ + Ô ) − 12 Ó −2 ( ÓÍ¤Ô 2 − Ói¯ ). An equivalence transformation of the form ÓÏ +  Ô vÏ , combined with the corresponding transformation of the independent variable and changed notation, yields: ;˜ÒÍ + ; 2 = â (Ë ), (13) Ï ÒÍ = ;˜Ï + ã (Ë )Ï −1 . The first equation of system (13) is the Riccati equation. Its general solution can be represented in terms of a fundamental system of solutions of the “truncated” linear equation:

ÏÒ˜Í Í Ò

= â (Ë )Ï ,

(14)

which coincides with (12) for ã ≡ 0. The second equation of system (13) is a Bernoulli equation. It can be integrated by quadrature for an arbitrary function = (Ë , ). Therefore, the general solution of equation (12) can be expressed in terms of a fundamental system of solutions of the linear equation (14). Note that in the general case, equation (12) admits no point groups.

; ;



If an operator admitted by equation (1) has no differential invariants of the first-order, then it is possible to apply the general factorization principle. Theorem 2. An arbitrary B th-order differential equation (1) can be factorized to the system of special structure H (  −  ) = 4   , H , H 
,  , H (  −  −1)‘  , (15)  ) , H = 5   , , 
,  , (Z º ( ¼ ) ≠ 0,

º

œ

 )  is a lowerprovided that equation (7) admits a formal operator (5) for which 5  , , 
,  , (Z order differential invariant on the manifold given by (7). The coordinate ` of this operator satisfies the linear equation with total derivatives: `EN N

H

H

H

+ c  [ ` ] N
+ ))) + c (  ) [ ` ] N ( 

N

N

)

= 0.

(16)

Equation (16) plays a crucial role in both the direct and inverse problems. It can be regarded as an equation for the determination of the coordinate of the canonical operator (if one knows the

Page 70

invariant H ). It can also be regarded as an equation for the determination of an invariant (if one knows the coordinate ` ). In the latter case, this is a first-order partial differential equation. Example 3. The third-order nonlinear equation

ÏÏ Ò˜Í Í Ò˜Í Ò

+

admits two operators X1 =

Ï

€

ë

Ï Ò˜Í Í Òì

,

2

−Ï

X2 =

− â (Ë )Ï

ÒÍ Ï Ò˜Í Í Ò

Ï Ï Ø X

Ë æ Ù

−2

2

€

=0

(17)

,

(18)

which can be found with the help of the direct algorithm, if the structure of the operator is specified by the second expression in (6). The first operator, X1 , is the usual point operator of scaling (the original equation is homogeneous) and provides the usual reduction of order of equation (17) by one. The second operator, X2 , is nonlocal. Let us construct differential invariants of the operator X2 . To this end, we should solve the equations:

€

€ € € Á € è 1 + » Ò [ Á ] € è 1 = 0, Á =€ Ï X Ï −2 Ë , Ï Ï ÒÍ æ Á € è 2 + » Ò [ Á ] € è 2 + » 2Ò [ Á ] € è 2 = 0. Ï Ï ÒÍ Ï Ò˜Í Í Ò Á € € After differentiation with respect to , the first equation in (19) becomes €è €è Ï Ï −2 Ë Ù 1 + … Ï −1 + Ï ÒÍ X Ï −2 Ë Ù † 1Ò = 0.€ æ æ Ø X Ï Ø ÏÍ

(19)

€

Let us show that this equation admits no solutions depending only on Ë , Ï , Ï ÒÍ , and è 1 Ï ÒÍ ≠ 0, i.e., there are no first-order differential invariants. The nonlocal expression  Ï −2 Ë depends on derivatives of arbitrarily high orders and æ can be regarded as an independent quantity. Therefore, the first equation (19) can be split and we obtain the system:

€

€ %

€

€

€

Ï è 1 + Ï ÒÍ è 1Ò Ï ÏÍ

€

Ï

= 0,

−1

€

€

è1 Ï ÒÍ

= 0.

It follows that è 1 Ï ÒÍ = 0. Let us find a second-order differential invariant. After differentiation with respect to becomes

€

−2

Ë

ۏ

€

€è 2 −1 Ï + Ï ÒÍ X Ï −2 Ë † + Ï Ò˜ … Í Í Ò X Ï −2 Ë æ Ù Ï æ Ù Ï ÒÍ Ø æ Ø Ë  −2 Splitting this equation with respect to the nonlocal € variable Ï € , we establish that equation, the nonlocal variable is canceled, €è € è æ Ï Ï Ø X

2

+

Ï

Hence, we find that

è2=;

=

Ï ˜ÒÍ Í Ò % Ï

Ï

2

+ ÏҘ ÍÍÒ

2

Ï Ò˜Í Í Ò

%

Á ,€ the second equation in (19) € è €Ù Ï Ò˜Í€ 2Í Ò = 0. è 2 % Ï ÒÍ = 0. In the remaining

= 0.

, and equation (17) is factorized to the system:

;˜ÒÍ + ; 2 = â (Ë ), ÏÒ˜Í Í Ò − ÏD; = 0.

^_

References for Subsection 0.6.2: R. L. Anderson and N. H. Ibragimov (1979), O. N. Pavlovskii and G. N. Yakovenko (1982), N. H. Ibragimov (1985), P. J. Olver (1986), V. F. Zaitsev (2001).

0.6.3. First Integrals (Conservation Laws)

. = .   , , 
,  , (  A function differential equation

−1)

. if the total derivative of the function if

. c  [ ]≡8

 is called a first integral (conservation law) of the ordinary

 −1)    =    , ,  , (? ( )

(1)

along the trajectories of equation (1) is zero or, equivalently,

 −1)  V (  ) −   , ,  , (?  −1)  W = 0,  , , 
,  , (?

(2)

where 8 is an integrating factor. From this definition it is clear that 8 = ºD(Ân −1) . ºœ The algorithm of finding a first integral is similar to that of finding an admissible operator. It is necessary to prescribe the desired structure of the first integral (or the integrating factor) and

Page 71

substitute it into the determining equation (2). Subsequent splitting with respect to lower derivatives (assumed to be independent variables) leads to the determining system. x3y'z|{}~­³´€ An arbitrary function of first integrals is also a first integral of the same equation. Therefore, having found a first integral depending on ("
 )  , one has to make sure that it is nontrivial; i.e., it cannot be represented as the product of first integrals depending on lower powers of the derivative.

x3y'z|{}~‚µ €

If the equation has , functionally independent first integrals, then its order can be reduced by , by successively excluding higher derivatives (see Example 3). For second-order equations 

 =  ( , , 
), (3) the determining equation (2) can be written in the form

N

.

+ 
N

 N

.

.

+  ( , , 
) N

N




N

.

= 0.

(4)

In this case, one can solve the direct problem (find for the given equation), as well as the inverse problem (find possible  for the given structure of the first integral). Example 1. Let us find all equations of the form

Ï Ò˜Í Í Ò

=

ê (Ë , Ï

)

(5)

admitting a first integral that is quadratic with respect to the first derivative:

4

=

( Ë , Ï )( ÏÒÍ )2 + ( Ë , Ï ) ÏÒÍ +

Ã

Ä

Å

( Ë , Ï ).

Then the left-hand side of the determining equation (4) is a cubic polynomial with respect to with respect to powers of Ï ÒÍ yields the system of four equations:

Ð Ã Ä Å

The solution of this system for

ê (Ë , Ï

where

Æ

=

ê

Ò

= 0, + = 0, + + 2 …ê + ê = 0.

Ä^ Åw Ä

Ò Ò

Ã

Ï ÒÍ

. The procedure of splitting

= 0,

is given by:

à −3  2 Æ (; ) + 12 à −2 … Ø Ã„Ã±ÍÒ˜Í Ò ; = à −1  2 Ï + 21 X ä à −3  2 æ Ë , )=

Æ (; ), Ã = Ã (Ë ), and = ä ä 4 = Ã ( Ï Ò Í )2 − ( Ã Í Ò Ï −

Ã

1 2 ±ÍÒ Ù 2

−

Ï

−

Ã

ÍÒ ä

Ã

1 ±ÍÒ 2

+

ä

†,

( Ë ) are arbitrary functions. The first integral has the form:

ä

)Ï

ÒÍ

+

1 4

Example 2. Consider the equation

Ã

−1

(

Ã

ÏÒ˜Í Í Ò

Í Ò )2 Ï

=



2

−

Ë Ï

1 2

à −1 Ã



.

−1 2

ÍÒ Ï ä

+

1 4

Ã

−1

ä

2

−2X

Æ (; ) ; . æ

Let us find its first integral, which is a cubic polynomial with respect to the first derivative:

4

=

Ã

( Ë , Ï )( ÏÒÍ )3 + ( Ë , Ï )( ÏÒÍ )2 +

Ä

Å

( Ë , Ï ) ÏÒÍ +

Ç

( Ë , Ï ).

In this case, the left-hand side of the determining equation (4) is a fourth-order polynomial in system consists of five equations: = 0, Ò + = 0,

Ã

Ð

^Ä  Ä + Åw + 3 Ë Ï −1  2 Ã Å Ò + ÇÈ + 2 Ë Ï −1  2 Ä Ç Ò +  Ë Ï −1  2 Å = 0. Ò

Ï ÒÍ

, and hence the determining

= 0, = 0,

Page 72

Solving this system, we obtain the first integral in the form:

4

= ( ÏÒÍ )3 − 6

Example 3. The equation

1 2

ÏÒ˜Í Í Ò˜Í Í Ò˜Ò

4

admits three first integrals:

4 4

1

=

2

=

3

=

ÏÒÍ ÏÒ˜Í Í Ò˜Í Ò

4

Ë Ë

4

1

−

2

−

Equating these expressions to independent constants equation (see 4.2.1.1).

^_



Ë Ï



 3  2 + 2

ÏÒÍ

=

+ 4 …Ï

…Ï −5  3

Ë

2 3



Ï Ò˜Í Í Ò )2 + 32 …Ï −2 3 , 3 ÏÏ Ò˜Í Í Ò˜Í Ò + 12 Ï ÒÍ Ï Ò˜Í Í Ò , 2 1 Ë 24 3 Í Í Ò − ( Ï Ò Í )2 . 1 + 2 ÏÏ Ò˜ 2 −

1 ( 2

 1,  2, 

3

.

and eliminating

Ï Ò˜Í Í Ò˜Í Ò

and

Ï Ò˜Í Í Ò

, we obtain a first-order

References for Subsection 0.6.3: P. J. Olver (1986), A. D. Polyanin and V. F. Zaitsev (1995).

0.6.4. Discrete-Group Method. Point Transformations Consider transformations of the class of ordinary differential equations

  =    , ,  , (  ( )

−1)

, a ,

(1)

whose elements are uniquely defined by a vector of essential parameters a. Any set of invertible transformations

 =  (a , ‡ ),

= (a , ‡ )

( g



−   ž g ≠ 0),

(2)

mapping each equation of class (1) into some (other) equation of the same class

‡ (g  ) =  a , ‡ ,  , ‡ (g 

−1)

, b ,

(3)

and containing the identical transformation is called a discrete point group of transformations admitted by the class (1). Transformation (2) maps any solution of equation (1) to a solution of equation (3). Therefore, knowing the discrete group of transformations for some class of equations and having a set of solvable equations of this class, one can construct new solvable cases. Point transformations (2) can be found by a direct method — namely, if one substitutes an arbitrary transformation of the form (2) into equation (1) and imposes condition (3), one arrives at a determining equation containing partial derivatives up to order B of the unknown functions  and and having variable coefficients depending on  , , 
,  , (  −1) . Since the functions  and do not depend on the derivatives, the determining equation can be “split” with respect to the “independent” variables 
,  , (  −1) , and we obtain an overdetermined system which is nonlinear, in contrast to that obtained by the Lie method (see Subsection 0.6.1). Example 1. For second-order equations

ÏÒ˜Í Í Ò

ê (Ë , Ï , Ï ÒÍ

=

, a),

(4)

the substitution of (2) into (4) yields (â

8 ãU¥

−ã

8 ⥠)ž¤ Í89Í 8 + ( â¥ãU¥i¥ − ãU¥žâ¥i¥ )(¤žÍ8 )3 + ( â 8 ãU¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥´â¥ 8 )(¤žÍ8 )2 + ( â¥ã 898 − ãU¥žâ 898 + 2 â 8 ãU¥ 8 − 2 ã 8 ⥠8 ) ¤žÍ8 + â 8 ã 898 − ã 8 â 898 = ( â 8 + ⥤žÍ8 )3 ê Ø

â ,ã , ã

8 + ãU¥ ¤ Í8 â 8 + ⥠¤ Í8

, a . (5)

Ù

Let us require that the transformed equation (5) belong to the class (4), i.e.,

¤žÍ89Í 8

=

ê

: ¤ ¤8

( , , žÍ , b).

(6)

Condition (6) imposed on the determining equation (5), i.e., the replacement of ¤ Í89Í 8 by the right-hand side of equation (6), leads us to the relation ( â 8 ãU¥ − ã 8 ⥠) ê ( : , ¤ , ¤žÍ8 , b) + ( â¥ãU¥i¥ − ãU¥´â¥i¥ )( ¤žÍ8 )3 + ( â 8 ãU¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2 ãU¥´â¥ 8 )( ¤žÍ8 )2 ã 8 + ãU¥¤ Í8 , a , (7) + ( â¥ã 898 − ãU¥´â 898 + 2 â 8 ãU¥ 8 − 2 ã 8 ⥠8 ) ¤žÍ8 + â 8 ã 898 − ã 8 â 898 = ( â 8 + ⥠¤žÍ8 )3 ê â , ã , 8 Ø â + ⥤ Í8 Ù

Page 73

¤ Í8 . Expanding the function ê into a series in powers of ¤ Í8 , we can represent ¼’ 4 Ë , Ï , [ â ], [ ã ] ì ( ¤ Í8 ) õ = 0, (8)

which contains the “independent” variable (7) in the form

õ

õ ë

=0

where the symbols [ â ] and [ ã ] indicate dependence on the functions â , ã and their partial derivatives involved in (7). The sum in (8) is finite if ê is a polynomial with respect to the third variable [for a polynomial of degree ≥ 4, both sides of the equation must be first multiplied by ( â + â Í ) −3 ]. Condition (8) is satisfied if the following equations hold:

5

8 ¥¤ 8 4 E= 0,

ô

õ

ÿ›ÿªÿ

= 0, 1, 2,

Example 2. Consider a special case of equation (4) with the right-hand side independent of the derivative

ÏÒ˜Í Í Ò Relation (7) has the form: (â

8 ãU¥

−ã

8 â¥

=

ê (Ë , Ï

:

, a).

(9)

¥ãU¥i¥ − ãU¥žâ¥i¥ )(¤žÍ8 )3 + ( â 8 ãU¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥´â¥ 8 )(¤žÍ8 )2 + ( â¥ã 898 − ãU¥žâ 898 + 2 â 8 ãU¥ 8 − 2 ã 8 ⥠8 ) ¤žÍ8 + â 8 ã 898 − ã 8 â 898 = ( â 8 + ⥤žÍ8 )3 ê

: ¤

Ï ÒÍ

) ê ( , , b) + ( â

( â , ã , a).

In this case, the sum (8) is finite and the determining system has the form:

â 8 â ¥ã 8 Uã ¥

It can be shown that for Consider the case â

¥

Since

â 8Í

â¥ãU¥i¥ − ãU¥žâ¥i¥ = â ¥ 3 ê ( â , ã , a), â 8 Uã ¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥žâ¥ 8 = 3 â 8 â ¥ 2 ê ( â , ã , a), â¥ã 898 − ãU¥žâ 898 + 2 â 8 ãU¥ 8 − 2ã 8 ⥠8 = 3 â 8 2 ⥞ê ( â , ã , a), â 8 ã 898 − ã 8 â 898 + ( â 8 ãU¥ − ã 8 ⥠) ê (: , ¤ , b) = â 8 3 ê ( â , ã , a).

(10)

≠ 0, solving system (10) is equivalent to solving the original equation (9). = 0. In this case, the first equation of the system holds identically and the system becomes

â 8 Í Uã ¥i¥ = 0, ãU¥´â 898 − 2 â 8 ãU¥ 8 = 0, â 8 ã 898 − ã 8 â 898 + â 8 ãU¥ê

( : , ¤ , b) = â 8 3 ê

(11) ( â , ã , a).

≠ 0, the first two equations yield

ã

: ¤

( , )=

É (: )¤ + Ê (: ), â8 Í =  [É (: )]2 .

(12)

Substituting (12) into the last equation of system (11) and “splitting” the resulting relation with respect to powers of the “independent” variable , we obtain a new system of (ordinary) differential equations. Solving this system, we find the unknown functions and , and finally, the desired discrete group of transformations. In order to give calculation details, one has to know the specific structure of the function ê ( Ë , Ï ), for in the general case it was only shown that any discrete point group of transformations of equation (9) for â = 0 consists of Kummer–Liouville transformations (12).

¤

É

Ê

N¥

Example 3. Consider the generalized Emden–Fowler equation:

 ËEϱ Here, a = { 5 , ´ , Ì } is the vector of essential parameters, and  ÏÒ˜Í Í Ò

=

by scaling the independent variable and the unknown function).

Ë

( ÏÒÍ ) .

(13)

is an unessential parameter (it can be made equal to unity

O

1 . First, we note that equation (13) admits a discrete group of transformations determined by the hodograph transformation, i.e., by passing to the inverse function:

Ë

¤

= ,

Ï

:

= ,

where

¤ = ¤ (: ).

Í

5 Í

(14)

´

This transformation is a consequence of the invariance of equation (13) with respect to the transformation Ë ÊÏ , , 3− , − (note that the hodograph transformation changes the sign of the unessential parameter ). Denoting the transformation (14) by , let us schematically represent its action on the parameters of the equation as follows:

ÌÍ

Ì  Í



©

Í Î$Î {´ , 5 , 3 − Ì } 5 ´ , Ì} Ï © yields the original equation. Double application of the transformation { ,

transformation

©

.



(15)

Page 74

Ì

O

2 . For

= 0, equation (13) is of the class (8), and the last equation of system (11) becomes

É 8 ÊhÍ8 + „‚ É 2 Ñ: Ði¤^Ò = ƒ 2 (ɐ¤ + Ê )± â E , (16) where Ó , Ô , and ‚ are the parameters of the transformed equation ¤ Í89Í 8 = ‚ƒ: Ð ¤ Ò , and â (: ) =  X [É (: )]2 : . æ ´ Let , Ô ≠ 0, 1, 2. Then relation (16) is possible only if Ê ( : ) ≡ 0. Splitting with respect to powers of ¤ leads us to the system: ÉÉ 89Í 8Í − 2(É 8 Í )2 = 0, (17) ‚ƒ:ÑÐ = ƒ 2 É ± +3 â E . By integration we find that É = : −1 , â = : −1 (to within unessential coefficients). Thus, we arrive at the transformation Ë = : −1 , Ï = : −1 ¤ , where ¤ = ¤ (: ). (18) Denoting the transformation (18) by Õ , let us schematically represent its action on the parameters of the equation: {5 , ´ , 0} Í,Î ÎÎÎ {− 5 − ´ − 3, ´ , 0} transformation Õ . (19) Double application of the transformation Õ yields the original equation. 3 O . Let Ì = 0 and ´ = 2. Then, Ô = 2 and the splitting procedure for equation (16) yields the system of three equations: É…É 89Í 8Í − 2(…É 8 Í )2 = 2ƒ 2 É 6 tÊ â E , ɃhÊ Í89Í 8 − 2…É 8 Í hÊ Í8 = ƒ 2 É 5 Ê 2 â E , ‚ƒ:ÑÐ = ƒ 2 É 5 â E . [

ÉɅ89Í 8Í − 2(Ʌ8 Í )2 ]¤ + ƒÉ ÊhÍ89Í 8

− 2…Í

Its solution gives us the transformation

Ë

:×Ö

= , { , 2, 0}

5

where we use the notation:



= (8

5

2

5

Ï = : õ¤ + ³° : T Í,Î ÎÎÎ {Ó , 2, 0}



+ 40 + 49)−1 2 ,

ô

=



−1 , 2

¤ ¤ :

transformation of the variables, = ( ); transformation of the vector of essential parameters;

Ó

Example 4. Likewise, for the class of equations

= â (Ë )ã (Ï ) (Ï

›

Ï Ò˜Í Í Ò we find two transformations:

©

^_

Õ

›

: {â , ã , } : {â ,

Ï

±

, 1}

ÍÏÎ0Î Í,Î ÎÎÎ

› % Ï ÒÍ )} â (: ), Ï ± , 1}

{ã , â , −( Ï {: ± −

−3

 5

1 [ (2 + 5) − 5], 2

=

ÒÍ

)3 (1

−1

ÒÍ

Ø

 5

= − ( + 2),

°

=

5

5

( + 2)( + 3)



.

)

transformation of the variables; see (14); transformation of the variables; see (18).

References for Subsection 0.6.4: V. F. Zaitsev and A. D. Polyanin (1994), A. D. Polyanin and V. F. Zaitsev (1995).

0.6.5. Discrete-Group Method. The Method of RF-Pairs The direct method (see Subsection 0.6.4) is unsuitable for finding nonpoint transformations of second-order equations (i.e., transformations containing derivatives), since the determining equation cannot be split into equations forming an overdetermined system. Therefore, instead of searching for B¨acklund transformations in the form of arbitrary functions  =  (a , ‡ , ‡ g
), = (a , ‡ , ‡ g
), one uses the superposition of some “standard” transformation containing the derivative and a point transformation which can be found by the direct method. The “standard” dependence on the derivative can be introduced by means of an RF-pair, which amounts to a transformation of successively increasing and decreasing the order of the equation (this transformation is not equivalent to the identity transformation). An additional point-transformation is necessary, since the equation obtained by an RF-pair is usually outside the original class.

Page 75

1 b . Suppose that any equation of the original class can be solved for the independent variable  :

 ( , 
, 

 ) =  . Termwise differentiation of this equation with respect to  yields the following autonomous equation:



  
+ N


 + N




 = 1,   N N N "
whose order can be reduced with the substitution  = H ( ). This pair of transformations is called N

a first RF-pair.

2 b . Suppose that any equation of the original class can be solved for the dependent variable :

 ( , 
, 

 ) = . Then, termwise differentiation of this equation with respect to  brings us to the following equation which does not explicitly contain :

N N



+ N





N




 

 + N

 N




 = 
.

The order of this equation can be reduced by means of the substitution
 = H ( ). This pair of transformations is called a second RF-pair. Example 1. Consider transformations of the class of generalized Emden–Fowler equations:

Ï Ò˜Í Í Ò

=

 ËEϱ

Ë

ÒÍ

(Ï

).

(1)

This class will be briefly denoted by the vector of essential parameters { 5 , ´ , Ì }. Application of the first RF-pair transforms this equation to ;˜NÍ Í  = (Ì − 1); −1 (;˜Í )2 + ´ Ï −1 ;˜Í + 5³ 1 Ï ± ; Ë −E −1 (;˜Í ) E −1 . (2) E E E E Now we have to find a point transformation that maps class (2) into class (1) (with another vector of parameters): ¤žÍ89Í 8 = ‚ƒ:ÑÐD¤^Ò (¤žÍ ) ” . (3)

Î

Note that in this case, the desired transformation does not map the given class into itself as in Subsection 0.6.4, but is a mapping of the equations classes (2) (1). Nevertheless, the method for finding transformations

Ï

: ¤

= â ( , ),

;

: ¤

=ã (, )

(â

8 ãU¥ − â¥ã 8

≠ 0)

is completely the same and involves solving the determining equation:

8 ⥠) ƒ‚ :ÑÐi¤^Ò (¤žÍ8 ) ” + ( â¥ãU¥i¥ − ãU¥´â¥i¥ )(¤žÍ8 )3 + ( â 8 ãU¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥´â¥ 8 )(¤žÍ8 )2 + ( â ¥ ã 898 − ãU¥žâ 898 + 2 â 8 ãU¥ 8 − 2ã 8 ⥠8 )¤ Í8 + â 8 ã 898 − ã 8 â 898 = Ì −ã 1 ( â 8 + ⥠¤ Í8 )(ã 8 + ãU¥ ¤ Í8 )2 ´ 8 1 − −1 2 +1 −1 + ( â + ⥤žÍ8 )2 ( ã 8 + ãU¥ ¤žÍ8 ) + 5³ â ± ã Ë E ( â 8 + ⥤žÍ8 ) E (ã 8 + ãU¥ ¤žÍ8 ) E . (4) â E E E E E Following the procedure set out in Subsection 0.6.4, we omit the general case â 8 â ¥ ã 8 ãU¥ ≠ 0 and consider transformations for which at least one of the above partial derivatives is zero. 1 O . Case ⥠= 0, ã 8 = 0. Equation (4) has the form ‚Ñâ 8 ãU¥ :ÑÐi¤^Ò (¤žÍ8 ) ” + â 8 ãU¥i¥ (¤žÍ8 )2 − ãU¥´â 898 ¤žÍ8 = Ì −ã 1 â 8 (ãU¥ )2 (¤ 8 )2 ´ 1 − −1 2 +1 −1 −1 â ± ã Ë E ( â 8 ) E (ãU¥ ) E (¤ 8 ) E . + ( â 8 )2 ãU¥¤ Í8 + 5³ â E E E E E E and for 5 ≠ 0, −1, • ≠ 1, 2 can easily be solved by splitting, â =: Ù , ã =¤ Ú . (â

8 ãU¥

−ã

1 +1

1 −2

Page 76

As a result, using an RF-pair, we obtain:

¤ Í8 ) 1 , Ï = : ± 1+1 , Ï ÒÍ = ¤ 2−1 Ë transformation of the variables, ¤ = ¤ ( : ); E ´ 5 − 1 1 { 5 , ´ , Ì } Î Û ÎÎÎÎ ÝÜ − ´ + 1 , Ì − 2 , 5ßÞ transformation of the vector of essential parameters. 2 O . Case â 8 = 0, ãU¥ = 0. Similar calculations bring us to the formulas: Ë = (ž¤ Í8 )− 1 , Ï = ¤ ± 1+1 , ÏÒÍ = : 2−1 transformation of the variables, ¤ = ¤ ( : ); Ë´ E ÎÎÎÎÎ 1 5 2 +1 ´ Ü 1 − Ì , − 5 + 1 , ´ Þ transformation of the vector of essential parameters. {5 , , Ì } Û Ë

=(

(5)

(6)

©

Transformation (6) can be obtained by successive application of transformation (5) and the hodograph transformation (see Subsection 0.6.4). The inverse transformations have a similar structure. For instance, the inverse of transformation (5) can be written (after changing notation) as follows:

Ë

¤

¤ 8 ±1 ,

where ¤ = ¤ ( : ). E Denoting the transformation (7) by à , let us schematically represent its action on the parameters of the equation: ´ { 5 , ´ , Ì } Î Û ÎÎÎÎ Ü 1 1− Ì , − 5 5 + 1 , 2 ´ + 1 Þ transformation à . Applying the transformation à three times, we obtain the original equation. =

1 +1

= ( žÍ )−

Ï

,

ÏÒÍ

=

: 1−1 Ë ,

(7)

(8)

©

It can be shown that all transformations which can be found from equation (4), without additional restrictions on the parameters of the original and the transformed equations, are obtained by superposition of the transformations and (see Subsection 0.6.4, Example 3), which form a group of order 6. The parameters of these equations are given in Figure 1. Example 2. Suppose that

Ì

à

= 0 in equation (1). Then, on the class of Emden–Fowler equations

Õ

Ï Ò˜Í Í Ò

=

5 ´

 ËEϱ

(briefly denoted by { ,

, 0} ),

(9)

one can define the transformation (see Subsection 0.6.4, Example 3). Therefore, in this case, the group considered in the previous example is prolonged to a group of order 12 (see Figure 2). This prolongation takes place each time the third component of the parameter vector becomes equal to zero. This happens, for instance, if = 1 in equation (9). In this case, the order of the group is equal to 24 (see Figure 3).

5

Example 3. The class of second-order equations

= â ( Ë ) ã ( Ï ) ( ÏÒÍ )

›

ÏÒ˜Í Í Ò

(10)

admits a discrete group of transformations similar to that for the generalized Emden–Fowler equation. Most simply, this group can be obtained by inverting the transformation (6). Thus, we seek the parameters of the transformation as functions of a single variable, Ë = (žÍ ), Ï = ( ), ÏÒÍ = ( ).

© Introducing a point generator

¤8

ä

 ¤

á :

(see Subsection 0.6.4), we find a discrete group of transformations relating the equations shown in Figure 4. The functions â 1 ( Ë 1 ), ã 1 ( Ï 1 ), 1 ( Ï ÒÍ 1 ) determine the original equation, while the corresponding functions for the transformed equations, â ( Ë ), ã ( Ï ), ( Ï ÒÍ ¼ ) with ô = 2, 3, are determined by the parametric formulas:

õ

õ



õ

â 2 (Ë

2)

=

ã 2 (Ï

2)

=

2)

=−

› 2 ( and

õ

›

›



õ

Ë

1,

1

â 1 (Ë

ã

1 [ ã 1 ( Ï 1 )]3

â 3 (Ë

3)

=

ã 1 (Ï

ã 3 (Ï

3)

=



3)

=

› 3 (

Ï

,

1)

1

1

æ

1

1)

æÏ æ

1 1



,

Ë

,

Ï

,

â æË1, 1



1 › 1æ ( 1) ,

2

=

X

2

=

X â 1 (Ë

2

=

1

ã 1 (Ï

3

=

X ã 1 (Ï

3

=

X

3

=

â 1 (Ë



1)

› 1 (æ  1

Ë

,

(11)

1)

1)



æ

Ï æ 1

1)

1,

,

(12)

1 ),

where õ = Ï ÒÍ ¼ , ô = 1, 2, 3. The above example allows us to eliminate “singular points” of the group of transformations defined by (7) for = −1, = −1, = 1, 2. For these values of the parameters, the form (10) and the transformations (11), (12) should be used.

´

^_

Ì

5

References for Subsection 0.6.5: V. F. Zaitsev and A. D. Polyanin (1994), A. D. Polyanin and V. F. Zaitsev (1995).

Page 77

Figure 1

Figure 2

Page 78

Figure 3

Figure 4

Page 79

Chapter 1

First-Order Differential Equations 1.1. Simplest Equations with Arbitrary Functions Integrable in Closed Form* 1.1.1. Equations of the Form â$ä ã = å ( æ ) Solution:**

= X¯ ( )  + L .

M

1.1.2. Equations of the Form

â$äã = å (â )

Solution:  = X

M +L .  ( ) Particular solutions: = #  , where #  are roots of the algebraic (transcendental) equation  ( #  ) = 0.

1.1.3. Separable Equations

â$äã = å ( æ )ç (â )

Solution: X

M = Xv ( )  + L . ( ) M Particular solutions: = #  , where #  are roots of the algebraic (transcendental) equation ( #  ) = 0. x3y'z|{}~/€ The equation of the form  1 ( ) 1 ( )
 =  2 ( ) 2 ( ) is reduced to the form 1.1.3 by dividing both sides by  1 1 .

1.1.4. Linear Equation Solution:

=L  Ž + Ž X

ç (æ )â$äã = å 1 ( æ )â + å 0 ( æ )  ( )  ,  −Ž 0 ( ) M

1.1.5. Bernoulli Equation

where

 ( ) = X

ç ( æ )â$äã = å 1 ( æ )â + å„è ( æ )â è

Here, B is an arbitrary number. The substitution ¨ ( ) = ¨2
= (1 − B )  ( )¨ + (1 − B )  ( ). 1  Solution: 1−

 1 ( )  . ( ) M

 = L  Ž + (1 − B )  Ž XZ − Ž   ( )  , ( ) M

where

1−

 leads to a linear equation:

 ( ) = (1 − B ) X

 1 ( )  . ( ) M

* Special cases of equations 1.1.1–1.1.5 for specific functions  ,  0 ,  1 ,   , and are not discussed in this book; such cases can readily be recognized by the appearance of equations investigated, and the solution can be obtained using the general formulas given in Section 1.1. ** Hereinafter we shall often use the term “solution” to mean “general solution.”

Page 81

1.1.6. Homogeneous Equation

â$äã = å (âêéwæ )

The substitution ‡ ( ) =   leads to a separable equation: ‡
 =  (‡ ) − ‡ . Solution: X

‡

M

= ln | | + L .

 (‡ ) − ‡

= #   , where #  are roots of the algebraic (transcendental) equation

Particular solutions: #  −  ( #  ) = 0.

1.2. Riccati Equation ë ( ì ) í­ï î = ð 2 ( ì ) í

2

+ ð 1(ì )í + ð 0(ì )

1.2.1. Preliminary Remarks For  2 ≡ 0, we obtain a linear equation (see Subsection 1.1.4); and for  0 ≡ 0, we have a Bernoulli equation (see Subsection 1.1.5 with B = 2), whose solutions were given previously. Below we discuss equations with  0  2 1 0. 1 b . Given a particular solution as: =

0 (  ) + ` (  ) ¬ L − X¯` (  )

0

=

0(

 ) of the Riccati equation, the general solution can be written

 2 ( )  ® ( ) M

−1

 , where ` ( ) = exp ™ X V 2  2 ( ) 0 ( ) +  1 ( )W M ¢ . ( )

To the particular solution 0 ( ) there corresponds L = F . Often only particular solutions will be given for the specific equations presented below in Subsections 1.2.2–1.2.8. The general solutions of these equations can be obtained by the above formulas. 2 b . The substitution

‡ ( ) = exp O − X



2

 M P

reduces the Riccati equation to a second-order linear equation:

 2 2 ‡

 + V  2 
− (  2 )
 −  1  2 W‡
 +  0  22 ‡ = 0. The latter often may be easier to solve than the original Riccati equation. Specific second-order linear equations are outlined in Section 2.1.

1.2.2. Equations Containing Power Functions 1.2.2-1. Equations of the form ( )
 =  2 ( ) 1.

ñóò

ôñ

=

2

+

õï

2

+  0 ( ).

+ö .

For  = 0, we have a separable equation of the form 1.1.2. For  ≠ 0, the substitution 'a = ' + ( leads to an equation of the form 1.2.2.4: g
=  2 + 'a . 2.

ñóò

ñ

2

=

–ô

2

ï

2

+ 3ô .

Particular solution: 3.

ñ óò

=

ñ

2

+

ô ï

2

+

õï

0

=  − 

−1

.

+ö .

This is a special case of equation 1.2.2.27 with ƒ = 0 and ý = 0.

Page 82

1.2. RICCATI EQUATION ã ( Ë ) Ï

4.

ñóò

ôñ

=

2

õ ïƒ÷

+

ÒÍ

=

â 2 (Ë )Ï

2

â 1 (Ë )Ï

+

+

â 0 (Ë

)

.

Special Riccati equation, B is an arbitrary number. 1 ¨2
Solution: = − ¨ , where ¨ ( ) = £  L 1 ø



Q

1 2



1£

O

,

"  P

+L

2

n

1 2



O

1£

,

"  S , P

, = 12 (B + 2); ø + ( H ) and n + ( H ) are the Bessel functions, B ≠ −2. For the case B = −2, see

equation 1.2.2.13. 5.

ñ óò

ñ

2

=

ôù ï ÷

+

–ô

–1

Particular solution: 6.

ñ óò

ôñ

=

2

õ ï 2÷

+

0

ï ÷

+ö

ï 2÷

2

.

=   . –1

. 1

  B +1 B ( ‰ =  −  leads to an equation of the form 1.2.2.38: ˆ‰ Š
+ ˆ‰ 2 + ‰ = ' ˆ + . B +1 B +1

For the case B = −1, see equation 1.2.2.13. For B ≠ −1, the transformation ˆ =

7.

ñ óò

ô ï ÷ ñ

=

2

õ ï –÷

+

–2

.

‡ M + L , where ‡ = 2 ‡ + ý±‡ + 1

Solution: £  ln  = X 8.

ñóò

ô ƒï ÷ ñ

=

2

õ ïƒú

+



2



+

B +1 B +1 2 b . For B = −1 and ¡

+ − ˆ  +1 .

ñóò

=

ñ

2

  

+1

B +1 , ý = £ .  

leads to a Riccati equation of the form 1.2.2.4:

≠ −1, the transformation





=  +

+1

, ¨

= −1 

leads to a Riccati

¨ 2+ −1 . +1 +1 ¡ ¡ = −1, the original equation is a separable equation. In this case we have the

¡

solution: ln | | = X 9.

+1



equation of the form 1.2.2.4: ¨ û
= 3 b . For B =



,

.

1 b . For B ≠ −1, the substitution ˆ =  

Š
=

+1

ï

+ ü (ô

M2 

+ õ )÷ ( ö

+

ï

+L . + ý )– ÷

–4

.

 +  1 , ‡ = [( (˜ + )2 + ( ( (˜ + )], where ¢ =  − ˜( , leads (˜ + ¢
M M M M to an equation of the form 1.2.2.4: ‡ Š = ‡ 2 + , ¢ −2 ˆ  . The transformation ˆ =

10.

ñóò

=

ô ƒï ÷ ñ

2

õÿþ ïƒú

+

Particular solution: 11. 12.

13.

ñ óò

= (ô

ï 2÷

+

õï ÷

–1

–1

– ôƒõ

2

 + . 0 = '

)ñ

2

ïƒ÷ +2ú

.

+ö .

¨ 
= ( ¨ The substitution = −1 ¨ leads to an equation of the form 1.2.2.6: I

2

+  2  + ' 

−1

.

( ô 2 ï + õ 2 )(ñ ó ò + ƒñ 2 ) + ô 0 ï + õ 0 = 0. The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.108: (  2  +  2 )‡

 + Œ (  0  +  0 )‡ = 0.

ï 2 ñ óò

=

ô ï 2ñ

Solution:

Œ

2

2

=



+ Π+  = 0.

Œ

+õ . −  2  O

   2  + L 2 Œ + 1 P

−1

, where Πis a root of the quadratic equation

Page 83

14.

ï 2 ñ óò

ï 2ñ

=

2

–ô

2

ï

+ ô (1 – 2 õ )ï

4

Particular solution: 15.

ï 2 ñ óò

=

ï 2 ñ óò

=

ô ï 2ñ

2

+

õ ïƒ÷

−1

=  + '

0

– õ ( õ + 1).

2

.

+ö .

The substitution ¨ = 

+ # , where # is a root of the quadratic equation !# leads to an equation of the form 1.2.2.35:  ¨u
=  ¨ 2 + (1 − 2 !# )¨ + '  . 16.

ï 2ñ

2

ô ï 2ú ( õ ïƒú

+

+ ö )÷ + 41 (1 – ù

The transformation ˆ = ' + + ( , ¨

form 1.2.2.4: ¨3Š
= ¨ 17.

(ö 2 ï

18.

ï 4 ñ óò

+ õ 2ï

2

óò

+ ô 2 )(ñ

Ė

+

) ˆ  . −2

¡ 2

)+

ô

0



1

¡

).

1−



= –ï

ô ï 2(ï

+

+

1− ¡ 2

¡

 −+

leads to an equation of the

= 0.

ñ

– ô 2. 1  = +

4 2



– 1)2 (ñ

óò

+



2

Ė

2

tan O )+

õï





+L 2

+ö

ï

P

. + s = 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.218:  2 ( − 1)2 ‡

 + Œ ( ' 2 + (˜ + ° )‡ = 0.

ï

20.

(ô

21.

ïƒ÷ +1 ñ óò

2

õï

+

+ ö )2 (ñó ò + ñ 2 ) +




= 0.

The substitution = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.234: (  2 + ' + ( )2 ‡

 + #*‡ = 0. =

ô ï 2÷ ñ

2

+

ö ïƒú

+ý .

The substitution ¨ =  

+ # , where # is a root of the quadratic equation !  # leads to an equation of the form 1.2.2.35:  ¨ 
=  ¨ 2 + (B − 2 !# )¨ + (˜ + . 22.

(ô

ïƒ÷

+ õ )ñó ò =

õÿñ

2

(ô

ïƒ÷

+

õ ïƒú

+

ô ïƒ÷

–2

2

− B±# + = 0,

M

.

0 = −1  .

Particular solution: 23.

− # + ( = 0,

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.179: ( ( 2  2 +  2  +  2 )‡

 + Œ 0 ‡ = 0.

Solution: 19.

+  (

2

=

2

2

+ ö )(ñó ò – ñ 2 ) +

ôù (ù

– 1) ïw÷

B/  −1 +   + ¡ 0 =−   + ' + + (

Particular solution:

–2 −1

+

õÿþ (þ

– 1) ïƒú

–2

= 0.

.

1.2.2-2. Other equations. 24.

ñóò

ôñ

=

2

+

õÿñ

+ö

ï

+ü .

¨2


=− ¨ ¨2

 =  ¨2
−  ( (˜ + , )¨  .

The substitution

25.

+

ô ï ÷

Particular solution:

0

ñ óò

=

ñ

2

+

ô ï ÷ ñ

–1

leads to a second-order linear equation of the form 2.1.2.12:

.

= −1  .

Page 84

1.2. RICCATI EQUATION ã ( Ë ) Ï

26.

27.

28.

ñóò

=

ñ

2

ñ óò

=

ñ

2

ñ óò

=

ñ

2

ô ƒï ÷ ñ

+

+

õ ïƒ÷

ï

+ (

ô ï ÷ ñ

+

ñóò

= –(ù + 1) ïw÷

ñ

0 2

ñ óò

ô ï ÷ ñ

=

2

+

+

õï ú ñ

ñóò

ô ƒï ÷ ñ

=

2

–ô

ñóò

= – ôù

ïƒ÷ –1 ñ

2

+

ñóò

ô ƒï ÷ ñ

=

2

+

35.

36. 37.

38.

39. 40.

ï ñóò

0

2

+

õï

+ö .

ñ

–ô

=.

ô ïƒ÷ +ú =  −

+ 0

+1 −1

õ^ö ï ú

ïƒú

. – ô„ö

= −( .

â 1 (Ë )Ï

+

â 0 (Ë

85

)

2

  + +( . 0 = '

+ õ )ñ – ö

   +  )−1 . 0 = ( + ö³ü 0

ï ÷

.

.

õÿþ ïƒú

+ ö )ñ +

ö ïƒú (ô ïƒ÷

õ ïƒú ñ

Particular solution: 34.

+

– õ 2.

ïƒ÷ ( õ ƒï ú

Particular solution: 33.

ï

ï ÷

– ôƒõ

Particular solution: 32.

2

The substitution = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.31: ‡

 − ( ƒ… + ý )‡
 + (  2 + ' + ( )‡ = 0.

Particular solution: 31.

â 2 (Ë )Ï

.

+  )ñ + ô

Particular solution: 30.

=

The substitution = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.45: ‡

 −   ‡
 + '   −1 ‡ = 0.

Particular solution: 29.

–1

ÒÍ

–1

ïƒú

ï –1 – õ^ö ïƒú +

= (˜  .

= ôñ 2 + õÿñ + ö ï 2  . The transformation a =  , ¨ = 

−



.

. – ô„ö

2

ïƒ÷ +2

.

¨ g
=  ¨ leads to a separable equation:  3

2

+( .

= ôñ 2 + õÿñ + ö ïƒ÷ . The transformation ˆ =  , ‰ =  −  leads to the special Riccati equation of the form 1.2.2.4:  ( B ‰ Š
= ‰ 2 + ˆ + , where = − 2.

ï ñóò ï ñóò





¡



= ôñ 2 + (ù + õ ïƒ÷ )ñ + ö ï 2÷ . ¨ 
=   The substitution = ¨   leads to a separable equation: 3

ï ñóò

= ï ñ 2 + ôñ + õ ïƒ÷ . The substitution = −‡
 ‡ ‡

 − ‡
 + '  ‡ = 0.

−1

( ¨

2

+  ¨ + ( ).

leads to a second-order linear equation of the form 2.1.2.67:

ï ñóò

+ ô 3 ï ñ 2 + ô 2 ñ + ô 1 ï + ô 0 = 0. The substitution  3 = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.64: ‡

 +  2 ‡
 +  3 (  1  +  0 )‡ = 0.

ï ñóò

= ô ïƒ÷ ñ 2 + õÿñ + ö ï –÷ . ¨ 
=  ¨ The substitution ¨ =   leads to a separable equation:  3

ï ñóò

= ô ïƒ÷ ñ 2 + þxñ – ôƒõ 2 ïƒ÷ +2ú . Particular solution: 0 = ' + .

2

+ (  + B )¨ + ( .

Page 85

41.

ï ñóò

=

ï 2÷ ñ

= +

Solution: 42.

43.

– ù )ñ + ï

+ (þ

2

ï ñóò

=

ô ƒï ÷ ñ

ï ñ óò

=

ô ï 2÷ ñ

2

−

  ++

 tan O

õÿñ

+

+ (õ

ï ÷

+

2

B +

ö ïƒú

ú

+L

¡

.

. .

P

The transformation ˆ =   −  , ‰ =   leads to a special Riccati equation of the form 1.2.2.4: − B − 2 . (B +  )‰ Š
= ‰ 2 + (˜ˆ  , where , = ¡ B + 2

– ù )ñ + ö .

For B = 0, this is a separable equation. For B ≠ 0, the solution is:

BIX 44.

ï ñ óò

=

ô ï 2÷ +ú ñ

 ¨

ï ÷ +ú

+ (õ

2

¨ =  +L , M ¨ + +(

2

– ù )ñ + ö

ï ú

õ 1 )ñ

ô 0ï

where

¨ =   .

.

¨ 
=   + + The substitution ¨ =   leads to a separable equation: 3

45.

(ô

46.

(ô

2

ï

õ

+

2 )(

ñ óò

+

Ė

2

) + (ô

1

ï

+

+

+

õ

0

−1

( ¨

2

+  ¨ + ( ).

= 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.108: (  2  +  2 )‡

 + (  1  +  1 )‡
 + Œ (  0  +  0 )‡ = 0.

ï

õï

+ ö )ñ„ó ò =

 (ô ñ

+

ñóò

+

ï ñ

– 2ô

)2 +  ( ôñ +

õï

)–

õï

+ .

The substitution a =  + ' leads to a first-order linear equation with respect to  =  (a ): ( ƒha 2 + ý…a + =/ + ˜( ) g
=    +( .

47.

2ï

2

= 2ñ

2

2ï

2

ñóò

= 2ñ

2

+ 3ï

ñ

– 2ô

Particular solution: 49. 50.

51.

52.

53.

ô ï 2ñ

2

=

ö ï 2ñ

2

ï 2 ñ óò

=

ô ï 2ñ

2

ï 2 ñ óò

=

ô ï 2ñ

2

ï 2 ñ óò

=

ö ï 2ñ

2

ï 2 ñ óò

=

ï 2 ñ óò

+

0

õï ñ

The substitution ¨ =  + (ô

ï

.

£  . 0 = 

Particular solution: 48.

2

ï

2

2

ï

. =  £  − 12  .

+ö .

¨ 
=  ¨ leads to a separable equation:  p

+

õ ï )ñ

+

 ï

+ s.

2

+

ï

2

+ (  + 1)¨ + ( .

+ .

The substitution ( = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.139:  2 ‡

 −  (  +  )‡
 + ( ( ƒ… 2 + ý± + = )‡ = 0. +

õï ñ

+

öï ÷

+

õï ñ

+

ö ï 2÷

The substitution  = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.132:  2 ‡

 − '‡
 +  ( (˜  + ° )‡ = 0. + s ïƒ÷ .

The substitution  = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.133:  2 ‡

 − '‡
 +   ( (˜  + ° )‡ = 0. + (ô

ïƒ÷

+ õ )ï

ñ

+

ï 2÷

+

ïƒ÷

+ . The substitution ( = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.146:  2 ‡

 − (   +  ) ‡
 + ( ( ƒ… 2  + ý±  + = )‡ = 0.

Page 86

1.2. RICCATI EQUATION ã ( Ë ) Ï

54.

55.

ï 2 ñ óò

ï 2÷

= (

ïƒ÷

+

+ )ñ

â 2 (Ë )Ï

=

+ õ )ï

ïƒ÷

+ (ô

2

ÒÍ

ñ

2

öï

+

+

â 1 (Ë )Ï

2

.

+

â 0 (Ë

87

)

¨ 
= (˜ 2 ¨ The substitution = −1 ¨ leads to an equation of the form 1.2.2.53:  2 2 ¨ (   +  ) + ƒ… 2  + ý±  + = . (ï

óò

– 1)ñ

2

+ (ñ

– 2ï

2

−

+ 1) = 0.

2Œ − 1

=

The substitution

ñ

2

Œ

 + 2

(

1−Œ

1

Œ

leads to an equation of the same form:

‡ ( )

− 1)‡
 + ( Œ − 1)(‡

2

− 2‡ + 1) = 0.

If Œ = B is a positive integer, then by using the above substitution, the original equation can be reduced to an equation of the same form in which Œ = 1, i.e., to an equation of the form 1.2.2.58 with  = 1,  = −1. 56.

ï

(ô

2

+ õ )ñó ò +

ñ

2

Particular solution: 57.

ï

(ô

2

+ õ )ñ

óò

+

ñ

0

2

õ

ï ñ

+

=−

 +ý  . ƒ

ï ñ

+ = 0.

+

+



( ô +  ) = 0.

 +ý 1  − leads to an equation of the same form: (  ƒ ‡ ( )  +ý O = −  ‡ 2 + (2  + ý )‡ + ƒ = 0. ƒ P =−

The substitution

58.

ï

(ô

2

+ õ )ñó ò + ñ

Solution: 59.

ï

(ô

2

+

õï

2

– 2ï

=  + OX + ö )ñó ò =



ñ

(ô

ï

2

+

õï

+ ö )ñó ò =

Particular solution:

ñ

61.

(ô

ï

2

2

+

õ 2ï

62.

(ô

2

ï

2

+

õ 2ï



+L

+

ï

−1

P

.

+ õ )ñ + ( – ô )ï

0

= − Œ  + # , where # =

2

+ (ô

ï

+  )‡
 +

– õ = 0.

+ (2

+ )ñ –

2

ï

2

2

1 2

+ .

 − ±A ü

 2 − 4“ − 4 Œ(  .

+ (õ – )ï +

wö

.

=Π .

0

+ ö 2 )ñó ò =

Particular solution:

M2

2

2

Particular solutions: 60.

+ (1 – ô ) ï

ñ

2

ñ

2

+ (ô

1

ï

+

õ 1 )ñ

– ( +

1

ï

+

õ 1 )ñ

+ô

– ô 2)ï

ô

1

+

õ 0ï

2

+ (õ

2

–

õ 1 )ï

+

wö

2.

=Π .

0

+ ö 2 )ñó ò =

ñ

2

+ (ô

0

ï

2

+ ö 0.

Let Œ and ý be roots of the system of the quadratic equations

Œ

2

+ Π(

1

−  2) + 

0

= 0,

ý

2

+ ýh 1 + (

0

− Œ(

2

= 0,

where the first equation is solved independently (in the general case there are four roots). If some roots satisfy the condition 2 Œ ý + Œ" 1 + ý… 1 +  0 − Œ" 2 = 0, the original equation possesses a particular solution: 0 = Œ  + ý .

Page 87

88 63.

FIRST-ORDER DIFFERENTIAL EQUATIONS ( ï – ô )( ï – õ )ñ„ó ò + ñ

2

+ ü (ñ +

ï

– ô )(ñ + ï – õ ) = 0.

To the case , = 0 there corresponds a separable equation. To , = −1 there corresponds a linear equation. For , ≠ −1 and , ≠ 0, with the aid of the substitution ,!‡ ( ) = + , ( +  ), we obtain the general solution:

+

õ 2ï

ï 3 ñ óò

=

ô ï 3ñ

2

ï 3 ñ óò

=

ô ï 3ñ

2

ï (ï

+ ô )(ñó ò +

64.

(ö

65.

66.

67.

2

ï

+, ( + − )  −  O =L + , ( +  −  )  − hP 1 1 + =L +, ( + − )  −

2

+ ô 2 )(ñó ò +

69.

70.

71.

72.

73.

2

) + (õ

1

ï

+ ô 1 )ñ + ô

0

if  =  .

= 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.179: ( ( 2  2 +  2  +  2 )‡

 + (  1  +  1 )‡
 + Œ 0 ‡ = 0. + (õ

ï

2

+ ö )ñ + s ï .

The substitution  = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.183:  3 ‡

 − ( ' 2 + ( )‡
 + ! °˜‡ = 0. + ï (õ

ï

+ ö )ñ + 

ï

+ .

The substitution  = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.186:  3 ‡

 −  ( ' + ( )‡
 +  ( ƒ… + ý )‡ = 0. 2

Ė

The substitution Œ  ( 2 +  )‡

 + ( ' 68.

Ė

if  ≠  ,

ï 2(ï

+ ô )(ñó ò +

2

= ‡
 + ( )‡

2

Ė

) + (õ

2

ï

+ ö )ñ + s ï = 0.

2

‡ leads to a second-order linear equation of the form 2.1.2.190:
 + Œ°˜‡ = 0.

) + ï (õ

ï

ï

+ ö )ñ + 

+

= 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.194:  2 ( +  )‡ 

 +  ( ' + ( )‡
 + Œ ( ƒ… + ý )‡ = 0.

ï

+ ö )( ï ñ ó ò – ñ ) – ñ 2 + ï − Solution: ln û û = L + 2 X  û + û

(ô

2

ï 2(ï

+

õï

û

û

+ ô )(ñó ò +

Ė

– 1)2 (ñó ò +

Ė

2

2

+

2

2

) + ï (õ

ï

2

2

= 0.

2

+ ' + (

M



.

+ ö )ñ + s = 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.219:  2 ( 2 +  )‡

 +  ( ' 2 + ( )‡
 + Œ°˜‡ = 0.

ô (ï

2

õ ï (ï

)+

2

– 1)ñ +

öï

2

+ý

ï

+ s = 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.227:  ( 2 − 1)2 ‡

 + ' ( 2 − 1)‡
 + Œ ( (˜ 2 +  + ° )‡ = 0. M

ïƒ÷ +1 ñ óò

=

ô ï 2÷ ñ

õ ƒï ÷ ñ

+

ö ïƒú

+ý .

The substitution ¨ =   + # , where # is a root of the quadratic equation !  # 2 −(  + B ) # + = 0, M u ¨
¨ ¨ 2 leads to an equation of the form 1.2.2.35:   =  + (B +  − 2 !# ) + (˜ + .

ï (ô ï

+ õ )ñó ò =

 ƒï ÷ ñ

2

The transformation a =   − , ( ˜H +  ).

+ ( – ôù

ï )ñ

+

ï –÷

. , H =  −  leads to a separable equation: [ƒ…a 2 + (ý + 'B )a + = ]H g
=

Page 88

1.2. RICCATI EQUATION ã ( Ë ) Ï

74.

75. 76.

ÒÍ

â 2 (Ë )Ï

=

2

+

â 1 (Ë )Ï

+

â 0 (Ë

89

)

ï 2 (ô ïƒ÷

– 1)(ñó ò + ƒñ 2 ) + ( ïƒ÷ + ) ï ñ + ïƒ÷ + s = 0. The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.254:  2 (   − 1)‡

 + (  +  )‡
 + Œ (&  + ° )‡ = 0.

( ô ïƒ÷ + õ ïƒú + ö )ñó ò = ö^ñ 2 – õ ïƒú Particular solution: 0 = −1  . ( ô ï ÷ + õ ï ú + ö )ñ Particular solution:

óò

=

ô ï ÷ –2 ñ

( ô ïƒ÷ + õ ïƒú + ö )ñó ò =  ï ñ Particular solution: 0 = − Œ .

78.

(ô

+

õï ú

+ ö )( ï

2

+

= .

0

77.

ï ÷

–1

+

2

– ñ ) + sï

ñ óò

ñ

–2

.

õï ú

–1

ñ

+ö .

ï sñ

– 

2

 (ñ

Particular solutions: 0 = A" £ Œ .

ïƒ÷

+ô

2

ï

–

2

ï

+ 

ï

s

.

) = 0.

1.2.3. Equations Containing Exponential Functions 1.2.3-1. Equations with exponential functions. 1.

ñóò

ôñ

=

2

+

õ ó

.



The substitution a = ž 2.

ñ óò

ñ

2

=

+

ô  ó

–ô

Particular solution: 3.

4.

5.

ñóò

=

ñ

2

ñóò

=

ñ

2

ñ óò

=

+

ô

+

ôñ

õ ó

+

+

0

+ 'a .



=   . 2

ó



.

+ö .

The substitution . = −‡
 ‡ leads to a second-order linear equation of the form 2.1.3.10:  ‡

 − ‡
 + . (   + ( )‡ = 0.

ñ

2

+

õÿñ

+ ô ( – õ )

ñóò

ñ

2

=

+

ô ó ñ

0

– ôƒõ

ñóò = ñ 2 + ô 2 ó (  ó

0

1.2.2.4: ¨3Š
= ¨

ñ óò

=

ñ

2

+

ô 8 ó

2

+ Π+

ó





–ô

 

2 2

ó

–

õ

2

2

+ (2 Œ )−2 ( ˆ

=.

+ õ )÷ –

−2



ˆ  .

õ 6 ó 2

.

.

+



1 4

2

.

+ , ¨

=

ö 4 ó

–

The transformation ˆ =  2  , ¨ = 

¨2Š
= ¨

ó

=   .

The transformation ˆ =  

8.

.

+ ö

õ ó

Particular solution: 7.

ó

 

2 2

2

The substitution . = −‡
 ‡ leads to a second-order linear equation of the form 2.1.3.5:   ‡

 + . (  +   + (  2  )‡ = 0.

Particular solution: 6.

leads to an equation of the form 1.2.2.35: Πa 
g = 

+ 'ˆ + ( ).

−2

1

O  −

Œ

2





−

Œ

2

 −

 P

leads to an equation of the form

.

 O

2Œ

−

1 leads to an equation of the form 1.2.2.3: 2P

Page 89

90 9.

10.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

ô  ó ñ

=

2

õ só

+

ñ óò

õ ó ñ

=

2

+ ô

ñ óò

ô  ó ñ

=

2

+

ó



ñ óò

ô ó ñ

=

2

+

õÿñ

+

Ė

ñóò

ó

 ñ

=

2



14.

ñóò

óñ

= – 

2

15.

ñóò

ô  ó ñ

=

2

+

ó

0

16.

17.

18.

19.

ñóò =À ô   ó ñ

2

+

õÿñ

ñ óò

ô (2 + )ó ñ

=

2

+ ( + Π)H + ( .

.



ô ( – )ó

.



−

= −Œ   . – ô

 

( – )



=  − . –



ö só

ó

.

õ ó

= −   .

+ [ õ

+ ý

ñ óò

ô  ó ñ

=

2

+

õÿñ

+

 

( + )

 The substitution ¨ = ž

–

ó

.

.

ö  ÷ ó

ó

ó

– ]ñ + ö

.

¨ 
=  (  leads to a separable equation: 3 + ý

 (2÷

+1)

ó

2

  ( ¨

+ )

g
= a 2

2

ñóò

ó (ñ

 

=

–

õ ó

ó

( ô

+

õ  ó

ó ( ô

ó + õ 

)2 + 0

2

+

=− .

ü ó ñ

¡

+ ö )(ñó ò – ñ ) + ô

Particular solution:

+

2

+ 'a

+

.



ñ

+ 'a

.

=   .

+ ö )ñó ò = 0

õ ó

2

+  ¨ + ( ).

 The substitution a =   leads to an equation of the form 1.2.2.52: ,!a 2 
g = a 2 (˜a  +1 + a 2(  +1) . M

Particular solution:

21.

0

+

ó

2

 The substitution a =   leads to an equation of the form 1.2.2.51: ! , a (˜a (  + ) J' + . M

Particular solution: 20.

  

ôƒõ ( + )ó ñ

Particular solution:

.

.

2 ( +2 )

 ñ

Particular solution:

−

=   .

0

+ ô

+ 'a

.

leads to a separable equation: H 
= !H

+

Particular solution:

õ ( +2 )ó 

ö – ó

0

ô  ó ñ

+

2

2

=   .

– ôƒõ

Particular solution: 13.

–ô

0

The substitution H = ž 12.

À

≠ 0.

 The substitution a =   leads to an equation of the form 1.2.2.4: , 
g =  Particular solution:

11.

ü

,

2



2



–þ

ó

2

+

 Œ   + '“     0 =−    +   +(

+

üIþ ó

õ 2   ó

.

= 0.

.

1.2.3-2. Equations with power and exponential functions. 22.

ñ óò

=

ñ

2

+

ô ï óñ

Particular solution:

+ ô 0



ó

.

= −1  .

Page 90

1.2. RICCATI EQUATION ã ( Ë ) Ï

23.

ñ óò

ô  ó ñ

=

2

ñóò

2

+

õÿù ïƒ÷

ñóò

ó

 ñ

=

2

ñóò

= – 

óñ

Particular solution: 27.

ñóò

ô ó ñ

=

– ôƒõ

2

ñóò

ô ƒï ÷ ñ

=

2

0

Particular solution: 29.

ñóò

ô ƒï ÷ ñ

=

2

Ė

+

ñóò

ô ƒï ÷ ñ

=

2

– ôƒõ

ñ óò

= –( ü + 1) ï

ñ

34.

2

0

+

0

–

õï ÷

ï ñ óò

=

ô  ó ñ

Solution: 36.

ï ñóò

=

ñóò

=

ñ

+

ô ï 2÷   ó ñ 2

+ 2 ô

 ¨

2

+

2

=   .

õ ó

+



Particular solution:

.

=   . =  −

−1

.

+ ö )ñ +

=  



ó

.

õ  ó

.

– ô

¨

õÿù ï ÷

ôƒõ 2 ï 2   ó −1

ïƒ÷  ó

–ô 0

ï ÷

– )ñ + ö

.

¨ 
=     (  ¨ leads to a separable equation: 3

+ (õ

2



+( .

 

–1

 

ó

2 2

=  





2

2

2

+  ¨ + ( ).

.

.

M

 +L

– ù )ñ + ö

M = XÈ  + ¨ +(

ï  ó

.

   +( . 0 = '

üIñ

.

.



= '   tan O X  

Solution: X 37.

2

ïƒ÷  2 ó

– ö )2 +

Particular solution: 35.

2

 The substitution ¨ = ž

ô  ó (ñ



ô ï  +1   ó ñ

0

.

–1

ïƒ÷  2 ó

2

ó 2 + (õ ï ÷  

ñ óò = ô ï ÷  2 ó ñ =

91

)

=   .

ñ óò = ô ï ÷ ñ 2 – ô ï ÷ ( õ  ó

ñ óò

â 0 (Ë

.

õÿù ïƒ÷

+

= '  .

ïƒ÷  ó ñ

Particular solution: 33.



– ôƒõ

0

Particular solution: 32.

+

.

ïƒ÷

–ô

=  − .

– ôƒõ

Particular solution: 31.

0

0

Particular solution: 30.

â 1 (Ë )Ï

= −Œ  − .

ïƒ÷  ó ñ

õ ó

+



ïƒ÷  ó ñ

Particular solution: 28.

ó  ï 2÷

2

ïƒ÷  – ó

+ ô 0

+ô

2

– ôƒõ

  . 0 = '

ô ƒï ÷ ñ

+

Particular solution: 26.

+

.

–1

Particular solution: 25.

2

M

2

ô ó ñ

=

õ – ó

â 2 (Ë )Ï

=

H  =  + L , where H = ž . !H + ŒH + 

Solution: X 24.

+

ÒÍ

−1

 



M

P

ó

. .

 + L , where ¨ =   .

.

.

Page 91

92 38.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

ô – ó ñ 2

=

ñóò

ô ƒï ÷ ñ

=

2

+

ôƒõ

+ ôƒõ

2

2

.

ï ñ

+

ïƒ÷  ó

2

.

2 2 =    J 2 tan O/XÈ    J 2  + L . M P

Solution: 40.

ï ñ

+

2 2 =    J 2 tan O Z X  − J 2  + L . M P

Solution: 39.

2

ï 4 (ñ ó ò

– ñ 2 ) = ô + õ exp( ü ï ) + ö exp(2 ü ï ). The transformations ˆ = 1  , ¨ = − 2 −  leads to a Riccati equation of the form 1.2.3.3: ¨ Š
= ¨ 2 +  +    Š + (  2  Š .

1.2.4. Equations Containing Hyperbolic Functions 1.2.4-1. Equations with hyperbolic sine and cosine. 1.

ñ óò

=

ñ

2

–ô

2

Particular solution: 2. 3.

= ñ 2 + ô sinh( Particular solution:

2

ñ óò

=

ï )ñ

=  cosh( Π ).

0

+ ôƒõ sinh( = − .

sinh(

ñóò

= [ ô sinh2 (

ï

0 2

sinh3 ( 0

0

ï

= coth( Π ).

6.

[ ô sinh( ï ) + õ ]ñ„ó ò = ñ 2 + ö sinh(

Particular solution: 0 = − .

7.

[ ô sinh(

ï

M

9. 10.

ñ óò

=

ñ

2

ï

)+

ï )ñ

–ý

–ô . 2

+ö

ý

sinh(

ï

).

ô

2

+  + cosh ï .

The transformation  = 2a , ƒ = −‡
 ‡ leads to the modified Mathieu equation 2.1.4.9: ‡ g

g − (  − 2  cosh 2a )‡ = 0, where  = −4ƒ…ý ,  = 2ƒ±= .

ñóò

= ñ 2 + ô cosh( Particular solution:

ñ óò = ñ + ô ï 2

ñóò

ï )ñ

0

+ ôƒõ cosh( = − .

coshú ( õ ï )ñ

Particular solution: 11.

0

).

sinh( ï ) = 0. Œ cosh( Œ  ) =− .  sinh( Œ  ) + 

) + õ ](ñ„ó ò – ñ 2 ) +

Particular solution:

ï

).

– ô sinh2 (

2

).

) – õ 2.

= cosh( Π ).

) – ]ñ

Particular solution:

8.

ï

+ ô sinh ú ( õ = −1  .

–

ï

sinh2 (

ñóò = ñ 2 + ô ï sinhú ( õ ï )ñ

Particular solution: 5.

)–ô

0

ï )ñ

ñ óò

Particular solution: 4.

ï

+ ô sinh(

= [ ô cosh2 (

ï

0

0

)–

+ ô cosh ú ( õ = −1  .

) – ]ñ

Particular solution:

ï

2

+

ô

+

ï

õ

2

.

).

– ô cosh2 (

ï

).

= tanh( Π ).

Page 92

1.2. RICCATI EQUATION ã ( Ë ) Ï

12.

2ñó ò = [ ô –

ô

+

Particular solution: 13.

ñóò

ñ

2

=

–

2

+

ï )]ñ

cosh( cosh÷ (

ô

) sinh–÷

ï

â 2 (Ë )Ï

=

2

– ô cosh(

Π .

1 2

= tanh 

0

+ô +

2

ÒÍ

–4

ï

(

â 1 (Ë )Ï

+

ï

â 0 (Ë

+

)

93

).

).

1 The transformation ˆ = coth( Œ  ), ¨ = − sinh2 ( Œ  ) −sinh( Œ  ) cosh( Œ  ) leads to an equation

of the form 1.2.2.4: ¨3Š
= ¨ 14.

ñ óò

ô

=

sinh(

ï )ñ

2

+Œ

ˆ  .

Œ

) cosh ÷ (

ï

+ õ sinh(

2

−2

The transformation ˆ = cosh( Œ  ), ¨

¨2Š
= ¨

15.

ñóò

2

ô

=

+ Œ

cosh(

−2

=

ˆ  .

ï )ñ

2

The transformation ˆ = sinh( Œ  ), ¨

¨2Š
= ¨

16.

2

+ Œ

ï

[ ô cosh(

−2

óò

=

Particular solution: 17.

[ ô cosh(

ï

ñ

leads to an equation of the form 1.2.2.4:

).



leads to an equation of the form 1.2.2.4:

–ý

2

+

öý

cosh(

ï

).

=− .

0

) + õ ](ñ„ó ò – ñ

Particular solution:

Œ

ï )ñ

+ ö cosh(

2

).



ï

=

ˆ  .

) + õ ]ñ

Œ

) sinh ÷ (

ï

+ õ cosh(

ï

0

M

cosh( ï ) = 0. Œ sinh( Œ  ) =− .  cosh( Œ  ) +  2

)+

ô

2

1.2.4-2. Equations with hyperbolic tangent and cotangent. 18.

ñóò

ñ

2

=

+

ô

Particular solution: 19.

ñ óò

ñ

2

=

+ 3 ô –

2

ñóò

ñ

2

=

+

ô ï

21.

[ ô tanh(

ï

Particular solution: 22.

ñ óò

ñ

=

2

+

ô

ñóò

ñ

2

=

–

2

ñóò

=

ñ

2

+

ô ï

ñ

2

0

ï )ñ

+ ö tanh(

0

cothú ( õ

Particular solution:

0

). –ý

2

+

öý

tanh(

ï

).

=− .

M

ï

).

=  coth( Π ).

+ 3 ô – ô ( ô + ) coth2 (

Particular solution: 24.

0

ï

= −1  .

– ô ( ô + ) coth2 (

Particular solution: 23.

=

+ ô tanhú ( õ

ï )ñ

0

óò

).

=  tanh( Œ  ) − Œ coth( Œ  ).

tanhú ( õ

) + õ ]ñ

ï

– ô ( ô + ) tanh2 ( 0

Particular solution:

).

=  tanh( Π ).

0

Particular solution: 20.

ï

– ô ( ô + ) tanh2 (

ï

).

=  coth( Œ  ) − Œ tanh( Œ  ).

ï )ñ

+

ô

= −1  .

cothú ( õ

ï

).

Page 93

94

FIRST-ORDER DIFFERENTIAL EQUATIONS

25.

[ ô coth( ï ) + õ ]ñ„ó ò = ñ 2 + ö coth(

Particular solution: 0 = − .

26.

ñ óò

ñ

2

=

Particular solution: 27.

ñóò

ñ

2

=

ô

+

0

õ

+

M

ï

tanh2 (

2

–2

2

)–2

ï )ñ

–ý

ï

coth2 (

0

ý

coth(

ï

).

).

= Πtanh( Π ) + Πcoth( Π ).

– 2 ôƒõ – ô (ô + ) tanh2 (

Particular solution:

+ö

2

ï

) – õ ( õ + ) coth2 (

ï

).

=  tanh( Π ) +  coth( Π ).

1.2.5. Equations Containing Logarithmic Functions 1.2.5-1. Equations of the form ( )
 =  2 ( ) 1.

= ô (ln ï )÷

ñóò

ñ

2

+

õÿþ ïƒú

Particular solution: 2. 3.

4.

ï ñóò

2

(ln ï )÷ .

= ' + .

+ 'a + ( .

= ï ñ 2 – ô 2 ï ln2 ( ï ) + ô . Particular solution: 0 =  ln(ý± ).

= B − 1:

ï ñóò ï ñóò

=

ï ñ

–ô

2

2

ï

ln2  (

ï ñóò

ï

)+

ôƒü

ln 

–1

 0 =  ln ( ý± ).

(

ï

).

+

1 2

leads to an equation of the form 1.2.2.3:

+

1 leads to an equation of the form 1.2.2.4: 2

= ô ïƒ÷ ñ 2 + õ – ôƒõ 2 ïƒ÷ ln2 ï . Particular solution: 0 =  ln  .

ï 2 ñ óò

= ï 2 ñ 2 + ô ln2 ï + õ ln ï + ö . The transformation ˆ = ln  , ¨ =  ¨2Š
= ¨ 2 + ˆ 2 + 'ˆ + ( − 1 . 4

ï 2 ñ óò

=

ï 2ñ

2

+ ô ( õ ln ï + ö )÷ + 14 .

The transformation ˆ =  ln  + ( , ¨

¨2Š
= ¨

2

ï

ï )(ñ óò

2

ln( ô

+ 

−2

ˆ  .

=

 

– ñ 2 ) = 1.

Particular solution:

0

= −[ ln(  )]−1 .

1.2.5-2. Equations of the form ( )
 =  2 ( ) 10.

2

= ôñ 2 + õ ln  ï + ö ln2  +2 ï . The substitution a = ln  leads to an equation of the form 1.2.2.6 with , g
=  2 + 'a  + (˜a 2  +2 .

8.

9.

ï 2ú

ï ñóò

Particular solution:

7.

– ôƒõ

+  0 ( ).

= ôñ 2 + õ ln ï + ö . g The substitution  =  leads to an equation of the form 1.2.2.1: 
g = 

5. 6.

0

–1

2

ñóò

=

ñ

2

+

ô

ln(

ï )ñ

Particular solution:

– ôƒõ ln( 0

ï

2

+  1 ( )
 +  0 ( ).

) – õ 2.

=.

Page 94

1.2. RICCATI EQUATION ã ( Ë ) Ï

11. 12.

ñóò

=

ñ

2

+

lnú ( õ

ô ï

ï )ñ

Particular solution:

ñóò

+1

ln ï

0

= ' ln  .

=

ô ƒï ÷ ñ

ïƒ÷

– ôƒõ

2

ñóò

= –(ù + 1) ïw÷

ñ

2

+

Particular solution: 14.

ñóò

= ô (ln ï )÷

ñ

2

ñ óò

= ô (ln ï )  (ñ –

ñóò

= ô (ln ï )÷

ñ

2

ï ñ óò

18.

ï ñóò

lnú (

ô

Solution: 19.

ï ñóò

20.

ï ñóò

=

ô ï 2÷

Solution: X 21.

22. 23.

ñ

M2

H

â 0 (Ë

95

)

+ õ ln ï + õ .

ñ

–1

.

(ln ï )ú

õ^ö

– ô„ö 2 (ln ï )÷ .

+ L , where H = 

!H + 

+ õ ln ï

+

= −( .

0

2

â 1 (Ë )Ï

– ô (ln ï )ú .

õÿù ï ÷

+

+

üIñ

+

ôƒõ 2 ï 2

= '  tan /XÈ 

= ô ïƒ÷ (ñ

Solution:

ï )ñ

+1

   +( . 0 = '

= ( ôñ + õ ln ï )2 .

=

+

).

ñ

.

– ö )2 +

+ õ (ln ï )ú

Solution: ln  = X

2

= ' ln  .

õï ÷

Particular solution: 17.

−1

(ln ï )÷

0

Particular solution: 16.

ï

(ln ï )ú

+1

=  −

ï

â 2 (Ë )Ï

=

+ õ ln ï + õ .

ñ

ô ïƒ÷

0

– ôƒõ

Particular solution: 15.

ln ú ( õ

ô

0 = −1  .

Particular solution: 13.

+

ÒÍ

Q

−1

+

lnú (

ï

+  ln  . ).

ln ( Π )  + L2S .

M

) –õ . 2

 1 +   =L . +  ln  B

(ln ï )ñ

 ¨

2

+ (õ

ïƒ÷

ln ï – ù )ñ + ö ln ï .

¨ = XÈ  M 2+ ¨ +(

−1

ln 

M

 + L , where ¨ =   .

= ô 2 ï 2 ñ 2 – ï ñ + õ 2 ln ÷ ï . The substitution  2 = −‡
 ‡ leads to a second-order linear equation of the form 2.1.5.27:  2 ‡

 + ‡
 + (  )2 ln *‡ = 0.

ï 2 ñ óò

( ô ln ï + õ )ñ„ó ò = ñ 2 + ö (ln ï )÷ Particular solution: 0 = − Œ . ( ô ln ï + õ )ñ ó ò = (ln ï )÷ ñ 2 + Particular solution: 0 = − Œ .

ñ

–

2

ö^ñ

–

2

+

wö

(ln ï )÷ .

(ln ï )÷ +

ö

.

1.2.6. Equations Containing Trigonometric Functions 1.2.6-1. Equations with sine. 1.

= ñ 2 +  + sin( ï ). The substitution 2a = 2 Œ  + leads to an equation of the form 1.2.6.14: Œ
g = ƒ

ñóò



2

+ ý + = cos a .

Page 95

96 2.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

ñ

–ô

2

=

2

+ ô sin(

Particular solution: 3.

ñ óò

=

ñóò

=

ñ

2

ñ

2

+ ö sin ÷ (

2

+

0

ï

)+ô

2

sin2 (

ï

).

(

ï

= −  cos( Œ  ). + ô ) sin–÷

ï

–4

+ õ ).

sin( Œ  +  ) ¨ sin2 ( Œ  +  ) , = + cot( Œ  +  )S leads to an equation The transformation ˆ = sin( Œ  +  ) sin(  −  ) Q Œ ¨ 3
¨ of the form 1.2.2.4: Š = 2 + #*ˆ  , where # = ( [ Œ sin (  −  )]−2 . 4.

ô

+

ï )ñ

sin(

Particular solution: 5.

ñóò

ñ

2

=

0

sinú ( õ

ô ï

+

ñóò

=

ï )ñ

sin(

2

0

ñóò

ô

=[ +

0

ï )]ñ

sin2 (

Particular solution: 9.

ñóò

= –( ü + 1) ï

ñ

2

0

ñ óò

sin  (

ô

=

ï

0

11.

ï ñóò

=

sinú (

ô ï

[ ô sin(

) + õ ]ñ„ó ò =

Particular solution: 13.

ï

[ ô sin(

).

1 2

2

=  −

−1

õï ÷

+

üIñ

ñ

2 0

0

ï

).

– ô (sin ï ) ú .

. – ö )2 +

õÿù ï ÷

–1

ï 2

sinú (

ï

   +( . 0 = '

2

).

.



= − cot( Œ  ).

+ ôƒõ

2

−1

+ ö sin(

=− .

.

).

sin + ( Π )  + L2S . M

ï )ñ

–ý

2

+ö

ý

sin(

ï

).

M

sin( ï ) = 0. Œ cos( Œ  ) =− .  sin( Œ  ) + 

) + õ ](ñ„ó ò – ñ 2 ) – ô

Particular solution:

Π +

1 4

– ô + ô sin2 (

+

ï

– ô – ô sin(

+

= '   tan    X   Q

Solution: 12.

ï )ñ

2

= tan 

+ )(ñ –

Particular solution:

.

).

ô ï +1 (sin ï )ú ñ

+

Particular solution: 10.

ï

2

= − cos( Œ  ).

ï )]ñ

2ñó ò = [ + ô – ô sin( Particular solution:

8.

ï

sin3 (

Particular solution: 7.

+ ô sinú ( õ

0 = −1  .

+

õ

)–

= − .

ï )ñ

Particular solution: 6.

ï

+ ôƒõ sin(

2

1.2.6-2. Equations with cosine. 14.

15.

ñ óò

=

ñóò

=

ñ

2

+  + cos ï .

The transformation  = 2a , ƒ = −‡
 ‡ leads to a Mathieu equation of the form 2.1.6.29: ‡ g

g + (  − 2  cos 2a )‡ = 0, where  = 4 ƒ…ý ,  = −2ƒ±= .

ñ

2

–ô

2

+ ô cos(

Particular solution:

0

ï

)+ô

2

cos2 (

ï

).

=  sin( Π ).

Page 96

1.2. RICCATI EQUATION ã ( Ë ) Ï

16.

17. 18. 19.

ÒÍ

=

â 2 (Ë )Ï

= ñ 2 + ô cos( ï )ñ + ôƒõ cos( Particular solution: 0 = −  .

ï

ñóò

= ñ 2 + ô ï cosú ( õ Particular solution:

ï )ñ

ñóò ñ óò

=

ï )ñ

cos(

2

+ ô cosú ( õ = −1  .

0

ï

cos3 (

+

ï

= sin( Π ).

0

=[ +

ô

ï )]ñ

cos2 (

Particular solution:

ñóò

= –( ü + 1) ï

ñ

2

=

cos  (

ô

ï

=

cosú (

ô

Solution:

[ ô cos(

=  −

−1

ï ÷

ï )ñ

2

+

üIñ

óò

Particular solution:

ñ

õÿù ï ÷

ï 2

cosú (

−1

+ ö cos(

=− .

2 0

M

+Œ

2

+

2

ï

).

).

–1

.

ï

).

+ cos ( Π )  + L2S . M

ï )ñ

–ý

2

+ö

ý

cos(

ï

).

cos( ï ) = 0. Œ sin( Œ  ) = .  cos( Œ  ) + 

– ñ ) – ô 2

0

2

– ô (cos ï )ú .

– ö )2 +

+ ôƒõ

= '   tan   /XÈ  Q

) + õ ](ñ

.

   +( . 0 = '

[ ô cos( ï ) + õ ]ñ„ó ò = Particular solution:

ï

0

ô ï +1 (cos ï )ú ñ

Particular solution:

ï ñ óò

= tan( Π ).

+ )(ñ – õ

ï

– ô + ô cos2 (

+

0

+

Particular solution:

ñ óò

2

97

)

).

ñóò

26.

â 0 (Ë

).

21.

25.

+

) – õ 2.

2ñó ò = [ + ô + ô cos( ï )]ñ 2 + – ô + ô cos( Particular solution: 0 = tan  12 Œ   .

24.

â 1 (Ë )Ï

= ñ 2 + 2 + ö cos÷ ( ï + ô ) cos–÷ –4 ( ï + õ ). The substitution Œ  = ŒH − z 2 leads to an equation of the form 1.2.6.3: 
‘ = ( sin  ( ŒH +  ) sin−  −4 ( ŒH +  ).

20.

23.

+

ñóò

Particular solution:

22.

2

2

1.2.6-3. Equations with tangent. 27. 28. 29.

30.

= ñ 2 + ô + ô ( – ô ) tan2 ( ï ). Particular solution: 0 =  tan( Œ  ).

ñóò

= ñ 2 + 2 + 3 ô + ô ( – ô ) tan2 ( ï ). Particular solution: 0 =  tan( Œ  ) − Œ cot( Œ  ).

ñóò

= ôñ 2 + õ tan ï ñ + ö . The substitution  = −‡
 ‡ ‡

 −  tan *‡
 + !(˜‡ = 0.

ñ óò

leads to a second-order linear equation of the form 2.1.6.53:

= ôñ 2 + 2 ôƒõ tan ï ñ + õ ( ôƒõ – 1) tan2 ï . The substitution ‡ = +  tan  leads to a separable equation of the form 1.1.2: ‡
 = ‡

ñ óò

2

+ .

Page 97

98

FIRST-ORDER DIFFERENTIAL EQUATIONS

31.

= ñ 2 + ô tan( ï )ñ + ôƒõ tan( Particular solution: 0 = −  .

32.

ñóò

=

ñ

2

+

tanú ( õ

ô ï

ï )ñ

ñ óò

= –( ü + 1) ï

ñ

2

ñ óò

tan÷ (

ô

=

ï )ñ

0

Particular solution: 35.

ñóò

ï

tan (

ô

=

0

ï ñóò

tanú (

ô

=

Solution: 37.

−1

tan÷

2

ï

. +2

õ ïƒ÷

2

+

üIñ

[ ô tan( ï ) + õ ]ñ ó ò = Particular solution:

ñ

).

)+

õ

tanú (

−1

ï

tan2 (

ï 2

2

–1

ï )ñ

+ ü tan(

=− .

–ý

2

.

.

ï

).

üý

+

õ

)+

tan + ( Π )  + L2S . M

2 0

.

õÿù ïƒ÷

+ ôƒõ

= '   tan    X   Q

2

– ö )2 +

   +( . 0 = '

ï )ñ

ï

(

õ

– ô (tan ï )ú .

=  tan( Π ).

+ )(ñ –

Particular solution: 36.

=  −

– ôƒõ

2

tan ú ( õ

ô

ô ï  +1 (tan ï )ú ñ

+

Particular solution: 34.

+

)–

0 = −1  .

Particular solution: 33.

ï

ñóò

ï

tan(

).

M

1.2.6-4. Equations with cotangent. 38.

= ñ 2 + ô + ô ( – ô ) cot2 ( ï ). Particular solution: 0 = −  cot( Œ  ).

ñ óò

39.

= ñ 2 + 2 + 3 ô + ô ( – ô ) cot2 ( ï ). Particular solution: 0 = Œ tan( Œ  ) −  cot( Œ  ).

40.

ñ óò

41. 42. 43.

ñ óò

= ñ 2 – 2 ô cot( ô Particular solution:

ï )ñ

0

= ñ 2 + ô cot( ï )ñ + ôƒõ cot( Particular solution: 0 = −  . = ñ 2 + ô ï cotú ( õ Particular solution:

ñóò = –( ü + 1)ï  ñ

2

ñ óò

=

cot (

ô

ï

ï ñ óò

=

ô

Solution: 46.

cotú (

0

+ ô cotú ( õ = −1  .

0

=  −

+ )(ñ –

−1

õï ÷

. – ö )2 +

   +( . 0 = '

2

+

üIñ

+

= '   tan   /XÈ  Q

[ ô cot( ï ) + õ ]ñ ó ò = Particular solution:

ï

ô ï +1 (cot ï )ú ñ

+

ï )ñ

) – õ 2.

ï )ñ

ñ óò

Particular solution: 45.

ï

ñ óò

Particular solution: 44.

+ õ 2 – ô 2. =  cot(  ) −  cot( ' ).

ñ

−1

+ ö cot(

=− .

2 0

ôƒõ 2 ï 2

). – ô (cot ï )ú .

õÿù ï ÷

–1

cotú (

ï

. ).

+ cot ( Π )  + L S . M

ï )ñ

–ý

2

+ö

ý

cot(

ï

).

M

Page 98

1.2. RICCATI EQUATION ã ( Ë ) Ï

ÒÍ

=

â 2 (Ë )Ï

2

â 1 (Ë )Ï

+

+

â 0 (Ë

99

)

1.2.6-5. Equations containing combinations of trigonometric functions. 47.

48.

ñóò

ñ

2

=

2

+

+ ö sin ÷ (

) cos–÷

ï

–4

ï

(

).

This is a special case of equation 1.2.6.3 with  = 0 and  = 2.

ñ óò

ô

=

ï )ñ

sin(

2

) cos÷

ï

+ õ sin(

(

The transformation ˆ = cos( Œ  ), ¨

¨2Š
= ¨

49.

ñóò

2

=

+ Œ

−2

2

cos÷ (

ô

+

ñóò

ô

=

ï )ñ

cos(

2

51.

ñ óò

2

=

+ Œ

sin(

−2

) sin ÷ (

ï

+ õ cos(

ï

2

ô ï ÷

+

Particular solution: 52.

53.

sin ÷

+1

(2 ï )ñó ò =

ôñ

0

ñóò

ñ

2

=

ñóò

ñ

2

=

ñ óò

ñ

2

=

ñóò

ñ

2

=

0

+ þxñ cot ï +

ï

+ õ cos2÷

ñóò

ñ

2

=

–2

+

ñóò

=

ñ

2

–

õ

2

2

tan2 (

wô

+

ˆõ

õ

2

(sin ï )2ú .

=

+ B 2

+1

H + .

1 2

2

–

sin(

ï )ñ

Particular solution:

 +L

2

+

0

= Œ cot( Œ  ) − Œ tan( Œ  ).

cot2 (

ï

+ Πsin(

−2

ï

) + ô cos2 (

ˆ  .

ï )ñ

=

ï

) + õ ( – õ ) cot2 (

) sin ÷ (

Πcos( Π )

– ô tan(

0 = 1 cos( Π ).

.

).

=  tan( Œ  ) −  cot( Œ  ).

tan2 (

2

ô

P

)–2

0

3 4

2

M

.

P

ï

2

ï

 +L M

+ 2 ôƒõ + ô ( – ô ) tan2 (

form 1.2.2.4: ¨3Š
= ¨

ñóò

2

cos2ú'ï .

The transformation ˆ = sin( Œ  ), ¨

59.

.

ï

= −  sin +  cot O/X sin + 

Particular solution: 58.

ï ÷

–ô

= −  cos +  cot O X cos+ 

Particular solution: 57.

leads to an equation of the form 1.2.2.4:

= −  cot  .

– þxñ tan ï +

Solution: 56.



).

– ñ tan ï + ô (1 – ô ) cot2 ï .

Solution: 55.

Œ

ï

(

. The substitution H = tan   leads to a separable equation: 2  sin(2 )H 
= !H

Particular solution: 54.

–1

= 1 cos( Π ).

sin2÷

2

ï )ñ

cos(

).

=

ˆ  .

ï )ñ

leads to an equation of the form 1.2.2.4:

Œ

– ô cos÷

ï )ñ

The transformation ˆ = sin( Œ  ), ¨

¨2Š
= ¨



0 = 1 cos( Π ).

Particular solution: 50.

). = −

ˆ  .

ï )ñ

sin(



ï

ï

+

ï

ï

).

).

sin( Π ) leads to an equation of the 2 cos2 ( Π )

).

Page 99

100

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.2.7. Equations Containing Inverse Trigonometric Functions 1.2.7-1. Equations containing arcsine. 1.

ñóò

ñ

2

=

+ (arcsin ï ) ÷

Particular solution: 2.

ñóò

ñ

2

=

(arcsin ï ) ÷

ï

+

ñóò

= –( ü + 1) ï

ñ

ñóò

+ (arcsin ï ) ÷ ( ï

2

ñ óò

= (arcsin ï ) ÷

ñ

2

ñ óò

= (arcsin ï ) ÷ = (arcsin ï ) ÷

ñ

2

ñóò

ñ

2

ï ñóò

+

= − .

õ ï ú

–

–

–1

= ý± + .

ïƒú

õ

ñ

2

+

– 

– õ )2 +

üIñ

= ( ô ï 2÷ ñ 2 + õ ïƒ÷ The substitution H =  

ï ñóò

– 1).

−1

2

ñ

+

ï 2ú

õÿþ ï ú

–1

.

(arcsin ï )÷ .

ôþ ïƒú

ˆõ 2 ï 2

+

= '  tan Œ"/XZ 

Q

ñ

(arcsin ï )÷ .

2

  + + . 0 = 

= (arcsin ï ) ÷

+1

(arcsin ï ) ÷

ï ú

+ þ

= (arcsin ï ) ÷ (ñ – ô

Solution: 9.

ôƒõ

0

Particular solution: 8.

ôñ

+

 + . 0 = '

Particular solution: 7.

.

0

Particular solution: 6.

 0 = 

− −1

Particular solution: 5.

+ (arcsin ï ) ÷ .

ñ

= −1  .

0

Particular solution: 4.

+ ô (arcsin ï ) ÷ .

2

= − .

0

Particular solution: 3.

–ô

ñ

–1

.

(arcsin ï ) ÷ .

(arcsin  ) 

M

 + L2S .

+ ö )(arcsin ï ) ú – ùñ . leads to a separable equation: H 
=  

ñ

−1

(arcsin  ) + ( !H

2

+ ˜H + ( ).

1.2.7-2. Equations containing arccosine. 10.

ñóò

ñ

2

=

+ (arccos ï )÷

Particular solution: 11.

ñóò

=

ñ

2

+

ï

0

ñóò

= –( ü + 1) ï

ñ

0

ñóò ñ óò

= − .

= (arccos ï )÷ = (arccos ï )÷

Particular solution:

+ (arccos ï ) ÷ .

ñ

= −1  .

 0 = 

.

ôñ

+ ôƒõ –

− −1

ñ

2

Particular solution: 14.

+ ô (arccos ï ) ÷ .

2

+ (arccos ï )÷ ( ï

2

Particular solution: 13.

–ô

(arccos ï )÷

Particular solution: 12.

ñ

+

ñ

0 2

= − .

–

õ ï ú

 + . 0 = '

õ

2

+1

ñ

– 1).

(arccos ï )÷ .

(arccos ï )÷

ñ

+

õÿþ ï ú

–1

.

Page 100

1.2. RICCATI EQUATION ã ( Ë ) Ï

15.

ñóò

= (arccos ï )÷

ñ

Particular solution: 16.

ñóò

0

ï ñóò ï ñóò

ñ

2

üIñ

+

Q

2

2

ï 2ú

+

ˆõ 2 ï 2

−1

â 2 (Ë )Ï

2

+

â 1 (Ë )Ï

+

â 0 (Ë

–1

.

(arccos ï )÷ .

(arccos  ) 

M

 + L2S .

– ùñ .

leads to a separable equation: H 
=  

The substitution H =  

101

)

(arccos ï )÷ .

ïƒú

+ ö )(arccos ï ) ú

õ ƒï ÷ ñ

+

=

– õ )2 + ôþ

= '  tan Œ"/XZ 

ï 2÷ ñ

= (ô

ïƒú

– 

  + + . 0 = 

= (arccos ï )÷

Solution: 18.

–1

= ý± + .

= (arccos ï )÷ (ñ – ô

Particular solution: 17.

ïƒú

+ þ

2

ÒÍ

−1

(arccos  ) + ( !H

2

+ ˜H + ( ).

(arctan  ) + ( !H

2

+ ˜H + ( ).

1.2.7-3. Equations containing arctangent. 19.

ñóò

ñ

2

=

+ (arctan ï ) ÷

Particular solution: 20.

ñ óò

ñ

2

=

+

0

ñóò

= –( ü + 1) ï

ñ

0

ñóò

0

= (arctan ï ) ÷

ñ

ñóò

0

= (arctan ï ) ÷

ñ

2

ñ óò

= (arctan ï ) ÷

ñ

2

Particular solution: 25.

ñóò

=  −

ï ñóò

= (arctan ï ) ÷

ï ñóò

= (ô

−1

õ ïƒú

+

õÿþ ï ú

2

+

Q

2

+

–

üIñ

õ ƒï ÷ ñ

õ

– 1).

(arctan ï ) ÷ .

2

+

+

õÿþ ïƒú

–1

.

(arctan ï )÷ .

ôþ ïƒú

ˆõ 2 ï 2

−1

ñ

ˆõ 2 ï 2ú

– õ )2 +

  + + . 0 = 

ñ

ñ

(arctan ï ) ÷ –1

= '  tan Œ"/XZ 

ï 2÷ ñ

–

= ' + .

ïƒú

+1

.

ôƒõ

= − .

= (arctan ï ) ÷ (ñ – ô

Solution: 27.

= −1  .

–

0

Particular solution: 26.

+ (arctan ï ) ÷ .

ñ

 + . 0 = '

Particular solution: 24.

(arctan ï ) ÷ .

ô

+

= − .

+ ôñ +

2

Particular solution: 23.

2

+ (arctan ï ) ÷ ( ï

2

Particular solution: 22.

–ô

(arctan ï ) ÷

ï

Particular solution: 21.

ñ

–1

.

(arctan ï )÷ .

(arctan  ) 

+ ö )(arctan ï ) ú

M

 + L2S .

– ùñ .

leads to a separable equation: H 
=  

The substitution H =  

−1

1.2.7-4. Equations containing arccotangent. 28.

ñóò

=

ñ

2

+ (arccot ï )÷

Particular solution:

0

ñ

–ô

2

+

ô

(arccot ï ) ÷ .

= − .

Page 101

102 29.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

ñ

2

=

(arccot ï )÷

ï

+

Particular solution: 30.

ñóò

= –( ü + 1) ï

ñ

ñóò

+ (arccot ï )÷ ( ï

2

 0 = 

− −1

= (arccot ï )÷

ñ

ñóò

= (arccot ï )÷

ñ

2

ñ óò

= (arccot ï )÷

ñ

2

Particular solution: 34.

ñóò ï ñóò

ïƒú

õ

ñ

2

= ( ô ï 2÷ ñ 2 + õ ï ÷ The substitution H =  

ï ñ óò

ˆõ 2 ï 2

+

= '  tan Œ" X  

Q

2

– õ )2 + ôþ

üIñ

+

– 1).

(arccot ï )÷ .

2

– ˆõ

  + + . 0 = 

= (arccot ï )÷

Solution: 36.

–1

= ' + .

= (arccot ï )÷ (ñ – ô

ñ

(arccot ï )÷

õÿþ ï ú

+ 0

Particular solution: 35.

ïƒú

– õ

 + . 0 = '

Particular solution: 33.

= − .

0

+1

.

+ ôñ + ôƒõ –

2

Particular solution: 32.

= −1  .

0

Particular solution: 31.

+ (arccot ï )÷ .

ñ

−1

ñ

ï 2ú

+

õÿþ ïƒú

–1

.

ïƒú

(arccot ï )÷ . –1

.

(arccot ï )÷ .

(arccot  ) 

M

 + L2S .

+ ö )(arccot ï ) ú – ùñ . leads to a separable equation: H 
=  

ñ

−1

(arccot  ) + ( !H

2

+ ˜H + ( ).

1.2.8. Equations with Arbitrary Functions ]

Notation:  =  ( ) and = ( ) are arbitrary functions;  ,  , B , and Πare arbitrary parameters. 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).

1.

ñóò

ñ

2

=

+

!„ñ

–ô

2

– ô! .

Particular solution: 2.

ñóò

!„ñ

=

2

+ ôñ – ôƒõ – õ

Particular solution: 3.

ñóò

=

ñ

2

+

= .

0

ï !„ñ

!

.

0

=.

0

= −1  .

+! .

Particular solution:

2

4.

= !„ñ 2 – ô ïƒ÷ !„ñ + ôù ïƒ÷ –1 . Particular solution: 0 =   .

5.

ñóò

ñóò

=

!„ñ

2

+ ôù

ïƒ÷

–1

ñóò

= –(ù + 1) ïw÷

ñ

Particular solution:

ï ÷ !

2 2

   . 0 = 

Particular solution: 6.

–ô

2

+

ïƒ÷ +1 „! ñ

 0 = 

− −1

. –! . .

Page 102

1.2. RICCATI EQUATION ã ( Ë ) Ï

7.

ï ñóò

!„ñ

=

2

=

Solution: 8.

9.

ï ñóò

=

ô ï 2÷ !

+ ùñ +

ï 2÷ „! ñ



2

=

!„ñ

2

ïƒ÷ !

+ (ô

+ "Iñ – ô

2

ñóò

=

!„ñ

2

+ "Iñ + ôù

ñóò

=

!„ñ

2

ïƒ÷ "Iñ

–ô

ñóò

=

ô ó ñ

2

ñóò

=

!„ñ

2

ñóò

=

!„ñ

2

ñóò

=

!„ñ

2

ó !„ñ

– ô

=

Solution: 16.

ñ óò

=

!„ñ

2

Ė



ñóò

=

– ! ( ô

ó

 !„ñ

2

ôù ïƒ÷

19.

ñóò

=

!„ñ

2

–1



=

!„ñ

2

ó "Iñ ï  ó

Particular solution:

 ( )( H 2 + !H +  ).

–ô

2

! ï 2÷

ô 2 ï 2 ÷ ("

.

– ! ).

.



.

.

ô  ó

+ õ )ñ +



=  

0



+ "Iñ + ô

+ 2 ô

−1

=   .

ó

0 2

0

if  < 0.

.

+ .

õ – ó ! ó"

– ô



.

–ô

=   .

ó

+ô



=   . –ô

if  > 0,

leads to a separable equation: H 
=  ( )( H

+ ô

ñóò

ó

+ ( ô! – )ñ +

– ô

if  < 0.





ñóò

2

if  > 0,

.

ó

0

!„ñ

+

  !

2 2

Particular solution: =

103

)

.

£    tan O £ lX      + L M  P  |  |   tanh O − |  | X     + L ü ü P M

ó

â 0 (Ë

=   .

0

Particular solution: 20.

Œ





+

= −  − .

–ô

The substitution H = ž 18.

õ!

ïƒ÷ "

#!

+

ô 2 ó !

+

–ô

+ ô

Particular solution: 17.

–1

0

ó

+ ô

+

ïƒ÷

0

Particular solution: 15.

= .

ô ó !„ñ

+

Particular solution: 14.

â 1 (Ë )Ï

leads to a separable equation: H 
=  

   . 0 = 

Particular solution: 13.

– ù )ñ +

0

+

Particular solution: 12.

+

.

   . 0 = 

Particular solution: 11.

2

– ô" .

!

Particular solution: 10.

â 2 (Ë )Ï

=

£ Ñ  tan O £  X   −1   + L M P |  |   tanh O − |  | XÈ  −1   + L ü ü M P

The substitution H =  

ñóò

ÒÍ

2

=  

! 2 

2

 

ó

2 2

2

ó

  !

2 2

ó ("

2

+ !H +  ).

.

– ! ).

.

.

Page 103

104 21.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

=

!„ñ

2

+

Solution:

=

ï ñ 

+ ô!



ó

2

.

£     2J 2 tan O £ lX    2J 2  2 |  |   J 2 tanh O − |  | XZ  ü ü

22.

= !„ñ 2 – ô tanh2 ( ï )( ô! + ) + ô . Particular solution: 0 =  tanh( Œ  ).

23.

ñ óò

24. 25. 26.

27. 28.

30.

if  < 0.

ï )(ô !

= !„ñ 2 – ô coth2 ( Particular solution:

0

+ ) + ô . =  coth( Œ  ).

= !„ñ 2 – ô 2 ! + ô sinh( ï ) – ô 2 ! sinh2 ( Particular solution: 0 =  cosh( Œ  ).

ñóò

ï ñóò

= !„ñ 2 + ô – ô Particular solution:

2

ï

).

(ln ï )2 . 0 =  ln  .

!

ï ñ óò

= ! (ñ + ô ln ï )2 – ô .  ( ) 1 Solution: +X  =L . +  ln   M = !„ñ 2 – ô ï ln ï Particular solution:

ñ óò ñóò

= – ô ln ï

ñ

2

+

!„ñ 0

ô! (ï

= sin( ï )ñ 2 + Particular solution:

ñóò

0

+ ô ln ï + ô . =  ln  .

ln ï – ï )ñ – ! . 1 . =  ( ln  −  )

cos( ï )ñ – ! .

0 = 1 cos( Π ).

!

= !„ñ 2 – ô 2 ! + ô sin( ï ) + ô 2 ! sin2 ( Particular solution: 0 = −  cos( Œ  ).

ñóò

31.

= !„ñ 2 – ô 2 ! + ô cos( ï ) + ô 2 ! cos2 ( Particular solution: 0 =  sin( Œ  ).

32.

ñ óò

33.

if  > 0,

ñóò

Particular solution: 29.

 +L M 2J 2 P   +L M P

ñóò

= !„ñ 2 – ô tan2 ( Particular solution:

ñóò

= !„ñ 2 – ô cot2 ( Particular solution:

ï ï

). ).

ï )(ô ! 0

– ) + ô . =  tan( Œ  ).

0

– ) + ô . = −  cot( Œ  ).

ï )(ô !

1.2.8-2. Equations containing arbitrary functions and their derivatives. 34. 35.

ñóò

= ñ 2 – ! 2 + !„ó ò . Particular solution:

ñ óò

= !„ñ 2 – !"Iñ + " Particular solution:

0

óò

= .

. 0

= .

Page 104

1.2. RICCATI EQUATION ã ( Ë ) Ï

36.

ñóò

= – !„ó ò

ñ

2

!"Iñ

+

ñóò

= " ( ñ – ! )2 +

!„ó ò

ñóò

! óò ñ "

=

2

" óò !

–

! 2 ñ óò

!„óò ñ

–

2

ñóò

!„óò ñ

=

2

+

ô ó !„ñ

Particular solution: 41.

ñ óò

!„ñ

=

2

+"

óò ñ 

=

Solution:

ñóò

ñ

2

=

! óò òó !

–

0

= 1  .

0

= .

0

= −  .

â 1 (Ë )Ï

â 0 (Ë

+

105

)

= .

0

ô ó

+

.

0 = −1  .

+ ô!

2

$

.

£   — tan O £  X   — ü

42.

+

+ " (ñ – ! ) = 0.

Particular solution: 40.

2

.

Particular solution: 39.

â 2 (Ë )Ï

.

Particular solution: 38.

=

–" .

Particular solution: 37.

ÒÍ

|  |  — tanh O − |  | X

ü

M

 +L   —

if  > 0,

M

P

 +L P

if  < 0.

.

Particular solution:


= −   .

0

1.2.9. Some Transformations ]

Notation:  , , and % are arbitrary composite functions of their argument, which is written in parentheses following the function name (the argument is a function of  ). 1.

2.

3.

ñóò

ñ

2

=

ô 2 ! (ô ï

+

+ õ ).

The transformation ˆ =  +  , ‡ =   leads to the equation ‡ Š
= ‡

ñóò

ñ

2

=

+

ï

–4

!

(1 

ï

).

The transformation ˆ = 1  , ¨ = −

ô ï öï

1 !IO ï ( ö + ý )4 The transformation

ñóò

=

ñ

2

+

2

4.

ï 2 ñ óò

=

2

2

+  (ˆ ).

+õ . +ý P

 +  , (˜ + M

ï 4 ! ( ï )ñ

+  (ˆ ).

¨ Š
= ¨ −  leads to the equation 3

¨ = 1 [( (˜ + )2 + ( ( (˜ + )], ¢ M M u ¨
¨ leads to a simpler equation: Š = 2 + ¢ −2  (ˆ ). ˆ =

2

where

¢ = M

− ˜( ,

+ 1.

The substitution ‡ = −



1 2

−



1

leads to the equation ‡
 = ‡

2

+  ( ).

Page 105

106 5.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò

=

ï 2ñ

2

+

ï 2÷ ! ( ô ï ÷

+ õ ) + 14 (1 – ù

The transformation ˆ =   +  , ¨

¨2Š
= ¨

6.

7.

ñóò

2

2

B



). 1−



+

1 − B −  leads to a simpler equation: 2 B

+ " ( ï )ñ + % ( ï ).

The substitution = −1 ¨

¨ 
= % ( )¨ leads to an equation of the same form: p

ñóò

1 4

ñ

2

=

ó

 2 ! ( 

+

ó

)–



2

ñóò

ñ

=

2

–

2

ó

1

Œ

 −



ô  ó + !IO ó ( ö  + ý )4 ö  ó  2

 2 2 2  2 ¨ = ( (  +  M ) + (   − M , ¢eŒ   2¢   leads to a simpler equation: ¨uŠ
= ¨ 2 + ( ¢eŒ )−2  (ˆ ).

10.

11.

 coth( ï )  . The transformation ˆ = coth( Œ  ), ¨ = − Œ equation: ¨3Š
= ¨ 2 + Œ −2  (ˆ ).

ñ

2

=

–

2

ñ

2

=

ï 2 ñ óò

=

–

ï 2ñ

2

2

ñ óò

ñ

2

=

+

2

+ ! ( ô ln ï + õ ) + 41 .

ï )!

+ sin– 4 (

(cot(

ï

The transformation ˆ = cot( Œ  ), ¨

¨2Š
= ¨

13.

ñ óò

ñ

=

2 2

+Œ

−2

 (ˆ ).

+

2

+ cos– 4 (

14.

−1

sinh2 ( Œ  ) −

ñóò

=

ñ

2

+Œ

−2

 (ˆ ).

2

+

2

+ sin– 4 (

ï

+ õ ) !IO

+

¢ = M

− ˜( ,

1 2

sinh(2 Π ) leads to a simpler

1 2

sinh(2 Π ) leads to a simpler



1 leads to a simpler equation: ¨uŠ
= ¨ 2

2

+

−2

 (ˆ ).

)). = − sin2 ( Œ  )

ï )u !  tan( ï

The transformation ˆ = tan( Œ  ), ¨

¨2Š
= ¨

where

ï )!

+ cosh– 4 (

1 The transformation ˆ =  ln  +  , ¨ =  12.

 (ˆ ).

ï )!

+ sinh– 4 (

 tanh( ï )  . The transformation ˆ = tanh( Œ  ), ¨ = Œ −1 cosh2 ( Œ  ) + equation: ¨ Š
= ¨ 2 + Œ −2  (ˆ ).

ñóò

−2

+õ . +ý P

4 The transformation

ñóò

− ( )¨ +  ( ).

 1 −  −  leads to a simpler equation: ‡ Š
= ‡ 2 + Œ 2

    +  ˆ = ,  (  + M

9.

2

.

The transformation ˆ = ž , ‡ = 8.

1

=

+ ( B )−2  (ˆ ).

= ! ( ï )ñ

2

Q Œ

+ cot( Π )S leads to a simpler equation:

) . = cos2 ( Π )

sin( sin(

ï ï

Q Œ

− tan( Œ  )S leads to a simpler equation:

+ô ) . +õ )P

sin( Œ  +  ) ¨ sin2 ( Œ  +  ) , = + cot( Œ  +  )S leads to a simpler sin( Œ  +  ) sin(  −  ) Q Œ 3 ¨
¨ 2 −2 + [ Œ sin(  −  )]  (ˆ ). equation: Š = The transformation ˆ =

Page 106

107

1.3. ABEL EQUATIONS OF THE SECOND KIND

1.3. Abel Equations of the Second Kind 1.3.1. Equations of the Form â!â$ä ã – â = å ( æ ) 1.3.1-1. Preliminary remarks. Classification tables. For the sake of convenience, listed in Tables 5–8 are all the Abel equations discussed in Section 1.3. Tables 5–7 classify Abel equations in which the functions  are of the same form; Table 8 gives other Abel equations. In Table 5, equations are arranged in accordance with the growth of the parameter . In Table 6, equations are arranged in accordance with the growth of the parameter  . ¡ In Table 7, equations are arranged in accordance with the growth of the parameter ° . The rightmost column of the tables indicates the equation numbers where the corresponding solutions are written out. TABLE 5 Solvable Abel equations of the form !
 − = °˜ + #* + , # is an arbitrary parameter

° ¡

arbitrary −7 −4 −5 2 −2 −2 −5 3 −5 3 −5 3 −7 5

−

Equation

2( + 1) ¡ ( + 3)2 ¡

15 4 6 12 0 2 −3 16 −9 100 63 4 −5 36

¡

°

Equation

1.3.1.10

−1

0

1.3.1.16

1.3.1.56 1.3.1.54 1.3.1.47 1.3.1.33 1.3.1.19 1.3.1.30 1.3.1.23 1.3.1.48 1.3.1.27

−1 2 −1 2 −1 2 −1 2 0 0 1 2 2 2

−2 9 −4 25 0 20 arbitrary 0 −12 49 −6 25 6 25

1.3.1.26 1.3.1.22 1.3.1.32 1.3.1.55 1.3.1.2 1.3.1.1 1.3.1.53 1.3.1.45 1.3.1.46

Solvable Abel equations of the form !
 −



 °

−1

−3

−1

−3

−1 −3 5 −5 11 −1 3 −1 3 −1 3 −1 3 −1 5 0 2

−3 −7 5 −13 11 −5 3 −5 3 −5 3 −5 3 −4 5 −1 2 3

arbitrary 2 +1 ¡ 2 4 ¡0 −5 36 −33 196 −3 16 −3 16 −3 16 15 4 −10 49 −2 9 4 9

TABLE 6 = °˜ + ƒh#*K + ý…#

ƒ

2

'& , # is an arbitrary parameter ý

Equation

1

−1

1.3.1.5

1

−1

1.3.1.13

1 arbitrary 286 # 3 arbitrary 3 5 6 13 # 5 arbitrary 2

−1 arbitrary −770 # 9 arbitrary −12 −12 −3 −7 # 20 arbitrary 2

1.3.1.7 1.3.1.62 1.3.1.69 1.3.1.61 1.3.1.40 1.3.1.15 1.3.1.60 1.3.1.68 1.3.1.3 1.3.1.14

Page 107

108

FIRST-ORDER DIFFERENTIAL EQUATIONS TABLE 7 Solvable Abel equations of the form ! 
− = °˜ + ./# ( ƒ… 1 J # is an arbitrary parameter

°

.

arbitrary ≠ 0 2( − 1) ¡ ( − 3)2 ¡

−1 4 −30 121 −12 49 −12 49 −12 49 −12 49 −12 49 −12 49 −12 49 −12 49 −12 49 −6 25 −6 25 −28 121 −2 9 −2 9 −2 9 −10 49 −4 25 −4 25 0 0 0 0 0 0 0 2 2 20

arbitrary 2 ( − 3)2 ¡

1 4 3 242 arbitrary 1 98 6 49 2 49 4 49 1 49 6 49 2 49 1 2 25 6 25 2 121 arbitrary arbitrary 6 2 49 arbitrary 1 50 arbitrary 1 B +2 B +2 1 2 arbitrary 2 2 arbitrary

ƒ 0

¡

( + 3)

¡

1 21 arbitrary 25 1 5 −10 5 −3 1 3 49 + 3 $ 2 2 5 0 0 0 4 0 7 0 1 1 1 −1 1 0 −10 10 0

4

¡

2

+ ý…# + =/#

J ),

2 −1 2



ý

=

Equation

arbitrary

0

1.3.1.2

3 ( + 3)

1.3.1.12

3 6 0 10 5 15 10 65 12 55 15 196 + 75 $ 16 6 4 15 arbitrary arbitrary 2 12 arbitrary 6 arbitrary arbitrary (B + 1)(B + 3) 2B + 3 0 3 0 30 30 arbitrary

1.3.1.17 1.3.1.29 1.3.1.53 1.3.1.25 1.3.1.38 1.3.1.24 1.3.1.31 1.3.1.52 1.3.1.28 1.3.1.58 1.3.1.64 1.3.1.20 1.3.1.39 1.3.1.51 1.3.1.3 1.3.1.26 1.3.1.11 1.3.1.57 1.3.1.22 1.3.1.59 1.3.1.32 1.3.1.36 1.3.1.34 1.3.1.35 1.3.1.37 1.3.1.4 1.3.1.1 1.3.1.50 1.3.1.49 1.3.1.55

2

+3 +9

¡

5 35 0 41 8 34 27 262 23 166 12 49 − 15 $ 2 19 7 106 arbitrary 0 1 61 0 49 0 2 2(B + 2) 2(B + 2) 2 4 arbitrary 19 31 0

¡

¡

Given below in this section are all solvable Abel equations known so far. The equations are arranged into groups, in which all solutions are expressed in terms of the same functions. Notation is given before each group. In most cases the solutions are presented in parametric form:

 =  1 (U , L ),

=

2(

U , L ),

where U is the parameter and L is an arbitrary constant.

Page 108

109

1.3. ABEL EQUATIONS OF THE SECOND KIND

TABLE 8 Other solvable Abel equations of the form ! 
−

=  ( )

Function  ( )

#* 

− ,$I  + ,$

−1

2

#*

9 625

−

Equation 1.3.1.6 (particular solution)

 

2 2 −1

−1

#

1.3.1.44

27  − #* J + 34 # 2  −1 J 3 − 625 # 4  −5 J 3 4 −5 J 3 6 − 25  + 75 #* 1 J 3 + 313 # 2  −1 J 3 − 100  3 # 6 − 25  +  1 J 3 +  + (˜ −1 J 3 +  −2 J 3 M (coefficients  ,  , ( , and are related by an equality) M − 21  + 7 # 2  123 −1 J 7 + 280 #* −5 J 7 − 400 # 2  −9 J 7  3 4

100

1 3

3 2

1.3.1.66 1.3.1.67 1.3.1.65 1.3.1.70

9

£ #*

,

£ 

1.3.1.63

+ $I + L

2

#

2

1.3.1.18

+ 4#

3 9  2 − 6 2  + £ 32 64  2 +  2

1.3.1.43

3 6 2 + 5  2  + £ 2 2 8 16  + 

1.3.1.21

6 2 + 9 # 3  + £ 2 8 16  + #

1.3.1.41

9 30 2 + 33 #  + £ 2 32 64  + #

1.3.1.42

−

# + $ exp(−2 # ) # [exp(2 # ) − 1]

 2 Π 2  2 Π 2

 

−  ( Œ + 1)   + Œ   





+  

1.3.1.8 1.3.1.9 +

1.3.1.73 (particular solution)



1.3.1.74 (particular solution) 1.3.1.75 (particular solution)

2  2 Πsin(2 Π ) + 2  sin( Π ) 1.3.1-2. Solvable equations and their solutions. 1. 2.

ññ óò

–ñ =
. Solution:  =

− # ln | + # | + L .

–ñ =
ï +( ,
≠ 0. Solution in parametric form:

ññóò

 = L exp O − X U

2

U

M

U

− U − #P

−

$ #

,

= L3U exp O − X

U

2

U

M

U

− U − #P

.

Page 109

110 3.

FIRST-ORDER DIFFERENTIAL EQUATIONS – ñ = – 92 ï +
+ ( ï –1 ) 2 . 1 b . Solution in parametric form with # > 0:

ññóò

 = 2¬

2 (2 , − 1) L3U  − ( , − 2)U − , − 1 ® , L3U  + U + 1

where # = 23  ( ,

+1 + , 2 L3U  + U ,  L3U + U + 1

= 23  3 J 2 (2 , − 1)( , − 2)( , + 1).

− , + 1), $

2

( , − 1)2 L3U 

= −6 

2 b . Solution in parametric form with # < 0:

 = ˆ [2 Œ  − + * − ( L 1 Œ − 3 L 2   ) sin  ÑU − (3 L 1   + L 2 Œ ) cos  ÑU ]2 , = 6ˆ , ( L 12 + L 22 )  2 − [ L 1 ( Œ 2 −   2 ) − 2 L 2  ·Œ ]  − + * sin  ÑU − [2 L 1  ·Œ + L 2 ( Œ 2 −   2 )]  − +* cos  ÑU/. , where ˆ =  (  − + * + L

sin  ÑU + L

1

2

cos  ÑU )−2 , # = 19  (3 

− Œ 2 ), $

2

=

2 27

Œ (9 

2

+ 5 Π2 ).

3 b . For the case # = 0, see equation 1.3.1.26. 4.

– ñ = 2
(ï 1) 2 + 4
+ 3
Solution in parametric form:

ññ óò

2

ï –1 ) 2 ).

 = 14  (3 A 2U01 )2 , where

0

5.

=

+

0




X


 X

=X

−

M

 = O ú






Here,

˜

where ¢

=

= 4 # + 1.

P 2





(U

2


























# = − 12  1 J 2 ,

+ U ),

= arctan U − L ,

²

+

= £ 1 + U 2,

+

=£ U

−

= £ 1 − U 2.

− 1,

2

.

,

= (U + 1) O ú

2 + U − # ) exp ” £ −¢ + U − # ) exp ” −

˜

J

−1 2

P

− $ÊO ú

2U + 1 arctan £ −¢ •

2 2U + 1 •

2 13

2U + 1 − £ ¢ (U + U − # ) ” 2U + 1 + £ ¢ • 2 2U + 1 L + 2 $ X exp ” £ arctan £ U −¢ −¢ • M 2 L + 2 $wX exp ” − U 2U + 1 • M 2




ú

J

(U






–3

−1 2





=

˜





2

1 01 2

U −1 U 1 −L , ² = ln M U 2 − 1 2 ûû U + 1 ûû û û 1 U 1+U = ln û −L , ² M 1−U 2 2 û 1 − U ûû û û

– ñ =
ï + ( ï –1 – ( 2 ï Solution in parametric form:

ññ óò

U

1+U

= A 01 ( ²

2U + 1 − £ ¢ L + 2 $wX¯” 2U + 1 + £ ¢ •

2 13

M

U

˜

J

1 2

P

.

if ¢

< 0,

if ¢

= 0,

if ¢

> 0,

if ¢

< 0,

if ¢

= 0,

if ¢

> 0,

Page 110

111

1.3. ABEL EQUATIONS OF THE SECOND KIND

6.

ññ óò

–ñ =


ï 

–1

Particular solution: 7.

+

ü/( 2 ï 2 –1 .

# =  − $I  −

0

.

,$

– ñ =
ï –1 –
2 ï –3 . Solution in parametric form:

ññóò

 = U

−1

(U − ln |1 + U | − L )1 J 2 ,

where # =  2 2. 8.

ï 

– ü/(

=

Q

1+U

1 (U − ln |1 + U | − L )1 J 2 − U (U − ln |1 + U | − L )−1 J 2 S , 2

U

ó

– ñ =
+ (4 –2 )65 . Solution in parametric form:

ññóò

£ U 2 + #2$ û # ln U + £ U 2 + #2$ + L û û û ûó û û

 = # ln û 9.

U

2

U

û û

(arctan U − L ) û ,

# V U + (U U

=

û

+1

1

) - −2 U − L ,

M

² + = £ 1 AIU +

2(þ

+ 1) ï +
(þ + 3)2 Solution in parametric form:

ññóò

–ñ =–

¡

−1 + +3P

¡

– ñ = – 92 ï + 6
2 (1 + 2
Solution in parametric form:

ññóò

2

 =# where «

=«

²

−1 2, 3 2 ,

J

J

ïƒú

2

−# .

− 1)(arctan U − L )W .

−4

«

−2

² =²

(²

2 −1

+1



 =



(

¡

− 3)2

−2

U

where # = −

2

= !« + +




−3

M

U −L ,

2 −1

O² + « + +

2 U , −1 P

¡

= −12 #

2

²

−4

«

−2

(²

2

« − 2U ),

−1 2 .

J

=

2

) J

+1 −1 2

> 0.

«ÈA 6U 1 J 2 )2 ,

[( −3) ² + « + +3U ]2 ,

¡

= X (1 AIU +

+ .

+3) ï

 1J ¡

1−

,

(þ

=

,0

 + =² + « + −U .

+1 ,

ï –1 ) 2 ),

2(þ – 1) ï 2
Vþ + (þ – 3)2 (þ – 3)2 Solution in parametric form:

ññóò – ñ

« + =« +

.

+3 U « + + −1

+1 where # = A2¡ O ¡ ± 2

12.

û

û

 = ¡

11.

û

In the solutions of equations 10–15, the following notation is used:

« + , - = X (1 AIU + 10.

+1

# ln U + £ U 2 + #2$ + L û û û £ U 2 + #2$ û

=U

û

– ñ =
(  2 )65 – 1). Solution in parametric form:

ññóò

 = # ln û ]

û,

)

1 2

¡



+(4þ

−3

U

−2

2

+3þ

+9)
+3þ

« + [A ( −1)U + ¡

+1

(þ

+3)


ï )

2 –1 2

W.

« + −2 « + +2U ² + ],

.

Page 111

112 13.

FIRST-ORDER DIFFERENTIAL EQUATIONS

 = 14.

2þ

+1ï +
ï –1 –
4þ 2 Solution in parametric form:

ññóò

–ñ =

«

J

1 2

U 1 J 2 ² 2+

=

,

 = 15.

ï

–3

.

U − V 1 T (2 + 1)U + +1 W ² 2+ « ¡ , 2  U 1 J 2 ² 2+ « 1 J 2 ¡

– ñ = 49 ï + 2
ï 2 + 2
Solution in parametric form:

ññóò

2

1 U 3#

ï

3

−1

 3,

2

1 U 9#

=

ï )

2 –5 3

−2

= −2 # , «

=« +

¡

, 3 2.

J

« 3 (U ²

3

−«

AIU 4 « 3 ).

3

.

 = U 1 J 2 « −3 J 2  3 J 2 , = 14 U 1 J 2 « 1 4J 3 where # = 24  , « = « −5 J 3 ,  =  −5 J 3 . ]

2

.

3 ï – ñ = – 16 + 5
ï –1 ) 3 – 12
Solution in parametric form:

ññóò



where

J 

J (

−3 2

−1 2

2

J «

−2 3

− 2U  − U

2

),

In the solutions of equations 16–18, the following notation is used:

16.

 = X exp(T"U 2 ) U − L , M

– ñ =
ï –1 . Solution in parametric form:

ññ óò

 = 

17.

−1

exp(T"U 2 ),

– ñ = – 41 ï + 14
( ï 1 ) 2 + 5
Solution in parametric form:

ññóò

1 16

2  V 3 A 8U exp(AU 2 )W ,

–ñ =A

2ô

7 ï 2 A

V exp(T"U 2 ) A 2UW ,

−1

= 

ññ óò

 = 18.

= 2U X exp(T U 2 ) U − L2SrA exp(T U 2 ). Q M

+ 3


2

ï –1 ) 2 ).

=  exp(AU 2 ) V (2U

2

# = T 2 2.

A 1)  exp(A"U 2 ) A*U W ,

where # = 14 £  .

2

. 8ô 2 Solution in parametric form:

 = A  (  )−1 ( 2 T 2  2 ), ]

where

=  (  )−1 [exp(T U 2 ) − 2  2 ].

In the solutions of equations 19–21, the following notation is used:

« = £ U (U + 1) − ln L  £ U + £ U + 1  , û û û û 19.

U +1 , U

² =

– ñ = 2 ï +
ï –2 . Solution in parametric form:

 =1−

ññóò

 = 13 ! «

20.

J U ,

−2 3

6 ï 2 – ñ = – 25 + 25
(2ï 1 ) Solution in parametric form:

ññ óò

 = U

−2

(5 ²u« − 3U )2 ,

2

J  U − ²u«  ,

= !«

−2 3 2 3

+ 19


+ 6


= 5 U

−3

2

U +1 ln L  £ U + £ U + 1  . û û U û û

where

ï –1 ) 2 ).

« [(2U + 3)« − 2U 2 ² ],

3 # = − 243 2  .

where

# = −£  .

Page 112

113

1.3. ABEL EQUATIONS OF THE SECOND KIND

21.

3ï 37 ï 2 + +ô 8 8 Solution in parametric form:

ññ óò

–ñ =

2

ô

7 16 ï

–

2 2

ô

+

2

.

 « 2 − 2U 2   = £ , 2 2 U « £  ]



=

2

4U 

4£ 2

2

−«

U « £ 

.

In the solutions of equations 22–25, the following notation is used:

.

22.

2

2

= A (U

4 ï – ñ = – 25 +
ï –1 ) 2 . Solution in parametric form:

. 2.

.

.

J

J

3 2 −9 8 , 3 4

.

−4 2 (14

.

U

−9

3

12 ï 1 – ñ = – 49 + 98
(25ï Solution in parametric form:

. .

.

−4 3 (21 2 4

4

= A (U

4

2

− 6U

+ 4 L3U − 3).

.

J

−4 3 ( 22 3

.

−U

3 ),

# =A

where

4 5

 £ 5 .

− 16

.

.

J

+ 34


)

2 2 3) ,

2 4 ),

2 –1 2

.

.

. .

−

ï )

+ 15


= 28  1 2

.

J

−1 2 −9 8 ( 32 3 4

.

−4 2 3 (4 3 2

+ 41
= 21 

U

+ 10


2

.

.

# = A 9 2 (10  )2 J 3.

where

).

.

− 3U

ï –1 ) 2 ).

. .

−4 2 4 (9 2 4 3

. T

2 2

2 4

T

. .

where

# = −3 £  .

. .

where

# = −8 £  .

2 3 ),

−8

2 2 3 ),

In the solutions of equations 26–29, the following notation is used: 1 3

8

= exp( £ 3 U ) + L sin U ,

= 2 exp( £ 3 U ) − L sin U + £ 3 L cos U ,

2

8

= 2 exp( £ 3 U ) − L sin U − £ 3 L cos U ,

– ñ = – 92 ï +
ï –1 ) 2 . Solution in parametric form:

=4

4

8 8

1 3

−

8

2 2.

ññóò

 = 3

8

8

−2 2 1 2,

8

= 2

5 ï – ñ = – 36 +
ï –7 ) 5 . Solution in parametric form:

8

−2 2 1 ( 2

−2

8 8

1 3 ),

where

# = 16(3  )3 J 2.

ññóò

 = 48  28.

2

2 2 2) ,

ññ óò

8

27.

.

.

= 9

12 ï 2 – ñ = – 49 + 49
(5ï 1 ) Solution in parametric form:

8

26.

.

= 4

ññóò

 = ]

J

−4 3 , 3

2

9 ï – ñ = – 100 +
ï –5 ) 3 . Solution in parametric form:

 =

25.

.

− 3U + L ,

ññóò

 = 10  24.

3

=U

3

ññóò

 = 5 23.

.

− 1),

8 5J 2 8 1

J

−5 4 , 4

12 ï 6 – ñ = – 49 + 49
(–3ï Solution in parametric form:

ññóò

 =

8

8 8

−4 2 (7 1 3

−2

8

2 2 2) ,

= 5

)

1 2

8

J 8

8

J

−1 2 −5 4 (8 13 1 4

+ 23


= −7 

+ 12


8 8

2

8 8

−

8 8

2 4 ),

where

# = (48 )2 J 5  2 .

ï –1 ) 2 ).

−4 2 1 2 (4 1 2

−4

8 8

2 1 3

+

8 28 2

3 ),

where

# = £  2.

Page 113

114 29.

FIRST-ORDER DIFFERENTIAL EQUATIONS 30 ï 3 – ñ = – 121 + 242
(21ï Solution in parametric form:

8

 = ]

)

ññóò

8 8

−6 1 (11 2 4

− 64

8

+ 35


1 2

3 2 1) ,

= −11 

ï )

8

+ 6


8

2 –1 2

8

−6 2 4( 4 1

−5

).

8 28

+ 32

4

2

8 38

where # = −32 £  .

2 ),

1

In the solutions of equations 30 and 31, the following notation is used: { = tanh(U + L ) + tan U , { = tanh(U + L ) − tan U , 91 9 2 9 = cosh U − sin( U + L ), 1 2 = sinh U + cos(U + L ), 3 = sinh U − cos( U + L ).

30.

3 ï – ñ = – 16 +
ï –5 ) 3 . 1 b . Solution in parametric form with # < 0:

ññóò

 = 8

{

J

−3 2 , 1

{

= 3

{ {

J

−3 2 (2 − 1

1 2 ),

2 b . Solution in parametric form with # > 0: 31.

 = 4

9 3 J 2 9 −3 J 2 , 1

9 −1 J 2 9 −3 J 2 9 (

= 3

2 4 49

1

2 1

2

12 ï – ñ = – 49 +
(–10 ï 1 ) 2 + 27
+ 10
1 b . Solution in parametric form with # < 0:

ññóò

 =  (10 − 7

{ {

1 2)

2

= 7

,

{ ({ 1

3 1

+3

−

9 9

2 3 ),

ï )

2 –1 2

{ {

2 1 2

2 b . Solution in parametric form with # > 0:

 = ]

9

9 9

−4 1 (7 2 3

9

− 5 12 ),

= −7 

9 −4 9 9 1

3 2( 2

−3

# = −12 8 J 3 .

where

# = 3 2 (4  )2 J 3 .

where

).

{

− 4 2 ),

9 9

2 2 3

+2

where

9 29

1 3 ),

# = −2 £  .

where

# =£  .

In the solutions of equations 32–43, the following notation is used: : L Á (U ) + L 2 n Á (U ) for the upper sign, Á = ™ 1ø L 1 ®Á (U ) + L 2 ; Á (U ) for the lower sign,

: : : : : : : : žÁ = U ( Á )
+ @ Á , = 1 J 3 , ’ 1 = U
+ 13 , ’ 2 = ’ 12 ACU 2 2 , ’ 3 = A 23 U 2 3 − 2 ’ 1 ’ 2 , * * where ø Á (U ) and n Á (U ) are the Bessel functions, and ®Á (U ) and ; Á (U ) are the modified Bessel

functions.

: : The solutions of equations 32–43 contain only the ratio
, where the prime :* denotes differentiation with respect to U . Therefore, for symmetry, function is defined in terms of two “arbitrary” constants L 1 and L 2 (instead, we can set, for instance, L 1 = 1 and L 2 = L ). x3y'z|{}~/€

32.

33.

34.

– ñ =
ï –1 ) 2 . Solution in parametric form:

ññóò

J :

−4 3

−2

– ñ =
ï –2 . Solution in parametric form:

ññóò

:  = 2 U 4 J 3

2

−1 2 ,

’

= U

−4 3

J :

−2

’ 2,

= A 3 U

−2 3

J :

−1

’

’ 12 ,

– ñ =
(ù + 2)[ ï 1 ) 2 + 2(ù + 2)
Solution in parametric form:

ññóò

 =

35.

 = U

:

:

Á−2 [ žÁ − ( @ + 1) Á ]2 ,

=

:

– ñ =
(ù + 2)[ ï 1 ) 2 + 2(ù + 2)
Solution in parametric form:

 =  Á −2 [U

2

:

2

ÁrA (2 − @ ) žÁ ] ,

where # = T/@ £  , @ =

1

B +2

’ 3,

+ (ù + 1)(ù + 3)


Á−2(  Á 2 −2 @

ññóò

−1 2

:

ÁžÁ!AhU

+ (2ù + 3)


# =T

where

2

:

ï )

2 –1 2

].

where # = @ £  , @ =

Á2 ),

ï )

# = −36 3 .

where

2 –1 2

 J

1 3 2 . 3

1

B +2

.

].

:

= A U 2  Á −2 [  Á 2 + 2(1 − @ ) ÁžžÁÑACU

2

:

Á2 ],

.

Page 114

115

1.3. ABEL EQUATIONS OF THE SECOND KIND

36.

– ñ =
ï 1) 2 + 2
2 + ( Solution in parametric form:

where $ 37.

ï –1 ) 2 .

ññóò

 =#

2

= (1 − @ 2 ) #

3

:

:
Á

Á−2 (U

:

− Á )2 ,

2

=#

:

:


Á−2 [U 2 ( Á )2 − ( @

:

2

TCU 2 ) Á2 ],

and the prime denotes differentiation with respect to U .

– ñ = 2
2 –
ï 1) 2. Solution in parametric form:

ññ óò

: :  =  ( 0
)−2 (U

0

: A 2 0
)2 ,

: : : = A U ( 0
)−2 [U ( 0
)2 + 2

0

:
: 0 AIU

2 0 ],

where # = £  and the prime denotes differentiation with respect to U . 38.

12 ï 6 – ñ = – 49 + 49
(ï 1) 2 + 8
Solution in parametric form:

ññóò

 = 3 ’ where # = 2 £ 3  ; 39.

and ’

−4

:

−6

(’

1

’

2

− 2’

2

− 7U

:

2 2

3)

+ 7


2

2

,

3 ï – ñ = – 16 + 3
ï –1 ) 3 – 12
Solution in parametric form:

+ 4


−4

= 5 U

ï )

2 –5 3

U

:

2

:

−

’

1

− 3’

2 1 ),

ï –1 ) 2 ).

:

−6

2 2

’ 2(’

−’

1

’ 3 ),

where

# = − £  2.

2

:

2 3 2

J

,

1 J and  3 J 2 are expressed in term of modified Bessel functions; # = 8  4 J 3 .

3 2

2

−1

:

J ’ J ’

2

.

−1 2 2 (2

U

:

−1 2 2 (3 2

’

3

’

1

T 3 ’ 22 ), 2 1

− 12 ’

’

2

A 4U

2

:

3

’ 1 ),

= 32  2 .

9 ï 15 7 ï 2 + T]õ 32 32 Solution in parametric form:

:

2

T

3õ

−1

2

7 64 ï 2 ] T õ

J ’ : = A 14   U −1 −3 J 2 ’

 = − 21 U 2

õ

−1 2 1 −1 2 1

−3 2

–ñ =

where 

−4 2 1 (3

’

2 3  3 J 2 − 2 3 J 2  3J 2 − U = : 4 U 3 J 2 33J J 22  31J J 22

J ’ : = T 18   U −1 −3 J 2 ’

ññ óò

2

.

–ñ =

where 

:

:

 33J J 22  = , : U 3 J 2 33J J 22

 = − 41 U

42.

2

3ï 37 ï 2 3õ 2 + A õ 2A 7 ï 2 8 8 16 A Solution in parametric form:

ññóò

2

= 28 U

),

ññóò

:

ï –1 ) 2 ).

2

are expressed in term of modified Bessel functions.

1

6 ï 6 – ñ = – 25 + 25
(2ï 1 ) Solution in parametric form:

where 41.

:

’

ññóò

 = U

40.

−4 2 1 (5 1

+ 5


−3 2

J ’ J ’

−3 2 2 −3 2 2

2

J

.

:

−1 2 (2 2 3 3 3 −1 2 2 (3 23 3

J

U

’

’

TCU

:

A 3 ’ 23 ), 3

’

3

− 3’

2 3 ),

= 6 2.

Page 115

116 43.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ññ óò

3 ï 3 7 – 32 32

–ñ =–

ï

+ô

2

2

15 ô

7 64 ï

+

2

+ô

2

Solution in parametric form:

J ’

−3 2 2

 = 12  ’ ]

−1 2 3 ( 3

’

−’

3 2 ),

1 24

=

.

2

J ’

−3 2 2

 ’

−1 2 3 (3 3

’

3 2

− 12 ’

A 4U

2

:

3

’ 3 ).

In the solutions of equations 44–52, the following notation is used:

<

«

1

=U

«

3

=

3

A (4 < ü

A (4 < ü

3

− 1) + 3U 2
− 1) A 2U'< 2 ,

3

«

« =U

4

2

ü

= U 2
A (4 <

3

− 1) + 2 < .

= < (U ) is given implicitly as follows: U = X

<

− L 2 . The upper A (4 < 3 − 1) ü sign in this formula corresponds to the classical elliptic Weierstrass function < = < (U + L 2 , 0, 1).

Here, the function

44.

ññóò

–ñ =


ï

2

9 625

–




–1

M

.

Solution in parametric form:

 = 5  U 2 I < T 45.

ññ óò

6 ï – ñ = – 25 +


ï

2

1 2

,

= U 2 «

4,

where

6 −1 . 125

# =A



.

Solution in parametric form:

 = 5 U 2 < , 46.

ññóò

–ñ =

ï

6 25


ï

+

2

= U 2 «

where

# =A

where

# =A

4,

6 −1 . 125



.

Solution in parametric form: = U 2 «

 = 5 !« 2 , 47.

ññóò

– ñ = 12 ï +


ï

)

–5 2

4,

6 −1 . 125



.

Solution in parametric form:

48.

ññóò

–ñ =

J

J «

 = -< 63 4

ï


ï

+

)

–5 3

J

J «

= -<

−4 7 , 3

−6 7

< «

−4 7 (14 2 3

−6 7

4

− 3),

where

# = T 147  7 J 2 .

.

Solution in parametric form:

 = 2 !« 49.

ññóò

J «

3 2 3

J

−9 8 , 4

= !«

– ñ = 2 ï + 2
(10 ï

)

1 2

J «

−1 2 3

+ 31


J

−9 8 (9 4

+ 30


«

2 3

T 16
ï )

2 –1 2

2 4 ),

where

2 2J 3 # = − 128 . 3  (2  )

).

Solution in parametric form:

 = -<

−2

where # = £  .

VU ü

A (4 <

3

2

− 1) − 3 <"W ,

= −2 U'<

−2

V< ü

A (4 <

3

− 1) A 2U'<

3

AIU!W ,

Page 116

117

1.3. ABEL EQUATIONS OF THE SECOND KIND

50.

– ñ = 2 ï + 2
(–10 ï 1 ) 2 + 19
Solution in parametric form:

ññóò

−2 2 (

 = !«

51.

«

1

− 6«

U

=X


2 4

M

+ 15


−«

2 1

ï )

+ 7«

2 –1 2

−4 3

2 4 (6

«


« 2 ),

1

# = −£  .

where

).

3 (7

= 56 !«

U

2

),

# = A 2£  .

where

+«

4 ),

# = T 3£  .

where

(incomplete elliptic integral of the second kind),

A (4U 3 − 1) A (4U 3 − 1), ® 1 = U (2 ®lTCU ü

−1

® 2=U

² + L ),

−1

( ²$®

® 3 = 4UÈ® 12 T ® 22 .

− 1),

1

12 ï – ñ = – 49 +
ï 1) 2. Solution in parametric form:

ññ óò

= −2  ( ® + L )−4 ( ®² + Lu² − 2U 2 ),

where

12 49

# =A

– ñ = 6 ï +
ï –4 . Solution in parametric form:

£ 7 .

ññóò

J

J

®

−3 5 −2 5 , 1

= U

– ñ = 20 ï +
ï –1 ) 2 . Solution in parametric form:

J

J

®

−3 5 −2 5 (5 1

²$® 1 − 2),

where

# = T 150  5 .

ññ óò

 = ³®

J

®

−4 3 2 2, 1

ï +
ï –7 . – ñ = 15 4 Solution in parametric form:

= −4 ³®

J

®

−4 3 2 ( 2 1

T 9UÈ® 12 ),

where

# = A 108 3 J 2.

ññóò

 = ³® 11 J

57.

2 2 3) ,

T 15 «

ü

 = U

56.

+ 106


3 2


 = 7 U 2 ( ® + L )−4 ,

55.

A 6«

ï –1 ) 2 ).

In the solutions of equations 53–60, the following notation is used:

² =

54.

−2 2 (

= A 44 -< 2 «

12 ï – ñ = – 49 + Solution in parametric form:

®

53.

)

1 2

− 5)2 ,

4

ññóò

−4 3 (28

2

= −2 !«

,

28 ï 2 – ñ = – 121 + 121
(5ï Solution in parametric form:

 = !« ]

2

ññóò

 =  (22 < 2 «

52.

2)

+ 30


®

J

= 12 ³  ®

J

J

® ® − 3® 12 ), ñ ñóò – ñ = – 4910 ï + 492
(4ï 1 ) 2 + 61
+ 12
2 ï –1 ) 2 ). 2 −3 8 , 3

®

−3 2 −3 8 ( 2 3 1 3

# =A

where

3 8 . 4



Solution in parametric form:

 =  (7 ²$® 1 − 3)2 ,

58.

= 14 ³® 1 [A (10U

12 ï 2 – ñ = – 49 + 49
(ï 1 ) 2 + 166
Solution in parametric form:

ññóò

 = ³®

−4 2 (42

UÈ® 12 T 5 ® 22 )2 ,

+ 55


= T 84 ³®

®

3

− 1) ®

ï )

2 –1 2

2 −4 1 2 (3

1

− ² ],

where

# =£  .

).

UÈ® 22 + ® 2 T 12U

®

2 2 1 ),

where

# =A £  .

Page 117

118 59.

FIRST-ORDER DIFFERENTIAL EQUATIONS 4 ï 1 – ñ = – 25 + 50
(7ï 1 ) Solution in parametric form:

ññóò

 = ³® 60.

+ 49


+ 6


2

ï –1 ) 2 ).

® ® − 16® 12 )2, = −5³® 1−4 ® 3 (A 3® 32 − ® 22 ® 3 + 8® 12 ® 3 ), 15 ï + 6
ï –1 ) 3 – 3
2 ï –5 ) 3 .

−4 1 (5 2 3

ññ óò

ññ óò

J

®

J

®

2 3 2 −3 4 , 1 2

3 ï – ñ = – 16 +

J

)

–1 3

J

®

J

®

−1 2 −1 2 −3 4 (2 1 2

= U


ï

The substitution  = U

+

( ï

)

–5 3

UÈ® 22 + ® 2 − 3U

.

J leads to an equation !
= − 3 U −5 J 2 + 9 U 2

−4

32

−2

− 32 * # U

− 32 $ ,

coincident with equation 1.3.3.13 when B = −1 2, ( = 0,  = 3 4,

ññóò

5 ï – ñ = – 36 +


ï

)

–3 5

( ï

–

)

–7 5

(

,

> 0.

The transformation  =  ¨ − 13 £ U + # $ 

J

−5 4

equation of the form 1.3.1.3:

63.

ññ óò

– ñ = ü (


ï

2

+

where # = − 31  (2  )1 J 3 .

®

2 2 1 ),

−3 2

*

62.

# = 8£  .

where

–ñ = 4 Solution in parametric form:

 = 2 U 3 J 61.

2

¨*¨
− ¨ = − 2 U + 2 # * 9 3$

( ï

2

1J 2  +  53 $  £ U‹ 1 J

5

= −3 $ .

leads to an

5 1J 2 1 £ U . 27 $¥P

+O

)–1 ) 2 .

=

+

5 6

=

,

= 3 # 2, 

M

4(  2 ¨

+  1 ¨ +  0) 4(  2 ¨ 2 +  1 ¨ +  0 ) , = ˆ + , where parameters 4# − ¨ 2 4# − ¨ 2 2  2 ,  1 , and  0 are found from the relations $ = 4#3 2 −  0 and L =  1 − 4  0  2 , leads to a Riccati equation: A/, ¨ Š
= (− 41 ˆ +  2 )¨ 2 +  1 ¨ + #*ˆ +  0 . The transformation  =

2

For L > 0, we can set  2 = 0,  1 = £ L , and  0 = − $ . In books by Zaitsev & Polyanin (1993, 1994) it is shown that the original equation is reducible to the degenerate hypergeometric equation.

64.

ññóò

12 ï 1 – ñ = – 49 + 3
( 49 +

The substitution  ( = 0,  = # , and

65.

66.

( )ï 1)

2

+ 3


2

4 ( 49 –

(

5 2

)+

15 4




3

1 ( 49 +

5 4

( )ï

)

–1 2

.

# ) leads to an equation of the form 1.3.3.13 with B = 3,  = 4 7, # − $ ): M ! Š
= (4ˆ 2 + 5 # )ˆ − V 48 ˆ 4 + 12 #  2 − $  ˆ 2 + 3 # 2 W˜ˆ 3 . 49 7 = ( + 54 = 12 ( 72

2

V (2 –
) ï 1 ) 3 – 3 ( (2
+ 1) + ( 2 (1 – 3
) ï –1 ) 3 –
>( 3 ï –2 ) 3 W . 2 1 2 ¨ −2 $ 2 (3 − 2 $ ¨ ) ¨ The transformation  = ¨ −3 , = ˆ + ¨ ShO + leads to a Riccati ¨ Q 5 ($ + 1) $ÈP equation:  2ˆ 2 − 25 $ 3 ˆ + 254 #2$ 6  ¨ Š
= $Iˆ ¨ 2 + ˆ − 25 $ 3  ¨ + 35 $ 2 .

ññ óò

6 ï – ñ = – 25 +

ˆ

2

– ñ = 34 ï – 32
The transformation

ññóò

 = # 3J 2 

J ,

−3 2

4 75

(

2

ï 1 ) 3 + 3
2 ï –1 ) 3 – 4

= 3#

J  ˆ´

3 2

2

−

3 25



2

27 625


4ï

)

–5 3

− 12  + 12  

leads to an equation of the form 1.3.1.46: ¨*¨uŠ
− ¨ =

.

J ,

−3 2 6 25

where

 =¨ ˆ 2−

6 25

ˆ 

−1

,

ˆ − ˆ 2.

Page 118

119

1.3. ABEL EQUATIONS OF THE SECOND KIND

67.

ññóò

6 ï – ñ = – 25 +

Denote # =

7 100


ï 1)

7 5

3

31 3

+


2ï

)

–1 3


4ï

100 3

–

 and perform the transformation 7 ¨  = ˆ 3J 2, = 20  + 87 ˆ − 35  − 507  2 ˆ

−1

)

–5 3



.

ˆ , ü

3 10

 .

ˆ =H −

21 20

ˆ =H −

where

3 As a result we obtain an equation of the form 1.3.4.30 with B = 17 , ( = − 10  :

68.

ññóò

10 ï – ñ = – 49 +

Denote # = 8

−2

VH −

13 5


2ï

3 10

  ¨ + 87 H 2 − 97 ! H + 17  2 W ¨ ‘
= − 21 ¨

)

–1 5

7 20

–


3ï

)

–4 5

2

+ 2H ¨ .

.

. The transformation

¨ = 4 U +  −3 J 5 − 7

leads to an equation of the form 1.3.1.64 with $ = −1 49: ¨*¨
− ¨ = − 12 U + 39  2 − 15  3 U −1 J 2 . £ U =  3

69.

ññ óò

1 16

*

ï

–ñ

 3J 5 −

5 112

49


ï )

33 2 –5 11 = – 196 + 286 – 770 3 9 = 85 −2 . The transformation

Denote #

J ,

−2 5



98


ï

)

3

–13 11

.



ññóò

   8J

J

£ U =

15 3 448

ï


(123 ï –1 ) 7 + 280
ï –5 ) 7 – 400
2 ï –9 ) 7 ).

11

−

14 11



−3 11

,

7 9

J ,

−7 4

=

11

−

39 2 56



*

–ñ = +

Denote # = 1  . The transformation 21 – 100

 =ˆ

2

 ¨ + 4ˆ + 75  +

35 −2 3



3 2 −1 50

 ˆ

ˆ

J ,

−3 4

where

leads to an equation of the form 1.3.4.30 with B = 3, ( = − 21 20  :

VH −

71.



784

¨ = 3 U +  −8 J 7

leads to an equation of the form 1.3.1.64 with $ = −1 49: ¨*¨
− ¨ = − 12 U + 39  2 − 15  3 U −1 J 2 . 49 98 784 70.

39 2 42

21 20

  ¨ + 4 H 2 − 7 !H + 3  2 W ¨ ‘
= 34 ¨

2

+ 2H ¨ .

−1

−

– ñ = ô ï + õ ïƒú . 1 b . For ≠ 3, the transformation

 ,

ññóò

¡

2

U =$

(

Q ¡

− 3)

2



+ 1S ,

¨ = 2( − 3)$ ¡

2

O ' +



2 2

+



+

P

leads to an equation

¨*¨
− ¨ = 2(¡ −1) ¦"U − $IU 1 J 2 + [2 − 3 −  ( − 3)2 ] $ 2 + [2 − +  ( − 3)2 ] $ 3 U * ¡ ¡ ¡ ¡ ¡ ( −3)2 ¡

2 b . Let ≠ 1 and  > −1 4. Denote ¡ (B + 2)(B + + 1) 1 −1  =− ¡ 2 , where B = B 1, 2 = O˜A £ ¡ − −3 . ¡ (2B + + 3) 2 P 1 + 4 ¡ Then the transformation



 −1 ˆ + 2B + + 3

J § .

−1 2

B +2 ¨ Oˆ ¨ Š
+ , B = B 1, 2 −1 P ¡ ¡ reduce the original equation to the classical Emden–Fowler equation ¨IŠª

Š = * # ˆ  ¨2+ , where 2 2B + + 3 # = O ¡ , which is discussed below in Section 2.3. −1 P ¡  =ˆ +

+2 −1

¨ ,

=

¡

+2 −1

Page 119

120 72.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ññóò

–ñ =–

(þ

Denote # = −

þ

+1 ï + + 2)2 2(



¡

, $ 2

+ 2)2 

¡


ï 2ú

=−

2 £ U = − (¡ + 2)   − + ¡

leads to the equation

+1

−1

( ï 3ú

+

2(

¡ ¡

2

+ 2)3 

+2

+ ¡

+1

¡

2

. . The transformation

' − + ,

¨ = 2(¡ + 1) U +  + + ¡ + 2  +2 ¡ ¡

¨*¨
− ¨ = 2¡ (¡ + 1) U +  + 'U * ( + 2)2 ¡

J

−1 2

(see Table 7 with ƒ = 0 in Subsection 1.3.1). 73.

ññ óò

–ñ =

ô 2  2 ó

– ô ( õ + 1) 

Particular solution: 74.

ññóò

–ñ =

0

ô 2  2 ó

76.

ññóò

– ñ = 2ô

2

ï  ó

+ ô

Particular solution: 75.

=  

0



=  

ï

sin(2



− . +

= −2  sin( Œ  ).

ññ óò

–

( ! + õ )2 ! ( ! ó ò )3

–ñ =

ô 2! óò ! óò òó


=   +

1

ññóò

ï

= (ô

+ õ )ñ + 1.

ï

ññ óò

= (ô

ññóò

= Oiô –

5.

1 2

−



2

.

óò òó

).

,

 + ,  


!

= ! ( ï ).

2


= −   +

 + .  


â!â$äã = å ( æ )â + 1

− ' leads to a Riccati equation with respect to  =  (ˆ ):

+ õ )–2 ñ + 1.

The substitution ˆ = −(  +  )−1 leads to an equation of the form 1.3.1.33: !
Š = + ( ˆ )−2 . 1 ñ ô ï P

+ 1. −  leads to a Bernoulli equation: ˆ Š
+ ˆ +  2 

The substitution ˆ = 4.

ï

) + 2 ô sin(

0

The substitution ˆ =  Š
= 12  2 + ' + ˆ .

3.

õ ó

 + + .  Œ

1.3.2. Equations of the Form

2.

+õ .

Particular solution:

Particular solutions:

1.

ó



ññ óò

= (ô

ï

+ õ )–1 )

2

= 3( ô

ï 3) 2 + 8ï

= 0.

ñ

+ 1. 2 The substitution H = (  +  )1 J

ññóò

2



)–1 )

2

ñ

2

leads to an equation of the form 1.3.1.2: !
‘ =

+ 1.

The substitution H = 12  (  1 J 2 + 8)1 J ! ‘
= − 2 H + 1  2 H 3. 9 5184 −1

+ 12 !  H .

2

leads to an equation of the form 1.3.1.10 with

¡

= 3:

Page 120

121

1.3. ABEL EQUATIONS OF THE SECOND KIND

6.

7.

ññóò

= (ô

ï –2 ) 3 – 2 ô –1 ï –1 ) 3 )ñ 3

ññóò

ô ó ñ

=

+ 1.

The substitution ˆ = 8.

ññóò

= ( ô

ó

+

Œ





 

õ – ó )ñ

 ¨

2

=ˆ −¨

2

¨ Š
= ˆ − ¨ 2. leads to a Riccati equation: 3  3 J 2 ˆ 2

leads to an equation of the form 1.3.1.16: !
Š = + 1.

The transformation ˆ = 9.

+ 1.

The transformation  =  3 J 2 ¨ 3 ,

+

+Œ ˆ ¨ − .

Œ



 −



−

Œ

    , ¨ =  leads to a Riccati equation: ¨3Š
= 

= ôñ cosh ï + 1. This is a special case of equation 1.3.3.75 with  = 0 and ( = 1.

ññóò

10.

= ôñ sinh ï + 1. This is a special case of equation 1.3.3.76 with  = 0 and ( = 1.

11.

ññóò

ññóò

ô

=

cos(

ï )ñ

+ 1.

The transformation  = −

‡
= −2U‡

*

12.

+ ( Œ ˆ )−1 .

ññóò

=

ô

2

1 8

+ ‡ − Œ U .

sin(

ï )ñ

2

+ 1.

The substitution  = ˆ +



2Œ

2

Œ

arctan

4‡

,

Œ

= U −

8 ‡ 16‡ 2 + Œ

2

leads to a Riccati equation:

leads to an equation of the form 1.3.2.11: !
Š =  cos( Œ ˆ ) + 1.

1.3.3. Equations of the Form

â!â$äã = å 1 ( æ )â + å 0 ( æ )

1.3.3-1. Preliminary remarks. With the aid of the substitution ˆ = Xv 1 ( )  , these equations are reducible to the form:

! Š
= +  (ˆ ),

M

where

 (ˆ ) =  0 ( )  1 ( ),

(1)

and by means of the substitution H = X¯ 0 ( )  they can be reduced to the form:

! ‘
= ( H ) + 1,

M

where ( H ) =  1 ( )  0 ( ).

(2)

Specific equations of the form (1) and (2) are outlined in Subsection 1.3.1 and 1.3.2, respectively. 1.3.3-2. Solvable equations and their solutions. 1.

2.

= ( ô ï + 3 õ )ñ + ö ï 3 – ôƒõ ï 2 – 2 õ 2 ï . The substitution =  2 a + ' leads to a linear equation with respect to  =  (a ): (−2a 2 + a + ( ) g
= aÖ +  .

ññóò

= (3 ô ï + õ )ñ – ô 2 ï 3 – ôƒõ ï 2 + ö ï . The substitution =  ¨ +  2 leads to a Bernoulli equation with respect to  =  (¨ ): (−¨ 2 +  ¨ + ( )
? = ¨  +  2 .

ññóò

Page 121

122 3. 4.

5.

6.

FIRST-ORDER DIFFERENTIAL EQUATIONS 2ññó ò = (7 ô ï + 5 õ )ñ – 3 ô 2 ï 3 – 2 ö ï 2 – 3 õ 2 ï . This is a special case of equation 1.3.3.11 with

= [(3 – þ ) ï – 1]ñ + (þ – 1)( ï 3 – ï 2 – ô ï ). ‘ The transformation  = ¨u H , = − H + −1 +  2 −  −  leads to an equation ¨*¨
= ¨ + !H + H + whose solvable cases are outlined in Subsection 1.3.1 (see Table 5).

ññóò

+ ï ( ô ï 2 + õ )ñ + ï = 0. The substitution H = − 21  2 leads to an equation of the form 1.3.2.1 with respect to !
‘ = (−2 !H +  ) + 1.

ññóò

+ ô (1 – ï –1 )ñ = ô 2 . Solution in parametric form:

9.

10.

11.

−1

+ ln « ),

= −  (U + 

−

*«

−1

),

 −*

« =X

where

– ô (1 – õ ï –1 )ñ = ô 2 õ . Solution in parametric form:

U

M

U

+L .

ññóò

 = 12  exp(T"U 2 )  8.

= ( H ):

ññ óò

 = −(U +  − * « 7.

= 3 2, , = 1 2.

¡

=T

,

1 2

−1



V 2U 2 p  A exp(T U 2 )W ,

where  = X exp(T"U 2 ) M

U

[(1 + 2ù )ï + ôù ]ñ – ù ï 2÷ ( ï + ô ). ¨ 1 ‘ The transformation  = , = −  +   +1 +   leads to a separable equation: ¨3
= ¨

ññ óò

ññóò

=

ï ÷

−1

U

+L .

–1

= ô ( ï – ù)õ ) ïƒ÷

H

–1

ñ

H

+ ö [ï

2

– (2ù + 1) õ

ï

+ ù (ù + 1) õ 2 ] ï

÷

2 –1

−

 − .

.

    O −  leads to an Abel equation of the form 1.3.1.2: The substitution ˆ =  B +1 P !
Š = + (B + 1)( −2 ˆ . = [ ô (2ù + ü ) ï  + õ ] ïƒ÷ –1 ñ + (– ô 2 ù ï 2  – ôƒõ ï + ö ) ï 2÷ –1 . The substitution =   (¨ +   ) leads to a Bernoulli equation with respect to  =  (¨ ): (B ¨ 2 −  ¨ − ( )
? = −¨  −   +1 .

ññóò

= [ ô (2þ + ü )ï 2  + õ (2þ – ü )]ïwú –  –1 ñ – ( ô 2 þ ï 4  + ö ï 2  + õ 2 þ ) ï 2ú –2  –1 . The transformation H =   , =  + (a +   + ' −  ) leads to a Riccati equation with respect to H = H (a ): (− a 2 + 2  − ( ) H g
= ,H 2 + ,!a›H + , . (1)

ññóò

¡ a 2+( The substitution H = ¡ ,

0

order linear equation:

¡ ¨2g
¨ , where ( 0 = ( − 2  ¡ , reduces equation (1) to a second-

( a 2 + ( 0 )2 ¨ g

g + (2

¡

The transformation ˆ =

a

a + (( 0

2

¡

+ , )a ( a 2 + ( 0 )¨ g
+ , 2 ¨ = 0.

(2)

¡

, ‡ = (1 − ˆ 2 ) žJ 2 ¨ , where “

) ü equation¡ 2.1.2.226: equation (2) to the Legendre

= −¡

2

¡

+,

, brings

(1 − ˆ 2 )‡ Šª

Š − 2ˆ‡ Š
+ [ @ ( @ + 1) − “ 2 (1 − ˆ 2 )−1 ]‡ = 0, where @ is a root of the quadratic equation @

2

+@ + ¡

−,

2

4

¡

2

2

−

,

2

¡

0

(

= 0.

Page 122

123

1.3. ABEL EQUATIONS OF THE SECOND KIND

12.

13.

14.

15.

ññ óò

+ 2@ – 3) ï + ù – 2@ + 3] ï

= [(þ

A.

= [ ô (ù – 1) ï + õ (2 + ù )] ï  –1 ( ô ï + õ )–  –2 ñ – [ ôù ï + õ ( + ù )] ï 2  –1 ( ô ï + õ )–2  –3 . 1 1 The substitution = ¨ +  S   +  (  +  )−  leads to an equation of the form Q  (  +  ) 1.3.4.5: (¨ +   +1 + '  )¨2
= [ B/  +  ( Œ + B )  −1 ]¨ .

ññóò

– ô V (þ – 1) ï + 1W ï –1 ñ = Solution in parametric form:

ô 2ï

ññóò

− 1)(U +

+1

+ 1)

U

U +L . M U + +1 + 1

=X

– ô (1 – õ ï –1 ) 2 )ñ = ô 2 õ Solution in parametric form:

ññóò

where

ø

:

–1

ï

(þ

« + ln O

U +

+ 1)( ï – 1).

+1

+1

U

«

+ −U =  V1+( U

,

P

¡

−1

)« W ,

ï –1 ) 2 .

2

L

:

−2

(

:
2 *),

= A  2 U

2

:

−2

V ( :
)2 A : 2 W ,

*

ø

U ) + L 2 n 0 (U ) for the upper sign, U ) + L 2 ; 0 (U ) for the lower sign, 0 ( U ) and n 0 ( U ) are the Bessel functions, and ® 0 ( U ) and ; 0 ( U ) are the modified Bessel

functions.

19.

1–2

M

 = T± 2 U

]

– 2@ + 3) ï – ù + @ – 2] ï

+ (ù – þ

2

= [ ô (2ù + 1) ï 2 + ö ï + õ (2ù – 1)] ï ÷ –2 ñ – (ù,ô 2 ï 4 + ô„ö ï 3 + ý ï 2 + õ^ö ï + ù)õ 2 ) ï 2÷ –3. Here,  ,  , ( , , and B are arbitrary numbers. M The substitution =   a +   +1 + '  −1 leads to a Riccati equation with respect to  =  (a ): (−B/a 2 + (˜a − + 2B± ) g
=  2 + aÖ +  .

where «

18.

+ @ – 1) ï

+ [(þ

ññ óò

(

17.

Añ

ˆ The transformation  = ¨ ¨ Š
, = #*ˆ  −- +2 ¨ + +- −1 −  2−- +  1−- leads to the generalized # ˆ  ¨2+ (¨2Š
)- , which is discussed in Section 2.5. Emden–Fowler equation ¨3Šª

Š = *

 = ¡

16.

–

™

L

1 0(

®

1 0(

+ 3( ô ï + õ )–2 ) 3 ï –7 ) 3 . 1 1  +  1 J 3 The substitution ¨ = + O leads to a separable equation for ¨ P  3  ¨2
=  −1 J 3 (  +  )−2 J 3 ( 1  − 3¨ 3 ). 9

ññ óò

= 3( ô

ï

=

3ññó ò = (–7


+ õ )–1 )

ï

3

ï –5 ) 3 ñ

+ 6s – 2 ) ï

) ñ

–1 3

+ 6( s ï – 1) ï

= ¨ ( ):

)

–2 3

+ 2(
ï + 5 )(–
ï + 3s + 4 ) ï 1 ) 3 ,
= s(3s + 4 ). ¨ −1 2J 3 The transformation  = (ˆ + Œ° ) , = ( + 4 Œ + 3 ° − #* ) leads to an equation of the form 1.3.4.10 with  = 1 3: [(ˆ + Œ° )¨ + (4 Œ + 3 ° )ˆ ]¨pŠ
= 23 ¨ 2 + 2(3 Œ + ° )¨ + 2ˆ .

In the solutions of equations 19 and 20, the following notation is used:

 = U (U + 1) − ln L  £ U + £ U + 1  , û û ü û û

=1−

+ 12 ô (6 ï – 1) ï –1 ñ = – 12 ô 2 ( ï – 1)(4ï – 1) ï Solution in parametric form:

ññóò

 = U

−2 2

,

–1

=  (1 − U

.

U +1 ln L  £ U + £ U + 1  . û û U û û

−2 2

− U 2

−2

).

Page 123

124

FIRST-ORDER DIFFERENTIAL EQUATIONS 1 2 ï – 21 ô (1 + 2 õ ï –2 )ñ = 16 ô (3 + 4 õ Solution in parametric form:

ï

ññ óò

20.

 = (˜U ]

−1

J ,

−1 2



–1

).

J (

−1 −1 −1 2

= − 14 !(˜U





2

− 4U! 2 ),

 = −( 2.

In the solutions of equations 21–23, the following notation is used:

8

8

3

ññóò

21.

1 14

ô (13ï

*

1

*

– 20) ï –9 ) 7 ñ

8 38 8

−2 2 3 ,

1

ô (ï

=  (4

8

*

– 1)( ï

2

1)

J 8

– 8) ï

J 8

8 38

8

−3 4, 3

1

= 

256 25

8 38 3

8 3 8  −2 J 7 8 1 4

−2 4 ,

=  (25

8

1

2

+7

+

256 25

{

J 8

3 3

2

3 ).

)

.

+7

8

2 4

8 38  1 4

.

–23 13

8

J

3 −5 13 −16 13 ( 42 3) 4

{

8 28

.

−15 7 3

8

J

8 38

+ 25

8

3 3

8 38

− 208

1 4 ).

In the solutions of equations 24–27, the following notation is used:

{

{

{ {

U + L ) cos U , 2 = tanh( U + L ) + tan U , 3 = tanh( U + L ) − tan U , 4 = 3 2 3, 9 9 9 9 9 9 92 1 = cosh U − sin(U + L ), 2 = sinh U + cos( U + L ), 3 = sinh U − cos( U + L ), 4 = 3 2 3 − 2 1. 1 = cosh(

1 + 15 ô (13ï – 18)ï –7 ) 5 ñ = – 154 ô 2 (ï – 1)(ï – 6)ï 1 b . Solution in parametric form with # < 0:

)

ññóò

 = −12

{

{

{

=  (12 2)−2 J

−3 2, 1

2 b . Solution in parametric form with # > 0:

 =6

9 2 9 −3 9 1 2

=  (6

3,

+ 12 ô (5 ï + 1) ï –1 ) 2 ñ = ô 2 (1 – ï Solution in parametric form:

ññ óò

 =

26.

)

− 64

–11 7

1 + 26 ô (19ï + 85)ï –18 ) 13ñ = – 263 ô 2 (ï – 1)(ï + 25)ï Solution in parametric form:

 = −25

25.

8

J

ññóò

23.

.

−2 7 −10 7 ( 32 2 3

−6 7

5 + 56 ô (23ï – 16)ï –9 ) 7ñ = – 563 ô 2 (ï – 1)(25ï – 32)ï Solution in parametric form:

 = − 256 25

*

)

–11 7

ññóò

22.

24.

3 – 14

+ = Solution in parametric form:

 = 64

]

8

£ £ £ = « + L sin( £ 3 U ), 2 = 2 « − L sin( 3 U ) + 3 L cos( 3 U ), 8 8
8
8 8 8 8 8 8
8
8 8 8 = 2 1 ( 2) − ( 1 ) 2 − 1 2 , 4 = 2 1 ( 3 ) − 5( 1 ) 3 + 1 3.

« = exp(3U ),

{ 2{ 1

2

9 29

1 3)

{

5

–9 5

{

J

−9 5 ( 13 1

J 9

1

3,

9 9 9

14 2 4 9 1 2 4,

{

2 1 2

+ 12 2 ).

).

−2 2 ,

= −

{

{

−3 3 2 ( 1

−

{ {

= 

28 9

{ 6{ 1

3



J

−2 5

{

= 

9 9  J 9 

−2 5 14 2 9 1 4

4 2

+ 4 2 ).

)

.

4 1

−

+

5 2 9 4

9 5



{

2 1 2

2 b . Solution in parametric form with # > 0:

 =

{ {

J 9 + 59 9 2 − 69 2 9 ). 2 3 1 3

3 + 35 ô (19ï – 14)ï 7 ) 5 ñ = – 354 ô 2(ï – 1)(9ï – 14)ï 1 b . Solution in parametric form with # < 0:

{ 4{

−5

−2 5 −9 5 3 ( 2 2

ññóò

 = − 289

.

9

5 9

{

2 3

−

+

28 9

{

3

9 9 

14 2 9 1 4

.

.

Page 124

125

1.3. ABEL EQUATIONS OF THE SECOND KIND

27.

3 + 10 ô (3ï + 7)ï –13 ) 10 ñ = – 15 ô 2(ï – 1)(ï + 9)ï 1 b . Solution in parametric form with # < 0:

)

ññóò

{ 4{

 =9

−2 3 ,

1

=  (9

{

–8 5

{

{

J

J

.

4 −3 10 −7 5 ( 32 1) 3

− 20

2 b . Solution in parametric form with # > 0:

9 49

2 2 4,

 = − 92 ]

9  J 9 J 9 

9 4 −3 10 −7 5 4 2 2

= −4  

2 4

−5

{

3

−9

9 29

1 4

{

4 1 ).

9 

9 4 2 2

+

.

In the solutions of equations 28–30, the following notation is used:

. .

2

28.

6

= A (U

2

= A (U

6

. . 2. 2 3

.

= A (U

4

− 6U

+ 12 L3U − 8 L

2

+ 27).

− 3U + L , 3

2

− 45U

4

)

−2 4 ,

=  (16

. . 2. 2 3

–9 5

.

J

32 27

. 2. 3

.

−3 6, 4

= A  (3

.

4)

J

−9 5

.

)

J 

−4 5 −2 5 5 3 6 4

3 + 14 ô (3ï + 11)ï –10 ) 7 ñ = – 141 ô 2 (ï – 1)(ï – 27)ï Solution in parametric form:

)

ññóò

. 3. 4

−2 6 ,

= T/ (3

.

4)

A 15

–9 5

.

J

J

−9 7

.

–13 7

.

J

2

+ 4 L3U − 3),

.

3 −2 5 ( 42 4)

3 + 20 ô (13ï – 8)ï –7 ) 5 ñ = – 201 ô 2 (ï – 1)(27ï – 32)ï Solution in parametric form:

−8 7 ( 62 6

. 2.

4

2

T 16

. .

2 2 3 ).

.

.

2 6

T 8

. 2. 3

6

27 4

A

.

3 4

.

.

T 28

. 2.

6

3

A 27

.

3 4 ).

In the solutions of equations 31–38, the following notation is used:

®

=X

² =

31.

+ 20 L3U

3

ññóò

 = T 27 ]

=U

1 + 10 ô (7ï – 12)ï –7 ) 5 ñ = – 101 ô 2 (ï – 1)(ï – 16)ï Solution in parametric form:

 = 30.

− 15U

4

3

ññóò

 = A 16

29.

.

− 1),

ü

U U (incomplete elliptic integral of the second kind), M A (4U 3 − 1) ü A (4U 3 − 1), ® 1 = U (2 ®·T\L −1 ² + L ), ® 2 = U −1 ( ²$® 1 − 1),

® 3 = 4UÈ® 12 T ® 22 , ñ ñóò – 21 ô (ï + 1)ï –7 ) 4 ñ

® 4 = ® 2 ® 3 − 8® 12 , ® 5 = 2² (® 1 2 ï ô ( – 1)(3ï + 5)ï –5 ) 2 .

+ L ) − U 2.

= 4 Solution in parametric form:

 = 16 U

®

−3 −1 1

² ,

= 13  

1 6

U ² 

J

J V (11U

®

−3 4 −1 4 1

3

− 2) ®

1

− 12 2 ² W.

The lower sign is taken in the notation adopted. 32.

– 21 ô ( ï + 1) ï –7 ) 4 ñ = 14 ô 2 ( ï – 1)( ï + 5) ï Solution in parametric form:

ññ óò

 = − 31 U

® ®

−2 2 1 2,

)

–5 2

= − 13  (UÈ® 1 )−1 J 2 

.

®

J

−3 4 1 (2 3 2



UÈ® 22 + ® 2 − 3U

®

2 2 1 ).

The lower sign is taken in the notation adopted.

Page 125

126 33.

FIRST-ORDER DIFFERENTIAL EQUATIONS 1 – 14 ô (4ï + 3)ï –8 ) 7 ñ = – 141 ô 2 (ï – 1)(16ï + 5)ï Solution in parametric form:

® ® ® ñ ñ óò + 16 ô (13ï – 3)ï –2 ) 3 ñ

3 −2 −1 2 16 1 2 3 ,

 =A

34.

1 – 28 ô (8ï – 1)ï –8 ) 7 ñ = Solution in parametric form:

ññ óò

 = TI®

36.

1 28

ô 2 (ï

® ®

−2 3 −1 1 3 4 ,

1 −1 ( 6

U

®

U

−1

( ® + L )−2 ® 5 ,

®

=A

® ® ®

–7

®

2 1 2 ).

®

A 7 ® 42 A 32 ®

®

2 1 4 ).

.

–7

−3

[(1 A 2U 3 )( ® + L ) − U 2 ² ].

.

 ( ® + L )4 ®

576 125

T 16 ®

.

J



®

2 2 3

® 42 A 4® 12 ® 4 ). A

)

®

−3 2 5 [ 5

1 + 42 ô (39ï – 4)ï –9 ) 7 ñ = – 421 ô 2 (ï – 1)(9ï – 16)ï Solution in parametric form:

)

–11 7

+ 5U

2

® 5 TCU 3 (®

+ L )2 ].

.

= 13  [16U 3 ( ® + L )2 ® 55 ]−2 J 7 [3 ®

−2 5 ,

A 7®

.

–9 7

ññ óò

2 5

+ 7U

2

® 5T

48U 3 ( ® + L )2 ].

In the solutions of equations 39–41, the following notation is used:

+ ô ( ï – 2) ï –1 ñ = 2 ô 2 ( ï – 1) ï Solution in parametric form:

ññóò

–1

−2

+ ô (1 – õ ï –2 )ñ = ô 2 õ ï –1 . Solution in parametric form:

,

= 2UIA exp(T"U 2 ).

.

+ ô (3 ï – 2) ï –1 ñ = –2 ô 2 ( ï – 1)2 ï Solution in parametric form:

ññóò

 = 12 exp(T"U 2 ) 

41.

J

= 36  ( ® + L )2 ²

=  V 1 T 2U

 = U exp(T U 2 )  , 40.

)

3 12 3 6 −1 7 (3 33 32 1 3 4

1 = − 32 s

 = X exp(T U 2 ) U + L , M

39.

®

– 52 ô (3 ï – 10)ï –4 ñ = 15 ô 2 ( ï – 1)(8ï – 5) ï Solution in parametric form: 5 24

®

–1 3

– 1)(32 ï + 3) ï

+ L )−1 ² ,

.

J



−8 3 3 ( 3 3 4

ññóò

 = A 16U 3 ( ® + L )2 ® ]

= − ³®

– ô (5 ï – 4)ï –4 ñ = ô 2 ( ï – 1)(3 ï – 1) ï Solution in parametric form:

 =A 38.

®

3 −2 3 4 ,

ññóò

 =A 37.

® ® ®

–9 7

13 12 6 2 −1 7 (3 32 16 1 2 3

1 = − 16  

= – 16 ô 2 ( ï – 1)(5ï – 1) ï Solution in parametric form:

 = T[®

35.

)

ññóò

–1

2

− U exp(T"U 2 ) 

−1

W.

. = 12  V 2 − exp(T"U 2 ) 

−2

‹Te

−2 −2



W.

ññóò

1

 = £  T 2

−1

,



= £  T 8

−1 −1



V exp(T U 2 ) − 2  2 W .

Page 126

127

1.3. ABEL EQUATIONS OF THE SECOND KIND

]

In the solutions of equations 42–52, the following notation is used:

«

1

=U

«

4

=U

3

ü

ü

A (4 < A (4 <

3 3

<

− 1) + 3U 2
− 1) + 2 < ,

«

5

«

=U

2

3

ü

= U 2
A (4 <

«

=

3

A (4 < ü

− 1) − 4U 2
3

«

= < (U ) is defined implicitly as follows: U = X

− 1) A 2U'< 2 ,

3

=U

6

A (4 < ü

3

− 1) − < .

<

− L . The upper sign − 1) ü in the above relations corresponds to the classical Weierstrass elliptic function < = < (U + L , 0, 1). Here, the function

42.

ññóò

–

1 4

ô (3ï

– 4) ï

) ñ

–5 2

1 4

=

ô 2(ï

– 1)( ï + 2) ï

–4

= − 12 U 

3 −1 2 ( 6

M

A (4 <

3

.

Solution in parametric form:

<

−2 −2

 = 23 U

« 6,

2 3

J



«

+ 2
2 6

«

− 3U

6

<

2 3

).

The upper sign is taken in the notation adopted. 43.

ññóò

+

(33 ï + 2) ï

ô

1 30

1 2 ï = – 30 ô ( – 1)(9ï – 4)ï

) ñ

–6 5

)

–7 5

.

Solution in parametric form:

44.

ññóò

–

1 8

J

 = 4U 2 < 3 «

−2 6 ,

– 8) ï

= – 18 ô 2 ( ï – 1)(3 ï – 4) ï

ô (ï

) ñ

–5 2

=A

1 3

 (4U 2 < 3 «

8 −1 5 (3 6) –4

+ 5
2 6

«

6

T 12U 2 < 3 ).

.

Solution in parametric form:

 = 43 «

1

«

−2 2 ,

= 14  

4 3

J

3 −1 2 ( 1



«

2 1

«

− 4«

1

«

+ 3«

2

3 2 ).

The lower sign is taken in the notation adopted. 45.

ññóò

+

(17 ï + 18) ï

ô

1 30

) ñ

–22 15

1 2 ï = – 30 ô ( – 1)(ï + 4)ï

)

–29 15

.

Solution in parametric form: −2 1

 = A 4« 46.

ññóò

–

1 13

ô (6ï

– 13)ï

3 2,

«

) ñ

–5 2

= A !«

J

−16 15 (4 1

« 2 )−7 J 15 ( «

1 2 ï = – 26 ô ( – 1)(ï – 13)ï

–4

2 1

− 5«

1

«

2

3 2 ).

T 4«

.

Solution in parametric form:

 =

13 6

«

−2 2

« 5,

1 = − 13  

13 6

J

3 −1 2 (2 5



«

2 5

«

A 13 « 2 «

5

− 6«

3 2 ).

The upper sign is taken in the notation adopted. 47.

ññóò

+

1 30

ô

(24 ï + 11) ï

) ñ

27 20

1 2 ï = – 60 ô ( – 1)(9ï + 1)ï

)

–17 10

.

Solution in parametric form:

 = 4«

3 2

«

−2 5 ,

= 13  (4 «

J

3 −7 20 2)

«

J

−13 10 (3 5

«

2 5

+ 20 «

2

«

5

− 12 «

3 2 ).

The upper sign is taken in the notation adopted.

Page 127

128 48.

FIRST-ORDER DIFFERENTIAL EQUATIONS – 52 ô (3 ï + 2) ï –8 ) 5 ñ = Solution in parametric form:

ññóò

49.

<

1 3

 =T

−1

51.

2

2 3

«

1 5

ô 2 (ï

–11 5

= T/ (3
J «

)

J «

−3 5 (3 4

)

–9 5

J «

−2 5 2

= !«

2 2

 = 2«

−3 4 , = 15 2 (

«

– 51 ô ( ï + 4) ï –8 ) 5 ñ Solution in parametric form:

ññ óò

<

−1 −3 3

J

ü

ô ï

−3 5 3

J «

A (4 <

)

− 1),

3

< 2ü A Q

(4 <

− 1) S

3

J

−3 5

Q

– ô (2 ï – 1)ï –5 ) 2 ñ = 21 ô 2 ( ï – 1)(3ï + 1) ï Solution in parametric form:

ññóò

 = 16 U

<

−1 −2

ü

4<

3

= −  V 6U 

− 1,

–4

4<

ü

2 3

)

«

3 4

+«

2 4

AC« 4 ).

2 3

«

2

2 2 ).

− 10 «

.

− 5«

2

14 < 3 «

3

2 3

«

2 2 ).

+ 4«

+ 2 A (4 <

ü

3

− 1) S .

.

J

−3 1 2

W

<

4<

ü

The upper sign is taken in the notation adopted.

]

TB< 2 «

.

−1

3

3 4

–3 5

J


.

−9 5 ( 4

9 −2 5 (2 4) –11 5

«

– 1)(3 ï + 7) ï

−2 5 1 3 3

= − 16 !«

−1 2

=  (4 «

.

J

2 −2 5 3)

– 1)(27 ï + 8) ï

−3 4 ,

«

– 1)(8ï + 1) ï

3 + 10 ô (13ï – 3)ï –4 ) 5 ñ = – 101 ô 2 (ï – 1)(27ï – 7)ï Solution in parametric form:

ññóò

 = 13 «

52.

ô 2 (ï

« 4,

– 56 ô (4 ï + 1) ï –7 ) 5 ñ = Solution in parametric form:

ññ óò

 = −«

50.

−2 3

«

1 5

3

− 1 + 2U'<

3

− 2U  .

2

− 1S ,

In the solutions of equations 53–55, the following notation is used:

53.

54.

ô (ï

Æ

1

– 6) ï –7 ) 5 ñ

+ L3U − 1,

ññóò + = Solution in parametric form: 1 5

ññóò

1 5

ô (21ï

2 5

2 2

 = A 3UÆ

Æ

+ 19) ï –7 ) 5 ñ

ô (ï 2

−1 3 ,

1

2 2

Æ

Æ

– 1)( ï

– 25

ô (ï

−2 3 ,

Æ

2

 =Æ

−2 1

Æ

+ 4) ï –9 ) 5 .

= A Æ

2 3,

A 1,

)

= − Æ

3

3

=U

A 3U + L .

−3 5 3

– 4) ï –9 ) 5 .

J Æ

J Æ

−2 5 1

–5 2

Æ

J V (1 A 5U 2 ) Æ

J Æ

2 −2 5 2)

– 1)(9 ï

2

– 3 ô ï –7 ) 4 ñ = 14 ô 2 ( ï – 1)( ï – 9) ï Solution in parametric form:

ññóò

2

=U

=  (3UÆ

+ = Solution in parametric form:

 = A/Æ

55.

2

= A"U

J

−4 5 2

−6 5 ( 3

Æ

2 3

+Æ

1

TeÆ

3

1

Æ

T 3UÆ

2 2

AçÆ

2 2

W.

2 1 ).

.

J Æ

−1 2 1

J

−3 2 ( 3

Æ

2 3

Æ

2 2

−Æ

2 1 ).

The lower sign is taken in the notation adopted.

]

56.

In the solutions of equations 56 and 57, the following notation is used:

ññóò

– ô V ( ü + 1) ï – 1W

ï –2 ñ

%  =X U 



−2

exp(T"U 2 ) U + L .

= ô ( ü + 1)( ï 2

Solution in parametric form: 2 − 2  = U  +1 exp(AU 2 ) % : , , +1 , −1 where ý = .

=

– 1) ï

Q

M

–2

, +1 2

.

U 

2 +1

exp(T"U 2 ) % −1 : A ( , + 1)U

,

Page 128

129

1.3. ABEL EQUATIONS OF THE SECOND KIND

57.

– ô [( ü – 2)ï + 2 ü – 3] ï Solution in parametric form:

ññ óò

2 2−

 = T 2U 58.

–

ñ

– 21 ô [(4 ü – 7)ï – 4 ü + 5] ï Solution in parametric form:

ññóò

:

)2 ’

−2

–

ñ

=

1 2

ô

2

:

)−3 J 5 ’

:
*

= U

.



(2 ü – 3)(ï – 1)2 ï

:

= 12  (U

,



J (2 ’

−7 5

:

+@

, @ =



2(1− ) 2− exp(

Q

= L 1 ®Á (U ) + L 2 ; Á (U ), ’ where modified Bessel functions.

]

1–2

=  (T 2 %  )1−  exp V T ( , −2)U 2 W U

 exp(AU 2 ) %  ,

 = (U

– 2)( ï – 1)2 ï

ô 2(ü

=

2

1–2



T"U 2 )+ O

2 A 4U 2 %  S . 2−, P

.

:

+5 ’ −U

2

:

2

),

1−, ; ® Á (U ) and 3 − 2,

; Á (U

) are the

In the solutions of equations 59 and 60, the following notation is used:

U M +L . U  +

Ä  =X 59.

– V (2ù – 1) ï – ôùlW ï –÷ –1 ñ = 1 b . Solution in parametric form:

ññóò

ù (ï

60.

= (  −  ) −  .

0

– V (ù + 1) ï – ôù W ï ÷ –1 ( ï – ô )–÷ 1 b . Solution in parametric form:

ññ óò

61.

–2

ñ

. −1

– ô V (2 ü – 3) ï + 1W ï –  ñ = Solution in parametric form:

ññóò

ô 2(ü

[(U  +  ) Ä  − U ].

ù ï 2÷ ( ï

=

= −   (  −  )− 

0

÷

= (U − !Ä  )− 

 = U Ä  −1 , 2 b . Particular solution:

–2

= U − Ä  

 = U Ä  −1 , 2 b . Particular solution:

– ô )ï

−1

−1

– ô )–2÷

–3

.

[U − (U  +  ) Ä  ].

.

– 2)[( ü – 1) ï + 1] ï



2(1– )

.

1 U +1 «  −2 (1 − , )(2 , − 3)(U + 1) 1−  + (2 − , ) «uS , (1 − , )(2 − , ) Q 2−   = U « − (1 − , )(U + 1) 1−  S , 1 − ,þQ

 =

where «

]

U

1

(U + 1) 1− 

M

U +L .

In the solutions of equations 62–66, the following notation is used:

²  = £ 1 AIU  62.

1

=X

+1 ,

«  = XÊ 1 AhU 

J

+1 −1 2



– ô V (ù + 2 ü – 3)ï + 3 – 2 ü/W Solution in parametric form:

ññóò

 =U

−1

2

« ´ ²  2−  ,

M

U +L ,

ï – ñ

= U 

−2

=

ô

  = ²  «  −U , 2

V (ù + ü – 1) ï

B +1  U « ´1−  ”!A 2− ,

+1

2

« ´ = X¥ 1 AhU 

+1

 

1 −2

– (ù + 2 ü – 3) ï + ü – 2W

M

« ´ − « ´ ² 2 + U ² 



2(1− ) 2−



•

ï

U +L . 1–2



.

.

Page 129

130 63.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ññ óò

–

ù

ô

ï

V (ù + 2) ï – 2W

–

÷

2 +1

÷ ñ

ô

=

ù

Solution in parametric form:



= A 2− 

 = A 2 « −1 ²  , 64.

2 ÷ +1 +4ï –2 ï – ÷ ñ = P ù ù +2 Solution in parametric form:

ññóò

ô

–

ù

O

B +2

65.

ññóò

+

ù

ô

B +1

ù (ù

3ù + 5 ï + +3 2

ù

ù

ññóò

– ôpO

ù

« 2   −2 ,

ù

+2

2

ï – ÷÷

–1 +1P

+



+1 − +3



= U

õ ïƒ÷ P ñ

+4 +3

ô

68.

ññóò

= ( ô

+ õ )ñ + ö

ññóò

= [ ô (2 + ) -

ó

ññ óò

= ( ô



ó

ó

– ôƒõ

ññóò

=

ññóò

=





+ õ ]

÷

3 +2

÷

.

2

+ ù – 4) ï – (ù – 1)(ù + 2)W

B

1  2 − U   + U  B +2 'B

O−

+1

D

÷

3 +2

÷

.



2 −1

Œ

ó



2( +2) +3

2 +3

− «  

–

ó

 ñ



.

 

2 2

 



ó

−   

= !(« 

õ

2

1

2

–

 2−U  Q  P

ù

2

+1

+ 2ù + 5 ï + ù +1

« 2 + (A 1) 

1 +3

ù

4 S +1

B +3 B +1

ï – ÷÷

+5 +3

.

U   S .

.

J O« ² 2 + 1 U .  1  1 B P

.

+ (– ô

ó

C 2

2

ó

– ôƒõ

+ ö )

2



ó

.

+

ôƒõ

(

ï

ó 

+ 1) 

+

õ

2

ï

ó (2ô ï

+ 2 ï + õ )ñ +

+ õ )ñ – 



2



ó (ô

2

ï

2

ï

+ ôƒõ

+ ö ).

 2D

ó (–ô ï

4

–õ

ï

2

−1

+

].

+ ' leads to an equation of the form 1.3.1.2: !
Š =

+

ô

(ù + 1) ï

¨ g
= a ¨ +  . = a ¨ +  leads to a linear equation: (−a 2 + a + ( )2

ó (2ô ï

2

–

« 2 , P

The substitution = ž (ˆ +  ) leads to a linear equation with respect to  (− Œ ˆ 2 + 'ˆ − ( ) Š
=  + ˆ . 71.

ï

leads to an equation of the form 1.3.3.10: !
‘ = [  (2“ +Œ ) H  +  ] H 

+ õ )ñ + ö [ ô

The substitution ˆ = 70.

2

The transformation  = ln ¨ , The substitution H =  (−  2 “±H 2  − ˜H! + ( ) H

69.

+ (ù

ñ

2

 = (˜U « +1 J1 ,

ó



ô 2 ï O ù + 1 + õ ïƒ÷ ù ù

=–

Solution in parametric form:

67.

2

1+

2(ù + 3) Q

Solution in parametric form:

66.

+ 2)

–

.

O

+1

V 2ï

2

ï

– 2 ï – ù + 1W

2

1  +1 − !«   ²   ( «  A 2 ²  ).

+1

− =  (2  )−  «     

=–

 =U 

ô

1

 = 2 'U −  « −2   , where  = A

V (ù + 1) ï

2

+ (Œ ˆ .

=  (ˆ ):

+ ö ).

 The substitution =   (ˆ +  2 ) leads to a Riccati equation with respect to  =  (ˆ ):
2 2  ˆ + 'ˆ + ( ) Š =  + ˆ . (− 

Page 130

131

1.3. ABEL EQUATIONS OF THE SECOND KIND

72.

ññ óò

ï )  ó ñ

+ ô (1 + 2 õ

= –ô

2

õ ï 2  2 ó

.

Solution in parametric form:

 =

2



where  = X U 73.

ññóò

exp(A"U 2 )  ,

−1



=−

U

2

V 2U 2 exp(A"U 2 ) pA 1W exp[2 exp(AU 2 )  ],

−2

exp(T"U 2 ) U + L .

M

– ô V 1 + 2ù + 2ù (ù + 1) ï WE

óñ

÷

( +1)

= –ô

2

ù (ù

+ 1)(1 + ù

ï )ï  2(÷

+1)

ó

.

Solution in parametric form:

 = O 2B/U  « +

= X (1 AIU 

where « 74.

ññóò

1 exp V (B + 1)U  « W , B +1P

+ ô (1 + 2 õ

) J

+1 −1 2

U + B±« exp V (B + 1)U  « W , 1 AIU  P

= U  O

U +L . M

ï 1 ) 2) exp(2 õ ï 1 ) 2)ñ

= –ô

2

õ ï 3 ) 2 exp(4 õ ï 1 ) 2 ).

Solution in parametric form: −4

 = (˜U

:

−2

2

’

= − !(˜U

,

−4

:

−2

(’

2

ACU

2

:

2

) exp(−2 'U

−2

:

−1

’ ).

Here,

 = (T ( )−1 J 2 ,

:
*

’ =U

:

:

+ ,

=

™

L L

ø

U ) + L 2 n 1 (U ) for the upper sign, U ) + L 2 ; 1 (U ) for the lower sign,

1 1(

®

1 1(

where ø 1 (U ) and n 1 (U ) are the Bessel functions, and ® 1 (U ) and ; functions. 75.

ññóò

= ( ô cosh ï + õ )ñ – ôƒõ sinh ï + ö .



The transformation a = −  sinh  , ˆ =  76.

ññóò

= ( ô sinh ï + õ )ñ – ôƒõ cosh ï + ö .



The transformation a = −  cosh  , ˆ =  77.

ññóò ññóò ññóò

=

ô ï

ï

cos(

2

ññóò

leads to a Riccati equation: 2( 'a + ( )ˆ g
= ˆ 2 +2aֈ +  .

?

,

?

= (ˆ + ¨ ) 

¨ Š
= ¨ + ˆ . leads to a linear equation: (−ˆ 2 + ˆ +  )2

?

=

ô ï

sin(

ï

2

, = (H + ¨

2

)

?

‘ leads to a Riccati equation: (−H 2 + !H +  )¨2
= ¨

2

+H .

)ñ + ï .

The substitution H = 12  80.

leads to a Riccati equation: 2( 'a + ( )ˆ g
= ˆ 2 +2aֈ −  .

= (2 ln2 ï + 2 ln ï + ô )ñ + ï (– ln4 ï – ô ln2 ï + õ ).

The transformation  =  79.

U ) are the modified Bessel

= (2 ln ï + ô + 1)ñ + ï (– ln2 ï – ô ln ï + õ ).

The transformation  =  78.

1(

2

‘ leads to an Abel equation of the form 1.3.2.11: !"
=  cos(2 ŒH ) +1.

)ñ + ï .

The substitution H = 12 

2

leads to an Abel equation of the form 1.3.2.12: !"
‘ =  sin(2 ŒH ) +1.

Page 131

132

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.3.4. Equations of the Form [ç 1 ( æ ) â + ç 0 ( æ )] âä ã = å 2 ( æ ) â

2+

å 1 ( æ )â + å 0 ( æ )

1.3.4-1. Preliminary remarks. With the aid of the substitution

¨ =O

+



0

« ,

1P

« = exp O − X

where





2 1

 , M P

(1)

these equations are reducible to a simpler form:

¨*¨ 
=  ( )¨ +  ( ), 1 0 where



1

= ¬ M

M

” 



0

+

1•





1

−2

1

0 2 ® « , 12 

0

= ”





(2) 0 1

−

0  1 02  2 + 3 « 12 1 •

2

.

Specific Abel equations of the form (2) are outlined in 1.3.1–1.3.3. In the degenerate cases with  0 ≡ 0 or  1 ≡ 0, the variables in equation (2) are separable. 1.3.4-2. Solvable equations and their solutions. 1. 2.

(
ñ + ( ï + ô )ñ„ó ò + (­ñ + ü Solution: # 2 + ,! 2 + 2( $I

ï

+ +

õ

= 0. + ' ) = L .

(ñ + ô ï + õ )ñ ó ò = ñ +  ï + . The substitution = ‡ −    −  leads to the equation

‡"‡
 = (  + ƒ )‡ + (ý − !ƒ ) + = − ˜ƒ

which is separable with  = − ƒ . For  ≠ − ƒ , the substitution ‡ = (  + ƒ )¨ leads to an equation of the form 1.3.1.1 or 1.3.1.2:

¨*¨ 
= ¨ + ¢

3.

4.

5.

6.

−2

−2

(ý − !ƒ ) + ¢

(= − ˜ƒ ),

where

¢ = +ƒ .

(ñ + ôƒü ï 2 + õ ï + ö )ñó ò = – ôñ 2 + 2 ôƒü ï ñ + þxñ + ü ( ü + õ – þ ) ï + s. The substitution = H + ,! leads to a Riccati equation with respect to  =  ( H ): (ñ +

[− !H


ïƒ÷

2

+ ô )ñó ò + ù
2, Solution: 2 +  + +1

+(

¡

ïƒ÷ –1 ñ +1

¡

− , ) H + ° − (, ] ‘
= ,! +

ü ïƒú

+ 2( #* 

+ +

õ

2

+ (  + , ) + H + ( .

= 0. + ' ) = L .

(ñ + ô ï ÷ +1 + õ ï ÷ )ñ ó ò = ( ôù ï ÷ + ö ï ÷ –1 )ñ . The substitution =   (¨ −  ) leads to a Bernoulli equation with respect to  =  (¨ ): [−B ¨ 2 + ( 'B + ( )¨ − ˜( ]
? = ¨  +    2.

ï ññóò

=

ôñ

2

+

õÿñ

+ö

ïƒ÷

+ s.

 The transformation ˆ =   , ¨ = −  −  leads to an equation ¨*¨3Š
= ¨ + #*ˆ + $Iˆ + , where  # = −  −2 ° , $ = −  −2 ( , = (  − B )  (see Subsection 1.3.1). ¡ −

7.

ï ññ óò

= –ùñ 2 + ô (2ù + 1) ï ñ + õÿñ – ô 2 ù ï 2 – ôƒõ ï + ö . The substitution = ¨ +  leads to a Bernoulli equation with respect to  (−B ¨ 2 +  ¨ + ( )
? = ¨  +  2 .

=  (¨ ):

Page 132

133

1.3. ABEL EQUATIONS OF THE SECOND KIND

8.

9.

10.

11.

2 ï ññó ò = (1 – ù )ñ 2 + [ ô (2ù + 1) ï + 2ù – 1]ñ – ô 2 ù ï 2 – õ ï – ù . The transformation  = ˆ 2 , = ˆa + ˆ 2 + 1 leads to a Riccati equation: (−B/a 2 + 2 B −  )ˆ g
= ˆ 2 + aֈ + 1. (
ï ñ –
]üIñ + ( ï – (mü )ñ„ó ò = =]ñ 2 + F ï ñ + ( ( – F}ü )ñ . The transformation  = ¨ + , , = ˆ ¨ leads to a linear equation with respect to ¨ [( L − # )ˆ 2 + c\ˆ ]¨2Š
= #*ˆ ¨ + $ .

[(3 ô ï + s)ñ + (4 + 3s) ï ]ñ„ó ò = 2 ôñ 2 + 2(3 + s)ñ + 2 ï . The substitution ¨ =  2 + (3 Œ + ° ) +  leads to an Abel equation of the form 1.3.3.3: 2¨*¨2
= (7  + 5  )¨ − 3  2 3 − 2 ( 2 − 3  2 , where  = ° + 2 Œ , ( = 12  (13 Œ + 6 ° ).



ï

[(4 ô

+ s)ñ + (4 + 3s) ï ]ñ

The substitution ¨ = 34  2¨*¨2
= (7  + 5  )¨ − 3 



12.

(2
ï ñ

14.

15. 16.

17.

ï

+ ôñ + õ The substitution [ #2H

13.

2

19.

2

óò

=

3 2

ôñ

2

+ 2(3 + s)ñ + 2 ï .

+ (3 Œ + ° ) +  leads to an Abel equation of the form 1.3.3.3: − 2 ( 2 − 3  2 , where  = ° + 2 Œ , ( = 18  (60 Œ + 25 ° ).

2 3

+ ö )ñ ó ò =
ñ 2 +
]ü 2 ï 2 + þxñ + ü ( ôƒü + õ – þ )ï + s. =H +! ,  leads to a Riccati equation with respect to  =  (H ): +(

− , ) H + ° − (, ] ‘
= 2 #3,!

¡

2

+ (2 #2H + , +  ) + !H + ( .

+ 1) ï þ –1 ï 1–þ 2 Sñó ò = ñ + ñ + . Q þ +3 2 þ +3 −1 1− ¡ 2+ ¡ The substitution ¨ = +  leads to an equation of the form 1.3.3.4: 2 +3 2ï

ñ

+ (1 – þ

2(þ

)
ñ –

¡ ¨*¨
= [(3 − ) − 1]¨ + ( − 1)(  ¡ ¡

3

2

−

−

),

where

 = # − 2( + 1)( + 3)−2 . ¡ ¡

(2 ôñ + õ ï )ñ ó ò = ô (2 – þ )ñ 2 + õ (1 – þ ) ï ñ + ö ï 2 +
ï ú +2 . The transformation H =   , ¨ = −#* + +  H 2 +  H − ( leads to a separable equation: ¡ ¡ ¨*¨ ‘
= (2 !H +  )( H 2 +  H − ( ).

ï

(ï ñ

+ï

¡

¡

+ ô ) ñ óò = ñ + ï ñ

¡

+õ . Solution: ( + ) +  +  = L ( ' −  2

2

2

)2 .

(2
ï ñ + ( ï 2 + õ )ñó ò =
ñ 2 + ü (
]ü + ( ) ï 2 + ö . The substitution = H + ,! leads to a Riccati equation with respect to  ‘ ( #2H 2 + ( − , )
= (2 #3, + $ ) 2 + 2 #2´ H  + . +( ï 2+ü The substitution

(


ï ñ

=  ( H ):

ï )ñ ó ò

= F ñ 2 + G ï ñ + H ï 2 + üIñ . = "H leads to a linear equation with respect to  =  ( H ): [( c

18.

= ¨ ( ):

− # )H

2

+ ( « − $ ) H +  ] ‘
= ( #2H + $ ) + ,

(
ï ñ + ( ï 2 + ü ï )ñ ó ò =
ñ 2 + ( ï ñ + (
]õ + ü )ñ + (mõ ï + õ ü . This is a special case of equation 1.3.4.22. Solutions: = L3 −  and #

(2
ï ñ + ( ï 2 + ü ï )ñ ó ò The substitution = [− #*ˆ 2 + ( L − $ )ˆ + c ]

=


ñ

2

+

= ï ñ

+

F ï

2

+

üIñ

–

=B ï

–
I

2

+ $I + , = 0.

– ü' .

ˆ + ý leads to a linear equation with respect to  =  (ˆ ): Š
= (2 * # ˆ + $ ) + 2#*ý + , .

Page 133

134 20.

21.

22.

FIRST-ORDER DIFFERENTIAL EQUATIONS (
ï ñ + ( ï 2 + ü ï )ñó ò =
ñ 2 + = ï ñ + F ï 2 + ( ü –
I )ñ – =B ï – ü' . The substitution = ˆ +ý leads to a linear equation with respect to  =  (ˆ ): [( L − $ )ˆ + c ] Š
= ( #*ˆ + $ ) + #*ý + , .

(
ï ñ +
]üIñ + ( ï 2 + (mü ï )ñó ò = =]ñ 2 + F ï ñ + ü ( F – ( )ñ . The transformation  = ¨ − , , = ˆ ¨ leads to a linear equation with respect to ¨ [( L − # )ˆ 2 + ( c − $ )ˆ ]¨2Š
= ( #*ˆ + $ )¨ − ,$ . (
ï ñ + ( ï 2 + ô Jacobi equation.

1

ï

õ 1ñ

+

+

ö 1 )ñ óò

=


ñ

2

+

( ï ñ

+ô

2

ï

+

õ 2ñ

ö

+

= ¨ (ˆ ):

2.

1 b . With the help of the transformation  =  ¯ + ƒ , = ¯ + ý , where ƒ and ý are the parameters which are determined by solving the algebraic system 2

#2ƒ…ý + $Cƒ

+  1ƒ +  1ý + (

1

2

#*ý

= 0,

+ $Cƒ…ý +  2 ƒ +  2 ý + (

= 0,

2

we obtain the equation




( #  ¯ ¯ + $  ¯ 2 +  ¯ 1  ¯ + ¯ 1 ¯ ) ¯  ¯ = # ¯ 2 + $  ¯ ¯ +  ¯ 2  ¯ + ¯ 2 ¯ ,

 where  ¯ 1 = 2 $Cƒ + #*ý +  1,  ¯ 2 = $Iý +  2 , ¯ 1 = #2ƒ +  1 ,  2 = 2 #*ý + $Cƒ +  2 . The transformation 

H = ¯  ¯ , = 1  ¯ leads to a linear equation:   [ ¯ 1 H

+ (¯ 1 − ¯ 2 ) H −  ¯ 2 ] ‘
= ( ¯ 1 H +  ¯ 1 ) + #2H + $ .

2

2 b . The original equation can be also rewritten in the form (


− )(B 3  +

¡

+ , 3 ) − 
(B 1  +

3

1

¡

+ , 1) + B 2 +

¡

2

+,

2

= 0.

The solution of this equation in parametric form can be obtained from the solution of the following system of constant coefficient linear differential equations:

using the formulas  (a ) = 

23.

(
ï ñ

+( ï

2 + ô ñ +õ ï

24.

25.

+ , 1 3, 2  2 + , 2 3, 3  2 + , 3 3,

and (a ) = 

1 2

¡

 . 2 3

− , ) H + ° − (, ] ‘
= ( #3, + $ )

¡

2

+ ( #2H + , +  ) + !H + ( .

(2
ï ñ + ( ï 2 + ôñ + õ ï + ö )ñ ó ò =
ñ 2 + ü (
]ü + ( ) ï 2 + ôƒüIñ + õ ü ï + s. The substitution = H + ,! leads to a Riccati equation with respect to  =  ( H ):

‘ + ° − (, )
= (2 #3, + $ )

( #2H

2

2

+(

2

+ (2 #2H + , +  ) + !H + ( .

(2
ï ñ –
]ü ï 2 + ôñ + õ ï + ö )ñó ò =
ñ 2 + þxñ + ü ( ôƒü + õ – þ )ï + s. The substitution = H + ! ,  leads to a Riccati equation with respect to  =  (H ): [ #2H

26.

3



+ ¡ 1+ ¡ 1+ 1

+ ö )ñó ò = ü/
ï ñ + ü/( ï 2 + þxñ + ü ( ôƒü + õ – þ ) ï + s. = H + ,! leads to a Riccati equation with respect to  =  ( H ):

The substitution [(

1 

(  1 ) g
= B 1  (  2 ) g
= B 2  (  3 ) g
= B 3 

− , ) H + ° − (, ] ‘
= #3,!

¡

2

+ (2 #2H + , +  ) + !H + ( .

(2
ï ñ + ( ï 2 + ôñ – ôƒü ï + õ )ñ ó ò =
ñ 2 + ü (
]ü + ( ) ï 2 + þxñ – þ ü ï + s. The substitution = H + ,! leads to a Riccati equation with respect to  =  ( H ): [ #2H

2

+(

¡

− , ) H + ° − , ] ‘
= (2 #3, + $ )

2

+ 2 #2H´ + !H + ( .

Page 134

135

1.3. ABEL EQUATIONS OF THE SECOND KIND

27.

ï ñ

(2


ï

+(

2

+

2

28.

ï ñ +( ï

[


õï

+


ñ

( )ï

+ ü (
]ü +

2

‘ + (  − , ) H + ° − (, ]
= (2 #3, + $ )

+ ( ü – 1)


2

+ ö )ñó ò =

2

õÿñ

+

+ ôƒü

2

ï

+ s.

= H + ,! leads to a Riccati equation with respect to  =  ( H ):

The substitution [ #2H

ôñ

ôñ

– (
]õ

ü + ( ô )ï ]„ñ óò

=
ñ

2

+ (2 #2H + , +  ) + !H + ( .

2

+(

ï ñ

– (
]õ + (

ôƒü )ñ

ï

+ ( ü – 1) (mõ

.

This is a special case of equation 1.3.4.22. Solution in parametric form:

a + #3L3a  , a +L

 =

'a − $çL3a  . a +L

=

The solution can be presented in implicit form as well:

L  (#

+ ö )ñ + (1 – ù ) ï

ï

29.

[( ô

30.

[( ï + ö )ñ + (ù + 1) ï

+ $I )  + [ # (  − ) + $ (  −  )] 

2

2

– ô (2ù + 1) ï + ô

1.3.4.8:

‘
2 H ¨*¨ = (1 − B )¨

32.

(

ñ

.

+ 2ï

2

− ' ) = 0.

The substitution ¨ =  +  leads to an equation of the form 1.3.4.8: 2!¨*¨
= (1 − B )¨ 2 + [  (2B + 1) + 2B − 1]¨ −  2 B 2 −  − B , where  = (2B − 1)  − ( . 

The transformation H =

31.

+ (2ù – 1) ï – ù ]ñ„ó ò = 2 ôñ

−1

ï

ï ñ

(2 ô

2

3B − 1 1 ¨ , B −1

=

2

ù ]ñ óò

=

+ 3)ï

ñ

2

3ù – 1

– õ (þ

2

2

ï

+ô

2

1

ï

+ ô 0 )ñ +

õ 2ï

2

+

õ 1ï

+

2

+ 2ï

− ˜H − B ,

ïwú

+ 2)ñ + ö

ñ

.

õ 0 ]ñ ó ò

=

ö 2ñ

2

 =

(3B − 1) ( + B (2B + 1) . B −1

.

The transformation H =  , ¨ = − (˜ + +1 +  ( + 1) ¡ equation: ¨*¨3‘
= ( + 1)2 (2 !H +  )( !H 2 + ˜H ). ¡

[( ô

ñ

3B − 1  B + leads to an equation of the form B −1 B −1

+ [  (2B + 1) H + 2B − 1]¨ −  2 ± B H

+ õ )ñ„ó ò = – ô (þ

2ù

+

2 2

+ (

ö 1ñ

+

ö

+ 1)

¡

leads to a separable

0.

This is a Riccati equation with respect to  =  ( ). The substitution  = −

(

2

leads to a second-order linear equation:

 2 ¨

− [(  2 )
+  1  2 ]¨
+  0  22 ¨ = 0,    33.

ï

[(12 ô

2 2

+ 1)ñ + 4 ö

ï

– 5õ

2

ï ]ñ óò

= –2 ï (3 ô

ñ

2 2

(

2

!© + ˜© ; +( 1 +( 0

2

+( 2

1

+

+( 2

0

¨2
¨ 

™ = 1, 2, 3.

+ 2 ö^ñ + 3 õ 2 ).

The substitution ¨ =  (3  2 2 + 2 ( + 3  2 ) leads to an Abel equation of the form 1.3.3.3: 2¨*¨2
= (7  + 5  )¨ − 3  2 3 − 2 ( 2 − 3  2 .

ï [(þ

34.



– 1)(


Solution: #* 35.

ï

– 7ô

ž© =

where



2

ï

2

(2 ô

ï ñ

ï

+ ( )ñ + þ + c\

2

+ õ )ñó ò = –4 ô

(F

ï

+ G ï + H )]ñó ò = [
(1 – ù )ï – (­ù ]ñ 2

+ «p + $

ï 2ñ

2

– 3õ

+  = L3  + .

ï ñ

The transformation H =  , ¨ = 2  ¨*¨2‘
= 2(2 !H +  )(2!H 2 + 2 ˜H − , ).

+

öï

2 2

2

+ü .

+ 2 '

2

+ [ F (2 – ù )ï

− (˜

2

−,

2

+

G

(1 – ù ) ï – H

ù ]ñ

.

leads to a separable equation:

Page 135

136 36. 37.

38.

FIRST-ORDER DIFFERENTIAL EQUATIONS ( ï ñ + ô ïƒ÷ + õ ï 2 )ñó ò = ñ 2 + ö ïƒ÷ + õ ï ñ . The transformation a =   , H =   −2 leads to a linear equation: ( ( − a ) H g
= (B −2)(!H + a +  ). (2 ô ïƒ÷ ñ + õ )ñó ò = – ô (3ù + þ )ïw÷ ñ 2 – õ (2ù + þ )ñ +
The transformation H =   , ¨ = − #*  + + + (B + )( !  H ( ¡ ‘
* ¨ ¨ 2 2 equation: = (B + ) (2 !H +  ) O !H + ˜H − . ¡ B + P

ï

ó

ññóò

ó

ïwú 2

+

ö ï –÷

. + ˜H ) − ( leads to a separable

¡ó

= –ùñ 2 + ô (2ù + 1)  ñ + õÿñ – ô 2 ùJ 2 – ôƒõ + ö . The transformation  = ln ˆ , = ¨ + ˆ leads to a Bernoulli equation with respect to ˆ = ˆ (¨ ): (−B ¨ 2 +  ¨ + ( )ˆ ?
= ¨ ˆ + ˆ 2 .

1.3.5. Some Types of First- and Second-Order Equations Reducible to Abel Equations of the Second Kind ]

Notation:  , , % ,  , Y , j , ` ,  , and 4

are arbitrary functions of their arguments.

1.3.5-1. Quasi-homogeneous equations. 1 b . Let us consider a quasi-homogeneous equation of the form

 ( Á ) Á

+1


+ ( Á ) + #*  = 0.

In the special case Œ = 0 this equation is homogeneous. Á , ¨ = #*  + ( H ) − @H! ( H ) leads to an Abel equation: The transformation H = 

¨*¨ ‘
= [−( Œ + @ )  + ‘
− @H! ‘
]¨ + Œ" ( − @H! ).

2 b . A quasi-homogeneous equation of the form

 ( Á ) Á


+ ( Á ) +    [ % ( Á ) Á +1 
+  ( Á )] = 0 Á , =  −   to an Abel equation:  can be reduced by the transformation H =  , [ (H ) − @H! ( H )] +  (H ) − @H!% (H ) . ‘
= Œ" (H ) 2 + Œ"% (H ) . +1

1.3.5-2. Equations of the theory of chemical reactors and the combustion theory. In the theory of chemical reactors and the combustion theory, one encounters equations of the form



 −  
=  ( ). The substitution ¨ ( ) = 
 leads to the Abel equation ¨*¨
− ¨ =  

are given in Subsection 1.3.1.

−2

 ( ), whose solvable cases

1.3.5-3. Equations of the theory of nonlinear oscillations. 1 b . Let us consider equations of the theory of nonlinear oscillations of the form



 + Y ( ) 
+ = 0.

The substitution H ( ) = 
leads to the Abel equation

HH
+ Y ( ) H + = 0,  which is reduced, with the aid of the substitution U = 12 (  − 2 ), to the following form: Yu A £  − 2U  HH
= (U ) H + 1, where (U ) = A . £  − 2U *

(1)

(2)

Specific cases of equation (2) are outlined in Subsection 1.3.2.

Page 136

137

1.3. ABEL EQUATIONS OF THE SECOND KIND

2 b . An equation of the theory of nonlinear oscillations of the form



 + ` ( 
) + = 0 can be reduced by the transformation H = 
 , ¨ = − − ` (
 ) to an Abel equation of the form (1): ¨*¨ ‘
+ `
‘ ( H )¨ + H = 0. 1.3.5-4. Second-order homogeneous equations of various types. 1 b . A homogeneous equation with respect to the independent variable has the form



 =  ( ) 
+  ( ). The substitution ¨ ( ) =  
leads to an Abel equation: ¨*¨3
= [ ( ) + 1]¨ +  ( ).  2 b . A generalized homogeneous equation  

 = (   ) 
+  −  −1  (   ) 

2

can be reduced by the transformation a =   , ‡ =   ( 
 + ,

) to an Abel equation:

‡"‡ g
= [ (a ) + 2 , + 1]‡ +  (a ) − ,!aÝ (a ) − , ( , + 1)a . To the Emden–Fowler equation, discussed in Section 2.3, there correspond (a ) = 0,  (a ) = #*a + , and , = +  +2 −1 . 3 b . A generalized homogeneous equation

 

 =  9 : O can be reduced by the transformation ‰ =




P

+ 


, ¨ =  9

−2

+2

4O

:

−1




P

to an Abel equation:

[  (‰ )¨ + 4 (‰ ) + ‰ − ‰ 2 ]¨ K
= [(ý − 1)‰ + ƒ + 2]¨ . To the generalized Emden–Fowler equation, discussed in Section 2.5, there correspond ƒ = B − 0 , ý = + 0 ,  (‰ ) = #*‰ - , and 4 (‰ ) = 0.

¡

1.3.5-5. Second-order equations invariant under some transformations. 1 b . An equation invariant under “dilatation–translation” transformation has the form

 The transformation



 =  9  :   ( 
) +  −2 ( 
). =  
, ‡ =   9 +2  :   leads  to an Abel  equation:
[  ( )‡ + ( ) + ]‡ û = (ý + ƒ + 2)‡ .

2 b . An equation invariant under “translation–dilatation” transformation has the form



 =  9  The transformation ˆ = 
 , ¨ =  9 :


  : *O −1

P

+ rO


 P

.

leads to an Abel equation:

[  (ˆ )¨ + (ˆ ) − ˆ 2 ]¨ Š
= [(ý − 1)ˆ + ƒ ]¨ .

Page 137

138

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.4. Equations Containing Polynomial Functions of í 1.4.1. Abel Equations of the First Kind âäã = å 3 ( æ )â 3 + å 2 ( æ )â 2 + å 1 ( æ )â + å 0 ( æ ) 1.4.1-1. Preliminary remarks. 1 b . If 0 = 0 ( ) is a particular solution of the equation in question, the substitution reduces it to an Abel equation of the second kind:

¨*¨ 
= −(3 

2 3 0

+ 2

2 0

+  1 )¨

2

− (3 

−

0

= 1 ¨

+  2 )¨ −  3 ,

3 0

which is discussed in Section 1.3. For  0 ( ) ≡ 0, we can choose

0

≡ 0 as a particular solution.

2 b . The transformation 2

ˆ = X  3«

M

 ,

‡ = O

+



3

2

P

3

−1

«

,

 2 « = exp X O 1 − 2  S , Q 3  3 PuM

where

brings the original equation to the normal form:

‡ Š
= ‡

3

+ ` (ˆ ),

1

` =

where

 3«

3

Q

 0+

1  M O 3  

M

2 3

P

−

 1  2 2  23 S . + 3 3 27  32

1.4.1-2. Solvable equations and their solutions. 1.

2.

3.

ñóò

=

ôñ

3

ñóò

= –ñ

3

ñ óò

= –ñ

3

+

õ ï –3 ) 2 .

This is a special case of equation 1.4.1.9 with B = −1 2. + 3ô

2

ï 2ñ

ï

+ (ô

6.

ï

3

+ô .

+ õ )ñ 2 .

= −1 ‡ leads to an Abel equation of the form 1.3.2.1: ‡"‡
 = (  +  )‡ + 1.

ñ óò

= –ñ

3

ï

+ (ô

+ õ )–2 ñ 2 .

= −1 ‡ leads to an Abel equation of the form 1.3.2.2: ‡"‡
 = (  +  )−2 ‡ +1.

The substitution 5.

3

The substitution = 1 ‡ +  leads to an Abel equation of the form 1.3.2.1: ‡"‡
 = 3 ‡ + 1.

The substitution 4.

– 2ô

ñóò

= –ñ

3

ñóò

=

ôñ

3

+ (ô

ï

+ õ )–1 )

ñ

2 2

.

The substitution = −1 ‡ leads to an Abel equation of the form 1.3.2.4: ‡"‡
 = (  +  )−1 J 2 ‡ +1. + 3 ôƒõ

ï ñ

2

–

õ

– 2 ôƒõ

ï

3 3

.

This is a special case of equation 1.4.1.10 with B = 0 and 7.

ñóò

=

ô ï ñ

3

+

õÿñ

2

.

ñóò

=

ô ï ñ

3

+ 3 ôƒõ

ï 2ñ

= 1.

leads to a separable equation: ‡
 = ‡

The substitution ‡ =  8.

¡

2

– õ – 2 ôƒõ

ï

3 4

3

+ '‡

2

+‡ .

.

This is a special case of equation 1.4.1.10 with B =

¡

= 1.

Page 138

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

9.

11.

= ô ï 2÷ +1 ñ 3 + õ ï –÷ –2 . The substitution ¨ =   +1 leads to a separable equation:  ¨3
=  ¨ 3 + (B + 1)¨ +  . For  = − 13 (B + 1) # −2 and  = 23 # (B + 1), the solution is written in parametric form:



18.

19.

20.

exp(−  ),

U*P

 = U − 13 ln |U + 13 | + L .

where

= ô ïƒ÷ ñ 3 + 3 ôƒõ ïƒ÷ +ú ñ 2 + ö ï ñ – 2 ôƒõ 3 ïƒ÷ +3ú + õ^ö ïƒú +  – õÿþ The substitution ¨ = + ' + leads to a Bernoulli equation: ¨ 
=   ¨

ï ñóò

17.

1

ï 1–ú

3

ñ

+1 3

–2

ú

− 3  2  

+2

+ ¨ .

–1

. +( (˜  −3  2  

+2

+ )¨ .

(9 ô + 2 + 9 õÿþ ïwú –1 )( ô ï 1–ú + õ )–  –2 . 1 1 3 − For Œ = , the substitution = O ¨ +   1− + +    leads to the Abel 3  (1 − )  + ' + P equation ¨*¨2
=¡ ¨ +  + ' + , which is discussed in Subsection 1.3.1. + õ )2 

–ï

3

ïƒú

ñóò

13.

16.

= − #þO 1 +

,

= ô ïƒ÷ ñ 3 + 3 ôƒõ ïƒ÷ +ú ñ 2 – õÿþ ïƒú –1 – 2 ôƒõ 3 ïƒ÷ +3ú . The substitution ¨ = + ' + leads to a Bernoulli equation: ¨ 
=   ¨

9ñó ò = – ïƒú ( ô

15.

B +1P

ñóò

12.

14.

139

ñóò

 = exp O 10.

Ï

= ô ï 4 ñ 3 + ( õ ï 2 – 1)ñ + ö ï . The substitution ¨ =  leads to a separable equation: ¨p
=  (  ¨

3

= ôñ 3 + 3 ôƒõ ïƒ÷ ñ 2 – õÿù ïƒ÷ – 2 ôƒõ 3 ï 3÷ . ¨ 
=  The substitution ¨ = + '  leads to a Bernoulli equation: u

ï ñóò

= ô ï 2÷ +1 ñ 3 + ( õ ï – ù )ñ + ö ï 1–÷ . ¨ 
=  ¨ The substitution ¨ =   leads to a separable equation: 3

ï ñ óò

= ô ï ÷ +2 ñ 3 + ( õ ï The substitution ¨ = 

÷

ï ñ óò

– 1)ñ + ö ï ÷ –1 . ¨ 
=   leads to a separable equation: p

ï 2 ñ óò

= ñ 3 – 3ô 2 ï 4 ñ + 2ô 3 ï 6 + 2ô ï 3. The transformation  = 1 ˆ , =    2 + 1 ¨ ¨*¨2Š
= 3 ˆ −2 ¨ + 1.

3

−1

+  ¨ + ( ). −1

¨

3

− 3  2  2 

−1

¨ .

+ ¨ +( .

( ¨

+  ¨ + ( ).

3

leads to an equation of the form 1.3.2.2:

= –( ô ï + õ ïƒú )ñ 3 + ñ 2 . The substitution = −1 ¨ leads to an equation ¨*¨u
= ¨ +  + ' + , which is discussed in Subsection 1.3.1.

ñóò

= (
ï 2 + ( ï + = )–1 ) The substitution = −1  ¨ ( #* 2 + $I + L )−1 J 2 .

ñóò

ñ

+ ñ 2. leads to an Abel equation of the form 1.3.1.63: ¨*¨I
= ¨

2 3

6  = – ï –16 ) 9  ô ï – 25 ñ Solution in parametric form:

)

ñóò

34 9 3

6 U 25 

 =A where The function

2

<

=T

,

<

«

1

+

2 27

 9ô

ï

125  ( !« 108

= U 2
«

ï 

)

ô

ï

–

2 25

1)

−25 18 7 9 7 18

2

–11 18

J U J < J

=U

ü

A (4 <

= < (U ) is defined implicitly by U = X

3

)

6 –61 18 . 25



–

−1 1

«

−

«

−1 2 (18


1

A 5 « 2 ),

− 1) + 2 < .

M

A (4 <

<

− 1) ü the formulas corresponds to the classical Weierstrass elliptic function 3

− L . The upper sign in

<

= < (U + L , 0, 1).

Page 139

140 21.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

= –ñ

3

ó

+ ô

 ñ

2

ñ óò

= –ñ

3

+ 3ô

ó

 3   ¨ + 1.

ñóò

ó

  ñ

= – 13

ó

– 2ô 3  3  1 = ¨ +  

  ñ

2 2

The substitution

23.

 leads to an Abel equation of the form 1.3.2.7: ¨*¨I
=   ¨ + 1.

= −1 ¨

The substitution 22.

.

–1 2

3

 

2 3

+

2 –

ó

+ ô



ó

.

¨*¨p
=

leads to an Abel equation of the form 1.3.2.7:

.

Solution in parametric form:



 = 24.

ñóò

=

ô 2 ó ñ

3

Œ

+

õ ó ñ

25.

ñóò = ô ó ñ

3

ô ó ñ

3

ñóò

=

ñóò

=

ô ó ñ

3

+ 3 ôƒõ + 3 ôƒõ

ñóò

=

ô ó ñ

3

+ 3 ôƒõ

The substitution ‡ = 29.

óñ

ñóò

= ô

ñóò

= [ ô + õ exp(2 ï

3

+ 3 ôƒõ

ñóò

= – 23 ô

ï

–1

‡"‡
 = ‡ + (6  )−1 .

33.

2

+ 

 

( + )

+^ ö ñ – 2ôƒõ  ó 3

óñ

2

+

õ^ö

3

+ '‡

 

( + )

 +  

 

óñ

2

3

ï 2)ñ

3

ó

– õ C



ó

+ 2 ôƒõ

  

2 ( +2 )

óñ

– õ C



ó

2

+ “ñ – 2 ôƒõ

2

+ [(3 ôƒõ 2 + ö ) 

M



+ ( ( − 3  2   )‡ .

  

3 ( +3 )

ó

 

( +2 )

ó

3

− 3  2  ( 

3

−  2  ( 

 ‡

+2 )

.

.

 leads to a Bernoulli equation: ‡
 =   ‡

 +  

Èô )]ñ

  

3 ( +3 )

3

 leads to a Bernoulli equation: ‡
 =   ‡

óñ

óñ

– 2 ôƒõ

+ ( ( + Œ )‡ + .

2

.

.  leads to a Bernoulli equation: ‡
 =    ‡



 +  

( + )

exp(2 ô

The substitution

32.

.

+ s]ñ + õ ( ôƒõ 2 + ö ) 

3

+ V “ −3  2  ( 

 

( +3 )

 leads to a Bernoulli equation: ‡
 =    ‡

3

ó +õ

 ‡

+2 )

.

.

  W˜‡

+2 )

(s – ) 

 + [(    ( +2 )



ó

. .

+ ° ]‡ .

+ ñ 2.

= −1 ‡ leads to an equation of the form 1.3.1.8: ‡"‡
 = ‡ −  −  exp(2  ).

The substitution 31.

 = U − 13 ln |U + 13 | + L .

where

leads to a separable equation: ‡
 = ‡

 

The substitution ‡ = 30.

 −Ž ,

ý – ó

+ ö^ñ +

( + )

The substitution ‡ = 28.



U P

 +  leads to a Bernoulli equation: ‡
 =   ‡

The substitution ‡ = 27.

−

ó + 3 ôƒõ ñ

The substitution ‡ = 26.

2

=‡  

The substitution

1

= −Œ O 1 +

,

= O

+

2

) exp(– ô

ï

2

).

exp(−  2 ) leads to an equation of the form 1.3.1.16:

P

= – ô exp(2ô

ñóò

= –ô

ï

–2

exp(2 ô −2

+ (1 – 2 ô

ï 3 )ñ = O

The substitution

‡"‡
 = ‡ + (9  )−1 

3

.

3

ï

ô ï

) exp(– ô ï 3). 2 The transformation ˆ =  2 , = O +  exp(−  3 ) leads to an equation of the form 3 ‡ P 1.3.1.32: ‡"‡ Š
= ‡pA 2(9  )−1 ˆ −1 J 2 .

ñ óò

ï 3 )ñ

1 2 ‡

4 3

+ (1 –

3

+ 2 ï (1 – ô ï 3 ) exp(– ô ï 3 ). 1 +  2 exp(−  3 ) leads to an equation of the form 1.3.1.33:

3 ‡

P

Page 140

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

34.

= – 29 ô –1 ï –1 ) 2 exp(2 ô ï 3 ) 2 )ñ Solution in parametric form:

ñóò

J :

−4 3

 = 'U where

−2

:

ø

J

2 −3 2 , 3

 =T



’

ø

−1

1

=U

:
*

’

−1 2 2 (

U

+

1 3

:

: ,

3

T\’ 1 ’ 2 ) exp  A ’

2

=’

2 1

ACU

4

– 9ô

:

2

2 3

U

2

,

−2

:

−3

’

3 1

,

J U ) + L 2 n 1 J 3 (U ) for the upper sign, J U ) + L 2 ; 1 J 3 (U ) for the lower sign, 1 J 3 ( U ) and n 1 J 3 ( U ) are the Bessel functions, and ® 1 J 3 ( U ) and ; 1 J 3 ( U ) are the modified Bessel =

™

L

L

J :

−2 3



141

ô ï –1 ) 2 (2ô ï 3 ) 2 – 1) exp(–ô ï 3 ) 2).

3 4

+

1 =− U

’ 12 ,

1 1 3(

®

1 1 3(

functions. 35.

3

Ï

ñóò

= –ô

ï

–3

(ï

2

ï

– ô )4 ) 3 exp O

Solution in parametric form:

 =

£ 2  , U

=−

2

ñ

3ô P

3 J U J

3

+

1 27 ô

2

ï 2 (ï

2

– ô )–13 ) 6 (2 ï

ï

£ 2 4(U + 1)  2 + 4U 2  − 3U 2 7 J 6 2(U + 1)  − U 2  −U )

2 3 2 3 (2

2

+ 27 ô 2 ) exp O –

4

exp O −



3U

2

P

ï

2

6ô P

.

,

where  = U − ln |1 + U | + L . 36.

ñóò

=

ôñ

3

+ õ cosh(

ï )ñ

The transformation a =

¨

37.

ñóò

2

− 2Œ a ¨ −  .

=

ôñ

3

+ õ sinh(

ï )ñ

The transformation a =

¨

38.

39.

40. 41. 42.

43.

2

− 2Œ a ¨ +  .

2

. 1

+

 Œ

sinh( Œ  ), ¨

= ž

cosh( Œ  ), ¨

= ž



¨ g
= leads to a Riccati equation: 2  3

2

. 1

+

Œ





¨ g
= leads to a Riccati equation: 2  3

= –ñ 3 + 3 ô 2 cosh2 ï ñ – 2 ô 3 cosh3 ï + ô sinh ï . The substitution =  cosh  + 1 ¨ leads to an Abel equation of the form 1.3.2.9: ¨*¨2
= 3  cosh  ¨ + 1.

ñ óò

= –ñ 3 + 3 ô 2 sinh2 ï ñ – 2 ô 3 sinh3 ï + ô cosh ï . The substitution =  sinh  + 1 ¨ leads to an Abel equation of the form 1.3.2.10: ¨*¨2
= 3  sinh  ¨ + 1.

ñ óò

= –ñ 3 + ô cos( ï )ñ 2 . The substitution = −1 ‡ leads to an Abel equation of the form 1.3.2.11: ‡"‡
 =  cos( Œ  )‡ +1.

ñóò

= –ñ 3 + ô sin( ï )ñ 2 . The substitution = −1 ‡ leads to an Abel equation of the form 1.3.2.12: ‡"‡
 =  sin( Œ  )‡ +1.

ñ óò

= –ñ 3 + 3 ô 2 cos2 ( ï )ñ + ô sin( ï ) + 2 ô 3 cos3 ( ï ). The substitution = −  cos( Œ  ) + 1 ¨ leads to an Abel equation of the form 1.3.2.11: ¨*¨2
= −3  cos( Œ  )¨ + 1.

ñóò

= –ñ 3 + 3 ô 2 sin2 ( ï )ñ + ô cos( The substitution =  sin( Œ  ) + 1 ¨ ¨*¨2
= 3  sin( Œ  )¨ + 1.

ñóò

ï

) – 2 ô 3 sin3 ( ï ). leads to an Abel equation of the form 1.3.2.12:

Page 141

142

]

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

44.

=

ô!„ñ

3

+ ODõ!"

2

+

= ¨

The substitution

" óò ñ " P

= ! ( ï ),

!

In equations 44–47, the following notation is used:

= " ( ï ),

"

= % ( ï ).

%

+ ö!" 3 .

¨ 
=  2 (  ¨ leads to a separable equation: p

+  ¨ + ( ).

3

ñóò

= !„ñ 3 + 3 !%„ñ 2 + (" + 3 !% 2 )ñ + !% 3 + "/% – %„òó . ¨ 
= (  ) ¨ +  (  )¨ The substitution ¨ = + % ( ) leads to a Bernoulli equation: p

45.

ñóò

46.

=

!

2(

" óò ô " + õ

Solution: X

¨

)3

ñ

3

! óò ñ !

+

+

!"Ióò

3

.

.

¨ 1 + L = ln | ´ +  |, where ¨ = . M 3− ¨ +1   ( ´ +  )

+ õL" ñ – " ó ò ñ – ! óò % + ! + " . +õ P ! –" " –!  +  −  −    Solution: | −  |  | − |  û − = L exp X (  − )2 %  S . û M  + û Q  + û

ñ óò

47.

ô! ô

= (ñ – ! )(ñ – " ) Oiñ –

û

û

1.4.2. Equations of the Form ( M 22 â 2 + M 12 æ0â + M 11 æ 2 + M

0)

âäã = N

â

22

2

+N

æ0â + N

12

11

æ

2

+N

0

1.4.2-1. Preliminary remarks. Some transformations. 1 b . For # 22 = 0, this is an Abel equation (see Subsection 1.3.4). For $  with respect to  =  ( ). 2 b . The transformation H =   ,  [( #

0

H − $ 0) + #

22

H 3 + (#

12

−$

= 22 )

−2

11

= 0, this is an Abel equation

leads to an Abel equation of the second kind: 

H 2 + (#

11

−$

12 )

H −$

11 ]

‘
= 2#

0

2

+ 2( #

22

H 2+#



12

H +#

11 )

.

3 b . The transformation  =  ¯ + ƒ , = ¯ + ý , where ƒ and ý are parameters, which are determined by solving the second-order algebraic system

#

22

2

ý

+#

12

ƒ…ý + #

11

ƒ

2

+#

0

= 0,

$

22

ý

2

+$

12

ƒ…ý + $

11

2

ƒ

+$

0

= 0,

leads to the equation [#

22 ¯

2

+#

 ¯+#

12 ¯



11 ¯

2

The transformation ˆ =  ¯ {[  2 ˆ

2

+ (

1

ý + # 12 ƒ ) ¯ + (2 # 11 ƒ + # 12 ý ) ¯ )] ¯ 
¯ = $ 22 ¯ 2 + $ 12  ¯ ¯ + $ 11  ¯ 2 + (2 $ 22 ý + $ 12 ƒ ) ¯ + (2 $ 11 ƒ + $ 12 ý ) ¯ .  ¯ , ¨ = 1  ¯ reduces this equation to an Abel equation of the second kind: + # 22 ˆ 3 + ( # 12 − $ 22 )ˆ 2 + ( # 11 − $ 12 )ˆ − $ 11 }¨ Š
= (  ˆ +  )¨ 2 + ( # ˆ 2 + # ˆ + # )¨ ,

+ (2 #

−  2 )ˆ −  1 ]¨

22

2

where 

1

= 2#

11

ƒ +#

4 b . The substitution

1

22

12

11

ý ,  1 = 2 $ 11 ƒ + $ 12 ý ,  2 = 2 # 22 ý + # 12 ƒ , and  2 = 2 $ 22 ý + $ 12 ƒ . = a +  , where parameter is determined by solving the cubic equation

12

(#

22

2+#

12

+#

11 )

−$

22

2−$

12

−$

11

= 0,

leads to an Abel equation of the second kind with respect to  =  (a ): [ ÆuaÖ + ( $ where Æ

22

= 2$

−# 22

22

)a 2 + $

+$

12

0

−#

− (2 #

0 22

]  g
= ( #

+#

22

2+#

12

+#

11 )



2

+ (2 #

22

+#

12 )

aÖ + #

22

a 2 + # 0,

12 ).

Page 142

Ï

143

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

1.4.2-2. Solvable equations and their solutions. 1. 2.

3.

4.

(
ñ 2 + ï 2 )ñ Solution: #

óò

= –2 ï ñ + ( 3 − $I 3 + 3(

ï 2

2

+ô . −  ) = L .

(
ñ 2 + ( ï 2 – ô 2 ( )ñó ò = =]ñ 2 + 2 ( The transformation  = ¨ +  , = ˆ ¨ ( #*ˆ 2 + $ )¨ + 2 !$ .

7. 8. 9. 10. 11. 12. 13. 14.

3

+ L3ˆ

2

¨ Š
= + $Iˆ )2

(
ñ 2 – 2
]ü ï ñ + (mü ï 2 )ñó ò = – (­ñ 2 + 2 (mü ï ñ –
]ü 3 ï 2 + ô . The substitution = H + ,! leads to a Riccati equation with respect to  =  ( H ): 2

+  ]


‘ = , ( $ − #3, )

2

+ #2H 2 .

(
ñ 2 + 2 ( ï ñ +
]ü 2 ï 2 )ñ ó ò = (­ñ 2 + 2
]ü 2 ï ñ + (mü 2 ï 2 + ô . The substitution = H + ,! leads to a Riccati equation with respect to  =  ( H ): [( $ − #3, ) H

6.

. leads to a linear equation: (− #*ˆ

(
ñ 2 + ( ï ñ + = ï 2 )ñó ò = F ñ 2 + G ï ñ + H ï 2 . Homogeneous equation. The substitution H =   leads to a separable equation: "H 
= ( #2H 2 + $CH + L )−1 [− #2H 3 + ( c − $ ) H 2 + ( « − L ) H +  ].

[−( #3, + $ ) H

5.

ï ñ

(
ñ 2 + ( ï ñ + = The substitution (
ñ 2 + 2 ( Solution: #

ï ñ

ï

2

+  ]


‘ = 2 , ( #3, + $ )

2

+ 2( #3, + $ ) H´ + #2H 2 .

+ ô )ñ ó ò =
]üIñ 2 + (mü ï ñ + =­ü ï 2 + õ . = H + ,! leads to a Riccati equation with respect to  =  ( H ): 2

‘ (  − , )
= ( #3,

2

+ $ç, + L )

+ F ï 2 + ô )ñ ó ò = – (­ñ 3 − «p 3 + 3( $I 2 + c\

2 2

– 2F +

2

ï ñ

2

+ (2 #3, + $ ) H´ + #2H

+G ï 2+õ . − ' ) = L .

(
ñ 2 – 2
ï ñ + ( ï 2 +
– ( )ñó ò = –
ñ 2 + 2 ( ï ñ – ( ï 2 + This is a special case of equation 1.4.2.21 with  = 1 and  = 1.




(
ñ 2 + 2
ï ñ + ( ï 2 +
– ( )ñó ò =
ñ 2 + 2 ( ï ñ + ( ï 2 –
+ This is a special case of equation 1.4.2.21 with  = 1 and  = −1.

(
ñ 2 – 4
ï ñ + ( ï 2 + 4
– ( )ñ ó ò = –2
ñ 2 + 2 ( ï ñ – 2 ( ï This is a special case of equation 1.4.2.21 with  = 1 and  = 2.

(

–

(

. .

+ 8


2

(
ñ 2 + 4
ï ñ + ( ï 2 + 4
– ( )ñó ò = 2
ñ 2 + 2 ( ï ñ + 2 ( ï 2 – 8
This is a special case of equation 1.4.2.21 with  = 1 and  = −2. (
ñ 2 – 6
ï ñ + ( ï 2 + 9
– ( )ñó ò = –3
ñ 2 + 2 ( ï ñ – 3 ( ï This is a special case of equation 1.4.2.21 with  = 1 and  = 3.

+ 27


2

(
ñ 2 + 6
ï ñ + ( ï 2 + 9
– ( )ñó ò = 3
ñ 2 + 2 ( ï ñ + 3 ( ï 2 – 27
This is a special case of equation 1.4.2.21 with  = 1 and  = −3. 2(
ñ 2 –
ï ñ + ( ï 2 +
– 4 ( )ñó ò = –
ñ 2 + 4 ( ï ñ – ( ï 2 + This is a special case of equation 1.4.2.21 with  = 2 and  = 1.




+ .

– 2( . + 2( . – 3( . + 3( .

– 4( .

Page 143

144

FIRST-ORDER DIFFERENTIAL EQUATIONS 2(
ñ 2 +
ï ñ + ( ï 2 +
– 4 ( )ñó ò =
ñ 2 + 4 ( ï ñ + ( ï 2 –
+ 4 ( . This is a special case of equation 1.4.2.21 with  = 2 and  = −1.

15.

( ôñ 2 – 2 õ ï ñ + ô ï 2 + ôƒõ 2 – ô 3 )ñó ò = – õÿñ 2 + 2 ô ï ñ – õ ï 2 + õ 3 – ô 2 õ . This is a special case of equation 1.4.2.21 with # = 1 and $ = 1.

16.

( ôñ 2 – 2 õ ï ñ – ô ï 2 + ôƒõ 2 + ô 3 )ñ ó ò = – õÿñ 2 – 2 ô ï ñ + õ ï 2 + õ 3 + ô 2 õ . This is a special case of equation 1.4.2.21 with # = 1 and $ = −1.

17.

( ôñ 2 – 2 õ ï ñ + 2 ô ï 2 + ôƒõ 2 – 2 ô 3 )ñó ò = – õÿñ 2 + 4 ô ï ñ – 2 õ ï 2 + This is a special case of equation 1.4.2.21 with # = 1 and $ = 2.

18.

õ

– 2ô

3

( ôñ 2 – 2 õ ï ñ – 2 ô ï 2 + ôƒõ 2 + 2 ô 3 )ñ ó ò = – õÿñ 2 – 4 ô ï ñ + 2 õ ï 2 + õ 3 + 2 ô This is a special case of equation 1.4.2.21 with # = 1 and $ = −2.

2

19.

(
ñ 2 + ( ï ñ + = The substitution

20.

[( c

ï

3

õ

.

+ ô )ñó ò = F ñ 2 + ü (2
]ü + ( – 2 F ) ï ñ + ü (–
]ü 2 + F}ü + = ) ï = H + ,! leads to a Riccati equation with respect to  =  ( H ):

− #3, ) H

(− !#*ˆ

.

2

2

+  − , ] ‘
= ( #3,

( ô
ñ 2 – 2 õ
ï ñ + ô( ï 2 + ôƒõ The transformation  = ¨ +  ,

21.

2

õ

+ ˜#*ˆ

2

2

2

2

+ $ç, + L )

= – õ
ñ 2 + 2 ô( ï ñ – õ( = ˆ ¨ +  leads to a linear equation: –ô




3

( )ñ ó ò

+ !$Iˆ − ˜$ )¨ Š
= ( !#*ˆ

2

+ (2 #3, + $ ) H´ + #2H

ï

2

+

2

+õ .

+ .

õ 3


–ô

2

õ(

.

− 2 ˜#*ˆ + !$ )¨ + 2  2 $ − 2  2 # .

2

1.4.3. Equations of the Form ( M 22 â 2 + M = N 22 â 2 + N 12 æ0â + N 11 æ 2 + N 2 â + N

12 1

æ

æ0â + M

11

æ

2

+M

2

â +M

1

æ )âäã

1.4.3-1. Preliminary remarks. 1 b . For # 22 = 0, this is an Abel equation (see Subsection 1.3.4). For $ with respect to  =  ( ).

11

= 0 this is an Abel equation

2 b . The transformation ˆ =   , ¨ = 1  leads to an Abel equation of the second kind: {[ #

2

ˆ 2 + (#

1

−$

2)

ˆ − $ 1 ]¨ + #

22

ˆ 3 + (#

12

−$

ˆ − $ 11 }¨ Š
= ( # 2 ˆ + # 1 )¨ 2 + ( # 22 ˆ 2 + #

ˆ 2 + (#

22 )

−$

11

12 )

12

ˆ +#

11 )

¨ .

3 b . In Item 3 b of Subsection 1.4.4, another transformation is given which reduces the original equation to an Abel equation of the second kind. 4 b . Dynamical systems of the second-order

M M

 a

=

.

( , ),

M M

a

= Æ ( , ),

(1)

which describe the behavior of simplest Lagrangian and Hamiltonian systems in mechanics, are often reduced to equations of the type in question if

.

( , ) =  ( , )( #

Æ ( , ) =  ( , )($

22 22

2 2

+# +$

12 12

 

+# +$

11 11

 

2

+#

2

2

+$

2

+#

 ), + $ 1  ), 1

(2)

where  =  ( , ) is an arbitrary function.

Page 144

Ï

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

145

In particular, dynamical systems (1) with functions (2) and  ≡ 1 arise in analyzing complex . and Æ are substituted by their Taylor-series equilibrium states. In this case, the functions expansions in the vicinity of the equilibrium state  = = 0 with the first and second order terms retained. Whenever a solution of the ordinary differential equation (#

22

2

+#

12



+#

11



2

+#

+#

2

is obtained in parametric form,  =  (‡ , L (1), (2) is determined by

 =  (‡ , L 1 ),

= (‡ , L

1

 ) 
= $

2

22

= (‡ , L

1 ),

1 ),

12



+$

11



2

+$

2

+$

1



the corresponding solution of system




‡ + L 2. M  (‡ , L 1 ), (‡ , L 1 ) 

.

a =X

1 ),

+$

The last relation defines an implicit dependence of the parameter ‡ on a , ‡ = ‡ (a , L 1 , L 2 ), and makes it possible to establish, with the aid of the first two formulas, the dependence of  and on a . 1.4.3-2. Solvable equations and their solutions. 1.

(ñ 2 – ï 2 + ôñ )ñó ò = ñ 2 – ï 2 + ô Solution in parametric form:

ï

.

 = a + L |a |−1  4g ,

2.

(ñ 2 – ï 2 + ôñ )ñ ó ò = 2ñ 2 – 2 ï Solution in parametric form:

= − a + L |a |−1 

4.

(ñ 2 – ï 2 + ôñ – ô ï )ñ ó ò = ñ 2 – ï Solution in parametric form: (ñ 2 – ï 2 + ôñ + 2 ô ï )ñó ò = ñ Solution in parametric form:

2

ï

– ôñ + ô

–ï

2

2

+ 2 ôñ +

(ñ 2 – ï 2 + ôñ + 2 ô ï )ñó ò = 2 ï Solution in parametric form:

ñ

– 2ï

2

+

4

g

(ñ 2 – ï 2 + ôñ – 2 ô ï )ñó ò = 4ñ Solution in parametric form:

2

– 6ï

ôñ

+ 2ï

ñ

(ñ 2 – ï 2 + ôñ + 3 ô ï )ñó ò = –ñ Solution in parametric form:

2

+ 4ï

ñ

– 3ï

 = 12 a + L |a |−1  −  J g ,

g.

= a + L |a |3 

ï

4

g

.

.

= −2a + L3a 2

2

.

+ 2ô

 = 13 a + L |a |2 J 3   J g ,

7.

ô ï

= − a + L 

,

g  = a + L3a −2  −  J ,

6.

g = L3a 2   J . .

 = a + L  2g ,

 = − a + L |a |3 

5.

g.

+ ôñ .

ñ

 = a + L3a 2   J g ,

3.

4

+ ôñ – 2 ô

ï

g  J .

−2 −

.

g = 23 a + L |a |2 J 3   J . 2

+ ôñ + 3 ô

ï

.

g = − 23 a + L |a |−1  −  J .

Page 145

146 8.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñ 2 – ï ñ + ôñ + ô ï )ñó ò = ï Solution in parametric form:

–ï

ñ

2

ôñ

+

ï

+ô

.

 = −a + L |a |−1   J g ,

9.

10.

(ñ 2 – ï ñ + ôñ + ô ï )ñ ó ò = ñ 2 – ï Solution in parametric form: (ñ 2 – ï ñ + ôñ – 2 ô ï )ñó ò = 3ñ Solution in parametric form:

ñ

g = a + L |a |−1   J .

+ 2 ôñ .

 = − a + L3a 2  g , 2

– 5ï

+ 2ï

ñ

2

g

= L3a 2  .

 = 12 a + L |a |1 J 2   J g ,

11.

(ñ 2 + ï ñ – 2 ï 2 + ôñ + ô ï )ñó ò = Solution in parametric form:

ñ

2

ï ñ

+

12.

(ñ 2 + ï ñ – 2 ï 2 + ôñ + ô ï )ñó ò = 2ñ Solution in parametric form:

–ï

2

(ñ 2 + ï ñ – 2 ï 2 + ôñ – ô ï )ñ Solution in parametric form:

óò

=

ñ

2

ï ñ

+

2

14.

15.

(ñ 2 + ï ñ – 2 ï 2 + ôñ – 2 ô ï )ñó ò = 5ñ Solution in parametric form: (ñ

2

– 2ï

ñ

+ï

Solution:  = 16.

2

+

ôñ )ñ óò

+



=

ôñ

L − ln | |

(ñ 2 – 2 ï ñ + ï 2 + ôñ + ô ï )ñ Solution in parametric form:

18.

ï

.

2

ï

– 2 ôñ + 2 ô

3

+ 2ï

ñ

2

.

g.

+ ôñ – 2 ô

ï

.

g = 12 a + L |a |3 J 4   J .

.

.

óò

= –ñ

 =− +ï

+ô

= −2 a + L 

– 7ï

2

+ 2ï



2 ln |a |

2 + ô ñ – 2ô ï )ñ óò



3 ln |a | = 2(ñ

–ï

ñ

2

+ L3a ,

(ñ 2 – 2 ï ñ + ï 2 + ôñ + 2 ô ï )ñó ò = –2(ñ Solution in parametric form:

(ñ 2 – 2 ï ñ

ôñ

+

 = 14 a + L |a |3 J 4   J g ,

 =− 17.

2

g  .

−2 9

g = − a + L | a |3   J .

– 2ï

 = a + L  3g ,

.

= −2 a + L3a

 = a + L | a |3   J g ,

13.

ï

+ 2ô

2

–ï

ñ

.

g = a + L | a |1 J 2   J .

– 2ï

 = a + L3a −2  9g ,

ï

+ ôñ – 2 ô

2

– 2ï

= +ï

ñ

+ L3a , 2

– 2ï

+ ôñ + ô

2

=

ñ

+

ï



2 ln |a |

2

ï

.

+ L3a .

ï

) + ôñ + 2 ô

.

2 + L3a . 3 ln |a | ) + ôñ – 2 ô

ï

.

Solution in parametric form:

 =



ln |a |

+ L3a ,

=

2 + L3a . ln |a |

Page 146

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

19.

(ñ 2 + 2 ï ñ + ï 2 + ôñ + 2 ô ï )ñó ò = –ñ Solution in parametric form:

 =L 20.

2

O a 1J 3 +

 =L 21.

óò

2

= –ñ

Oa 3 +

4a

óò

2

ÃP

2

–ï

ñ

– 2ï

–ï

ñ

2

a,

= –ñ

– 2ï

2

+ L3a ,

(ñ 2 + 2 ï ñ – 3 ï 2 + ôñ + ô ï )ñó ò = 3ñ Solution in parametric form:

– 2ï

(ñ 2 + 2 ï ñ – 3 ï 2 + ôñ + ô ï )ñó ò = Solution in parametric form:

ñ

+ 2ï

2

–ï

ñ

– 3ï

ñ

 = a + L |a |−1  8g ,

24.

(ñ 2 + 2 ï ñ – 3 ï 2 + ôñ + 2 ô ï )ñó ò = Solution in parametric form:

ñ

 = a + L |a |−3 

25.

(ñ 2 – ï 2 + ôñ + õ ï )ñó ò = ñ 2 – ï Solution in parametric form:

2

+

16

õÿñ

g,

+ô

+  = (  −  )a + L |a |−  −   4g ,

26.

(ñ 2 – ï ñ + ôñ + õ ï )ñ ó ò = ñ 2 – ï Solution in parametric form:

ñ

(ñ 2 + ï ñ – 2 ï 2 + ôñ + õ ï )ñó ò = Solution in parametric form:

ñ

2

+ï

(ñ 2 – 2 ï ñ + ï 2 + ôñ – ôƒõ ï )ñó ò = õ (ñ Solution in parametric form:

 =

ÃP

ôñ

 ≠ 0.

+ L3a ,

 ≠ 0.

ô ï

2

.

ï

+ô

.

g = − 21 a + L3a 2   J .

ï

– ôñ + 3 ô

2

– 3ï

ñ

2

+ 3ô

ï

.

ï

1  + L3a ,  − 1 ln |a |

g.

. 16

g.

.

 +  4g −   ,

ñ

– 2ï

= L |a | 2

 + g   ,

– 2ï

ñ =

ï

+ ( õ – ô )ñ + 2ô

+ï

2

)+

ôñ

 ≠.

 ≠ 0.

– ôƒõ

1  + L3a ,  − 1 ln |a |

.

 +  9g −   ,

= 2(  − 2  )a + L |a |− 2 

2

8

= −3 a + L |a |−3 

+ ( ô + õ )ñ .

+  = (2  −  )a + L |a |− 2  −   9g ,

28.

+

a,

= (  −  )a + L |a |− 

+  = − 'a + L |a |    g ,

27.

2

4a

2

= −3 a + L |a |−1 

+ 2ï

2

Oa 3 +

 ≠ 0.

.

4a 3 +L 3

– 2 ôñ +

2

2

= −L

2

1−

–ï

ñ

 = 12 a + L3a 2   J g ,

23.

3

= −L

ï

+ ôñ – ô

2

.

4a 2 + L3a , 5 P

O a 1J 3 +

2

= −L

147

ô ï

+ 2 ôñ +

2

4a 3 1− +L 3

3

(ñ 2 + 2 ï ñ + ï 2 + ôñ – 2 ô ï )ñ Solution in parametric form:

 =L 22.

4a 2 + L3a , 5 P

(ñ 2 + 2 ï ñ + ï 2 + ôñ – ô ï )ñ Solution in parametric form:

– 2ï

2

Ï

ï

 ≠ 2 .

.

 ≠ 1.

Page 147

148 29.

30.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñ 2 + 2 ï ñ – 3 ï 2 + ôñ + õ ï )ñó ò = Solution in parametric form: (ñ – 3 ï ñ 2

+  = (3  −  )a + L |a |− 3  −   + 2ï

2

+ ô ñ + õ ï )ñ óò

31.

(ñ + 3 ï ñ

32.

16

+ 2ï

– 4ï

2

+ ô ñ + õ ï )ñ ó ò

= ñ + 3ï ñ

– (
+ 1)ï

2

g,

33.

+ ÿõ ñ – 2 õ ï ] ñ óò

# +3

+ L |a | O

(ñ 2 – 2
ï ñ +
2 ï 2 + õÿñ – Solution in parametric form:

36.

38.

– 4ï

2

+ ( õ – 3 ô )ñ + 4ô ï

 + −  

= (
+ 4)ñ – (
+ 6) ï ñ


ñ

a,

 ≠ 3 .

.

 +  −g +   ,

2

 ≠ −2  . .

25

g,

= #3L

3

= (2 –

2

+

õÿñ

– 2õ

ï

.

# ≠ −3.

õï

.

2( # − 1) 3 a + L 2a ,  ≠ 0. 3 )ñ 2 – 2 ï ñ +
ï 2 + õÿñ –
]õ ï .

1+




#*a

g

 ≠ 4 .

+ 2ï

+2 2a g + L |a | O +3   J , # +3 O – 2
2 ï ñ +
3 ï 2 + õÿñ –

2


]õ ï ]ñ óò

a

+ (3 ô + õ )ñ – 2 ô ï

g,

=

= 2

2

16

g

+ L3a 2   J , = + L3a 2   J , # ≠ 1. 1−# 1−# (ñ 2 – 2
ï ñ +
2 ï 2 + ôñ + õ ï )ñó ò =
(ñ 2 – 2
ï ñ +
2 ï 2 ) + ( ô
+ ô + õ )ñ – ô
ï . Solution in parametric form:  +  + (1 − # )2 2 (1 − # )2 2  = L 2 ¬ a  O +  + a ® + L3a , = #3L 2 ¬ a  O +   + a ® +3 L a, (2 − # )  +  (2 − # )  +  where  +  ≠ 0 and (2 − # )  +  ≠ 0. [ñ 2 – (
+ 2) ï ñ + (
+ 1) ï 2 + Solution in parametric form:

 =

37.

õ ï )ñ ó ò

 + −  

.

= 2(2  +  )a + L |a | 2 

 Jg,

 =

35.

O

+2 +3

2( # − 1) 3 a +L  =L 1+ 3 [ñ 2 – 2
ï ñ + (2
– 1) ï 2 + õÿñ – Solution in parametric form: 3

34.

a

ï

+ ( õ – 2 ô )ñ + 3ô

= 4(  − 4  )a + L |a |− 4 

Solution in parametric form:

 =

+ 2ï

2

25

2

= 3(  − 3  )a + L |a |− 3 

2

+  = (4  −  )a + L |a |− 4  −  

[ñ +
ï ñ

– 3ï

ñ

g,

+  = (2  +  )a + L |a | 2  +   −g ,

Solution in parametric form: 2

2

= ñ – 3ï ñ

Solution in parametric form: 2

ñ

a

+ L |a | O 

–

( −1)

Jg,

O

1−# [
ñ 2 + ï ñ – (
+ 1)ï 2 + õÿñ + Solution in parametric form: (
ñ 2 + ( ï ñ + = The substitution [− #2H

õ ï ]ñ óò

 = a + L | a |2 O

= (


g  J ,

#*a

1−# + 1)ñ

+


ï

–ï

ñ

+

( −1)

+ L |a | O  2

2

O


ï

–

= − a + L | a |2 O

+1

2

õÿñ


]õ ï

–

Jg, +

.

# ≠ 1.

õÿñ

+

õï

.

g  J .

= F ñ 2 + G ï ñ + H ï 2 + üIñ . = "H leads to a linear equation with respect to  =  ( H ): 2

3

+

+ (c

ü ï )ñ óò

+1

=

ï ñ

= –


− $ )H

2

+ ( « − L ) H +  ] ‘
= ( #2H

39.

(
ñ + ( ï ñ

40.

(
ñ 2 + 2 ( ï ñ +
]ü 2 ï 2 + ôñ + õ ï )ñó ò = (­ñ 2 + 2
]ü 2 ï # , 2. This is a special case of equation 1.4.3.57 with L = 3

2

+= ï

ï


]õ ï ]ñ óò

õÿñ

2

– (­ñ – = ï ) ñ óò

=F ñ +G ï ñ 2

2

+ $CH + L ) + , .

+

 (=

ñ

+

–

G )ñ

.

The transformation  = ¨ + ƒ , = ˆ ¨ leads to a linear equation:
[− #*ˆ 3 + ( c − $ )ˆ 2 + ( « − L )ˆ ]¨ Š = ( #*ˆ 2 + $Iˆ + L )¨ + ƒsL .

(mü 2 ï

2

+

õÿñ

+

ôƒü 2 ï

.

Page 148

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

41.

(
ñ

2

(
ñ

2

(
ñ

2

+ 2(


]ü 2 ï

2


]ü 2 ï

2

ï ñ

+

ï ñ

+

ï ñ

– (mü

+

õ ï )ñ óò

ôñ

+

ôñ

– ôƒü

(­ñ

=

+ 2
]ü

2

This is a special case of equation 1.4.3.62 with L

42.

+ 2(

+

ï )ñ óò

=

= #3, .

(­ñ

+ 2(

ï

2

+

ôñ

õ ï )ñ óò

+

=

(­ñ

(
ñ

2

(
ñ

2

(
ñ

2

(
ñ

2

(
ñ

2

(
ñ

2

+ 2(

ï ñ

– (mü

ï

ï ñ

– (mü

ï

2

+

õ ï )ñ óò

ôñ

+

ôñ

– ôƒü

=

(­ñ

= #3, .

2

+ 2(

2

+

ï )ñ óò

+ 2
]ü

ï ñ

+

+ 2
]ü

ï ñ

+

+ 2
]ü

ï ñ

+

– 2
]ü

ï ñ

+

= ï

2

+ ôñ +

õ ï )ñ óò

=

= ï

2

+ ôñ +

õ ï )ñ óò

=

= ï

2

ï )ñ óò

+ ôñ – ôƒü

2

+

+ 2
]ü

2

ï ñ

–
]ü

3

ï

2

+ ôƒüIñ +

ï

2

= − $ç, .


]üIñ

2


]üIñ

2

(mü ï

2

ôñ

+

+

õ ï )ñ óò

2

This is a special case of equation 1.4.3.59 with 50.

(
ñ

2

– 2
]ü

ï ñ

+

(mü ï

2

ôñ

+

+

õ ï )ñ óò

51.

[ñ

2

+ 2


ï ñ


2ï

+

2

Solution in parametric form ( # ≠ 2, $ 2

 =L 52.

[ñ

2

– 2


ï ñ

2

Solution in parametric form ( $ 2

 =L 53.

[2 ñ

2

–(
+3) ï

ñ

¬ aQP −

≠ 2, , ≠ 0):

# −1 2 a ® + L3a , ( $ − 2) ,

+(
+1) ï

2

+ (­ñ –
>(

2

ï ñ

+

=­ü ï

2

+

ôƒüIñ

2

+ þxñ – þ

2

ñ

=­ü ï

+ 2 (mü

ï ñ

–
]ü

3

ï

2

+

õÿñ

+ 2 (mü

ï ñ

–
]ü

3

ï

2

+

ôƒüIñ

+


2ï

2

+

) – (


2

.

ôƒü 2 ï

.

õü ï

.

+

2

= , .

¡

ï ñ

+

+ 2
]ü

= − #3L

– 1) üIñ + (
– ( ) ü =
(ñ 2 – 2


+ ((

õÿñ

ü ï

+ þxñ – þ

ü ï

+ 1) (­ñ +

.

ï

.

õü ï

.

+ ôƒü +

.

2


(


– 1) (

ï

.

– 1) ü

ï

.

≠ 0):

# +1 2 ¬aO + a ® + L3a , ( # − 2) $


2ï

+

+

=.

¡

– 1) (­ñ – 2
>( ï ]ñ„ó ò = –
(ñ 2 + 2
ï

+ (


2

= #3, .

= – (­ñ

This is a special case of equation 1.4.3.59 with

=­ü ï

= #3, .

= – (­ñ

.

+

+ 2
]ü 2

õü ï

ï ñ

= #3, .

ü ï .

2

2

+ 2
]ü

= − $ç, .

3

.

ï

+ ôƒü

–
]ü

2


]üIñ

õÿñ

ï ñ

+ 2
]ü

This is a special case of equation 1.4.3.61 with $

49.

+ þxñ – þ

ï

2

=

2

+

3

This is a special case of equation 1.4.3.62 with $

48.

ôƒüIñ

–
]ü

This is a special case of equation 1.4.3.57 with $

47.

(mü 2 ï

õü ï

+

ï ñ

This is a special case of equation 1.4.3.61 with L

46.

+

2

2

(­ñ

=

(mü 2 ï

+

ï ñ

149

+ 2
]ü

=.

¡

This is a special case of equation 1.4.3.62 with L

45.

2

2

2

This is a special case of equation 1.4.3.58 with 44.

2

ï ñ

+ 2
]ü

2

This is a special case of equation 1.4.3.61 with L

43.

2

Ï

ï ]„ñ óò ï ñ +
2ï = #3L

ï ]ñ ó ò

# +1 2 ¬aO + a ® + L3a . ( # − 2) $

2

= (
+1)ñ

2

2

¬ aQP − 2

– 1) üIñ –
( (

) + (
>(

# −1 2 a ® + L3a . ( $ − 2) ,

–(3
+1) ï

ñ

+2


ï

2

+ (­ñ –
>(

ï

.

Solution in parametric form:

 =

a

1−#

+ L |a |−1 

−

P Jg,

=

#*a

1−#

+ L |a |−1 

−

P Jg,

# ≠ 1.

Page 149

150 54.

FIRST-ORDER DIFFERENTIAL EQUATIONS [2ñ 2 –(3
+1) ï ñ +(3
–1) ï 2 + (­ñ –
>( Solution in parametric form:

a

 = 55.

1−#

+ L | a |3 

(
ñ 2 + ( ï ñ + = The substitution

ï

(
ñ 2 + 2( ï ñ

$

ln |a |

2

ñ +ï

2

ï

+ (­ñ –
>(

)–
(
– ( )ñ + ( (
– ( ) ï .

+ L3a .

2

+ !H .

2

+ 2 $ç, + L )

2

+ [2( #3, + $ )H + , +  ] + #2H

2

+ $ç, )

2

2

+ [2( #3, + $ ) H + , +  ] + #2H

2 + ô ñ + õ ï )ñ óò

2

= (­ñ 2 + 2 ]
ü 2ï ñ

+ ( , +  ) + #2H + (mü 2 ï

¡

ï

(
ñ 2 + 2 ( ï ñ + = The substitution

− , ] H´


‘ = 2 , ( #3, + $ )

2

)ï .

+ !H .

– 1)ñ

2

ï

2

+ !H .

2

+

− , ] H´ ‘
= ( #3,

¡

2

+ 2 $ç, + L )

2

ï ñ

+

Solution: ƒ

2

+ ý±

2

+ 2( #3, + $ ) H´ + #2H

+ !H .

+ ôñ + õ ï )ñ ó ò = (­ñ 2 + 2
]ü 2 ï ñ + ü (–
]ü 2 + (mü + = ) ï 2 + ôƒüIñ + õ = H + ,! leads to a Riccati equation with respect to  =  ( H ): 2

2

+ 2 $ç, + L )

2

a

1−,

ï

+ L |a | O 

P Jg,

=

2

+ [2( #3, + $ )H + , +  ] + #2H

2

+

2

)ï .

+ ôñ – ôƒü ï )ñó ò = (­ñ 2 +2
]ü 2 ï ñ + ü (–
]ü 2 + (mü + = ) ï 2 + þxñ – þ = H + ,! leads to a Riccati equation with respect to  =  ( H ):

 = [
( ñ

)ï .

+ !H .

+ [2( #3, + $ ) H + , +  ] + #2H

,!a

1−,

+ L |a | O 

P Jg,

– 1) ü

ï

2

.

+ !H .

2

+ [2 –
( ü + 1)] ï ñ + (
]ü – 1) ï 2 + (­ñ – (mü ï }ñó ò = (
– ü )ñ 2 + [2 ü –
( ü + 1)] ï ñ + (
Solution in parametric form:

{(


2

ï

2

+ þxñ + ü ( ôƒü + õ – þ = H + ,! leads to a Riccati equation with respect to  =  ( H ):

(
ñ 2 +2 ( ï ñ + = The substitution

.

# ≠ 1.

+ [(2 #3, + $ )H + , +  ] + #2H

‘ − , ] H´
= , ( $ − #3, )

¡

( $ − #3, ) H 2  ‘
= ( #3,

64.

2

− , ] H´ ‘
= ( #3,

¡

+]
ü 2ï

[( $ − #3, ) H

63.

=

–2 ï

2

(
ñ 2 – 2
]ü ï ñ + (mü ï 2 + ôñ + õ ï )ñó ò = – (­ñ 2 + 2 (mü ï ñ –
]ü 3 ï 2 + þxñ + ü ( ôƒü + õ – þ The substitution = H + ,! leads to a Riccati equation with respect to  =  ( H ):

[( $ − #3, ) H +

62.

+ L3a ,

2

P Jg,

ï

+2


(
ñ 2 + 2 ( ï ñ – (mü ï 2 + ôñ + õ ï )ñ ó ò = (­ñ 2 + 2
]ü 2 ï ñ –
]ü 3 ï 2 + þxñ + ü ( ôƒü + õ – þ The substitution = H + ,! leads to a Riccati equation with respect to  =  ( H ):

The substitution

61.

#

ln |a |


‘ = ( #3, 2 + $ç, + L )

[−( #3, + $ ) H +

60.

+ L | a |3 

ñ

(
ñ 2 + 2 ( ï ñ + = ï 2 + ôñ + õ ï )ñ ó ò = (­ñ 2 + 2
]ü 2 ï ñ + ü (–
]ü 2 + (mü + = ) ï 2 + õÿñ + ôƒü The substitution = H + ,! leads to a Riccati equation with respect to  =  ( H ):

[( $ − #3, ) H +

59.

1−#

–(
+3) ï

+ ôñ + õ ï )ñó ò =
]üIñ 2 + (mü ï ñ + =­ü ï 2 + ùñ + ( ôƒü + õ – ù )ï . = H + ,! leads to a Riccati equation with respect to  =  ( H ):

[( $ − #3, ) H +  − , ] H´ ‘
= ( #3,

58.

#*a

=

2

2

(B − , ) H´

57.

P Jg,

= (3–
)ñ

[
(ñ 2 –2 ï ñ + ï 2 )–
(
– ( )ñ + ( (
– ( ) ï ]ñ„ó ò = ( (ñ Solution in parametric form:

 = 56.

ï ]ñ óò

ü ï

.

ü ï

.

ï

.

+ !H .

+

(­ñ

– (mü

, ≠ 1.

) + (2  –
2  )ñ + ( –
>(4 ) ï ]ñ„ó ò + ( ( ñ 2 +  ï ñ + ï 2 ) + ( –
>(4 )ñ + (2 – ( + =  2 − #2. − $C.  + . = L exp(− # − I $  ).

2

2

 )ï

= 0.

Page 150

Ï

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

65.

]

The substitution

+
1 ï )ñó ò = ( 22 ñ 2 + ü (2
22 ü +
12 – 2 ( 22 ) ï + ü (–
22 ü 2 + ( 22 ü +
11 ) ï 2 + ( 2 ñ + ü (
2 ü +
= H + ,! leads to a Riccati equation with respect to  =  ( H ):

[( $

2

(


22

ñ

2

+

−#

22




12

ï ñ

22

+

, )H + $




11

ï

−#

2

2


2ñ

+

, ] H´ ‘
= ( #

, 2+#

12

(


ô 2ñ

2

ôƒõ ï ñ

– 2


+


]õ 2 ï

2

RT
ôñ

–

+ =ô

R

=

ñ 1

ï

.

2

H .

RT(mõ ï

.

–(

2)

2

, + # 11 ) + [(2 # 22 , + #

22

In equations 66–70, the following notation is used:

66.

151

12 )


]õ

H +#

2

– ô(

R ô ( ï )ñ óò 2 ( ñ 2 – 2ô (mõ ï ñ ­

, + # 1 ] + #

S

≠ 0,

(mõ 2 ï

+

2

H 2+#


]õ

+ ô( .

RT
]õÿñ

+

= –

22

Solution in parametric form:

#

 = 67.

[ ü/


ô 2ñ

2

–

ü/S¦ô ï ñ

+

üô(mõ ï

+ L3a ,

ln |a | – @U
=

2

ôñ

ü/


$

=

+ L3a .

ln |a |

+ (@3ô( – R ) ï ]ñ„ó ò ôƒõÿñ 2 – ü/SÈõ ï ñ + ü/(mõ

2

ï

Solution in parametric form:

3  = #*a + L |a |- +1   g ,

68.

ô 2ñ

2

R )ï ñ

– ô ( ü/S –

 = #*a + L |a |  69.

( ü/
3 ñ 2 –2 ü/
2 ( ï ñ + ü/
>( 2 ï Solution in parametric form:

 = #3L 70.

V ü/


ñ

3 2

– 2 ü/


2

( ï ñ

3

Å +

2 3

+1

2

exp O

–ô

2

0 , ¢CaP

ü/
>( 2 ï

= $Ia + L |a | 

ñ + ƒô õ ï )ñ óò = /ü
2 ­ ( ñ

, ¢Ca 3 + 1 + L

2

a,

ñ

ï ]ñ óò

= $çL

3

2

Å

+

+1

exp O 2

+ @U


2

Solution in parametric form ( 0 ≠ 1):

 = #3L

2

Oa - +1 +

, ¢

a 2 + L3a , 0 −1 P

1.4.4. Equations of the Form ( M 22 â 2 + M = N 22 â 2 + N 12 æ0â + N 11 æ 2 + N 2 â + N

= $çL

2

Oa - +1 +

æ0â + M 1æ + N

12

2

11

.

+ @U
]õÿñ – @U(mõ

ï

.

ï

.

ï

.

0 . ¢CaP

ï ñ + /ü ( 3 ï

, ¢Ca 3 + 1 + L

ô ñ – (@U
]õ + R )ï WNñóò =/ ü
2 (­ñ 2 – 2 ü/
>( 2 ï ñ + /ü ( 3 ï 2

ü/(mõ 2 ï

–2 ü/
>( 2 3

ï

R )ñ

3 g .

= $Ia + L |a |- +1  

+ õ ( üô( – R ) ï 2 + @U
ôñ – @3ô( = ô ( ü/(mõ + R )ñ 2 – õ ( ü/S + R ) ï Solution in parametric form:

[ ü/


+ @U(mõ

– (@U
]õ +

2

+ (@3ô(

2

2

– ôƒõÿñ + õ

2

a.

–

R )ñ

– @U(mõ

, ¢

a 2 + L3a . 0 −1 P

æ 2+M

2

â +M

1

æ + M 0 )âäã

0

1.4.4-1. Preliminary remarks. Some transformations. 1 b . With # 22 = 0, this is an Abel equation (see Subsection 1.3.4). With $ equation with respect to  =  ( ). See Subsection 1.4.2 for the case # 2 = # 1 = $ 2 = $ 1 = 0. See Subsection 1.4.3 for the case # 0 = $ 0 = 0.

11

= 0, this is an Abel

Page 151

152

FIRST-ORDER DIFFERENTIAL EQUATIONS

2 b . The transformation  =  ¯ + ƒ , = ¯ + ý , where ƒ and ý are parameters, which are determined by solving the second-order algebraic system # 22 ý 2 + # 12 ƒ…ý + # 11 ƒ 2 + # 2 ý + # 1 ƒ + # 0 = 0,

$

22

ý

2

+$

12

ƒ…ý + $

11

ƒ

2

+$

2

ý +$

1

ƒ +$

0

= 0,

leads to the equation (1) ( # 22 ¯ 2 + # 12  ¯ ¯ + # 11  ¯ 2 +  2 ¯ +  1  ¯ ) ¯ 
¯ = $ 22 ¯ 2 + $ 12  ¯ ¯ + $ 11  ¯ 2 +  2 ¯ +  1  ¯ , where  2 = 2 # 22 ý + # 12 ƒ + # 2 ,  1 = 2 # 11 ƒ + # 12 ý + # 1 ,  2 = 2 $ 22 ý + $ 12 ƒ + $ 2 ,  1 = 2 $ 11 ƒ + $ 12 ý + $ 1 . The transformation ˆ =  ¯  ¯ , ¨ = 1  ¯ reduces equation (1) to an Abel equation of the second kind: {[  2 ˆ 2 + (  1 −  2 )ˆ −  1 ]¨ + # 22 ˆ 3 + ( # 12 − $ 22 )ˆ 2 + ( # 11 − $ 12 )ˆ − $ 11 }¨ Š


= (  2 ˆ +  1 )¨ 2 + ( # 22 ˆ 2 + # 12 ˆ + # 11 )¨ . 3 b . The substitution = H +  , where the parameter is determined by solving the cubic equation ( # 22 2 + # 12 + # 11 ) − $ 22 2 − $ 12 − $ 11 = 0, leads to an Abel equation of the second kind with respect to  =  ( H ): [( ÆpH + ² ) + ( $ 22 − # 22 ) H 2 + ( $ 2 − # 2 ) H + $ 0 − # 0 ]
‘ = ( # 22 2 + # 12 + # 11 ) 2 + [(2 # 22 + # 12 ) H + # 2 + # 1 ] + # 22 H 2 + # 2 H + # 0 , where Æ = 2 $ 22 + $ 12 − (2 # 22 + # 12 ), ² = $ 2 + $ 1 − ( # 2 + # 1 ). 1.4.4-2. Solvable equations and their solutions. 1. 2.

3.

4.

5.

6.

7.

8.

( ô ï + õÿñ + ö )2 ñ ó ò = (  ï + ñ + )2 . This is a special case of equation 1.7.1.6 with  ( H ) = H


ñ 2+( ï ñ (

– (­ñ + ü ï

–$  ü )ñ óò = ] = ñ 2+F ï ñ

−2

.

+ ( ü – F )ñ . The transformation  = ¨ + ƒ , = ¨ ˆ leads to a linear equation: [− #*ˆ 3 + ( L − $ )ˆ 2 + c\ˆ ]¨ Š
= ( #*ˆ 2 + $Iˆ )¨ + , .

(
ñ 2 + 2
ï ñ + ( ï 2 +
– ( )ñó ò =
ñ 2 + 2 ( ï ñ + F ï 2 + 2( ( – F ) ï + F The transformation  = ¨ + 1, = ˆ ¨ − 1 leads to a linear equation: (− #*ˆ 3 − #*ˆ 2 + $Iˆ + c )¨ Š
= ( #*ˆ 2 + 2 #*ˆ + $ )¨ + 2( $ − # ). (
ñ 2 – 2
ï ñ + ( ï 2 +
– ( )ñó ò = The transformation  = ¨ − 1, = ˆ ¨ (− #*ˆ 3 + #*ˆ 2 + $Iˆ + L

–
ñ 2 + 2 ( ï ñ + = ï 2 + 2( ( + = ) ï +
− 1 leads to a linear equation: )¨ Š
= ( #*ˆ 2 − 2 #*ˆ + $ )¨ + 2( # − $ ).

(
ñ 2 + 2
ï ñ + ( ï 2 +
– ( )ñó ò =
ñ 2 + 2 ( ï ñ + = ï 2 + 2( = – ( ) ï –
The transformation  = ¨ − 1, = ˆ ¨ + 1 leads to a linear equation: (− #*ˆ 3 − * # ˆ 2 + $Iˆ + L )¨ Š
= ( #*ˆ 2 + 2 #*ˆ + $ )¨ + 2(# − $ ). (
ñ 2 – 2
ï ñ + ( ï 2 +
– ( )ñó ò = The transformation  = ¨ + 1, = ˆ ¨ (− #*ˆ 3 + #*ˆ 2 + $Iˆ + L

–
.

–
ñ 2 + 2 ( ï ñ + = ï 2 – 2( ( + = ) ï +
+ 1 leads to a linear equation:
)¨ Š = ( #*ˆ 2 − 2 #*ˆ + $ )¨ + 2( $ − # ).

+= .

+

=

.

+= .

(
ñ 2 –2
ï ñ + ( ï 2 +
– ( )ñ ó ò = =]ñ 2 +2 ( ï ñ + F ï 2 –2(
+ = )ñ –2( ( + F ) ï +2
+ = + F . The transformation  = ¨ + 1, = ˆ ¨ + 1 leads to a linear equation:
[− #*ˆ 3 + (2 # + L )ˆ 2 + $Iˆ + c ]¨ Š = ( #*ˆ 2 − 2 #*ˆ + $ )¨ + 2( $ − # ). (2
ñ 2 – 2
ï ñ + ( ï 2 + 2
– 4 ( )ñ ó ò = –
ñ 2 + 2 ( ï ñ + F ï 2 – 2( ( + 2 F ) ï +
This is a special case of equation 1.4.4.34 with ƒ = 2, ý = 1, and L = − # .

+ 4F .

Page 152

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

9.

3

− 2 #*ˆ

2


+ $Iˆ + L )¨ Š = ( #*ˆ

2

3

+ 2 #*ˆ

2

+ $Iˆ + L )¨ Š
= ( #*ˆ

2

= )ï

+

2

[− #*ˆ

3

+ ( L − 4 # )ˆ

2

+ 8


+ $Iˆ + 2 $ ]¨ Š
= ( #*ˆ

2

+ 4 = – 16
.

+ 4 #*ˆ + $ )¨ + 2 $ − 8 # .

12.

(2
ñ 2 + 2
ï ñ + ( ï 2 + 2
– 4 ( )ñó ò =
ñ 2 + 2 ( ï ñ + F ï 2 + 2( ( – 2 F )ï + 4 F This is a special case of equation 1.4.4.34 with ƒ = 2, ý = −1, and L = # .

13.

(2
ñ

14.

(
ñ 2 + 2 ( ï ñ +
]ü 2 ï 2 + ôñ + õ ï + þ )ñó ò = (­ñ 2 + 2
]ü This is a special case of equation 1.4.4.27 with L = #3, 2 .

15. 16. 17.

+ 2
ï ñ – ( ï 2 + 2
+ 4 ( )ñó ò =
ñ 2 – 2 ( ï ñ – F ï 2 + 2( ( – 2 F ) ï –
This is a special case of equation 1.4.4.34 with ƒ = −2, ý = 1, and L = − # . 2

ï ñ

+ (mü

2

2

ï ñ

+ (mü

2

2

ï ñ

+ (mü

2

ï ñ

–

2

ï ñ

–
]ü

3

2

ï ñ

–
]ü

3

2

(
ñ 2 + 2 ( ï ñ +
]ü 2 ï 2 + ôñ + õ ï + þ )ñ ó ò = (­ñ 2 + 2
]ü This is a special case of equation 1.4.4.32 with L = #3, 2 . (
ñ 2 + 2 ( ï ñ +
]ü 2 ï 2 + ôñ – ôƒü ï + õ )ñ ó ò = (­ñ 2 + 2
]ü This is a special case of equation 1.4.4.31 with L = #3, 2 . (
ñ 2 + 2 ( ï ñ – (mü ï 2 + ôñ + õ ï + ö )ñó ò = (­ñ 2 + 2
]ü This is a special case of equation 1.4.4.28 with =  .

2

¡

18. 19. 20. 21.

(
ñ 2 + 2 ( ï ñ – (mü ï 2 + ôñ + õ ï + þ )ñó ò = (­ñ 2 + 2
]ü This is a special case of equation 1.4.4.32 with L = − $ç, . (
ñ 2 + 2 ( ï ñ – (mü ï 2 + ôñ – ôƒü ï + õ )ñó ò = (­ñ 2 + 2
]ü This is a special case of equation 1.4.4.31 with L = − $ç, .

(
ñ 2 + 2
]ü ï ñ + = ï 2 + ôñ + õ ï + þ )ñó ò =
]üIñ 2 + 2
]ü This is a special case of equation 1.4.4.27 with $ = 3 # , . (
ñ 2 + 2
]ü ï ñ + = ï 2 + ôñ + õ ï + þ )ñ ó ò =
]üIñ 2 + 2
]ü This is a special case of equation 1.4.4.32 with $ = #3, .

22.

(
ñ 2 + 2
]ü ï ñ + = ï 2 + ôñ – ôƒü ï + õ )ñó ò =
]üIñ 2 + 2
]ü This is a special case of equation 1.4.4.31 with $ = #3, .

23.

(
ñ

– 2
]ü ï ñ + (mü ï 2 + ôñ + õ ï + ö )ñó ò = – (­ñ 2 + 2 (mü This is a special case of equation 1.4.4.29 with =  . 2

+= .

− 4 #*ˆ + $ )¨ + 2 $ − 8 # .

+ 4
ï ñ + ( ï 2 + 4
– ( )ñ ó ò = =]ñ 2 + 2 ( ï ñ + 2 ( ï 2 + 4( = – 2
)ñ + 2 ( The transformation  = ¨ + 1, = ˆ ¨ − 2 leads to a linear equation:

(
ñ

– 8
.

+ 4 #*ˆ + $ )¨ + 2 $ − 8 # .

(
ñ 2 – 4
ï ñ + ( ï 2 + 4
– ( )ñó ò = –2
ñ 2 + 2 ( ï ñ + = ï 2 – 2(2 ( The transformation  = ¨ + 1, = ˆ ¨ + 2 leads to a linear equation: (− #*ˆ

11.

153

(
ñ 2 + 4
ï ñ + ( ï 2 + 4
– ( )ñó ò = 2
ñ 2 + 2 ( ï ñ + = ï 2 – 2( = – 2 ( ) ï + = The transformation  = ¨ + 1, = ˆ ¨ − 2 leads to a linear equation: (− #*ˆ

10.

Ï

ï

+ =­ü

2

ï ñ

+ =­ü

2

ï ñ

+ =­ü

ï ñ

–
]ü

+ õÿñ + ôƒü

2

ï

+ s.

2

+ ôƒüIñ + õ

ü ï

+ s.

2

+ þxñ – þ

ü ï

+ s.

2

ï

ï ï ï ï

3

ï

– 4F .

2

ï


]ü 3 ï

ï ñ

2

ï

–
.

+

õÿñ

+

ôƒü 2 ï

+ s.

2

+ ôƒüIñ + õ

ü ï

+ s.

2

+ þxñ – þ

ü ï

+ s.

2

+ õÿñ + ôƒü

2

ï

+ s.

2

+ ôƒüIñ + õ

ü ï

+ s.

2

+ þxñ – þ

ü ï

+ s.

ï

+ s.

2

+ õÿñ + ôƒü

2

¡

Page 153

154 24.

FIRST-ORDER DIFFERENTIAL EQUATIONS (
ñ

ï ñ

– 2
]ü

2

ï

+ (mü

ï

+ ôñ + õ

2

+ ö )ñó ò = – (­ñ

This is a special case of equation 1.4.4.29 with 25.

(
ñ

2

(
ñ

+ 2 $I ¨ + L3

2

ï ñ

+(

2

ï

+=

2

ï ñ += ï

+2 (

(
ñ

2

2

2

[( $ − #3, ) H 2

+(

(
ñ

2

2

[( $ − #3, ) H

2

(
ñ

ï ñ

+ 2(

2

+(

[( $ − #3, ) H (
ñ

2

+2 (

2

ï –V ( 

2

+G

2

+

= ï

ï ñ +H ï

2

+ ,¯ ¨ ,

where

+ üIñ – GI

=
]üIñ

2

2

ï ñ

+ (mü

+ =­ü

ï

2

– ü' .

, ¯ = , + 2 $Iý .

+ ùñ + ü ( ôƒü + õ – ù )ï + s.

2

2

+ [(2 #3, + $ )H + , +  ] + #2H

+2
]ü

2

ï ñ + ü (– ]
ü

2

+ (mü + = ) ï

2

2

+ !H +

+ õÿñ + ôƒü

2

.

¡

ï

+s.

+(

2

ï + ö )ñ óò

= (­ñ

2

2

2

+ [2( #3, + $ )H + , +  ] + #2H

+2
]ü

2

ï ñ –]
ü 3ï

2

+ !H +

¡

.

)ï +s.

+ þxñ + ü ( ôƒü + õ – þ


‘ = ( #3, 2 + $ç, ) 2 + [2( #3, + $ )H + , +  ] + #2H 2 + !H + ( .

+ ôñ + õ

ï + ö )ñ óò

= – (­ñ

2

+2 (mü

ï ñ –]
ü 3ï

2

) ï +s.

+ þxñ + ü ( ôƒü + õ – þ


‘ = , ( $ − #3, )

− , ) H + ° − (, ]

¡

2

+ ôñ + õ

ï + ö )ñ óò = ­ ( ñ

2

+2
]ü

2

2

2

+ ( , +  ) + #2H

ï ñ +m ( ü 2ï

2

+ !H + ( . ) ï +s.

+ þxñ + ü ( ôƒü + õ – þ

2

+

ôñ

– ôƒü = (­ñ

ï 2

+ õ )ñ + 2
]ü

óò

2

ï ñ

2

2

+ [2( #3, + $ )H + , +  ] + #2H

+ ü (–
]ü

2

+

(mü

+

= )ï

2

+ !H + ( .

ü ï

+ þxñ – þ

+ s.

= H + ,! leads to a Riccati equation with respect to  =  ( H ): − , ) H + ° − , ] ‘
= ( #3,

¡

2

+ ôñ + õ

ï +þ )ñ óò = ­ ( ñ

2

2

+ 2 $ç, + L )

+2
]ü

2

2

2

+ 2( #3, + $ ) H´ + #2H

ï ñ + ü (– ]
ü

2

+ (mü + = ) ï

2

+ !H +  .

+ ôƒüIñ + õ

ü ï

+s.

= H + ,! leads to a Riccati equation with respect to  =  ( H ):

+° −

¡

+

ï ñ

+

Solution: ƒ

2

+ ý±

2

óò

+ «p ¨ + u

‘ − , ) H + ° − (, ]
= 2 , ( #3, + $ ) ¡

The substitution

[
( ñ

)ñ

+ s.

= H + ,! leads to a Riccati equation with respect to  =  ( H ):

ï ñ += ï

[( $ − #3, ) H

2

+(

The substitution

33.

ü ï

= H + ,! leads to a Riccati equation with respect to  =  ( H ):

ï ñ +]
ü 2ï

+2 (

+þ

2

ï + þ )ñ óò = ­ ( ñ

+ ôñ + õ

2

ï ñ +m ( ü ï

–2
]ü

ï

− , ) H + ° − (, ]

¡

The substitution

32.

+ ôƒüIñ + õ

= H + ,! leads to a Riccati equation with respect to  =  ( H ):

[−( #3, + $ ) H

31.

2

‘

The substitution

30.

ï

= H + ,! leads to a Riccati equation with respect to  =  ( H ):

The substitution

(
ñ

+ ôñ + õ

ï ñ –m ( ü ï

+2 (


+ ,¯  )¨  = $ ¨

+ (  − , ) H + ° − , ]
¡ = ( #3, 2 + 2 $ç, + L )

2

[( $ − #3, ) H

29.

3

= , .

¡

= (­ñ

, ] ‘
= ( #3, 2 + $ç, + L ) ¡

The substitution

28.

ï +I
 2)ñ óò

–
]ü

= H + ,! leads to a Riccati equation with respect to  =  ( H ):

[(B − , ) H + ° − (
ñ

2

+ ôñ + õ

2

The substitution

27.

– 2
Iñ + ü

ï ñ

The substitution ¨ = − ý leads to an equation of the form 1.4.3.38: (# ¨

26.

ï ñ += ï

+ 2(

2

+ 2 (mü

2

ï

, ] ‘
= ( #3, 2 + 2 $ç, + L )

2

+ [2( #3, + $ ) H + , +  ] + #2H

2

+ !H +

) + (
>S + 2  )ñ + (
BW +  ) ï +
B + S ]ñ„ó ò + ( ( ñ 2 +  ï ñ + ï 2 ) + ( (XS +  )ñ + ( (4W + 2 ) ï + (4 +

¡

.

2

+ = 

2

+>

+  + . = L exp(− #

W

= 0.

− $I ).

Page 154

Ï

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

34.

–  2 ( )ñó ò = =]ñ 2 + 2 ( ï ñ + F ï 2 – 2 (
+ = )ñ – 2( F + ( ) ï +  The transformation  = ¨ + ƒ , = ˆ ¨ + ý leads to a linear equation:

( 
ñ

2

– 2


[− ƒh#*ˆ 35.

(


22

ñ

2

ï ñ

+




3

12

+

2

 2


+

ï ñ

F

+

2

(2


− 2ý…#*ˆ + $ )¨ + 2( ƒh$ − ý

2

+




11

=

).

# ).

2

ñ

0.

, ) H 2 + ( $ 2 − # 2 , ) H + $ 0 − # 0 , ]
‘ = ( # 22 , 2 + # 12 , + # 11 ) 2 + [(2 # 22 , + # 12 ) H + # 2 , + # 1 ] + # 22 H 2 + # 2 H + # 0 . ¾ ¾ (
22 ñ 2 +
12 ï ñ +
11 ï 2 +
2 ñ +
1 ï +
0 )ñó ò = ( 22 ñ 2 + ( 12 ï ñ + ( 11 ï 2 + ( 2 ñ + ( 1 ï + ( 0 . Here, #2© , $3© , and # 1 are arbitrary parameters, and the other parameters are defined by the [( $

22

−#

22

relations:

#

ƒ − 2 # 22 ý , 2 + # 22 ý 2 − # 1 ƒ , 0 = − # 11 ƒ 2 = (2 # 11 − $ 12 ) ƒ + ( # 12 − 2 $ 22 ) ý + # 1 , 1 = −2 $ 11 ƒ − $ 12 ý , 2 + ( $ 12 − 2 # 11 ) ƒ…ý + ( $ 22 − # 12 )ý 0 = $ 11 ƒ 2

# $ $ $

= −#

12

( ƒ , ý are arbitrary parameters). The transformation  = ¨ + ƒ , [− #

22

ˆ 3 + ($

where , = 2 #

11

22

ƒ +#

−# 12

12 )

ˆ 2 + ($

=ˆ ¨ +ý 12

−#

11 )

2

−#

1

ý

leads to a linear equation:

ˆ +$

11 ]

¨ Š
= (#

22

ˆ 2+#

12

ˆ +#

11 )

¨ +, ,

ý + # 1.

1.4.5. Equations of the Form ( M 3 â = N 3 â 3 + N 2 æ0â 2 + N 1 æ 2 â + N 1.

2

2

+
2 ñ +
1 ï +
0 )ñó ò = ( 22 ñ 2 + ü (2
22 ü +
12 – 2 ( 22 ) ï + ü (–
22 ü 2 + ( 22 ü +
11 ) ï 2 + ( 2 ñ + ü (
2 ü +
1 – ( 2 ) ï + ( = H + ,! leads to a Riccati equation with respect to  =  ( H ):

+

ï

+ $Iˆ + c ]¨ Š
= ( ƒh#*ˆ

2

+ (2ý…# + L )ˆ

The substitution

36.

( ï

155

æ0â 2 + M 1 æ 2 â + M 3 + Z â + Z æ 0æ 1 0 ï ï ï ï ï 3 2 2 3 (ñ – ñ +  ô ñ + õ )ñ óò = ñ – + õÿñ + ô . 3+M

2

0

æ 3 + Y 1 â + Y 0 æ )âäã

Solution in parametric form (  ≠ 0): −1

 =L

2.

(ñ 3 – ï ñ 2 – ï 2 ñ + ï 3 + ôñ )ñ Solution in parametric form:

 =L

3.

− − a |a | 2    −g + 12  L |a |− 2    g ,

−1

−1

sign a 

óò

= –ñ

3

+

ï ñ

2

J g + 1  L |a |  1 J g , 8

(ñ 3 + ï ñ 2 – ï 2 ñ – ï 3 + ôñ + õ ï )ñó ò = –ñ Solution in parametric form (  ≠ −  ):

3

− 4a 2  = a + L |a |  +  exp O − ,  +  P

4.

(ñ 3 + ï ñ 2 – 2 ï 2 ñ + 2 ôñ + ô Solution in parametric form:

 =L

−1 −1 −1

a



ï )ñ óò

= –ñ

3

=L

+

J g + 1  L3a 2  1 J g , 3

+

–ï

ï 2ñ

=L

ñ

2

− − a |a | 2    −g − 12  L |a |− 2    g .

−1

–ï

3

−1

+ï

ï

+ô

sign a 

2

ñ

+ï

3



−1

+

J g − 1  L |a |  1 J g . 8

õÿñ

+

= a − L |a |  +  exp O − −

ï ñ

.

2

+ 4ï =L

2

– 4ï

ñ

−1 −1 −1

a



3

ô ï

.

4a 2 .  +  P

– ôñ + 4 ô

ï

.

J g − 2  L3a 2  1 J g . 3

Page 155

156 5.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñ 3 + ï ñ 2 – 5 ï 2 ñ + 3 ï 3 + ôñ + Solution in parametric form: −1

 =L 6.

−1

a |a | 

1 32

−1

= –3ñ

– 3ï

3

L |a |  1 J g ,

ô ï )ñ óò

−1

sign a 

ï )ñ óò

= 2ï

= –2ñ

3

ñ

– 6ï

3

2

ñ

2

−2  27 a 2  = a + L |a |   +  exp − S , Q 2(  +  )

9.

10.

−1

  

3 − − 2( − ) −

a |a |

g  

−

1 16 (

ô ï )ñ óò

3

= –ñ

12.

3

ï )ñ óò

= 3ñ

 = a + L3a 2 exp O − 13.

(ñ 3 – 4 ï ñ 2 + 5 ï 2 ñ – 2 ï 3 + 2 ôñ – 3 ô Solution in parametric form:

 =L

14.

−1

−1

sign a 

3

−1

sign a 

−1



3

3

2a P

ï )ñ óò

ñ

3 32

L |a |  1 J g .

ï

– ôñ + 4 ô

−1

ï

.

.

ï

.

J g − 2  L |a |  1 J g . 9



ï

.

27 a 2 S . Q 2(  +  )

ñ

+ 9ï

ñ

2

a |a |

2

– 3ï

L

    g +

2

– 3ï

2

ñ

3

= 3ñ

– 8ï

=L 3

−1

– 15 ï

=L

ñ

2

ñ

−1

2

– ôñ +

3

.

3 −   −  ) L |a | 2(  −  )  g .

ô ï

.

–ï

3

+

õÿñ

+

ô ï

.

− 4a 3 O | a |  +  + . 2  +  P

– 8ï

3

+ 10 ï

sign a 

ñ

3 16 (

ï

£ 2a 4 + 1.

2

3

–ï

ñ

2

£ 

+ 20 ï

2

– (2 ô – õ )ñ + 3 ô

3

3 − − 2( − ) −

−1

2

– 9ï

ñ

= a + 2 L3a 2 exp O − = 2ñ

J g + 1  L |a |  1 J g , 8

– ôñ + 3 ô

+ ( õ – ô )ñ + 2ô

= L3a − L

,

ï )ñ óò

2

– 3ï

– 14 ï

2

ï ñ

– 3ï

J g + L |a |  1 J g ,

(ñ 3 – 5 ï ñ 2 + 7 ï 2 ñ – 3 ï 3 + ôñ – 2 ô Solution in parametric form:

 =L

+

− 4a 3 O | a |  +  + , 2  +  P

(ñ 3 – 4 ï ñ 2 + 4 ï 2 ñ + ôñ – ô Solution in parametric form:

3

3

J g + 2  L |a |−1  1 J g . 3

sign a 

= L3a T

(ñ 3 + 3 ï ñ 2 + 3 ï 2 ñ + ï 3 + ôñ + õ ï )ñ ó ò = –ñ Solution in parametric form (  ≠ −2  ):

 = L3a + L

+ 8ï

=L

L 2  = L3aA £ £ 2a 4 + 1, 

11.

−1

Jg −

– ôñ + 4 ô

3

=L

−1

3

−2

3 −   −  ) L |a | 2(  −  )  g ,

(ñ 3 + 3 ï ñ 2 + 3 ï 2 ñ + ï 3 – ôñ + Solution in parametric form:

a |a | 

– 4ï

ñ

– 4ï

ñ

– 9ï

ñ

−1

sign a 

2

−1

2

= a − 2 L |a |   +  exp −

(ñ 3 + 3 ï ñ 2 – ï 2 ñ – 3 ï 3 + ôñ + õ ï )ñó ò = –ñ Solution in parametric form (  ≠  ):

 =L

−1

=L

+ 6ï

+ 15 ï

2

+ 2ï

2

J g + 1  L |a |  1 J g , 9

(ñ 3 + 3 ï ñ 2 – 4 ï 3 + ôñ + õ ï )ñó ò = –2ñ Solution in parametric form (  ≠ −  ):

ñ

=L

J g − 1  L |a |−1  1 J g , 3

−1

(ñ 3 – 3 ï 2 ñ + 2 ï 3 + 2 ôñ + ô Solution in parametric form:

 =L 8.

Jg +

(ñ 3 + 2 ï ñ 2 – ï 2 ñ – 2 ï 3 + 2 ôñ + Solution in parametric form:

 =L 7.

−1

sign a 

ô ï )ñ óò

−1

+ 21 ï

sign a 

+ 2 ôñ – 2 ô



2a 2 P

– 4ï

ñ

2

ï

.

. 3

J g + 2 L |a |  1 J g .

−1

2

ñ

– 9ï

3

ï

.

ï

.

+ 3 ôñ – 4 ô

+ 2 ôñ – 3 ô

J g + 3  L |a |  1 J g . 8

Page 156

Ï

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

15.

(ñ 3 – 5 ï ñ 2 + 8 ï 2 ñ – 4 ï 3 + ôñ + õ ï )ñó ò = 2ñ Solution in parametric form (  ≠ −  ):

– 10 ï

3

2 + a2  = a + L |a |  +   exp S , Q 2(  +  )

16.

(ñ 3 + 5 ï ñ 2 + 3 ï 2 ñ – 9 ï 3 + ôñ + õ ï )ñ Solution in parametric form (  ≠ −  ):

óò

18.

−1

= –3ñ

– 15 ï

3

 = L3a"A 19.

−1

−1

−1

  −g   −

24.

1 27 (

ñ

=L

−1

3

ñ

−1 −1 −1

a



3

2

– 12 ï

a |a | 

−1

ñ

3

+ 3ï

ñ ñ

2

2

J g − 1  L |a |−1  1 J g , 8

( ï ñ 2 – 2 ü ï 2 ñ + ü 2 ï 3 + ôñ – ô ï )ñ ó ò = Solution in parametric form ( , ≠ 1):

ñ

3

  = a + L |a | exp − S , Q 2( , − 1)a 2

Q 2(  +  )

+ 18 ï

2

– 9ï

ñ

ï

.

ï

.

S .

+ ( õ – 2 ô )ñ + 3ô

3



+ 24 ï

2

L

2

2

32 a 2 .  +  P + (4 ô + õ )ñ – 3 ô

3

– 16 ï

ñ

3

ï

– ôñ + ô

.

£ 2a 4 + 1.

£ 

ñ

=L

−1

– 2ü

2

+ 6ï

2

2

3

ï ñ

– 3ï

a

2

– 3ï −1

+

2

– 8ï

ï

.

  g   .

2 − −2

 − 2  ) L |a | 

+ (3 ô + õ )ñ – 2 ô

ï

.

2 + 2 + a |a |− 3  +2   −g − 2(3  + 2  ) L |a | 3  +2   g .

−1 −1 −1

=L

– ( ô – õ )ñ + 2ô

ñ

+ 18 ï

=L

ñ

3

2 27 (

2

2

– 8ï

ñ

  −g   +

+ï

ñ

a |a |−  +2   −g .

2 − −2

a |a |− 

ñ

−1

–ï

−1

=L

+ 3ï

=L

+ 3ï

a2

+ 27 ï

ñ

.

3 + 3 + a |a |− 4  +2   −g − 3(  + 12  ) L |a | 4  +2   g .

ñ

– 13 ï

3

J g + 1  L3a 2  1 J g , 8

(3ñ 3 + ï ñ 2 – 3 ï 2 ñ – ï 3 + ôñ )ñó ò = Solution in parametric form: −1

3

2

ï

+ (3 ô + õ )ñ – 2 ô

3

+ ( ô + õ )ñ .

2

  g   ,

(3 ñ 3 – ï ñ 2 – 3 ï 2 ñ + ï 3 + ôñ )ñó ò = –ñ Solution in parametric form:

– 9ï

= L3a"A

2 − −2

 − 2  ) L |a | 

2

– 11 ï

£ 2a 4 + 1,

2 + 2 + a |a |− 3  +2   −g − (3  + 2  ) L |a | 3  +2   g ,

 =L

ñ

a |a |−  +2   −g + (  + 2  ) L |a |  +2   g ,

2 − −2

a |a |− 

 =L

23.

2£ 

= 2ñ

(2 ñ 3 – 9 ï ñ 2 + 13 ï 2 ñ – 6 ï 3 + ôñ + õ ï )ñó ò = 3ñ Solution in parametric form (  ≠ − 32  ):

 =L

22.

3

(2 ñ 3 + 3 ï ñ 2 – 3 ï 2 ñ – 2 ï 3 + ôñ + õ ï )ñó ò = –ñ Solution in parametric form (  ≠ 2  ):

 =L

21.

2

   exp

−3

(2 ñ 3 – 3 ï ñ 2 + ï 2 ñ + ôñ + õ ï )ñó ò = ñ 3 – ï Solution in parametric form (  ≠ −2  ):

 =L

20.

L

óò

– 8ï

ñ

= a − 3 L |a |   +  exp O −

3 + 3 + a |a |− 4  +2   −g − (  + 12  ) L |a | 4  +2   g ,

(ñ 3 – 6 ï ñ 2 + 12 ï 2 ñ – 8 ï 3 – ôñ + ô ï )ñ Solution in parametric form (  > 0):

2

2 + +

(ñ 3 – 6 ï ñ 2 + 11 ï 2 ñ – 6 ï 3 + ôñ + õ ï )ñó ò = 2ñ Solution in parametric form (  ≠ − 21  ):

 =L

+ 16 ï

2

= a + 2 L |a | 

−3  32 a 2  = a + L |a |   +  exp O − ,  +  P

17.

ñ

157

a |a | 

+ô

3



3

ü 2ï 2ñ

.

J g − 1  L3a 2  1 J g . 8

+

−1

ï

ô ï

.

J g + 1  L |a |−1  1 J g . 8 +

üôñ

– üô

ï

.

 S . Q 2( , − 1)a 2

= a + , L |a | exp −

Page 157

158 25.

26.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñ 3 – 3 ü ï ñ 2 + 3 ü 2 ï 2 ñ – ü 3 ï 3 – ôñ + ô ï )ñó ò = Solution in parametric form (  > 0, , ≠ 1):

(ñ

3

– 3ü

ï ñ

2

ï 2ñ

3

+ 3ü

2

, −1£ 2 2a + 1, 2£ 

 = L3a"AeL 2

–ü

ï

27.

3

– ( ü + 2) ï

ñ

+

1 2

ï ñ

( , − 1)3 a 3S , ( , − 3)  − 2 

3

3

ï

4

3

  + + + 

Q

−1

J g +  L |a |  1 J g , ( , − 1)2

=L

−1

sign a 

ü 2ï

−1

3

ñ

3

– (2 ü + 1)ï

ñ

ï

– ôñ + ô

.

ï

.

3

ï

.

üô ï

.

ï

.

+ ( ü + 1) ôñ – 2 üô

2

ñ

–

ü 3ï

3

+

üôñ

–

 S . Q 2( , − 1)2 a 2

= a + , L3a 2 exp −

2

(
ñ 3 + ï ñ 2 –
ï 2 ñ – ï 3 + ôñ + õ ï )ñ ó ò = 1 b . Solution in parametric form with  ≠ 0:

3

( , − 1)3 a 3S . ( , − 3)  − 2 

2

+ ü ( ü + 2) ï 2 ñ – ü 2 ï 3 + ôñ + õ ï ]ñó ò = üIñ 3 – ü (2 ü + 1) ï ñ 2 + ü 2 ( ü + 2) ï 2 ñ – ü Solution in parametric form (  ≠ −  , , ≠ 1):

[ñ

ï

J g +  , L |a |  1 J g . ( , − 1)2

– ü ( ü – 4) ï 2 ñ + ü 2 ( ü – 2) ï 3 + ôñ – ô ï ]ñó ò = (2 ü – 1)ñ 3 – ü (4 ü – 1) ï ñ 2 + ü 2 (2 ü + 1) ï Solution in parametric form ( , ≠ −1):

[ñ

– ( ü + 2) ï

sign a 

4

+ [( ü + 1) ô + õ ]ñ – üô

|a | 

2

−1

–ü

3 2

, −1£ 2 2a + 1. 2£ 

2

= L3a + , L

ï ñ

+ 3ü

2

–ü

 ( , − 3), , ≠ 1):

+ ( , − 1)3 2  = a + L |a |  +   exp a S , Q 2(  +  )

30.

2

= L3aAe, L

  = a + L3a 2 exp − S , Q 2( , − 1)2 a 2

29.

– 3ü

+ (2 ü + 1)ï 2 ñ – ü ï 3 + 2 ôñ – ( ü + 1) ô ï ]ñ„ó ò = üIñ 3 – ü ( ü + 2) ï ñ 2 + ü (2 ü + 1)ï 2 ñ – Solution in parametric form ( , ≠ 1):

[ñ

 =L 28.

Q

  + + + 

|a | 

3

ô ñ + õ ï )ñóò = üIñ 3 – 3 ü 2 ï ñ 2 + 3 ü 3 ï 2 ñ

3

Solution in parametric form (  ≠

 = L3a + L

üIñ

ï

3

3

+ [( ü + 1) ô + õ ]ñ – üô

  + ( , − 1)3 2 + exp a S .  Q 2(  +  )

ñ

= a + , L |a |  3

+


ï ñ

2

–ï

2

ñ

–


ï

3

+

õÿñ

+ô

ï

.

− − − − a |a | 2   |a + 1|  O 2  − 14 L |a |  2  |a + 1| 2  O , − − − − = L −1 a |a | 2   |a + 1|  O 2  + 14  L |a |  2  |a + 1| 2  O .

 =L

−1

2 b . Solution in parametric form with  = 0:

 =L 31.

−1

a |a | O

−1 −1 2



J g − 1  L |a | 1−2O  1 J g , 8

=L

−1

a |a | O

−1 −1 2



J g + 1  L |a | 1−2O  1 J g . 8

(
3 ñ 3 +
2 ï ñ 2 +
1 ï 2 ñ +
0 ï 3 + ô ï )ñó ò = ( 3 ñ 3 + ( 2 ï ñ 2 + ( 1 ï 2 ñ + ( 0 ï 3 + ôñ . This is a special case of equation 1.7.1.13 with ² + ( , ) =  . The transformation a =   , ‡ =  2 leads to a linear equation:

. .
P (a ) − a O (a )]‡ g = 2 O (a )‡ + 2  , . (a ) = # a 3 + # a 2 + # a + # and . (a ) = $ a 3 + $ a 2 + $ a 3 2 1 0 3 2 1 P O [

where

.

+$

0.

Page 158

â ( Ë , Ï ) Ï ÒÍ

1.5. EQUATIONS OF THE FORM

32.

33.

– 4) ï 2 ñ – (
– 2) ï 3 + ôñ – ô = –(
– 2)ñ 3 – (
– 4) ï ñ Solution in parametric form:

[
ñ

[
ñ

+ 2) ï

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

+ (


3

+ 1)ï

+ 3(


3

ñ

2

– (




 = a + L |a |1− O exp O

ñ

2

2

– 4(
– 3)ï 3 + ôñ – ô ï ]ñ = –(2
– 3)ñ 3 – 6(
– 2) ï ñ

+ 12 ï

2

+ 2) ï

+ (


ñ

óò

2

+ 12 ï

ñ

2

8a 2 P

ñ

2


ï

+



= a − L |a |1− O exp O

,

8a 2 P

ï ]ñ óò

159

3

ô ï

.

ï

.

– ôñ +

.

ï

+ 8


3

– 2 ôñ + 2 ô

Solution in parametric form:

 = a + L |a |1− O exp O



18a P 2

= a − 2 L |a |1− O exp O

,



18a 2 P

.

1.5. Equations of the Form ð ( ì , í ) í ï î = ë ( ì , í ) Containing Arbitrary Parameters 1.5.1. Equations Containing Power Functions 1.5.1-1. Equations of the form 
 =  ( , ). 1.

2.

7

=
ñ + ( ï –1 ) 2 . The substitution ¨ = 2 # ¨ + 2 $C# −2  −1 J 2 .

ñóò ñóò

=

7

ñ




( ï

+

Let # = A 2 

−1

–1

4.

5.

=  [2U2Aç (U )]2 ,

 (U ) = exp(T"U 2 ) X exp(T"U 2 ) U + L2S Q M

where

7

−1

£

leads to an Abel equation of the form 1.3.1.33:

ñóò

−1

.

¨*¨p
=

7

=ô ñ + õ ï + ö ïƒú . The substitution ¨ = 2  −1 £ leads to the Abel equation ¨*¨ 
= ¨ + 2  is discussed in Subsection 1.3.1 (see Table 5).

ñóò ñóò

¨*¨p
=

.

=
ñ + ( ï –2 . The substitution ¨ = 2 # ¨ + 2 $C# −2  −2 .

ñ óò

=

ôñ ÷

÷ õ ï 1–÷

+

Solution: X 6.

leads to an Abel equation of the form 1.3.1.32:

£  , $ = T 4  (  > 0). Solution in parametric form:

 =  (U ), 3.

£

−1

=


ñ

s

–

(

−2

( ' + (˜ + ), which

.

¨ ¨ ¨  + M 1 ¨ +  = ln | | + L , where =   1−  À ï .

‘ The transformation  = (¨
)1 J' , Emden–Fowler equation:

1 −1

.

À

À Œ = ($ # )1 J , leads to the generalized = Œ (¨u H )1 J , where

¨ '‘

‘ = − Œ", H °$

−

À

À

1

1−

¨

 −1 (¨ ‘
)  ,

which is discussed in Section 2.5 (in the classification table, one should search for the equations satisfying the condition B + + 1 = 0).

¡

Page 159

160 7. 8.

FIRST-ORDER DIFFERENTIAL EQUATIONS = ( ô ï + õÿñ + ö )÷ . This is a special case of equation 1.7.1.1 with  (ˆ ) = ˆ  .

ñ óò

= ô ïƒú Solution:

ñóò X

9. 10. 11. 12.

÷ ¿÷ ú ñ ÷

– –

õ ïƒú

+

.

¨  1 J  ln | |, ¨  −M Œ ¨ + 1 + L = O hP

ñ óò = ô ï ÷ –1 ñ ú

+1

where

¨ =O 

+ õ ï ÷' –1 ñ úI +1 .

1

J

hP

 −+

−1

, Œ = ¡



+1

O

 1 J  .  P ·

This is a generalized homogeneous equation of the form 1.7.1.3 with  (ˆ ) = ˆ + 'ˆ  .

7

À

7

= ô ï ñ ñ + õ ïƒú ñ + ö ï s ñ . This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) = ' + , % ( ) = (˜ , and B = 1 2.

ñóò

À

= ô ï ñ 1+÷ + õ ïƒú ñ + ö ï s ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) = ' + , and % ( ) = (˜ .

ñóò

= ï ÷ –1 ñ 1–ú ( ô ï ÷ + õÿñ ú )  . This is a special case of equation 1.7.1.7 with  (ˆ ) = ˆ  .

ñ óò

1.5.1-2. Other equations. 13.

14. 15.

ï ñ óò

=

ôñ

+

õ

ü

ñ

2

+ö

ï

2.

The substitution ¨ =   leads to a separable equation:  ¨ 
= (  − 1)¨ +  £ ¨ = ñ + ô ïƒ÷ –ú ñ ú + õ ïƒ÷ –  ñ  . ¨ 
=   The substitution =  ¨ leads to a separable equation: p

ï ñóò

( ôñ

÷

+

õ ï )ñ ó ò

17. 18.

19.

20.

ï (ï ñ ÷

+( .

(  ¨2+ +  ¨  ).

= 1.

Solution:  =    OL + lX 16.

−2

2

  − 

M P

.

+ ô )ñó ò + õÿñ = 0. Solution: B… −  = lÝL  J  +   .

ï (ô ñ ú

+ þ )ñó ò = ñ V õ ïƒ÷ (  –1) ñ ú  – ù·W . This is a special case of equation 1.7.1.16 with  (ˆ ) = ˆ , (ˆ ) = 1, % (ˆ ) = 'ˆ  , and , = B .

( ô ï ÷ + õ ï 2 + ö ï ñ )ñ ó ò = ü ï ÷ + õ ï ñ + ö^ñ 2 . The transformation a =   , H =   −2 leads to a linear equation with respect to H = H (a ): ( , − a ) H g
= (B − 2)( !H +  + (˜a ). ( ôñ ÷ + õ ï 2 + ö ï ñ )ñó ò = üIñ ÷ + õ ï ñ + ö^ñ 2 . The transformation a =   , H =   −2 leads to a linear equation with respect to H = H (a ): a  ( , − a ) H g
= (B − 2)(a  H +  + (˜a ). ( ô ïƒ÷ + õÿñ ÷ + ï )ñó ò =  ï ñ ÷ –  +  ïƒú ñ ÷ –ú + ñ . The transformation a =   , H =   −1 leads to a linear equation: ( ƒ…a  −  + ý±a  − + − 'a  (B − 1)( 'a  +  ) H + B − 1.

+1

− a ) H g
=

Page 160

â ( Ë , Ï ) Ï ÒÍ

1.5. EQUATIONS OF THE FORM

21.

22. 23. 24. 25. 26.

( ô ï ÷ + õÿñ ÷ +
ï 2 + ( ï ñ )ñ ó ò =  ï  ñ ÷ –  +  ï ú ñ ÷ –ú +
ï ñ + (­ñ 2 . The transformation a =   , H =   −2 leads to a linear equation: ( ƒ…a  −  + ± ý a  − + − 'a   (B − 2)( 'a +  ) H + (B − 2)( $Ia + # ).

28.

30. 31.

33. 34.

− a ) H g
=

(  ï + ñ + ) ÷ ñó ò = ( ô ï + õÿñ + ö )÷ . This is a special case of equation 1.7.1.6 with  (ˆ ) = ˆ  .

( ô ï ÷ + õÿñ ú )ñ ó ò = ï ÷ –1 ñ 1–ú . This is a special case of equation 1.7.1.7 with  (ˆ ) = 1 ˆ . ( ôñ ú + õ Solution:

ï ÷

+ s)ñ



óò

+

+

 ï 

+

õÿù ï ÷ –1 ñ

+  = 0.

!Y ( ) + ƒ…j ( ) + '  +1

+1 ¡ | | ln

≠ −1,

if if

¡ ¡

= −1,

j ( ) =

+°



 

+ ý± = L , +1

, +1 ln | |

if , ≠ −1, if , = −1.

( ô ï 2 ñ ÷ + õ ï ñ ú + ö^ñ  )ñó ò = ñ[ + ñ\ + . This is a Riccati equation with respect to  =  ( ). ( ô ïƒ÷ ñ ú + ï )ñó ò = õ ï ñ ÷ +ú –  + ñ . The transformation a =   , H =   + + (B + − 1)( a + H + 1).

ï (ô ï ÷ ñ ú

+  )ñ

óò

+ ñ (õ

ï ÷ ñ ú

leads to a linear equation: a + ( 'a  −  − a ) H g
=

−1

+  ) = 0.

( 9  : )P ý − B±ƒ  − B± (    )O Solution: + = L , where # = ¡ , $ = ¡ . # $  ý − ˜ ƒ  ý − ˜ ƒ

ï (ô ù ï  ñ ÷ +

+ s)ñó ò + ñ ( õÿþ Solution: ,  + ,! + − ° (

ïƒú + ñ 

)−  = L .

( ô ï ÷ ñ ú +
ï 2 + ( ï ñ )ñ ó ò = õ ï  ñ ÷ +ú The transformation a =   , H =   + + (B + − 2)( a + H + $Ia + # ).

¡

32.

+1

[( ô ï + õÿñ ) ÷ + õÿñ ]ñó ò = ö ( ô ï + õÿñ )ú – ôñ . This is a special case of equation 1.7.1.15 with  (ˆ ) = ˆ  , (ˆ ) = 1, and % (ˆ ) = (˜ˆ + .

¡

29.

161

[( ô ï + õÿñ ) ÷ + õ ï ]ñó ò = ö ( ô ï + õÿñ )ú – ô ï . This is a special case of equation 1.7.1.14 with  (ˆ ) = ˆ  , (ˆ ) = 1, and % (ˆ ) = (˜ˆ + .

where Y ( ) = 27.

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

+ s) = 0. – −2



+
ï ñ + (­ñ 2 . leads to a linear equation: a + ( 'a  −  − a ) H g
=

À

( ôþ ïƒ÷ ñ ú –1 + õÿñ  )ñó ò + ôù ïƒ÷ –1 ñ ú + ö ï s = 0. This is a special case of equation 1.7.1.19 with  ( ) =   and ( ) = (˜ .

( ô ïƒ÷ ñ ú + õ ï ñ  )ñó ò = ñ s +  . This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5).

ï (ô ïƒ÷ – ñ ú

+ þ )ñó ò = ñ ( õ ï  ÷ –  ñ ú – ù ). This is a special case of equation 1.7.1.16 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

Page 161

162 35. 36.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ï (ô ï ÷ ñ ú –

+ þ )ñ ó ò = ñ ( õ ï  ÷ ñ  ú –  – ù ). This is a special case of equation 1.7.1.17 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

À

( ô ïƒ÷ +1 ñ ú –1 + õ ïƒ÷' +1 ñ úI –1 )ñó ò = ö ïƒ÷ s ñ ú s . This is a special case of equation 1.7.1.3 with  (ˆ ) = (˜ˆ ( ˆ + 'ˆ  )−1 .

37.

( ô ï ÷ + õÿñ ú )  ñ ó ò = ö ï ÷ –1 ñ 1–ú . This is a special case of equation 1.7.1.7 with  (ˆ ) = (˜ˆ −  .

38.

”

ï

1

+ô

+

3 1



ï

2

–ô

ñóò

•

3 2

– ñç”



1



+

3 1

2

23 •

= 0,

where 12 = ( ï + ô )2 + ñ 2 , 22 = ( ï – ô )2 + ñ 2 . This is the equation of force lines corresponding to the Coulomb law in electricity.  +  − Solution:  1 + 2 =L . 39. 40. 41.

&

– ñ = ( ô ï

ï ñóò

&

1

2

+ ï ). This is a special case of equation 1.7.1.24 with  (‡ ) = ‡ J + ÿõ ñ  )( ñ ñóò

ï ñ óò

– ñ = ( ô ï  + õÿñ  )(ññ ó ò – ï ). This is a special case of equation 1.7.1.25 with  (‡ ) = ‡ J + ï = ( ô ï + õÿñ  )( ï ñó ò – ñ ). This is a special case of equation 1.7.1.24 with  (‡ ) = ‡

ññóò

−

2

and ( , ¨ ) =   +  ¨  .

2

and ( , ¨ ) =   +  ¨  .

J 2 and ( , ¨ ) = (  +  ¨  )−1 .

1.5.2. Equations Containing Exponential Functions 1.5.2-1. Equations with exponential functions. 1.

ñóò

=

ô]

+õ . =−

Solution: 2.

ñóò

=

ô]

+

Solution:

ó ñ óò = B
 ] +D

õ ó = 

1

Œ

ln OL 

−





−

.





hP

.



− ln L −  X exp(   )  S .

Q

M

3.

–ô . Š This is a special case of equation 1.7.1.2 with  (ˆ ) = #  , B = 1, and  = 0.

4.

ñóò

= ô + ] + õ  . Á  and ( ) =     . This is a special case of equation 1.7.2.5 with  ( ) =  

ó

ó

5.

ñ óò

ó

ó

= ô  + ] + õ – ] .  Á This is a special case of equation 1.7.2.8 with  ( ) =   , ( ) = 0, and % ( ) =    .

ó

ó

ó

6.

= ô 2 ^ –_ ] + õ ^ + ö ^ –_ ] . This is a special case of equation 1.7.2.9 with  (ˆ ) = ˆ + ( .

7.

= ô ^ + ] + õ _ + ö` – ] . ¨ 
= Œ  9  ¨ The substitution ¨ = ž leads to a Riccati equation: 3

ñóò ñ óò

ó

ó

ó

2

  + Œ  : ¨ + (Œ  ; .

Page 162

1.5. EQUATIONS OF THE FORM

8.

( ô

]

õ ó )ñ óò

+

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

= 1.

Solution:  =    − ln L −  X exp(    ) 9.

( õ

^ ]

( ô

ó ] +_

+ õ )ñ

( ô

^ ó

õ _ ] )ñ óò

óò

+ ö )ñ

163

 D

=

ó + ]

Q

^ ]

– ô

S. M

.

Š

This is a special case of equation 1.7.2.13 with  (ˆ ) = ( , (ˆ ) = 1, and % (ˆ ) =  .

10. 11.

óò

=ö .

¨ 
= ý + ( (   The substitution ¨ ( ) = + ý± leads to a separable equation: p +

ó  ^ –_ ]

=

(

^ ó +` ]

+ ô )ñó ò +

õ ó +_ ]

+  )−1 .

.

This is a special case of equation 1.7.2.9 with  (ˆ ) = ˆ

12.

?

+ ô

−1

.

= 0.

This is a special case of equation 1.7.2.15 with  ( ) =  ;  and ( ) =  

Á .

1.5.2-2. Equations with power and exponential functions. 13.

ñóò

=

ô]

+

õ ïƒ÷

.

1 b . Solution in parametric form with B ≠ −1: 1 +1

 =U 

=

,

 ' U   U − ln L − U!S . X U −  +1 exp O B +1 Q B +1 B + 1 P3M

2 b . Solution in parametric form with B = −1,  ≠ −1:

 = * ,

= − ln OL 

−

*

−

  * .  +1 P

3 b . Solution in parametric form with B = −1,  = −1:

14.

ñóò

=

ôñ

–1

+

 = * ,

õ ó

= −U − ln( L − U ).

.

Solution in parametric form:

 = ln( #2« where  = T 2 $ 15.

2

,  = A"#

ô ó + ]

+

=

ô ƒï ÷  ]

ñóò

=

ñóò

=

ñóò

=

ñóò

−1

õ ïƒ÷

.

+

õ ó

.

ô ƒï ÷  ]

+

õ ïƒú

.

ô ƒï ÷  ]

+

õ ïƒú  – ]

−1

) TçU 2 ,

= $ [2 A exp(T"U 2 ) «

17. 18.

],

$ , « = X exp(T"U 2 ) U + L . M

This is a special case of equation 1.7.2.5 with  ( ) =   16.

−1

Á  and ( ) = '  .

Á This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) =   . This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) = ' + . . This is a special case of equation 1.7.2.8 with  ( ) =   , ( ) = 0, and % ( ) = ' + .

Page 163

164

FIRST-ORDER DIFFERENTIAL EQUATIONS

ó

ó

19.

= ö (ñ + ô  + õ )÷ – ô  . This is a special case of equation 1.7.2.10 with  (ˆ ) = (˜ˆ  .

20.

ñóò

ñ óò

= ( ô] +

õ ï –  )1 ) 

.

Solution in parametric form:

 = exp ¦"U − 21.

,

23. 24.

where

 (U ) = X

, U . M , (  +    * )−1 J' + 1

1V  (U ) + L2W!§ ,

where

 (U ) = X

, U . M , (  +   −  * )1 J' − 1

ó

= ( ôñ  + õ )1 )  . Solution in parametric form:

ñ óò

 =  (U ) + L , 22.

=  (U ) + L ,

1V  (U ) + L W § ,

= exp ¦U +

,

= ô ï ÷ –1   ÷ ] + õ ï ú –1   ú ] . This is a special case of equation 1.7.2.2 with  (ˆ ) = ˆ 

ñ óò ñóò

=

ô ïƒ÷ –1  ^ ]

+

õ ïƒ÷¿ú

–1

 ^ ú ]

−1

+ 'ˆ +

+1

.

This is a special case of equation 1.7.2.4 with  (ˆ ) = ˆ + 'ˆ + .

ó

ó

= ô  ÷ ñ ÷ +1 + õ –  . This is a special case of equation 1.7.2.1 with  (ˆ ) = ˆ 

ñ óò

ó

+1

ó

+ .

= ô ^ ñ ú +1 + õ ^ ÷ ñ ÷¿ú +1 . This is a special case of equation 1.7.2.3 with  (ˆ ) = ˆ + 'ˆ  .

26.

ñóò

28.

.

.

25.

27.

−1

ñóò

=

ô ÷ ó ñ ÷

+1

+

õ ú ó ñ ú

+1

. This is a special case of equation 1.7.2.1 with  (ˆ ) = ˆ 

+1

ó

+ 'ˆ +

= ô ïƒ÷ ñ  + õ ïƒ÷  ^ ñ  +1 – ñ . This is a special case of equation 1.7.2.7 with  ( ) =   , (ˆ ) =  + 'ˆ , and

ñóò

ñ óò = ô ó ñ 1+÷

ó + õ  ñ

ó + ö ñ 1–÷

.



This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) =  

ó

ó



¡

= 1.

, and % ( ) = ( 

Á .

29.

= ô ñ 1+÷ + õ  ñ + ö ïƒú ñ 1–÷ .   This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) =    , and % ( ) = (˜ + .

30.

ñóò

ñóò

=

ô ï  ñ 1+÷

+

õ ó ñ

+

ö ïƒú ñ 1–÷

.

 This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) =   , and % ( ) = (˜ + .

ó

ó

31.

= ô  ñ 1+÷ + õ ï ú ñ + ö ñ 1–÷ .   This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) = ' + , and % ( ) = (   .

32.

ñóò

33.

ñ óò

=

ô ó ñ 1+÷

+

õ ïƒú ñ

+

ö ï  ñ 1–÷

.

 This is a special case of equation 1.7.1.4 with  ( ) =    , ( ) = ' + , and % ( ) = (˜  . = ô ï ÷ +   ] + õ ï ÷¿ú +   ú ] – ù . This is a special case of equation 1.7.2.6 with  ( ) =  

ï ñ óò

−1

and (ˆ ) = ˆ + 'ˆ + .

Page 164

1.5. EQUATIONS OF THE FORM

34. 35. 36. 37.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

165

ó

( õÿñ + )ñ ó ò = ö D +  ] – ôñ . Š This is a special case of equation 1.7.1.15 with  (ˆ ) = Œ , (ˆ ) = 1, and % (ˆ ) = (  .

ï ññóò

= ô ïƒ÷ ] – ùñ . This is a special case of equation 1.7.2.11 with  (ˆ ) = ˆ and ƒ = 1.

ï ñ 2 ñ óò

= ô ïƒ÷ ] – ùñ 2 . This is a special case of equation 1.7.2.12 with  (ˆ ) = ˆ and ƒ = 1.

ó

( ôñ ÷ + õ )ñ ó ò = 1. 1 b . Solution in parametric form with B ≠ −1:

  U  U − ln L − U!S , X U −  +1 exp O B +1 Q B +1 B + 1 PuM 2 b . Solution in parametric form with B = −1,  ≠ −1:  =



 = − ln OL  −  * −



 +1

 * , P

=U 

1 +1

.

= * .

3 b . Solution in parametric form with B = −1,  = −1:

38.

( ô] +

õ ï )ñ óò

= 1.

a

Solution in implicit form:  = 39.

= * .

 = −U − ln( L − 'U ),

( ô] + õ ï 2 )ñó ò = 1. Solutions in parametric form:

 =−

: 1 U (ln )
, * 2

 L   +   1−   (L +  )

= ln O

U

2

4 "P

,

if  ≠ 1, if  = 1.

:

=L

ø

1 0(

U ) + L 2 n 0 (U )

and

: : 1 U 2 U (ln )
, = ln O − , = L 1 ® 0 (U ) + L 2 ; 0 (U ), * 2 4 "P where ø 0 (U ) and n 0 (U ) are the Bessel functions, and ® 0 (U ) and ; 0 (U ) are the modified Bessel functions.  =−

40.

( ô] + õ ï –1 )ñó ò = 1. Let  = A # $ ,  = T 2#

2

. Solution in parametric form:

 = # V 2UÑA exp(T U 2 )  (U )W ,

ó

ó

= ln V $ç (U )W T‹U 2 ,

where

 (U ) = X exp(T U 2 ) U + L S Q M

41.

( D +  ] + õ ï )ñó ò = öD +  ] – ô ï . Š Š This is a special case of equation 1.7.1.14 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) = (  .

42.

( D +  ] + õÿñ )ñó ò = öD +  ] – ôñ . Š Š This is a special case of equation 1.7.1.15 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) = (  .

43.

ó

−1

.

ó

ó

( ô ^ ñ ú + õ )ñ ó ò = ñ . This is a special case of equation 1.7.2.3 with  (ˆ ) = ( ˆ +  ) −1 .

Page 165

166 44. 45. 46.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ó

ó

(  ^ ñ ÷ + ô )ñ ó ò + õ  +_ ] + ô = 0. Á This is a special case of equation 1.7.2.15 with  ( ) =  and ( ) =   .

ó

(  ^ ñ ÷ + ô )ñó ò + õ ïƒú  _ ] + ô = 0. This is a special case of equation 1.7.2.15 with  ( ) =  and ( ) = ' + .

ó

ó

(  ^ ñ ú + þ ï )ñ ó ò = ñ ( õ ^ ÷ ñ ÷¿ú –  ï ). This is a special case of equation 1.7.2.17 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

47.

ï (ƒï ÷  ^ ]

+ ñ )ñó ò = õ ïƒ÷¿ú  ^ ú ] – ùñ . This is a special case of equation 1.7.2.16 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ + .

48.

(ô

ïƒ÷ ]

+

õ ï   ] )ñ óò

=

]

.

This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5).

49.

( ô ïƒ÷ ] + õ ï ñ ú )ñó ò =   ] . This is a Bernoulli equation with respect to  =  ( ).

50.

( ô ïƒ÷ ñ ú + õ ï ] )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ).

51.

(ô

52.

( ôþ ïƒ÷ ñ ú –1 + õ )ñó ò + ôù ïƒ÷ –1 ñ ú + ö = 0.  This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = (  .

53. 54. 55. 56.

ïƒ÷ ñ ú

+ õ ï ñ  )ñó ò = ] . This is a Bernoulli equation with respect to  =  ( ).

ó

( ôþ ï ÷ ñ ú –1 + õ ] )ñ ó ò + ôù ï ÷ –1 ñ ú + ö = 0. This is a special case of equation 1.7.1.19 with  ( ) =   and ( ) = ( .

ó

( ôþ ïƒ÷ ñ ú –1 + õÿñ  )ñó ò + ôù ïƒ÷ –1 ñ ú + ö = 0.  This is a special case of equation 1.7.1.19 with  ( ) =   and ( ) = (  .

ó

ó

[( ô ï + õÿñ ) ÷ + õ ^ ]ñ ó ò = ö ( ô ï + õÿñ )ú – ô ^ . This is a special case of equation 1.7.2.14 with  (ˆ ) = ˆ  , ( ) = 1, and % (ˆ ) = (˜ˆ + . [( ô ï + õÿñ ) ÷ + õ ^ ] ]ñó ò = ö ( ô ï + õÿñ )ú – ô ^ ] . This is a special case of equation 1.7.2.13 with  (ˆ ) = ˆ  , ( ) = 1, and % (ˆ ) = (˜ˆ + .

1.5.3. Equations Containing Hyperbolic Functions 1. 2. 3.

= ô cosh( ƒñ ) + õ cosh( b ï ). This is a special case of equation 1.7.2.18 with  ( ) = 0, ( ) =  , and % ( ) =  cosh( @! ).

ñóò

= ô sinh( ƒñ ) + õ sinh( b ï ). This is a special case of equation 1.7.2.18 with  ( ) =  , ( ) = 0, and % ( ) =  sinh( @! ).

ñ óò

= ô ïƒ÷ cosh( ƒñ ) + õ ïwú . This is a special case of equation 1.7.2.18 with  ( ) = 0, ( ) =   , and % ( ) = ' + .

ñóò

Page 166

1.5. EQUATIONS OF THE FORM

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

= ô ï ÷ sinh( ƒñ ) + õ ï ú . This is a special case of equation 1.7.2.18 with  ( ) =   , ( ) = 0, and % ( ) = ' + .

ñ óò

= ôñ 1+÷ + õÿñ + ö sinh( ï )ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =  , ( ) =  , and % ( ) = ( sinh( Œ  ).

ñóò

= ôñ 1+÷ + õ sinh( ï )ñ + ö^ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =  , ( ) =  sinh( Œ  ), and % ( ) = ( .

ñóò

= ñ cosh ï ( ôñ ÷¿ú sinh÷ –1 ï + õÿñ ú ). This is a special case of equation 1.7.2.22 with  (ˆ ) = ˆ  + 'ˆ .

ñóò

= ñ sinh ï ( ôñ ÷¿ú cosh÷ –1 ï + õÿñ ú ). This is a special case of equation 1.7.2.24 with  (ˆ ) = ˆ  + 'ˆ .

ñóò

= ( ô ï ÷ cosh ñ + õ ) coth ñ . This is a special case of equation 1.7.2.25 with  (ˆ ) = ˆ +  .

ï ñ óò ï ñóò

= ( ô ïƒ÷ sinh ñ + õ ) tanh ñ . This is a special case of equation 1.7.2.23 with  (ˆ ) = ˆ +  .

( ôñ ú cosh ï + õ )ñó ò = ñ ú +1 sinh ï . This is a special case of equation 1.7.2.24 with  (ˆ ) = ˆ ( ˆ +  ) −1 . ( ôñ ú sinh ï + õ )ñ„ó ò = ñ ú +1 cosh ï . This is a special case of equation 1.7.2.22 with  (ˆ ) = ˆ ( ˆ +  ) −1 .

( ô ïƒ÷ + õ ï coshú ñ )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5). ( ô ï ÷ + õ ï tanhú ñ )ñ ó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ).

( ô ïƒ÷ + õ ï coshú ñ )ñó ò = cosh  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

16.

( ô ïƒ÷ + õ ï tanhú ñ )ñó ò = tanh ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

17.

( ôþ

ïƒ÷ ñ ú

–1

18.

( ôþ

ïƒ÷ ñ ú

–1

19.

( ô ï ÷ ñ ú + õ ï )ñ ó ò = cosh ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

20.

167

+ õ )ñó ò + ôù

ïƒ÷ –1 ñ ú

+ ö sinh  (

+ õ )ñó ò + ôù

ïƒ÷ –1 ñ ú

+ ö tanh (

ï

) = 0.

This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( sinh  ( Π ).

ï

) = 0.

This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( tanh  ( Π ).

( ô ïƒ÷ ñ ú + õ ï )ñó ò = tanh ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

Page 167

168 21.

FIRST-ORDER DIFFERENTIAL EQUATIONS ( ô ïƒ÷ coshú ñ + õ ï )ñó ò = sinh  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

22.

( ô ïƒ÷ tanhú ñ + õ ï )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ).

23.

( ôþ

ïƒ÷ ñ ú

–1

( ôþ

ïƒ÷ ñ ú

–1

24.

+ õ sinh 

ñ )ñ óò

+ ôù

ïƒ÷ –1 ñ ú

+ ö = 0.

+ õ tanh

ñ )ñ óò

+ ôù

ïƒ÷ –1 ñ ú

+ ö = 0.

This is a special case of equation 1.7.1.19 with  ( ) =  sinh 

and ( ) = ( .

This is a special case of equation 1.7.1.19 with  ( ) =  tanh 

and ( ) = ( .

1.5.4. Equations Containing Logarithmic Functions 1. 2. 3. 4. 5. 6.

= ñ (  ï + þ ln ñ +  ). This is a special case of equation 1.7.2.3 with  (ˆ ) = ln ˆ + ý .

ñóò

= ô ï ÷ –1 ñ  ú +1 (ù ln ï + þ ln ñ ). This is a special case of equation 1.7.1.3 with  (ˆ ) = ˆ  ln ˆ .

ñóò

= ô ïƒ÷ ñ ln2 ñ + õ ïƒú ñ ln ñ + ö ï ñ . This is a special case of equation 1.7.3.1 with  ( ) =   , ( ) = ' + , and % ( ) = (˜  .

ñóò

ï ñóò

= ( ñ + ù ln ï ) ú +  . + This is a special case of equation 1.7.2.4 with  (ˆ ) = ln ˆ + ý .

ï ñóò

= ñ (ù ln ï + þ ln ñ ). This is a special case of equation 1.7.1.3 with  (ˆ ) = ln ˆ .

þ ï ñóò

=

ln ï + þ

ô ï s ñ  (ù

ln ñ ) – ùñ .

This is a special case of equation 1.7.1.5 with  ( ) = 7. 8. 9.

 ¡



À −1

and (ˆ ) = ln ˆ .

( ï D + õ )ñ ó ò = ñ ï D –1 + ö (ln ñ – ln ï ). This is a special case of equation 1.7.1.12 with  (ˆ ) =  , (ˆ ) = ( ln ˆ , and % (ˆ ) = 1.

ï ( ñ

+  )ñ ó ò = ù ln ï + (  – ù )ñ . This is a special case of equation 1.7.2.16 with  (ˆ ) = ý , (ˆ ) = 1, and % (ˆ ) = ln ˆ .

ï (ô

+ þ ï  )ñ ó ò = ñ ( õÿù ln ï + õÿþ ln ñ – ù ï  ). This is a special case of equation 1.7.1.16 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) =  ln ˆ .

10.

ï (ô

+ þxñ  )ñ ó ò = ñ ( õÿù ln ï + õÿþ ln ñ – ùñ  ). This is a special case of equation 1.7.1.17 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) =  ln ˆ .

11.

( ôþ

12.

( ô ln ñ + õ

ï ÷ ñ ú

–1

+ õ )ñ

óò

+ ôù

ï ÷ –1 ñ ú

+ ö ln  (

ï

) = 0.

This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( ln  ( Π ).

ï )ñ ó ò

= 1.

Solution:  =    ”

 

X

 −  M

+L

•

−

 

ln .

Page 168

1.5. EQUATIONS OF THE FORM

13. 14.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

(ln ñ )ñ ó ò = ñ ( ô ï ÷' ñ  + õ ï ÷ ñ ) – ùñ ln ñ . This is a special case of equation 1.7.3.7 with  (ˆ ) = ˆ  + 'ˆ and

169

ï

ï (ô

+þ

ïƒ÷ ñ ú

ln ñ )ñ„ó ò = ñ ( õ

– ù ln ñ + ö ).

¡

= 1.

This is a special case of equation 1.7.3.9 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) = 'ˆ + ( .

15. 16. 17. 18.

19.

( ô ïƒ÷ + õ ï lnú ñ )ñó ò = ln  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

ï (ô ï ÷ ñ ú

+ þ ln ï )ñ ó ò = ñ ( õ ï ÷' ñ úI – ù ln ï ). This is a special case of equation 1.7.3.10 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

ï (ô ƒï ÷ ñ ú

+þ

ïƒ÷' ñ I ú 

ln ñ )ñó ò = ñ ( õ

– ù ln ñ ).

This is a special case of equation 1.7.3.9 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

ï ÷ ñ ú

( ôþ

–1

+ õ ln 

ñ )ñ ó ò

+

ôù ï ÷ –1 ñ ú

+

ö

= 0.

This is a special case of equation 1.7.1.19 with  ( ) =  ln 

(ô

lnú

ïƒ÷

ñ

+

õ ï )ñ óò

and ( ) = ( .

= ln  ( Ė ).

This is a Bernoulli equation with respect to  =  ( ).

20.

( ô ïƒ÷ lnú ñ + õ ï ln  ñ )ñó ò = ñ s . This is a Bernoulli equation with respect to  =  ( ).

1.5.5. Equations Containing Trigonometric Functions 1.

ñóò

=

ñóò

= sin( ô



cos( ôñ ) +  cos( õ

ï

).

This is a special case of equation 1.7.4.11 with  ( ) = ƒ , ( ) = 0, and % ( ) = ý cos( ' ). 2.

ï

) cos( õÿñ ) + cos( ô

ï

) sin( õÿñ ).

This is a special case of equation 1.7.1.1 with  (ˆ ) = sin ˆ and ( = 0.

3.

= ô tan( õ ï ñ ). The solution is given by the relation:

ñ óò

?

X

0

exp  a  cos  £   a  1 2 2

M

a = L exp  '  , 1 2

2

where ¨ =





.

4.

= õ ï ÷ cos( ôñ ) + ö ï ú . This is a special case of equation 1.7.4.11 with  ( ) = '  , ( ) = 0, and % ( ) = (˜ + .

5.

ñóò

6. 7.

ñ óò

=

õ ïƒ÷

sin( ôñ ) + ö

ïwú

.

This is a special case of equation 1.7.4.11 with  ( ) = 0, ( ) = '  , and % ( ) = (˜ + .

= ñ cos ï ( ôñ ÷¿ú sin÷ –1 ï + õÿñ ú ). This is a special case of equation 1.7.4.4 with  (ˆ ) = ˆ  + 'ˆ .

ñóò

= ñ sin ï ( ôñ ÷¿ú cos÷ –1 ï + õÿñ ú ). This is a special case of equation 1.7.4.3 with  (ˆ ) = ˆ  + 'ˆ .

ñ óò

Page 169

170 8.

9. 10. 11.

FIRST-ORDER DIFFERENTIAL EQUATIONS sin2 ñ cos2 ñ + õ . cos2 ï sin2 ï This is a special case of equation 1.7.4.14 with  (ˆ ) = ˆ + 'ˆ

ñ óò

=

ô

= ôñ 1+÷ + õÿñ + ö sin( ï )ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =  , ( ) =  , and % ( ) = ( sin( Œ  ).

= ôñ 1+÷ + õ sin( ï )ñ + ö^ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =  , ( ) =  sin( Œ  ), and % ( ) = ( .

ñóò

ï ñóò

+

ô

sin( õ

ï

+ ö^ñ ) = 0.

B = 2: 2 ¨2
−  ¨

13.

14.

15. 16. 17. 18. 19. 20. 21. 22. 23.

.

ñóò

The substitution ¨

12.

−1

ï ñóò

=  tan

2

' + (

2 + 2( !  ( − 1)¨ − '

leads to a Riccati equation of the form 1.2.2.35 with 2

= 0.

= ô ï 2 tan( õÿñ ) + ñ . ¨ 
=  tan( ' ¨ ). The substitution =  ¨ leads to an equation of the form 1.5.5.3: I = ô ïƒ÷ cos2 ñ + õ cos ñ sin ñ . This is a special case of equation 1.7.4.8 with  (ˆ ) = 12 ( ˆ +  ).

ï ñóò

= ô ï ÷ sin2 ñ + õ cos ñ sin ñ . This is a special case of equation 1.7.4.7 with  (ˆ ) = 12 ( ˆ +  ).

ï ñ óò

= ô ï ú sin  ñ cos2–  ñ – ù sin 2ñ . This is a special case of equation 1.7.4.18 with  ( ) =  +

ï ñ óò

−2

´

−1

and (ˆ ) = ˆ  .

(1 + tan2 ñ )ñ ó ò = ô tanú +1 ñ + õ tan ñ + ö ï ÷ tan1–ú ñ . This is a special case of equation 1.7.4.19 with  ( ) =  , ( ) =  , and % ( ) = (˜  . ( ôþ ï ÷ ñ ú –1 + õ )ñ ó ò + ôù ï ÷ –1 ñ ú + ö sin  ( ï ) = 0. This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( sin  ( Œ  ).

( ôþ ï ÷ ñ ú –1 + õ )ñ ó ò + ôù ï ÷ –1 ñ ú + ö tan ( ï ) = 0. This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( tan  ( Œ  ).

( ô ïƒ÷ ñ ú + õ ï )ñó ò = cos ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5). ( ô ïƒ÷ ñ ú + õ ï )ñó ò = tan ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

( ôñ ú cos ï + õ )ñó ò = ñ ú +1 sin ï . This is a special case of equation 1.7.4.3 with  (ˆ ) = ˆ ( ˆ +  ) −1 .

( ôñ ú sin ï + õ )ñó ò = ñ ú +1 cos ï . This is a special case of equation 1.7.4.4 with  (ˆ ) = ˆ ( ˆ +  ) −1 . ( ô ïƒ÷ + õ ï cosú ñ )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ).

Page 170

1.5. EQUATIONS OF THE FORM

24. 25. 26. 27. 28. 29. 30.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

171

( ô ï ÷ + õ ï cosú ñ )ñ ó ò = cos  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

( ôþ ï ÷ ñ ú –1 + õ cos ñ )ñ ó ò + ôù ï ÷ –1 ñ ú + ö = 0. This is a special case of equation 1.7.1.19 with  ( ) =  cos 

( ô ïƒ÷ cosú ñ + õ ï )ñó ò = cos  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

and ( ) = ( .

( ô ïƒ÷ + õ ï tanú ñ )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ). ( ô ïƒ÷ + õ ï tanú ñ )ñó ò = tan ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

( ôþ ïƒ÷ ñ ú –1 + õ tan ñ )ñó ò + ôù ïƒ÷ –1 ñ ú + ö = 0. This is a special case of equation 1.7.1.19 with  ( ) =  tan 

( ô ïƒ÷ tanú ñ + õ ï )ñó ò = tan ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

and ( ) = ( .

1.5.6. Equations Containing Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions 1. 2. 3.

= ô ïƒ÷ ] + õ lnú ï . + This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) =  ln  .

ñóò

= ô ln÷ ( b ï )  ] + õ ï ú . This is a special case of equation 1.7.2.5 with  ( ) =  ln  ( @! ) and ( ) = ' + .

ñ óò

= ô ] ( ƒñ + ln ï )ú . + This is a special case of equation 1.7.2.2 with  (ˆ ) =  ln ˆ .

ñ óò

ó

4.

= ô –  ( ï + ln ñ )ú . + This is a special case of equation 1.7.2.1 with  (ˆ ) =  ln ˆ .

5.

ñóò

= ôñ ln2 ñ + õÿñ ln ñ + ö ñ .  This is a special case of equation 1.7.3.1 with  ( ) =  , ( ) =  , and % ( ) = (   .

6.

ñóò

ñóò

ó

ó

= ôñ ln2 ñ + õ ñ ln ñ + ö^ñ .  This is a special case of equation 1.7.3.1 with  ( ) =  , ( ) =   , and % ( ) = ( .

7.

= ô] sin ï + õ tan ï . This is a special case of equation 1.7.5.6 with  (ˆ ) = ˆ +  .

8.

ñóò

= ( ô sin ñ + õ ) tan ñ . This is a special case of equation 1.7.5.4 with  (ˆ ) = ˆ +  .

9.

ñóò

ñóò

ó

ó

ó

= ô sin2 ñ + õ – cos2 ñ . This is a special case of equation 1.7.5.8 with  (ˆ ) = 12 ( ˆ +  ˆ ).

Page 171

172 10.

11.

12.

13.

14.

15.

FIRST-ORDER DIFFERENTIAL EQUATIONS cos÷ (

ï ) ]

õï ú

ñ óò

=

ô

ñóò

=

ô ƒï ÷  ]

+ õ cosú (

ñóò

=

ô ƒï ÷  ]

+ õ tanú (

ñóò

=

ô

ñóò

=

ñóò

=

+

.

This is a special case of equation 1.7.2.5 with  ( ) =  cos  (“/ ) and ( ) = ' + .

ï

).

This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) =  cos + (“/ ).

ï

).

This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) =  tan + (“/ ). tan÷ (

ï ) ]

+

õ ïƒú

.

This is a special case of equation 1.7.2.5 with  ( ) =  tan  (“/ ) and ( ) = ' + .


B

ó

cos( ôñ ) +

ó

(4 

sin( ôñ ) +

ó


B-

.

  The substitution ¨ = tan( 12  ) leads to a linear equation: ¨ 
= !$   ¨ + !#   .

ô

sin(

ï

ï

) sinh( ƒñ ) + õ cos(

) cosh( Ė ).

This is a special case of equation 1.7.2.18 with  ( ) =  sin(“/ ), ( ) =  cos(“/ ), and % ( ) = 0. 16.

17.

ñóò

=

ôñ

ln2 ñ +

õÿñ

ln ñ + ö sin ÷ (

óò

ô

tan1+ú

ï )ñ

.

This is a special case of equation 1.7.3.1 with  ( ) =  , ( ) =  , and % ( ) = ( sin  ( Œ  ). (1 + tan2 ñ )ñ

=

ñ

+ õ tan ñ + ö



ó

tan1–ú

ñ

.



This is a special case of equation 1.7.4.19 with  ( ) =  , ( ) =  , and % ( ) = ( ! . 18.

ó

( ô

cos ñ + õ )ñó ò = cot ñ .

This is a special case of equation 1.7.5.5 with  (ˆ ) = ( ˆ +  ) −1 . 19.

ó

( ô

sin ñ + õ )ñ

óò

= tan ñ .

This is a special case of equation 1.7.5.4 with  (ˆ ) = ( ˆ +  ) −1 . 20.

( ô] cos ï + õ )ñó ò = tan ï . This is a special case of equation 1.7.5.6 with  (ˆ ) = ( ˆ +  ) −1 .

21.

( ô

]

sin ï + õ )ñ

óò

= cot ï .

This is a special case of equation 1.7.5.7 with  (ˆ ) = ( ˆ +  ) −1 . 22.

(

^ ó ñ ÷

+

ô )ñ óò

+

õ _ ]

lnú

23.

(

^ ó ñ ÷

+

ô )ñ óò

+

õ _ ]

cosú

24.

(

^ ó

ï

+ ô

= 0.

+ This is a special case of equation 1.7.2.15 with  ( ) =  and ( ) =  ln  .

ï

+

ô

= 0.

This is a special case of equation 1.7.2.15 with  ( ) =  and ( ) =  cos +  . cos÷

ñ

+ ô )ñó ò +

õ _ ]

cosú (

ï

) + ô

= 0.

This is a special case of equation 1.7.2.15 with  ( ) = cos 

and ( ) =  cos + ( Π ).

Page 172

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

) = 0 CONTAINING ARBITRARY PARAMETERS

173

1.6. Equations of the Form c ( ì , í , í ï î ) = 0 Containing Arbitrary Parameters 1.6.1. Equations of the Second Degree in â0ä ã 1.6.1-1. Equations of the form  ( , )(
 )2 = ( , ). 1.

(ñó ò )2 =

ôñ

õï

+

2

.

See equation 1.6.3.43. 2.

3.

(ñó ò )2 =

ñ

ï

+ô

2

õï

+

+ö .

The substitution ¨ = 2 ¨*¨2
− ¨ = 4  + 2  . ü

(ñ

óò

ôñ

)2 =

3

+

õÿñ

+ 

+ö . 3

Solution:  = L¥A\X (  4.

(ñó ò )2 =

ôñ

+

7 õ ï

2

+ ' + ( leads to an Abel equation of the form 1.3.1.2:

+ ( )−1 J

+

2

.

M

.

See equation 1.6.3.26. 5.

6.

(ñó ò )2 =

ôñ

+

7 õ ï

+ö ,

ïƒú

–

ô

The substitution  ¨ = 2  ¨*¨2
− ¨ = ˜ −2  −1 J 2 . ü

(ñó ò )2 =

ñ

+ô

+1

The substitution ¨

þ

2(þ

=2

Q

≠ 0. +  £  + ( leads to an Abel equation of the form 1.3.1.32:

+1 ï + 3)2 +  +

+1

2

+õ .

−

¡

+1  + 3)2

2( 2( + 1) ¡
* ¨ ¨ ¨  − =− ¡ form 1.3.1.10:  + 2 ( ¡ ( + 3)2 7.

(ñó ò )

8.

ï (ñ óò

2

= ƒñ + ô ï

2

+ õ ïƒú

¡

+1

2

+ S

+ 1)

J

1 2

leads to an Abel equation of the

+ .

+ö .

For Œ ≠ 0, the substitution Œ ¨ = 2( Œ +  2 + ' + +1 + ( )1 J 2 leads to the Abel equation ¨*¨2
− ¨ = 4 Œ −2  + 2 Œ −2 ( + 1) + , which is outlined in Subsection 1.3.1 (see Table 5). ¡ Special cases of the original equation are equations 1.6.1.1–1.6.1.6. )2 =

ô ï ñ

+õ .

See equation 1.6.3.32. 9.

10.

ï (ñ óò

)2 =

ô ï ñ

+

õï

The substitution  ¨ ¨*¨2
− ¨ = −2 ( −2 

ñ 2 (ñ ó ò

)2 =

ô ï 2ñ

2

+ö , =2  . ü

−2

ô

≠ 0. +  + (˜

−1

leads to an Abel equation of the form 1.3.1.33:

+õ .

See equation 1.6.3.34. 11.

ñ 2 (ñ ó ò

)2 =

ô ï –2 ) 5 ñ

2

+õ .

See equation 1.6.3.28.

Page 173

174 12.

FIRST-ORDER DIFFERENTIAL EQUATIONS ( ôñ

2

+

õ ï )(ñ óò

)2 = 1.

See equation 1.6.3.44. 13.

ï

(ô

2

+

õÿñ )(ñ óò

)2 =

ï 2ñ

.

See equation 1.6.3.46. 14.

ï ñ

(ô

+ õ )(ñ„ó ò )2 = ñ .

See equation 1.6.3.33. 15.

ï ñ 2 (ñ ó ò

ôñ

)2 =

2

+

õï

.

See equation 1.6.3.45. 16.

ï 2ñ

(ô

2

+ õ )(ñó ò )2 =

ï

2

.

See equation 1.6.3.35. 17.

7

(ô

ñ

õ ï )(ñ óò

+

)2 = 1.

See equation 1.6.3.27. 18.

ï 2 ñ 3 ) 5 + õÿñ )(ñ óò

(ô

)2 =

ï 2ñ

.

See equation 1.6.3.29. 19.

(ñ

óò

ô ]

+õ .

(ñó ò )2 =

ô

õ ó

(ñó ò )2 =

ôñ

)2 =

See equation 1.6.3.3 with , = 2.

20.

+

.

See equation 1.6.3.4 with , = 2.

21.

2

õ ó

+

.

See equation 1.6.3.8 with , = 2.

22.

ï 2 (ñ óò

)2 =

ô ï 2  ]

+õ .

See equation 1.6.3.9 with , = 2. 23.

( ô] +

õï

2

)(ñó ò )2 = 1.

See equation 1.6.3.9 with , = −2.

24.

( ô

óñ

2

óò

+ õ )(ñ

)2 =

ñ

2

.

See equation 1.6.3.8 with , = −2.

25.

(ñó ò )2 =

ôñ

+ õ ln ï .

See equation 1.6.3.13. 26.

27.

(ñ

óò

)2 =

Ė

+

ô

ln ï + õ ,

The substitution Œ ¨ = 2 £ Œ ¨*¨2
− ¨ = 2 Œ −2  −1 .

( ô ln ñ + õ

ï )(„ñ óò

≠ 0. +  ln  +  leads to an Abel equation of the form 1.3.1.16:

)2 = 1.

See equation 1.6.3.14.

Page 174

1.6. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY PARAMETERS

175

1.6.1-2. Equations of the form  ( , )(
 )2 = ( , )
 + % ( , ). 28.

(ñó ò )2 + ôñó ò + õÿñ = 0. Solution in parametric form:

' = −2a −  ln a + L ,

29.

30.



= −a 2 − a .

(ñó ò )2 + ôññó ò = õ ï + ö . We differentiate the equation with respect to  , take as the independent variable, and assume ˆ = 
 to obtain a linear equation with respect to = (ˆ ): (ˆ 2 −  )
Š + ˆ + 2ˆ 2 = 0. (ñ

óò

)2 +

ô ï ñ óò

+

õÿñ

+

öï

2

= 0.

2 g The transformation  =  , =  2 ‡ leads to an autonomous equation:  ‡ g
+ 2‡ + 12   = 1 2 1 4  − ( − '‡ . Having extracted the root and carried over the terms 2 ‡ + 2  from the left-hand

side to the right-hand side, we obtain a separable equation of the form 1.1.2.

31.

32. 33.

34.

35.

36.

= ï ñó ò + ô ï 2 + õ (ñó ò )2 + ö^ñó ò + ý , ô ≠ 0. Differentiating with respect to  and changing to new variables a = "
 and ¨ (a ) = −2  , we arrive at an Abel equation of the form 1.3.1.2: ¨*¨ g
− ¨ = −4 'a − 2 !( .

ñ

(ñ ó ò )2 + ( ô ï + õ )ñ ó ò – ôñ + ö = 0, Solutions: = (  +  ) L +   L 2 + (

ô

≠ 0. and 4 

(ñ ó ò )2 + ( ôñ + õ ï )ñ ó ò + ôƒõ ï ñ = 0. This equation can be factorized: (
 +   = L  −  and = − 21 ' 2 + L .

= 4 ( − (  +  )2 .

)(
 + ' ) = 0. Therefore, the solutions are:

(ñó ò )2 + ô ï 2 ñó ò + õ ï ñ = 0. The transformation H = ln  , ‡ =  −3 leads to an equation independent implicitly of H : (‡
‘ )2 + (  + 6‡ )‡
‘ + (3  +  + 9‡ )‡ = 0. Rewriting the latter equation to solve for ‡
‘ , we obtain a separable equation of the form 1.1.2. )2 – ññó ò – ï = 0. Solution in parametric form:

ô (ñ óò

ï (ñ óò

a V L +  ln  a + £ a 2 + 1  W ,  = £ 2 a +1

= a −

 a

.

)2 – ôññó ò + õ = 0. 1 b . For  ≠ 1, the solution in parametric form is written as:

 = L3a  +



2 − 1

2 b . For  = 1, the solution is: 37.

−1

ï (ñ óò

a 2,

a = a 2 +  ,

where

, =

= L3 +  L .

1 .  −1

)2 + ôññó ò + õ ï = 0. 1 b . For  ≠ −1, the solution in parametric form is written as:

 = L3a |(  + 1)a 2 +  | There are two singular solutions:

= A"

ü



+2 − 2( +1)



,

−  (  + 1).

=−

 2 (a +  ).  a

Page 175

176

FIRST-ORDER DIFFERENTIAL EQUATIONS 2 b . For  = −1, the solution in parametric form is written as:

 = L3a exp O − 38.

ï (ñ óò

)2 – ññ

óò

a2 , 2 P

= O a +



a P

.

+ ôñ = 0.

Solution in parametric form:

 = L (a −  ) exp(−a  ),

39.

40.

There is a singular solution:

= 0.

ï (ñ óò

)2 – ññ

+

ñ (ñ óò

)2 +

óò

+ô

ï 2 ñ óò

+

õÿñ óò

ö

ô

= 0,

= L3a 2 exp(−a  ).

≠ 0.

We divide the equation by 
and differentiate with respect to  . Passing to the new variables a = 
 and ¨ (a ) = −2 , we arrive at an Abel equation of the form 1.3.1.33: ¨*¨pg
− ¨ = !(˜a −2 .

ô ï ñóò

+

õÿñ

= 0.

Solution in parametric form:

a + (  + a 2 ) = 0,

(a 2 +  +  ) + = a  J ( 

L



+ )

,

where

¡

=

 + 2 . 2(  +  )

There two singular solutions = A" £ −  −  corresponding to the limit LZD = 0 is also a singular solution. 41.

ï (ñ óò

)2 + ( ô – ñ ) ñ

óò

õ

+

F . In addition,

= 0.

Solutions: L ( L3 − +  ) +  = 0 and ( −  )2 = 4 ' . 42.

ô ï (ñ óò

)2 + ( õ

ï

– ôñ + ü )ñó ò –

, L

õÿñ

= 0.

Solution: = L3 + . In addition, there is a singular solution which can be written in L +  parametric form as:

 =− 43.

ô ï (ñ óò

)2 – ( ôñ +

õï

– ô – õ )ñ„ó ò +

45.

ï (ñ óò

)2 + ôññó ò +

,!a . a + 

= 0.

õ ƒï ÷ ñ ú

L ( +  ) L − 

+ ' −  −  )2 − 4 '

and ( 

= 0.

= 0.

g The substitution  =  leads to an equation of the form 1.6.1.68: ("
g )2 +  !
g +   ( 

ï 2 (ñ ó ò

)2 – (2 ï

ñ

ô ï 2 (ñ óò

2

A ü

óò

+ñ

 + L

ï ññóò

)2 – 2 ô

Solutions:

+ ô )ñ

= L

Solutions: 46.

õÿñ

= a +

,

Differentiating with respect to  and factorizing, we obtain (2  "
 −  − ' +  +  )

 = 0. Equating both factors to zero and integrating, we arrive at the solutions: = L3 +

44.

,

( a +  )2

2

2

+1)

g + = 0.

= 0. = − 41 

and

+ñ

2

– ô ( ô – 1) ï

+ 

2

= L3

1+

2

−1

.

= 0.

 , where , = ü

(  − 1)  .

Page 176

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

47.

(ô

2

– 1) ï

2

(ñó ò )2 + 2 ï

ññóò

–ñ

+ô

2

2

ï

) = 0 CONTAINING ARBITRARY PARAMETERS

2

177

= 0.

Solution in parametric form:

 = L (a 2 + 1)−1 J 2  a + £ a 2 + 1  48.

ï 2 (ñ ó ò

)2 + ( ô

ï 2ñ

3

+ õ )ñ

óò

ññóò

–ï

+ ôƒõÿñ

3

−1

J

= a +  £ a 2 + 1.

,

= 0.

The equation can be factorized: (
 +  3 )( 2 
 +  ) = 0. Equating each of the factors to zero, we obtain the solutions: = A (2  + L )−1 J 2 and =   + L .

49.

(ï

2

50.

(ï

2

51.

(ï

2

– ô )(ñó ò )2 – 2 ï

2

= 0.

Solving for , differentiating with respect to  , and setting ¨ ( ) =
 , we obtain a factorized equation: ( ¨3
− ¨ )( 2 ¨ 2 +  2 −  ¨ 2 ) = 0. Equating each of the factors to zero, we arrive at the solutions: 1 2 2 = ( −  − L 2 ) and +  2 =  ( ≠ 0). 2L – ô 2 )(ñó ò )2 + 2 ï

ññóò

+ñ

2

= 0.

The equation can be factorized: ( 
 +  
 + )( 
 −  
 + ) = 0. Equating each of the factors to zero, we obtain the solutions: ( +  ) = L and ( −  ) = L .

óò

+ ô )(ñ

)2 – 2 ï

ññ óò

+ñ

2

õ

+

= 0.

Differentiating with respect to  , we obtain a factorized equation: [( Therefore, the solutions of the original equation are: =L 52.

ï 3 (ñ óò

)2 +

ï 2 ñ ñóò

ô ï ñ (ñ óò

=L

)2 – ( ôñ

2 1

where L

+L

2 2

+  = 0;

'

2

+

2

+  )
 − 

]

 = 0.

+  = 0.

+ ô = 0.

Solutions: L3 53.

 + L 2,

1

2

2

2

õï

+

 + 2

and 

2

+ ü )ñó ò +

õï ñ

= 4 . = 0.

This differential equation represents an equation of curvature lines of a surface defined by the relation #* 2 + $ 2 + LuH 2 = 1, where  = #2$ ( L − $ ),  = #2$ ( # − L ), , = L ( $ − # ). Solutions: ( L −  ) 54.

ñ 2 (ñ óò

)2 + 2 ô

ï ññóò

Solutions:

55.

( ô – õ )ñ

2

= L ( L −  )

+ (1 – ô )ñ 2

(ñó ò )2 – 2 õ

2

+ 

ï ññóò

2

2

+

ô ï

2

−, L

2

+ ( ô – 1) õ = 0.

and

−  = (  − 1)( + L )2 + ôñ

2

–

õï

2



2

and

= '

2

2

+ 

A 2 £ − , − , .

2

−  = 0.

– ôƒõ = 0.

Solutions:

 56.

( ôñ – ï

2

2

+

)(ñó ò )2 + 2 ï

Solution: ( L

2

= L3 +  −

ññóò

–ñ

+  )2 = 4 

.

2

 − L 4

2

and (  −  )

2

− '

2

= ( −  )  .

= 0.

Page 177

178 57.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñ 2 – ô 2 ï 2 )(ñó ò )2 + 2 ï ññó ò + (1 – ô 2 ) ï Solution in parametric form:

2

= 0.

L3a  = £ 2 , a +1

58.

( ôñ – õ ï )2 [ ô 2 (ñó ò )2 + õ 2 ] – ü 2 ( ôñó ò + õ )2 = 0. We solve the equation for  − ' and differentiate with respect to  . Setting ¨ ( ) = 
, we obtain a factorized equation with respect to ¨ ( ): (  ¨ −  )[(  2 ¨ 2 +  2 )3 J 2 A\, ¨2
] = 0. Equating each of the factors to zero and integrating, we arrive at the solutions: ( ' − L )2 + ( 

59.

60.

(ï

ñóò

(ï

ñóò

63.

64.

ïƒ÷ +1 ñ óò

+ ùñ )2 + ô =

L

B

2 2

B

ï 2÷

+ ùñ )2 – ô

+

+2

õ

+

67.

− ' = A/, £ 2.

(ñó ò )2 – õ = 0. 2

+ 'B

−2

.

(ññó ò + ï )2 = ô 2 ( ï 2 + ñ 2 )  [(ñó ò )2 + 1]. This equation splits into two equations of the form 1.8.1.4 with  (‡ ) = A"‡ J 2 :

! 
+  = A  (

ô ï 2÷ (ï ñ óò

2

+

2

) J

2


 2 ü ( ) + 1.

+ þxñ )2 + õ ï 2ú ( ï ñó ò + ùñ )2 = 1. Solution: = L 1  −  + L 2  − + . Here, the constants L 2 2 ( L 1 + L 2 )(B − )2 = 1.

( ï ñó ò

1

and L

2

are related by the constraint

¡

– ñ ) = ô ( ï 2 + ñ 2 )  (ññó ò + ï )2 . This equation splits into two equations of the form 1.7.1.20 with  (‡ ) = A ‡ J 2 : 2

2

2

2

+


) J 2 (!  +  ).

( ï ñó ò – ñ )2 = ô 2 ( ï 2 + ñ 2 )  [(ñó ò )2 + 1]. This equation splits into two equations of the form 1.8.1.3 with  (‡ ) = A"‡ J 2 :

 
− = A  (

66.



and

= 0.

 
− = A  (

65.

2

 − + L .

£ L = L3 −  A

Solutions:

62.

− L )2 = ,

( ï ñó ò + ô )2 – 2 ôñ + ï 2 = 0, ô ≠ 0. 2 2 The substitution 2  −  = ‡ leads to the equation ‡"‡
 −  (‡ −  )+  2 = 0. Further assuming ‡ −  =  ¨ ( ), we obtain ( ¨ +  )¨3
+ ¨ 2 + 1 = 0. Taking ¨ to be the independent variable, we arrive at a linear equation whose solution is:  = (¨ 2 + 1)−1 J 2 V L −  ln  ¨ + £ ¨ 2 + 1  W .

Solution: 61.

L

= L − £ . a 2+1

2

+

2

) J

2


 2 ü ( ) + 1.

( ï ñó ò + ñ + 2 ô ï )2 = 4( ï ñ + ô ï 2 + õ ). The substitution ‡ =  +  2 +  leads to a separable equation: ‡
 = A 2 £ ‡ .

( ô 2 ï + õ 2 ñ + ö 2 )(ñó ò )2 + ( ô 1 ï + õ 1 ñ + ö 1 )ñó ò + ô 0 ï + õ 0 ñ + ö 0 = 0. The Legendre transformation  = ‡ g
, = aև g
− ‡ (
 = a ) leads to a linear equation: [  (a ) + aÝ (a )]‡ g
= (a )‡ + % (a ),

where  (a ) =  2 a 2 +  1 a +  0 , (a ) =  2 a 2 +  1 a +  0 , and % (a ) = − ( 2 a 2 − ( 1 a − ( 0 .

Page 178

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

68.

(ñ

óò

)2 +

ôññ óò

= 0.

ñóò

+ñ =

¡

ô 3ó (ñ óò

ñóò

– ñ )2 .

= 12   L

Solution: 70.

179

 ≠ 2, solving for 
and performing the substitution ¨ = ´ + −2 , we arrive at a 2− separable equation: ¨ 
= Œ ¨ + ¡  ‹A £  2 − 4  ¨  ¨ (see Subsection 1.1.2). 2 2 b . With = 2, solving the original equation for "
 , we obtain a separable equation: ¡ £ 2 
2  =  − ‹A  − 4    . 1 b . With

69.

õ  ó ñ ú

+

) = 0 CONTAINING ARBITRARY PARAMETERS

2

ó

= (ñó ò 2 – ñ 2 )( ô





− 12 L 

+

õ –ó

−



.

).

This is a special case of equation 1.8.1.56 with  (‡ ) = ‡ . 71.

72.

ô 2 ó (ñ óò

+ ñ )2 +

õ 2_ ó (ñ óò

+



Ė

)2 = 1.

 Solution: = L 1  −  + L 2  −: . Here, the constants L constraint ( L 12 + L 22 )(ý − Œ )2 = 1.

ñóò

= ô (ñó ò 2 – ñ 2 ) cosh ï .

ñóò

= ô (ñó ò 2 – ñ 2 ) sinh ï .

1

and L

are related by the

2

This is a special case of equation 1.8.1.58 with  (‡ ) = ‡ . 73.

This is a special case of equation 1.8.1.59 with  (‡ ) = ‡ . 74.

ô (ñ óò

cosh ï – ñ sinh ï )2 + õ (ñó ò sinh ï – ñ cosh ï )2 = 1.

= L 1 sinh  + L Solution: constraint L 12 + L 22 = 1.

75.

(ñó ò )2 – ï

ññóò

Solutions: 

76.

+ñ

2

cosh  . Here, the constants L

ln( ôñ ) = 0.

= exp( L3 − L

2

1

and L

2

are related by the

= exp( 41  2 ).

) and 

) cos ï .

ñóò

= ô (ñó ò 2 + ñ

ñóò

= õ (ñó ò 2 + ñ 2 ) sin ï .

2

2

This is a special case of equation 1.8.1.64 with  (‡ ) = ‡ . 77.

This is a special case of equation 1.8.1.65 with  (‡ ) = ‡ and  = 0. 78.

ô (ñ óò

cos ï + ñ sin ï )2 + õ (ñó ò sin ï – ñ cos ï )2 = 1.

Solution: = L 1 sin  + L L 12 + L 22 = 1. 79.

ô (ñ óò

2

cos  . Here, the constants L

cosh ï – ñ sinh ï )2 + õ [(ñ

óò

)2 – ñ 2 ] +

ö

1

and L

ô (ñ óò

cos ï + ñ sin ï )2 + õ [(ñó ò )2 + ñ

Solution: L 12 +  ( L

2 1

2

are related by the constraint

1

and L

2

are related by the constraint

= 0.

Solution: = L 1 sinh  + L 2 cosh  . Here, the constants L constraint L 12 +  ( L 12 − L 22 ) + ( = 0. 80.

2

] + ö = 0.

= L 1 sin  + L 2 cos  . Here, the constants L + L 22 ) + ( = 0.

1

and L

2

are related by the

Page 179

180

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.6.2. Equations of the Third Degree in

â0äã

1.6.2-1. Equations of the form  ( , )( 
)3 = ( , ) 
+ % ( , ). 1. 2.

3. 4. 5. 6. 7.

(ñó ò )3 + ô ï + õÿñ + ö = 0. This is a special case of equation 1.8.1.9 with  (¨ ) = ¨

( ô ï + õÿñ + ö )(ñ„ó ò )3 =  ï Dividing both sides by  of the form 1.7.1.6 with 

9. 10. 11. 12.

.

+ ñ + . +  + ( and raising to the power 1/3, we finally arrive at an equation (¨ ) = ¨ −1 J 3 .

)3 + õÿñó ò = ï . This is a special case of equation 1.8.1.7 with  (¨ ) =  ¨

ô (ñ óò ô (ñ óò

)3 + õÿñó ò = ñ . This is a special case of equation 1.8.1.8 with  (¨ ) =  ¨

3

+ ¨ .

3

+ ¨ .

)3 + ï ñó ò = ñ . This is a special case of equation 1.8.1.10 with  (¨ ) =  ¨

ô (ñ óò

3

.

)3 + õ ï ñó ò = ñ . This is a special case of equation 1.8.1.11 with  (¨ ) =  ¨ and ( ) =  ¨

ô (ñ óò

(ñó ò )3 – ô ï ñó ò + ï 3 = 0, Solution in parametric form:

ô

3

.

≠ 0.

 = 8.

3

a

a 3+1

,

=L +



4a 3 + 1 . 6 (a 3 + 1)2 2

(ñó ò )3 – ô ï ññó ò + 2 ôñ 2 = 0. Differentiating with respect to  and eliminating , we obtain a factorized equation with respect to ¨ ( ) = 
: [2(¨2
)2 −  ¨2
+  ¨ ](9¨ −  2 ) = 0. Equating each of the factors to 1 zero and integrating, we find the solutions: = 14 L ( − L )2 and = 27  3 .

ô ï (ñ óò

)3 + õÿñó ò = ñ . This is a special case of equation 1.8.1.11 with  (¨ ) =  ¨

ô ï 3 ) 2 (ñ óò

)3 + 2 ï ñó ò = ñ . Solution: = 2 L £  + L

3

.

)3 + ï ñó ò = ñ . This is a special case of equation 1.8.1.15 with  (¨ ) =  ¨

ô ïƒ÷ (ñ óò

ô 3ó (ñ óò

and (¨ ) =  ¨ .

3

)3 + õ (ñó ò + ñ ) + ö = 0.  Solution: = L  − + ( L 3 − ( ) 

−1

3

.

.

1.6.2-2. Equations of the form  ( , )(
 )3 = ( , )(
 )2 + % ( , )
 + & ( , ). 13.

)3 + õ (ñó ò )2 = ï . This is a special case of equation 1.8.1.7 with  (¨ ) =  ¨

ô (ñ óò

3

+ ¨

2

.

Page 180

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

14. 15.

ô (ñ óò

)3 + õ (ñó ò )2 = ñ . This is a special case of equation 1.8.1.8 with  (¨ ) =  ¨

(ñó ò )3 + ô (ñó ò )2 + õÿñ + ôƒõ ï + Solution in parametric form:

ý

2

ln(a +  ) + L ,

In addition, there is a singular solution: 16.

17. 18. 19.

20.

21. 22. 23. 24.

25.

26. 27.

2

.

= − ' − a 3 − a 2 − .



M

M

2

+ õ ï (ñ ó ò

 = L + 32  a 2 + 2 'a + ( ln |a |,

= a 3 + 'a 2 + (˜a − .

M

) ) =ñ . This is a special case of equation 1.8.1.11 with  (¨ ) =  ¨ 3

+ ¨

= −  −  .

ô (ñ óò ) + õ (ñ óò ) + ö^ñ óò = ñ + ý . Solution in parametric form: ô (ñ óò

3

= 0.

2 ' = −3a 2 + 2 a − 2 

3

181

) = 0 CONTAINING ARBITRARY PARAMETERS

2

ô ï (ñ óò

)3 + õ (ñó ò )2 = ñ . This is a special case of equation 1.8.1.11 with  (¨ ) =  ¨

2

and (¨ ) =  ¨

3

.

3

and (¨ ) =  ¨

2

.

( ï 2 – ô 2 )(ñó ò )3 + õ ï ( ï 2 – ô 2 )(ñó ò )2 + ñó ò + õ ï = 0. The equation can be factorized: (
 + ' )[(
 )2 ( 2 −  2 )+1] = 0, whence we find the solutions: = − 21 ' 2 + L and = A arcsin(  ) + L .

ô (ñ óò

ó

+ ñ )3 + õ 3 (ñ ó ò )3 + ö = 0.  Solution: = L 1 − L 2  − . Here, the constants L L 13 + L 23 + ( = 0.

1

and L

2

are related by the constraint

( ï ñ ó ò – ñ )3 + ôñ + õ ï = 0. This is a special case of equation 1.8.1.16 with  (¨ ) = 1, (¨ ) =  , % (¨ ) =  , and B = 3.

( ï ñó ò – ñ )3 + ôññó ò + õ ï = 0. This is a special case of equation 1.8.1.16 with  (¨ ) = 1, (¨ ) =  ¨ , % (¨ ) =  , and B = 3. ( ï ñó ò – ñ )3 + ô ï ñó ò + õÿñ = 0. This is a special case of equation 1.8.1.16 with  (¨ ) = 1, (¨ ) =  , % (¨ ) =  ¨ , and B = 3. cosh ï – ñ sinh ï )3 + õ (ñ ó ò sinh ï – ñ cosh ï )3 + ö = 0. = L 1 sinh  − L 2 cosh  . Here, the constants L 1 and L Solution: constraint L 13 + L 23 + ( = 0.

ô (ñ óò

cosh ï – ñ sinh ï )3 + õ (ñ ó ò 2 – ñ 2 ) + ö = 0. Solution: = L 1 sinh  + L 2 cosh  . Here, the constants L constraint L 13 +  ( L 12 − L 22 ) + ( = 0.

ô (ñ óò

cosh ï – ñ sinh ï )3 = õ (ñó ò 2 – ñ 2 ) cosh ï . This is a special case of equation 1.6.4.21 with B = 3 and

ñóò (ñ óò

sinh ï – ñ cosh ï )3 = õ (ñó ò 2 – ñ 2 ) sinh ï . This is a special case of equation 1.6.4.22 with B = 3 and

1

and L

2

are related by the

2

are related by the

= 1.

ñóò (ñ óò

¡ ¡

= 1.

Page 181

182 28.

29. 30. 31.

FIRST-ORDER DIFFERENTIAL EQUATIONS cos ï + ñ sin ï )3 + õ (ñó ò sin ï – ñ cos ï )3 + ö = 0. Solution: = L 1 sin  − L 2 cos  . Here, the constants L 1 and L L 13 + L 23 + ( = 0.

ô (ñ óò

2

are related by the constraint

cos ï + ñ sin ï )3 + õ (ñó ò 2 + ñ 2 ) + ö = 0. This is a special case of equation 1.6.4.24 with B = 3 and , = 1.

ô (ñ óò

cos ï + ñ sin ï )3 = õ (ñó ò 2 + ñ 2 ) cos ï . This is a special case of equation 1.6.4.25 with B = 3 and

ñóò (ñ óò

= 1.

sin ï – ñ cos ï )3 = õ (ñ ó ò 2 + ñ 2 ) sin ï . This is a special case of equation 1.6.4.26 with B = 3 and

¡

ñ ó ò (ñ ó ò

= 1.

¡

1.6.3. Equations of the Form (â$ä ã ) d = å ( â ) + ç ( æ ) 1.6.3-1. Some transformations. 1 b . In the general case, the equation ( 
)  =  ( ) + ( )

(1)

can be reduced with the aid of the transformation a = X [ ( )]1 J'  , ‡ = X [  ( )]−1 J' M M same form (‡ g
)  =  (‡ ) + 4 (a ), where functions 

=  (‡ ) and 4

to the

= 4 (a ) are defined parametrically by the following formulas:

 (‡ ) = 4 (a ) =

1

 ( )

,

1 , ( )

‡ = X [  ( )]−1 J' a = X [ ( )]1 J'

,

M M

 .

2 b . Taking as the independent variable, we obtain from equation (1) an equation of the same class for  =  ( ): (
)−  = ( ) +  ( ). 3 b . The equation




=  £

+ ( )

 , = 1,  =  ¨ can be reduced with the aid of the substitution ( ) = 2  −1 £ 2  −2 ( ), which is outlined in Subsection 1.3.1. 4 b . The equation 
= −1 + ( ) ( , = 1,  = is an alternative form of representation of the Abel equation !"
 Subsection 1.3.2. À 5 b . The equation 
=  + ( )  , = 1,  = 

£



to the Abel equation ¨*¨3
− ¨

−1

=

)

= ( ) + 1, which is outlined in

À 

can be reduced, with the aid of the substitution  ¨

= − ù ( )  followed by raising both sides M

À of the equation to the power of 1 ° , to an equation of the class in question: ( ¨ 
)1 J =  ¨ + X (  )  .

M

Page 182

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

6 b . The equation

( 
)2 = 

183

) = 0 CONTAINING ARBITRARY PARAMETERS

( , = 2,  =  ,  ≠ 0) ¨ can be reduced with the aid of the substitution  = 2 £  + ( ) to an Abel equation of the second kind:

+ ( )

¨*¨ 
= ¨ + Y ( ),

where

Y = 2

−2


( ),

which is outlined in Subsection 1.3.1. 7 b . The equation

( 
)1 J

2

=

+ ( )

( , = 1 2,  = 

)

can be reduced by squaring both sides and performing the substitution H =  equation: H 
= !H 2 + 
.

+ ( ) to the Riccati

For some specific functions = ( ), the solutions of the latter equation are given in Section 1.2. 8 b . The equation

( 
)1 J

2

J + ( )

1 2

=

( , = 1 2,  = 

J )

1 2

can be reduced by squaring both sides and performing the substitution equation of the second kind:

ˆˆ 
=  exp(− 21  2  ) !ˆ + 12 exp(−  2  )

= exp(  2  )ˆ

2

to an Abel

2

(see Subsection 1.3.3). 9 b . The equation

( 
)−1 J

2

=  ( ) + 

( , = −1 2, =  )

can be reduced by squaring both sides and performing the substitution  =  ( ) +  to a Riccati equation: 
=  2 + 
.





For some specific functions  =  ( ), the solutions of the latter equation are given in Section 1.2. 1.6.3-2. Classification tables and exact solutions. For the sake of convenience, Tables 9–13 given below list all the equations outlined in Subsection 1.6.3. The five tables classify the equations in which functions  and are of the same form. The right-most column of a table indicates the numbers of the equations where the corresponding solutions are given. After the tables follow the equations—they are arranged into groups so that the solutions of the equations within each group are expressed in terms of the functions indicated before the groups as a notation list. 1.

(ñó ò )  =


ñ

s

+

(

.

Solution:  = X ( # 2.

(ñó ò )  =




+

4.

(ñó ò )  =

+ $ )−1 J'

.


B]

+

(

Ÿ




Solution:

+

(4

 +L . M

.

Solution:  = X ( #   + $ )−1 J'

(ñó ò )  =

+L .

M

= X ( # + $I )1 J'

Solution: 3.

( ïe

À

ó

.



= X ( # + $  )1 J'

M

M

+L .

 +L .

Page 183

184

FIRST-ORDER DIFFERENTIAL EQUATIONS

À

TABLE 9 Solvable equations of the form ( 
)  = #

°

&

Equation

,

°

&

Equation

arbitrary

arbitrary (° ≠ , )

,° , −°

1.6.3.7

−1

1

−1

1.6.3.23

arbitrary

arbitrary

0

1.6.3.1

−1

1 2

1.6.3.42

1.6.3.6

−1 2

1

1.6.3.17

1.6.3.2

−1 2

1 arbitrary ( ° ≠ −1, 0)

1 arbitrary (& ≠ −1, 0)

1.6.3.38

,

,

−

arbitrary

1− , 0

1+ , arbitrary

arbitrary

1

1

1.6.3.5

1 2

1

−2

−1

−2

1.6.3.46

1 2

1

−1

1.6.3.37

−2

−1

1

1.6.3.33

1

−1

−2

1.6.3.20

−2

1.6.3.29

1

−1

−1 2

1

1.6.3.27

1

−1

1

1.6.3.22

−2

1.6.3.30

−1

1.6.3.11

−1

1.6.3.16

−2

−2 5 1 2

−2

2

−2

1.6.3.35

1

−2

2 arbitrary ( ° ≠ 0) arbitrary ( ° ≠ −2, 0)

1

1.6.3.44

1

1 2 1 2

1

1.6.3.10

1

1 2

−1 2

1.6.3.24

2

1.6.3.15

1

1 2

1

1.6.3.41

−1

−2

−1

1.6.3.21

2

−2

−1

−2

1.6.3.31

2

−2

−2 5

1.6.3.45

−1

1 2

−1

−2

2

1.6.3.36

2

−2

2

1.6.3.34

−1

−1

1.6.3.12

2

1

−1

1.6.3.32

1.6.3.40

2

1

1 2

1.6.3.25

2

1

2

−2

−1 −1

−1 −1

1 2

−1 2 −1 2

−1

1 2

1.6.3.39

1.6.3.28

1.6.3.26 1.6.3.43

(ñó ò )  =
ñ + ( ï . Solution in parametric form:

 = X ( #*U 1 'J  + $ )−1 U + L , M 6.

Ÿ

,

arbitrary ( , ≠ −1, 1)

5.

+ $I





(ñó ò )  =
ñ 1–  + ( ï – 1+  , Solution in parametric form:

 = l” X where # = 



− 1+

 

 

+L M • = U 1 'J  + ý



#

1

¬ U − $ÈX ( #*U 1 'J  + $ )−1 U − $çL2® . M

| ü | ≠ 1.

U

− 1−

=

 ý…$ , $ = 



1+

+1

,

= /”U − ý X

U

− ýhL M • = U 1 'J  + ý

 

−1

,

  ¬  ( , − 1) ="® .  ( , + 1)

Page 184

1.6. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

185

) = 0 CONTAINING ARBITRARY PARAMETERS

TABLE 10 Solvable equations of the form Ÿ (
 )  = #  + $I

TABLEÀ 11 Solvable equations of the form  (
 )  = # +$ 

,

&

Equation

,

arbitrary

−,

1.6.3.9

arbitrary

,

1.6.3.8

arbitrary

0

1.6.3.3

arbitrary

0

1.6.3.4

−1

−1

1.5.2.40

−1

arbitrary

1.5.2.37

−1

1

1.5.2.38

1 2

1

1.6.3.18

−1

−1 2

2

1.5.2.39

1

−1

1.5.2.14

1

1.6.3.19

1

arbitrary

1.5.2.13

7.

8.

Equation

−1

1.5.2.8

1

1.5.2.2



(ñó ò )  =
ñ s + ( ï  –s , À Solution in parametric form:

( ñ óò )

s

 =L 

−

=L 

™

ó +4 ( 

=
ñ 

Equation

(
 )−2 = # ln + $I

1.6.3.14

(
 )−1 = # ln + $I (
 )2 = # + $ ln 

ü

≠ s.

À

1.5.4.12 1.6.3.13

À

1  , −° O# + $IU −À   − U!® ° À P 1  , −° U exp X¬ O# + $IU −   − !U ® ° P

−1

M −1

M

Uƒ¢ , À  − ƒU ¢ .

. Solution in parametric form: 1 ( # + $  −  * )1 J' − S

Q

1

 = exp ¦ U −

(ñó ò )–1 =


ñ

s

+

Solution:  = 

( ï

=X .

P  O# X

Q

X

,

(ñó ò )  =
B] + ( ï –  . Solution in parametric form:

,

X

Q

−1

1 ( $ + #   * )−1 J' + S

−1

Q

,

+L

P

−1

,

M

M

L

U +

,

À

M

U +L , M

1 ( # + $  −  * )1 J' − S

1 ( $ + #   * )−1 J' + S

 −P 

−1

,

1

= exp ¦U +

10.

Form of equation

exp ™ X¬

 =X

9.

Equation

TABLE 13 Solvable equations containing logarithmic functions

TABLE 12 Solvable equations of the form  (
 )  = #  + $ 

,

°

M

U −

§ .

L

,

,

§ ,

U +L .

.

Page 185

186

]

FIRST-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 11–14, the following notation is used:

11.

=
ñ 1 ) 2 + ( ï –1 . Solution in parametric form:

2

=  [2U2AC exp(T"U 2 )] ,

(ñó ò )–1 =
ñ –1 + ( ï 1 ) 2 . Solution in parametric form:

2

 =  [2U2AC exp(T"U 2 )] ,

13.

= ˜ exp(T U 2 ),

(ñó ò )2 =
ñ + ( ln ï . Solution in parametric form:

where # = 4 

−2

# = A 2

where

# = T 4 , $ = A 2 1 J 2 

2

2

.

−1

.

,

 , $ = T 4 ˜# .

(ñ ó ò )–2 =
ln ñ + ( ï . Solution in parametric form:

2

 = , [2U2AC exp(T"U 2 )] A 4 ln( ˜ ) − 4U 2 . , where # = T 4 !$ , $

]

 J , $ = T 4 .

−1 1 2

where

= f, [2U2Aç exp(T U 2 )] A 4 ln( ! ) − 4U

 = ! exp(T U 2 ), 14.

.

ñóò

 = ! exp(T"U 2 ),

12.

−1

 = X exp(T"U 2 ) U + L2S Q M

= 4 

−2

= ˜ exp(T U 2 ),

.

In the solutions of equations 15–19, the following notation is used:

:

=

™

ø

Á (U ) + L 2 n Á (U ) for the upper sign, L 1 ®Á (U ) + L 2 ; Á (U ) for the lower sign, L

1

where ø Á (U ) and n Á (U ) are the Bessel functions, and ®Á (U ) and ; Á (U ) are the modified Bessel functions. x3y'z|{}~/€ : : : The solutions of equations 15–19 contain only the ratio
= (ln )
. Therefore, * * for the sake of symmetric appearance, two “arbitrary” constants L 1 and L 2 are indicated in the : definition of function (instead, we can set, for instance, L 1 = 1 and L 2 = L ). 15.

(ñó ò )–1 =
ñ s + ( ï 2 , s ≠ –2, s ≠ 0. Solution in parametric form:

 = U where @ = 16.

1 ° +2 , # =T  ° +2 2

−2 −1−

(ñ ó ò )1 ) 2 =
ñ + ( ï e ,

≠ –1, Solution in parametric form:

Á V U (ln : )
+ @W , À

= 'U

*

, $

=−

° +2 2



−1 −1



2

Á ,

.

≠ 0.

Á U (ln : )
+ @2A & + 1 U 2 S , * Q 2& 1 J 2 1 (& + 1)  & + 1 −Ÿ where @ = , # =  −1 − S , $ =T  ˜# . & +1 Q 2 2&  = U 2 Á ,

= 'U

2

Page 186

1.6. EQUATIONS OF THE FORM

17.

ê ( Ë , Ï , Ï ÒÍ

(ñó ò )–1 ) 2 =
ñ s + ( ï , s ≠ –1, s ≠ 0. Solution in parametric form:

:  = U 2 Á U (ln À Q 1 ° +1 − , # =T  $ , where @ = ° +1 2°

18.

° +1 2 Á U S , = 'U 2 , * 2° ( ° + 1)  1 J 2 S . $ =  −1 − Q 2

)
+ @*A

ó

(ñó ò )1 ) 2 =
ñ + (4 . Solution in parametric form:

: =  V U (ln )
A

 = ln( U 2 ), where @ = 0, # =  19.

1J 2  − 12   , $ = T

−1

1 −1 2



(ñó ò )–1 ) 2 =
B] + ( ï . Solution in parametric form:

*

where @ = 0, # = T

1 2



−1

$ , $ =

−1

1 2 2

U W,

*

:  =  V U (ln )
A

]

187

) = 0 CONTAINING ARBITRARY PARAMETERS

˜# .

1 2

U 2W ,

= ln( 'U 2 ),

1J 2  − 21   .

In the solutions of equations 20–35, the following notation is used:

:

=

™

L

ø

J U ) + L 2 n 1 J 3 (U ) for the upper sign, 1 J 3 ( U ) + L 2 ; 1 J 3 ( U ) for the lower sign,

1 1 3(

L

1

®

where ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; 1 J 3 (U ) are the modified Bessel functions; : : : : ’ 1 = U
+ 13 , ’ 2 = ’ 12 ACU 2 2 , ’ 3 = A 23 U 2 3 − 2 ’ 1 ’ 2 .

*

x3y'z|{}~/€

:

:

:

The solutions of equations 20–35 contain only the ratio
= (ln )
. Therefore, * * for the sake of symmetric appearance, two “arbitrary” constants L 1 and L 2 are indicated in the : definition of function (instead, we can set, for instance, L 1 = 1 and L 2 = L ).

20.

21.

22.

23.

=
ñ –1 + ( ï –2 . Solution in parametric form:

ñ óò

J :

 = U

−4 3

 = U

−2 3

−2

’ 2,

(ñó ò )–1 =
ñ –2 + ( ï –1 . Solution in parametric form:

J :

−1

J :

−2 3

= 'U

−1

’

−1 2

’ 3,

where

# = 2

J :

−2

’ 2,

where

# =T

’

−1 2

−1

’ 1,

= 'U

−4 3

J :

−2

’ 2,

where

# =T

’ 2,

= 'U

−2 3

J :

−1

’ 1,

where

# = 2

’ 3,

=
ñ –1 + ( ï . Solution in parametric form:

= 'U

−4 3

−1 2

2 3

 , $ =T

2 3

 .

 , $ = 2  2 

−1

.

ñóò

J :

 = U

−2 3

 = U

−4 3

(ñ ó ò )–1 =
ñ + ( ï –1 . Solution in parametric form:

J :

−2

2 −1 2 , 3



−2



$ = 2

 , $ =T

−2

.

2 2 −1 . 3

 

Page 187

188 24.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò


ñ 1 )

=

2

( ï

+

)

–1 2

.

Solution in parametric form:

25.

(ñó ò )–1 =

J :

−2

)

+

−4 3

 = U


ñ

–1 2

2 1,

’

= 'U

( ï 1)

2

−8 3

J :

−4

’

2 2,

where

# = 2

J :

−8 3

26.

(ñó ò )2 =


ñ

−4

( ï 1)

+

2 2,

’

J :

−2

’

2 1,

where

# =T

2

27.

(ñ

óò

J :

−4 3

)–2 =

−2

2

+

( ï

J :

−4

J :

−4

−8 3

= 'U

’

2 2

(’

2 2

4 2 3

A

U

:

−8 3

 = U 4 3

28.

(ñó ò )2 =


ñ

–2

J $ , $ = 4 



−1 2

( ï

+

2 3

J , $ = 2 1J 2 

−1 2



−1

.

3

’ 1 ,

−2

# = 4

where

 , $ =T

J ˜# .

4 −1 2 3



.

Solution in parametric form:

where # = T



.

’ 12,


ñ 1 )

−4 3

= 'U

Solution in parametric form:

 = U

J .

2 −1 2 3

.

Solution in parametric form:

 = U

 J , $ =T

−1 1 2

)

–2 5

−2

4 3

A

2

U

:

3

’ 1 ),

= 'U

−4 3

J :

−4 3

= 'U

−2

2 1,

’

.

.

Solution in parametric form:

 = U

(ñ

óò

)–2 =

J ’ 5J 2, 1

−5 2

J  $ , $ =

4 −2 5 2 3

where # = T 29.

J :

−5 3



)


ñ

–2 5

( ï

+

–2

J :

−2

(’

A

2 2

4 3

U

2

:

3

’ 1 )1 J 2 ,

J 

16 −8 5 2 . 25



.

Solution in parametric form:

 = U

30.

ñ óò

=

J

16 2 −8 5 , 25

where # =


ñ 1 )

 

2

( ï

+

–2

J :

−4 3

−2

$ =T

(’

2 2

31.

(ñó ò )–1 =


ñ

2

–2

+

’

U

32.

(ñó ò )2 =

J :

−4 3

( ï 1)


ñ

( ï

+

’

–1

= 'U 2

2

’

−1 2 ,

’ 1 )1 J 2 ,

= 'U

J :

−5 3

J ’

−5 2

J

5 2 1 ,

J :

−4 3

−2

−2 2

’

 J

−2 2

’ 32 ,

where

# =A

4 −1 1 2 , 3

2

−1 2 ,

where

# = −4 , $ = A



$ = −4  .

.

’ 32 ,

= 'U 4 J

3

:

’

 J 

4 1 2 −1 . 3

.

Solution in parametric form:

:  = U 4 J 3

3

.

−1 2 ,

−2

:

J # .

 

Solution in parametric form:

 = U

2

4 2 −2 5 3

Solution in parametric form:

:  = U 4 J 3

4 3

A

= 'U

J :

−4 3

−2

’

−2 2 2 ( 3

’

− 4’

3 2 ),

where

# =

16 −2 9



 , $ = 4 ˜# .

Page 188

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

33.

(ñó ò )–2 =
ñ –1 + ( ï . Solution in parametric form:

 = U

34.

J :

−4 3

’

J

−1 2 , 2

= 'U

J :

−2 3

−1

= 'U 4 J

3 2 ),

− 4’

J :

−2 3

−1

(ñ ó ò )–2 =
ñ 2 + ( ï –2 . Solution in parametric form:

 = U ]

−2 2 2 ( 3

’

(ñ ó ò )2 =
ñ –2 + ( ï 2 . Solution in parametric form:

:  = U 2 J 3 ’

35.

−2

’

’

−1 2 2 ( 3

J

= 'U 2 J

−1 2 3 1 2 , 2 ( 3 −4 2)

’

’

’

3

:

2

J

− 4’

3

:

−1 2 ,

’

’

3 1 2 , 2)

J

−1 2 , 2

# = 4˜$ , $ =

where

where # = 4  2  2 $ , $

² =



Æ =







1 2

an L 2 . One of = A 1), while the

(ñ ó ò )–1 =
ñ –2 + ( ï 2 . Solution in parametric form:

where # = −

−2

1 Te@ 2

2

−1

²

 , $ = − 21 

= 'U 2 ,

Æ ,

−1 −1



@ = £ |1 − 4 #2$ |,

.

(ñó ò )1 ) 2 =
ñ + ( ï –1 . Solution in parametric form:

 = U 2 ,

= 'U

−2

O²

−1

Æ −

(ñó ò )–1 ) 2 =
ñ –1 + ( ï . Solution in parametric form:

 = U 39.

.

L 1 U Á + L 2 U −Á for the upper sign, L 1 sin( @ ln U ) + L 2 cos( @ ln U ) for the lower sign, L 1 ln U + L 2 for @ = 0, Á Á (1 + @ ) L 1 U + (1 − @ ) L 2 U − for the upper sign, ( L 1 − @ L 2 ) sin( @ ln U )+( L 2 + @ L 1 ) cos( @ ln U ) for the lower sign, L 1 ln U + L 1 + L 2 for @ = 0.

 = U

38.

−2

$ = 4 2  2 # .

 

The expressions of ² and Æ contain two “arbitrary” constants L them can be fixed to set it equal to any nonzero number (for example, we can set L other constant remains arbitrary.

37.



16 −4 2 . 9

=

16 2 −4 , 9

# =

where

16 9

In the solutions of equations 36–46, the following notation is used:

x3y'z|{}~/€

36.

189

) = 0 CONTAINING ARBITRARY PARAMETERS

−2

O²

−1

Æ −

1 Te@ 2

2

P

,

1 Te@ 2

2

,

where

# =

−1

= 'U 2 ,

where

# =

1 Te@ 2

P

=
ñ –1 + ( ï –1 ) 2 . Solution in parametric form:

O−

J



1 2

2 rP

2

, $

=

˜$ , $ = 

2

1 Te@ 2

−1

O−



2 "P

˜# .

J

1 2

.

ñ óò

 = U 2 ² 2 ,

= 'UÆ ,

where

# = (−1 Aç@ 2 )



2

2

,

$ =

J .

−1 2

Page 189

190 40.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñó ò )–1 =
ñ –1 ) 2 + ( ï –1 . Solution in parametric form: = 'U 2 ²

 = UÆ , 41.

= 'U 2 Æ

46.

J ,

−1 2

# = 

$ = (−1 Aç@ 2 )

,

where

# = 2(−1 Ae@ 2 ) 

2

,

= 'U ² ,

where

# = 4

(ñó ò )2 =
ñ + ( ï 2 . Solution in parametric form: = 'U 2 [ Æ

2

− (−1 Ae@ 2 ) ²

2

− (−1 Aç@ 2 ) ²

2

(ñ ó ò )2 =
ñ –2 + ( ï –1 . Solution in parametric form:

 = U 2 ²

2

 = U [ Æ

2

,

= 'U [ Æ

2

],

− (−1 Aç@ 2 ) ²

J

2 1 2

]

,

,



2

.

2

−2

= 'U ² ,

where

# = (−1 Ae@ 2 ) 

]

J

= 'U 2 ²

2

−2

.

−1

.

 , $ = (−1 Ae@ 2 ) 

# = 16

2 1 2

$ = 4

$ = 2(−1 Aç@ 2 )  1 J 2 

where

− (−1 Aç@ 2 ) ²

(ñó ò )–2 =
ñ –1 + ( ï –2 . Solution in parametric form:

2

−2

 J ,

−1 1 2

],

(ñó ò )–2 =
ñ 2 + ( ï . Solution in parametric form:

 = U 2 [ Æ

45.

2

(ñ ó ò )–1 =
ñ + ( ï 1 ) 2 . Solution in parametric form:

 = U ² ,

44.

where

=
ñ 1 ) 2 + ( ï . Solution in parametric form:

 = U 2 Æ

43.

,

ñóò

 = U ² ,

42.

2

−2

−2

˜# .

$ , $ = 16 

−2

.

,

where

# = (−1 Ae@ 2 ) 

−1 2

−1 2

,

where

# =  2

= (−1 Aç@ 2 )  2 

−1

−1

, $

 $ , $ =

 .

# .

1.6.4. Other Equations 1.6.4-1. Equations containing algebraic and power functions with respect to
 . 1.

2.

= ï ñó ò + ô ï 2 + õ ñ óò + ö , ô ≠ 0. ü Differentiating the equation with respect to  and changing to new variables a =
 and ¨ (a ) = −2  , we arrive at an Abel equation of the form 1.3.1.32: ¨*¨pg
− ¨ = − 'a −1 J 2 .

ñ

ï ñóò

– ï = ( ô ï + õÿñ ) (ñ ó ò )2 + 1. ü This is a special case of equation 1.8.1.5 with  (‡ ,  ) = ‡ + ' .

3.

+ ï = ( ô ï + õÿñ ) (ñ ó ò )2 + 1. ü This is a special case of equation 1.8.1.6 with  (‡ ,  ) = ‡ + ' .

4.

ñ

ññóò

=

ï ñóò

+ ô (ñó ò )÷ .

= 3 L  + L  . In addition, there is a singular solution: !#  −1 B  = −(B − 1)  −1 , B ≠ 1.

Solution:

= #* 



−1

, where

Page 190

1.6. EQUATIONS OF THE FORM

5. 6. 7.

8.

ê ( Ë , Ï , Ï ÒÍ

= ï ñ ó ò + ô ï ÷ (ñ ó ò )ú . This is a special case of equation 1.8.1.15 with  (¨ ) =  ¨p+ .

ñ

= ô ïƒ÷ (ñó ò )2÷ + 2 ï ñó ò . This is a special case of equation 1.8.1.51 with  (¨ ) =  ¨  .

ñ

= ï ñó ò + ô ï 2 + õ (ñó ò )2 + ö (ñó ò )ú +1 + ý , ô ≠ 0. Differentiating the equation with respect to  and passing to the new variables a =
 and ¨ (a ) = −2  , we arrive at the Abel equation ¨*¨ug
− ¨ = −4 'a − 2!( ( + 1)a + , whose solvable ¡ case are outlined in Subsection 1.3.1.

ñ

ô (ñ óò )÷

+ õ (ñó ò )ú = ï . 1 b . Solution in parametric form with B ≠ −1,

 = a  + 'a + ,

 =

ô (ñ óò )÷

 a

+ + 'a ,

B  a B −1

−1

 = L +  ln |a | +

11.

B +1

a

+1



+

¡

¡

ô ï (ñ óò )÷

 ¡

¡

¡

 ¡

¡

¡

+1

+1

a+

a+

+1

+1

.

.

≠ 1:

¡ ¡



+

≠ −1:

= L +  ln |a | +

−1

2 b . Solution in parametric form with B = 1,

10.

B

=L +

+ õ (ñó ò )ú = ñ . 1 b . Solution in parametric form with B ≠ 1,

 =L +

≠ −1:

¡

2 b . Solution in parametric form with B = −1,

9.

191

) = 0 CONTAINING ARBITRARY PARAMETERS

a+

−1

+ = a  + 'a .

,

≠ 1:

¡

−1

a+

−1

= a + 'a + .

,

+ õÿñ (ñó ò )ú + ö (ñó ò )  = 0. This is a special case of equation 1.8.1.12 with  (‡ ) = ‡  , (‡ ) = '‡ + , and % (‡ ) = (˜‡  . = ô ï ÷ ( ï ñ ó ò – ñ )ú . ¨ g
, The Legendre transformation  = u 2 ¨ + 2 ¨
a = ( g ) .

¨ g
− ¨ =a2

1 b . Solution in parametric form with

≠ −B , B ≠ −1:

ñ óò

 =O

a

 P



1

1

+B a  a O + L2® ¬ ¡ B +1  P

−

¡

+

+ +

1

,

2 b . Solution in parametric form with 1

 =

1

¡

= 1, B = −1:

−

+ + +

.

= −B , B ≠ −1:

¡

1

¡

1

a  B a  = LȬ aO − 1® exp ¬ aO ® .  P · B + 1 ·P ≠ −B , B = −1:

+  V  (1 − ) ln |a | + L3W 1− + , ¡ a

4 b . Solution with

1

1− a  +B a  ¡ a O =¬ − L2®u¬ ¡ a O + L2® 1+B  P B +1  P

a  B a   = L­O exp ¬ aO ® ,  P · B + 1 ·P 3 b . Solution in parametric form with

(
 = a ) leads to a separable equation:

= L3 



= aÖ − V  (1 −

¡

) ln |a | + L3W

1 1−

+ .

−1 .

Page 191

192 12.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

ô  ó (ñ óò

+ñ =

=   L 

Solution: 13.

14.

1 2

ô 1 ïƒ÷ ú 1

1

(ï

ñóò

+ù

=

+B

1

ñ ó ò (ñ ó ò + ô ï )÷

ô (ñ óò

L

−

2

ñ )ú

−



.

ö

+

= 0.

ô 2 ïƒ÷ ú

+

1

2

 − 1 −

1

2−B 1 + 1 +  2L 2

L

2

(ï

+

1

ö

ñóò

+ù

1

ñ )ú

and L

1

2

õ

+

22.

23.

1

1

≠

and L

ù

2.

2

are related by the

= 0.

õ  ó (ñ óò )

ô  ó (ñ óò

ô  _ ó (ñ óò

– ñ )  + õ [(ñ 1 2

1



2

óò

ö

+ −





= 0.

. Here, the constants L

)2 – ñ

2

1 2

−

= L 1 − L 2 L 2 )  + ( = 0.

+ Ė )  +

]÷ +

õ ÷ ` ó (ñ óò



ö

1

and L

2

are related by the constraint

= 0.

. Here, the constants L + ñ )÷ + ö = 0,

1



and L

2

L 1 −:  L 2 −  −   ; . Here, the constants L = −ý = −ý constraint L 1 + L 2 + ( = 0.

ô (ñ óò ô (ñ óò

=

cosh ï – ñ sinh ï ) ÷ + õ (ñó ò sinh ï – ñ cosh ï )  +

ö

1

cosh ï – ñ sinh ï ) ÷ = õ (ñó ò 2 – ñ

2

ñóò (ñ óò

sinh ï – ñ cosh ï ) ÷ = õ (ñó ò 2 – ñ

2

)ú

and L

2

are related by the

= 0. 1

and L

2

are related by the

1

and L

2

are related by the

cosh ï – ñ sinh ï ) ÷ + õ (ñó ò 2 – ñ 2 )  + ö = 0.

ñóò (ñ óò

are related by the constraint

≠ .

Solution: = L 1 sinh  + L 2 cosh  . Here, the constants L constraint L 1 +  ( L 12 − L 22 )  + ( = 0. 21.

ù

= 0,

Solution: = L 1 sinh  − L 2 cosh  . Here, the constants L constraint L 1 + L 2 + ( = 0. 20.

are related by the constraint

2

 −  2 . Here, the constants L

2

2−B +  = 0.

2

B

óò 2 + 2ôñ )ú

+ õ (ñ

+ ñ )÷ +

Solution:

19.

− 12 L 

The contact transformation m = 
 +  , n = (
 )2 + 2  , n o
= 2
 , where n = n (m ), leads to a separable equation: n o
= −2m −  ( ˜n + + ( ). The inverse contact transformation:  = 12  −1 (2m − n o
), = 18  −1 [4 n −( n o
)2 ], 
 = 12 n o
.

Solution: L 2 +  ( L 18.



.

+ ùñ )ú

ñóò

Solution: = L 1−L L 1 + L 2 + ( = 0. 17.

–1



= − L 1   + L 2 . Here, the constants L Solution:   ( L 1 B ) +  ( L 2 B ) + + ( = 0.

constraint  1 L

16.

−1

+ õ (ï

ô ï ÷ + (ñ óò )

Solution:

15.

– ñ )

cosh ï .

The contact transformation m = (
 )2 − 2 , n = 
 cosh  − sinh  , n o
= leads to a separable equation: 2 'm + n o
= n  . )ú

cos ï + ñ sin ï ) ÷ + õ (ñ

Solution: = L 1 sin  − L L 1 + L 2 + ( = 0.

2

óò

cosh  (
 )−1

1 2

sinh  (
 )−1

sinh ï .

The contact transformation m = (
 )2 − 2 , n = 
 sinh  − cosh  , n o
= leads to a separable equation: 2 'm + n o
= n  .

ô (ñ óò

1 2

sin ï – ñ cos ï )  + ö = 0.

cos  . Here, the constants L

1

and L

2

are related by the constraint

Page 192

1.6. EQUATIONS OF THE FORM

24.

25.

26.

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY PARAMETERS

cos ï + ñ sin ï ) ÷ + õ (ñ ó ò 2 + ñ 2 )  + ö = 0. Solution: = L 1 sin  + L 2 cos  . Here, the constants L L 1 +  ( L 12 + L 22 )  + ( = 0.

ô (ñ óò

1

and L

2

193

are related by the constraint

cos ï + ñ sin ï ) ÷ = õ (ñ ó ò 2 + ñ 2 )ú cos ï . The contact transformation m = (
 )2 + 2 , n = 
 cos  + sin  , n o
= to a separable equation: 2 'm + n o
= n  .

ñ ó ò (ñ ó ò

sin ï – ñ cos ï ) ÷ = õ (ñ ó ò 2 + ñ 2 )ú sin ï . The contact transformation m = (
 )2 + 2 , n = 
 sin  − cos  , n o
= to a separable equation: 2 'm + n o
= n  .

ñ ó ò (ñ ó ò

1 2

cos  (
 )−1 leads

1 2

sin  (
 )−1 leads

1.6.4-2. Equations containing exponential, logarithmic, and other functions with respect to 
.

ï

27.

= ô exp( ƒñ„ó ò ) + õ exp( “ñó ò ). This is a special case of equation 1.8.1.7 with  (¨ ) =  exp( Œ ¨ ) +  exp(“ ¨ ).

28.

ñ

29. 30.

= ô exp( ƒñ„ó ò ) + õ exp( “ñó ò ). This is a special case of equation 1.8.1.8 with  (¨ ) =  exp( Œ ¨ ) +  exp(“ ¨ ). = ï ñ ó ò + ô ï ÷ exp( ƒñ ó ò ). This is a special case of equation 1.8.1.15 with  (¨ ) =  exp( Œ ¨ ).

ñ

= ô ï exp( ƒñ„ó ò ) + õ exp( “ñó ò ). This is a special case of equation 1.8.1.11 with  (¨ ) =  exp( Œ ¨ ) and (¨ ) =  exp(“ ¨ ).

ñ

ï

31.

= ô sinh( ƒñ„ó ò ) + õ sinh( “ñ„ó ò ). This is a special case of equation 1.8.1.7 with  (¨ ) =  sinh( Œ ¨ ) +  sinh(“ ¨ ).

32.

= ô sinh( ƒñ ó ò ) + õ sinh( “ñ ó ò ). This is a special case of equation 1.8.1.8 with  (¨ ) =  sinh( Œ ¨ ) +  sinh(“ ¨ ).

33. 34.

ñ

= ï ñó ò + ô ïƒ÷ sinh ú ( ƒñó ò ). + This is a special case of equation 1.8.1.15 with  (¨ ) =  sinh ( Œ ¨ ).

ñ

= ô ï sinh( ƒñ„ó ò ) + õ sinh( “ñ„ó ò ). This is a special case of equation 1.8.1.11 with  (¨ ) =  sinh( Œ ¨ ) and (¨ ) =  sinh(“ ¨ ).

ñ

ï

35.

= ô cosh( ƒñ„ó ò ) + õ cosh( “ñ„ó ò ). This is a special case of equation 1.8.1.7 with  (¨ ) =  cosh( Œ ¨ ) +  cosh(“ ¨ ).

36.

= ô cosh( ƒñ„ó ò ) + õ cosh( “ñ„ó ò ). This is a special case of equation 1.8.1.8 with  (¨ ) =  cosh( Œ ¨ ) +  cosh(“ ¨ ).

37. 38.

ñ

= ï ñó ò + ô ïƒ÷ coshú ( ƒñó ò ). + This is a special case of equation 1.8.1.15 with  (¨ ) =  cosh ( Œ ¨ ).

ñ

= ô ï cosh( ƒñ„ó ò ) + õ cosh( “ñ„ó ò ). This is a special case of equation 1.8.1.11 with  (¨ ) =  cosh( Œ ¨ ) and (¨ ) =  cosh(“ ¨ ).

ñ

Page 193

194 39.

FIRST-ORDER DIFFERENTIAL EQUATIONS ln ñ

+ï

óò

ñ óò

ôñ

+

+

õ

= 0.

1 b . Solution in parametric form with  ≠ 0,  ≠ −1: 1

 =

+ L3a − 

a

1 = − (a + ln a +  ).

1 +1 ,



2 b . Solution in parametric form with  = 0:

 =−

ln a + 

a

= L + (  − 1) ln a +

,

1 (ln a )2 . 2

3 b . Solutions with  = −1:

40. 41.

42. 43.

ï

= L3 + ln L +  + ln ñ„ó ò + ô (ñó ò + ñ )  +

õ

and

= ln(−1  ) +  − 1.

= 0.

This is a special case of equation 1.8.1.41 with  (‡ , ¨ ) = ln ‡ +  ¨  +  .

ñ

=

ï ñóò

+ô

ï

ñ

=

ï ñóò

+ô

ïƒ÷

=

ô

2

+ õ ln ñó ò + ö ,

ô

≠ 0.

Differentiating the equation with respect to  and changing to new variables a =
 and ¨ (a ) = −2  , we arrive at an Abel equation of the form 1.3.1.16: ¨*¨pg
− ¨ = −2 'a −1 . ln ú ( ƒñó ò ).

+ This is a special case of equation 1.8.1.15 with  (¨ ) =  ln ( Œ ¨ ).

ï

sin( ƒñ„ó ò ) + õ sin( “ñ„ó ò ).

This is a special case of equation 1.8.1.7 with  (¨ ) =  sin( Œ ¨ ) +  sin(“ ¨ ).

44.

ñ = ô sin(ƒñ„óò ) + õ sin( “ñ„óò ). This is a special case of equation 1.8.1.8 with  (¨ ) =  sin( Œ ¨ ) +  sin(“ ¨ ).

45.

ñ

=

ï ñóò

ñ

=

ô ï

=

ô

46. 47. 48. 49. 50. 51.

+ô

ïƒ÷

sin ú ( ƒñó ò ).

This is a special case of equation 1.8.1.15 with  (¨ ) =  sin + ( Œ ¨ ). sin( ƒñ„ó ò ) + õ sin( “ñ„ó ò ).

This is a special case of equation 1.8.1.11 with  (¨ ) =  sin( Œ ¨ ) and (¨ ) =  sin(“ ¨ ).

ï

cos( ƒñ„ó ò ) + õ cos( “ñó ò ).

This is a special case of equation 1.8.1.7 with  (¨ ) =  cos( Œ ¨ ) +  cos(“ ¨ ).

ñ

=

ô

cos( ƒñ„ó ò ) + õ cos( “ñó ò ).

ñ

=

ï ñ óò

ñ

=

ô ï

This is a special case of equation 1.8.1.8 with  (¨ ) =  cos( Œ ¨ ) +  cos(“ ¨ ). +ô

ï ÷

cosú ( ƒñ

óò

).

This is a special case of equation 1.8.1.15 with  (¨ ) =  cos + ( Œ ¨ ). cos( ƒñ

óò

) + õ cos( “ñ

óò

).

This is a special case of equation 1.8.1.11 with  (¨ ) =  cos( Œ ¨ ) and (¨ ) =  cos(“ ¨ ).

ï ñ óò

ï ÷

tanú ( ƒñ

óò ). This is a special case of equation 1.8.1.15 with  (¨ ) =  tan + ( Œ ¨ ). ñ

=

+ô

Page 194

1.7. EQUATIONS OF THE FORM

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY FUNCTIONS

195

1.7. Equations of the Form ð ( ì , í ) í ï î = ë ( ì , í ) Containing Arbitrary Functions ]

Notation:  , , and % are arbitrary composite functions whose argument, indicated after the function name, can depend on both  and .

1.7.1. Equations Containing Power Functions 1.

= ! ( ô ï + õÿñ + ö ). In the case  = 0, we have an equation of the form 1.1.1. If  ≠ 0, the substitution ‡ ( ) =  +  + ( leads to a separable equation: ‡
 =  (‡ ) +  .

ñóò

2.

= ! (ñ + ô ïƒ÷ + õ ) – ôù ïƒ÷ –1 . The substitution ‡ = +   +  leads to a separable equation: ‡
 =  (‡ ).

3.

ñóò

ñóò

=

ñ ! (ƒï ÷ ñ ú ï

).

Generalized homogeneous equation. The substitution H =   + "H 
= B±H + H! ( H ).

leads to a separable equation:

¡

4.

5.

ñ óò = ! (ï )ñ 1+÷

+ " ( ï )ñ + % ( ï )ñ 1–÷ . ¨ 
= B… ( )¨ The substitution ¨ =  leads to a Riccati equation: 3

ñóò

=–

ù ñ þ ï

+ñ

 ! (ï )" (ƒï ÷ ñ ú

The substitution H =   + 6.

ñóò

=I ! O

ô ï  ï

2

+ B ( )¨ + B…% ( ).

).

leads to a separable equation: H 
=

+ õÿñ + ö . + ñ + P

¡



 − ´   ++ +  ( ) H +

−1

( H ).

= − (˜ý (ƒ − ´= 1 b . For ¢ = ý − ˜ƒ ≠ 0, the transformation  = ‡ + , =  (‡ ) + leads to ¢ ¢ an equation: ‡ + '  
=  O . ƒ…‡ + ý± P Dividing both the numerator and denominator of the fraction on the right-hand side by ‡ , we obtain a homogeneous equation of the form 1.1.6. 2 b . For ¢ = 0 and  ≠ 0, the substitution  ( ) =  +  + ( leads to a separable equation of the form 1.1.2: '  
=  +  O . ý± + = − (˜ýCP 3 b . For ¢ = 0 and ý ≠ 0, the substitution  ( ) = ƒ… + ý + = also leads to a separable equation: ' + (˜ý − =  
= ƒ + ýh*O .

ý±

7. 8.

P

= ï ÷ –1 ñ 1–ú ! ( ô ï ÷ + õÿñ ú ). ¨ 
=      +  + leads to a separable equation: 3 The substitution ¨ = 

ñ óò

ñ ÷ ñóò

+ ô ïƒ÷ + " ( ï ) ! (ñ ÷ +1 + ô The substitution ¨ =  +1 +  

ïƒ÷

+1

−1

[ B + 

¡

 (¨ )].

+1

) = 0. ¨ 
+ (B + 1) ( )  (¨ ) = 0. leads to a separable equation: 3

Page 195

196 9.

FIRST-ORDER DIFFERENTIAL EQUATIONS

÷ ! (ñ

[ï

)+ï

" (ñ )]ñ óò

= % (ñ ).

This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5). 10.

[ï

2

+

ï ! (ñ

óò

) + " (ñ )]ñ

= % (ñ ).

This is a Riccati equation with respect to  =  ( ) (see Section 1.2). 11.

= [ ! ( ï )ñ + " (ï )] (ñ – ô )(ñ – õ ).

ñóò

The substitution ‡  ( ) − ( ). 12.

Q

ñ

ü = ( −  ) ( −  ) leads to a Riccati equation: A 2‡
 = [  ( ) + ( )]‡

ñ

+ï

!IO ï P

2

D-%IO ï Sñóò P

=3 " O

ï

ñ P

+ñ

ñ ï D –1 I % O ï

P

2

−

.

The substitution = a leads to a Bernoulli equation with respect to  =  (a ): [ (a )− aª (a )] g
=  (a ) + % (a )  +1 . 13.

ï h ú (ï , ñ )]ñ óò . Darboux equation. Here,

[g

÷ (ï , ñ

)+

i ÷ (ï , ñ

=

)+ñ

h ú (ï , ñ

).

 and Æ  are homogeneous polynomials of order B , and ² + is a homogeneous polynomial of order . Dividing the Darboux equation by   leads to an ¡ equation of the form 1.7.1.12. 14.

ï

[ ! (ô

+

õÿñ

õ ï " (ô ï

)+

+

õÿñ )]„ñ óò

ï

= % (ô

+

õÿñ

ï " (ô ï

)–ô

+

õÿñ

).

The substitution a =    +  leads to a linear equation with respect to  [  (a ) + % (a )] g
=  (a ) +  (a ). 15.

[ ! (ô

ï

+

õÿñ

õÿñ" (ô ï

)+

õÿñ

ï " (ƒï ÷ ñ ú

)]ñó ò = ñ [ % ( ïƒ÷

ñ ú

)–ù

 " (ƒï ÷ ñ ú

)]ñó ò = ñ [ % ( ïƒ÷

ñ ú

)–

The substitution a =  +  − ´ (a ) + % (a ). 16.

17.

18.

19.

ï [ ! (ƒï ÷ ñ ú

)+þ

ï [ ! (ƒï ÷ ñ ú

) + þxñ

= % (ô

) – ôñ" (ô

ï

+

õÿñ

).

leads to a linear equation with respect to = (a ): [  (a )+ % (a )]
g =

ï " (ƒï ÷ ñ ú

)].

) – ùñ

 " (ƒï ÷ ñ ú

)].

ü!

ñ

The transformation a =   + , H =  −  leads to a linear equation with respect to H = H (a ): a [B… (a ) + % (a )]H g
= − ,  (a ) H − , (a ). ¡ ¡ The transformation a =   + , H = −  leads to a linear equation with respect to H = H (a ): a [B… (a ) + % (a )]H g
= − , % (a ) H + ,!B (a ). ¡ À ï s ï ïw÷ ú ï s ïƒ÷ ú [s ! (

ñ

) – þj" (

ñ

)]ñó ò = ñ [ù#" (

The transformation a =   + , ¨ =   [ ! (ñ ) + ôþ

ï ÷ ñ ú

–1

Solution: X¯ ( ) 20.

õÿñ )]„ñ óò

ï

+

+

=  (a ):

ï ñóò

– ñ = ! (ï

2

M

]ñ

+ " (ï ) +

M

(

)].

¨ g
= ¨ (¨ ). leads to a separable equation: aª (a )u

ôù ï ÷ –1 ñ ú

+ Xw ( )  +  

= 0.

+ =L .

+ ñ 2 )(ññó ò + ï ).

Setting  = & (a ) cos a , a = XÈ& −1  (& 2 ) & + L .

^_

óò

ñ

= & (a ) sin a and integrating, we obtain a solution in implicit form:

M

Reference: G. W. Bluman, J. D. Cole (1974, p. 100).

Page 196

â ( Ë , Ï ) Ï ÒÍ

1.7. EQUATIONS OF THE FORM

21.

= ã ( Ë , Ï ) CONTAINING ARBITRARY FUNCTIONS

197

ï ñóò

– ñ = ! ( ï 2 – ñ 2 )(ññó ò – ï ). Setting  = & (a ) cosh a , = & (a ) sinh a and integrating, we obtain a solution in implicit form: a = − X & −1  (& 2 ) & + L .

M

22.

23.

[ ï ! ( ï 2 + ñ 2 ) + ñ" ( ï 2 + ñ 2 ) + % ( ï 2 + ñ 2 )](ññ ó ò + ï ) = ï ñ ó ò – ñ . The transformation  = & cos Y , = & sin Y leads to an equation of the form 1.7.4.11 with respect to Y = Y (& ): Y
Ÿ =  (& 2 ) cos Y + (& 2 ) sin Y + & −1 % (& 2 ).

[ ï ! ( ï 2 – ñ 2 ) + ñ" ( ï 2 – ñ 2 ) + % ( ï 2 – ñ 2 )](ññ ó ò – ï ) = ï ñ ó ò – ñ . 1 b . For  > , the transformation  = & cosh Y , = & sinh Y leads to an equation of the form 1.7.2.18 with respect to Y = Y (& ): Y
Ÿ = −  (& 2 ) cosh Y − (& 2 ) sinh Y − & −1 % (& 2 ). 2 b . For  < , the transformation  = H sinh j , = H cosh j leads to an equation of the form 1.7.2.18 with respect to j = j ( H ): j ‘
= −  (− H 2) sinh j − (− H 2) cosh j − H −1 % (− H 2 ).

24.

ï ñóò

–ñ =

! ï

2

+ñ

2

Setting  = & (a ) cos a ,

ï

ï ñóò

–ñ =

! ï

2

–ñ

2

 "I”

Setting  = & (a ) cosh a ,

! ( ï , ñ )ñ ó ò

+ " ( ï , ñ ) = 0,

,

2

ñ

ï

2

a

(cos a , sin a )

ï

ï

,

 (& 2 ) & +L . & M

=ù

ñ

ï

(ññó ò – ï ).

–ñ –ñ ü ü = & (a ) sinh a and integrating, we obtain the solution:

ù 26.

2

M

ù 25.

ï

(ññó ò + ï ). +ñ +ñ 2• ü ü = & (a ) sin a and integrating, we obtain the solution:

 "I”

2

2

M

a

2

(cosh a , sinh a ) where

k ï k

!

•

2

 (& 2 ) & =L . & M

+ù

=

k

ñ k

"

.

Exact differential equation.    ( , a ) a + X (a , ) a = L , where  Solution: X 0 0 M M 0

0

and

0

are arbitrary numbers.

1.7.2. Equations Containing Exponential and Hyperbolic Functions 1. 2. 3.

4.

ó

ñóò

ó

=  –  ! (  ñ ).  The substitution ‡ = ž 

leads to a separable equation: ‡
 =  (‡ ) + Œ ‡ .

=  ] ! (  ] ï ). The substitution ‡ = ž   leads to a separable equation: ‡
 = Œ ‡

ñ óò

ó

2

 (‡ ) + ‡ .

= ñ! (  ^ ñ ú ). This equation is invariant under “translation–dilatation” transformation. The substitution  H =  9 + leads to a separable equation: H 
= ƒhH + H! ( H ).

ñóò ñóò

=

ƒï ÷ ï ! ( ^] 1

¡

).

This equation is invariant under “dilatation–translation” transformation. The substitution H =    9  leads to a separable equation: "H 
= B±H + ƒhH! ( H ).

Page 197

198 5.

6.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

= ! (ï )

ñ óò

=–ï

ñóò

=–

+ " ( ï ).

]

The substitution ‡ =  −   leads to a linear equation: ‡
 = − Œ ( )‡ − Œ" ( ).

ù

÷ ]

+ ! ( ï )" ( ï

).

The substitution H =    leads to a separable equation: H 
=  ( ) Hž ( H ). 7.

þ



ñ

+ñ

 ! ( ï )" (  ^ ó ñ ú

 The substitution H =  9 +

leads to a separable equation:

H 
= 8. 9. 10.

11.

).

ñóò

= ! ( ï ) ] + " ( ï ) + % ( ï )

ñóò

=

ñóò

= ! (ñ +

ñóò

=–

exp

¡ –

]

.

Q

ƒ

(1 − , ) S  ( ) H

¡

 ++ +

−1

( H ).

The substitution ‡ = ž leads to a Riccati equation: ‡
 = Œ" ( )‡

ó ó  ^ –_ l] ! (ô  ^

õ _ ]

+

2

+ Œ ( )‡ + Œ"% ( ).

).

  The substitution ¨ =   9 +   :  leads to a separable equation: ¨ 
=  9 [ !ƒ + 'ýh (¨ )].

ô ó

+ õ ) – ô

ó

.

 ¨ 
=  (¨ ). The substitution ¨ = +  ž +  leads to a separable equation: u

ù

 ï

+

! (ï ÷  ^ ]

)

ï ñ

.

The substitution a =    9  leads to a linear equation with respect to −B + ƒs (a ). 12.

ñ óò

ù

 ï

=–

+

! (ï ÷  ^ ]

ï ñ

)

2

[ ! (ô

ï

[ ! (ô

ï

õ ^ ]m" (ô ï

+

õ ^ ó " (ô ï

+

õÿñ

)+

õÿñ

)+

)+

ô ]ñ óò

+

2

aª (a )
g =

.

The substitution a =    9  leads to a Riccati equation: ƒ 13.

= (a ): ƒ

õÿñ )]ñ óò

= % (ô

õÿñ )]ñ óò

= % (ô

ï

+

2

aª (a )
g = −B

õÿñ

) – ô

^ ]n" (ô ï

+

õÿñ

) – ô

^ ó " (ô ï

+

õÿñ

).

õÿñ

).

2

+ ƒs (a ).

The transformation a =  +  , H =  − 9  leads to a linear equation with respect to H = H (a ): [  (a ) + % (a )] H g
= − ƒs% (a ) H + ƒh´ (a ).

14.

+

The transformation a =  +  , H =  9 [  (a ) + % (a )] H g
= − ƒs (a ) H − ƒs (a ).



15.

[

^ ó ! (ñ

Solution: X 16.

17.

ï [ ! (ƒï ÷  ^ ]

_ ]" (ï

+

 −:   ( ) M

)+

ô 

−

ï

+

leads to a linear equation with respect to H = H (a ):

= 0.

+ X  − 9 ( )  −   − 9

) + ñ" ( ïƒ÷  ^ ] )] ñ óò

M

= % ( ïƒ÷  ^ ]

 −:  =L .

) – ùñ" ( ïƒ÷

 ^ ]

).

The substitution a =    9  leads to a linear equation with respect to a [B… (a ) + ƒs% (a )] g
= −B (a ) + % (a ). [ ! (

^ ó ñ ú

)+þ

ï " ( ^ ó ñ ú

)]ñó ò = ñ [ % ( 

^ ó ñ ú

)–

ï " ( ^ ó ñ ú

)].  The substitution a =  9 + leads to a linear equation with respect to  a [ ƒs (a ) + % (a )] g
= (a ) +  (a ).

¡

=

(a ):

=  (a ):

¡

Page 198

1.7. EQUATIONS OF THE FORM

18. 19. 20.

21.

22. 23. 24. 25.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY FUNCTIONS

ñ óò

= ! ( ï ) sinh( ƒñ ) + " ( ï ) cosh( ƒñ ) + % ( ï ).

ñóò

= ! ( ï ) sinh2 ( ƒñ ) + " ( ï ) cosh2 ( ƒñ ) + % ( ï ) sinh(2 ƒñ ) + s( ï ).

ñóò

= ! ( ï ) sinh 

Ė

+ " (ï )  .

ñóò

= ! ( ï ) cosh 

Ė

+ " (ï )  .

ñ óò

=

! (ñ ú

sinh ï ).

The substitution ‡ =   leads to a Riccati equation: 2‡
 = Œ (  + )‡ The substitution ¨ = tanh( Œ The substitution ¨ = Œ 
( ). The substitution ¨ = Œ 
( ). coth ï

ñ

=

ñóò

=

ñóò

=

ï

–1

! (ñ ú

+ 2 Œ"% ¨ + Œ ( − ° ).

¨ 
= Œ" ( ) cosh ¨ + + ( ) leads to an equation of the form 1.7.2.18: I

tanh ñB! ( ïƒ÷ sinh ñ ).

tanh ï

2

¨ 
= Œ" ( ) sinh ¨ + + ( ) leads to an equation of the form 1.7.2.18: I

The transformation a =   , H = sinh

ñ

+ 2 Œ"% ‡ + Œ ( −  ).

¨ 
= Œ (  + ° )¨ ) leads to a Riccati equation: p

The transformation a = sinh  , H = +

ñóò

2

199

cosh ï ).

leads to an equation of the form 1.7.1.3: a›H g
=

¡

H! (a›H ).

leads to an equation of the form 1.7.1.3: B/a›H g
= H! (a›H ).

The substitution a = cosh  leads to an equation of the form 1.7.1.3: a
g =  (a + ).

ï

–1

coth ñB! ( ïƒ÷ cosh ñ ).

leads to an equation of the form 1.7.1.3: "H 
= H! (  H ).

The substitution H = cosh

1.7.3. Equations Containing Logarithmic Functions 1.

ñóò

= ! ( ï )ñ ln2 ñ + " ( ï )ñ ln ñ + % ( ï )ñ .

ñóò

=

The substitution ‡ = ln 2.

ï –1 ñ ú

+1

! (ñ ú

leads to a Riccati equation: ‡
 =  ( )‡

ln ï ).

The substitution a = ln  leads to an equation of the form 1.7.1.3: 3.

ñóò

=

ï –÷ –1 ñ ! (ïƒ÷

ln ñ ).

The substitution H = ln 4.

ñóò

=

ï –1  ]l! ( ]

ln ï ).

6.

ñ –ó ! ( ó

ñóò

=

ñóò

= –ù

+ ( )‡ + % ( ).


g = a

[a +  (a + )].

H  (  H ) leads to an equation of the form 1.7.1.3: H 
= .    H

The substitution a = ln  leads to an equation of the form 1.7.2.4: 5.

2


1 g = [a    (a   )]. a

ln ñ ).

  ( H )
The substitution H = ln leads to an equation of the form 1.7.2.3: H  = H .   H

ï –1 ñ

ln ñ + ñ! ( ï )" (ïw÷ ln ñ ).

The substitution ¨ ( ) =   ln

¨ 
=    ( ) (¨ ). leads to a separable equation: u

Page 199

200

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

7.

=–

ù ñ þ ï

+

ñ! (ï ÷ ñ ú ï ln ñ

)

.

The transformation a =   + , H = ln 2 aª (a ) H g
= −B±H +  (a ).

¡

8.

ñóò

=–

ù ñ þ ï

¡

+

ñ! (ï ÷ ñ ú ) . ï (ln ñ )2

The transformation a =   + , H = ln

9.

leads to a linear equation with respect to H = H (a ):

leads to a Riccati equation:

ï [ ! (ƒï ÷ ñ ú

¡

2

aª (a ) H g
= −B±H 2 +  (a ). ¡

) + þ ln ño" ( ïƒ÷ ñ ú )]ñó ò = ñ [ % ( ïƒ÷ ñ ú ) – ù ln ño" ( ïƒ÷ ñ ú )]. The transformation a =   + , H = ln leads to a linear equation with respect to H = H (a ): a [B… (a ) + % (a )]H g
= −B (a ) H + % (a ).

¡

10.

ï [ ! (ƒï ÷ ñ ú

) + þ ln ï " ( ïƒ÷ ñ ú )]ñó ò = ñ [ % ( ïƒ÷ ñ ú ) – ù ln ï " ( ïƒ÷ ñ ú )]. The transformation a =   + , H = ln  leads to a linear equation with respect to H = H (a ): a [B… (a ) + % (a )]H g
= (a ) H +  (a ).

¡

¡

1.7.4. Equations Containing Trigonometric Functions 1. 2. 3. 4. 5. 6. 7. 8.

9.

10.

= ñ ú +1 sin ï H (ñ ú cos ï ). This is an equation of the type 1.7.4.3 with  (ˆ ) = ˆž (ˆ ).

ñ óò

= ñ ú +1 cos ï H (ñ ú sin ï ). This is an equation of the type 1.7.4.4 with  (ˆ ) = ˆž (ˆ ).

ñóò

= ñ tan ï ! (ñ ú cos ï ). The substitution a = cos  leads to an equation of the form 1.7.1.3: a
g = −  (a + ).

ñóò

= ñ cot ï ! (ñ ú sin ï ). The substitution a = sin  leads to an equation of the form 1.7.1.3: a
g =  (a + ).

ñóò

= ï –1 tan ñB! ( ïƒ÷ sin ñ ). The transformation a =   , H = sin

ñóò

= ï –1 cot ñB! ( ïƒ÷ cos ñ ). The transformation a =   , H = cos

ñóò

= ï –1 sin 2ñB! ( ïƒ÷ tan ñ ). The transformation a =   , H = tan

ñóò

= ï –1 sin 2ñB! ( ïƒ÷ cot ñ ). The transformation a =   , H = cot

ñóò

ñ

leads to an equation of the form 1.7.1.3: B/a›H g
= H! (a›H ). leads to an equation of the form 1.7.1.3: B/a›H g
= − H! (a›H ). leads to an equation of the form 1.7.1.3: B/a›H g
= 2 H! (a›H ). leads to an equation of the form 1.7.1.3: B/a›H g
= −2 H! (a›H ).

! (ñ ú tan ï ). sin 2 ï The substitution a = tan  leads to an equation of the form 1.7.1.3: 2a
g =  (a + ).

ñ óò

=

ñ

! (ñ ú cot ï ). sin 2 ï The substitution a = cot  leads to an equation of the form 1.7.1.3: 2a
g = −  (a + ).

ñóò

=

Page 200

1.7. EQUATIONS OF THE FORM

11. 12.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY FUNCTIONS

= ! ( ï ) cos( ôñ ) + " ( ï ) sin( ôñ ) + % ( ï ). The substitution ‡ = tan(   2) leads to a Riccati equation: 2‡ 
=  ( % −  )‡ 2 +2 ´ !‡ +  (  + % ).

ñ óò

= ! ( ï ) cos2 ( ôñ ) + " ( ï ) sin2 ( ôñ ) + % ( ï ) sin(2 ôñ ) + s( ï ). The substitution ‡ = tan(  ) leads to a Riccati equation: ‡
 =  ( + ° )‡

ñ óò

2

+ 2 % ‡ +  (  + ° ).

13.

= ! (ñ + ô tan ï ) – ô tan2 ï . The substitution ‡ = +  tan  leads to a separable equation: ‡
 =  +  (‡ ).

14.

sin 2ñ ! (tan ï tan ñ ). sin 2 ï The transformation a = tan  , H = tan

15.

16.

17.

18. 19. 20.

21.

22.

23.

201

ñóò ñóò

=

= cot ï tan ñB! (sin ï sin ñ ). The transformation a = sin  , H = sin

ñ óò

! (ï

leads to an equation of the form 1.7.1.3: a›H g
= H! (a›H ).

" (sin ï sin ñ ). cos ñ The substitution ¨ ( ) = sin  sin leads to a separable equation: ¨ 
= sin 3 ( ) (¨ ).

ñóò

= – cot ï tan ñ +

leads to an equation of the form 1.7.1.3: a›H g
= H! (a›H ).

)

sin 2ñ + cos2 ñB! ( ï )" (tan ï tan ñ ). sin 2 ï ¨ 
= tan 3 ( ) (¨ ). The substitution ¨ ( ) = tan  tan leads to a separable equation: p

ñóò

=–

= –ù ï –1 sin 2ñ + cos2 ñB! ( ï )" ( ï 2÷ tan ñ ). The substitution ¨ ( ) =  2  tan leads to a separable equation: ¨ 
=  2   ( ) (¨ ).

ñóò

(1 + tan2 ñ )ñó ò = ! ( ï ) tanú +1 ñ + " ( ï ) tan ñ + % ( ï ) tan1–ú ñ . The substitution ‡ = tan + leads to a Riccati equation: ‡
 =  ( )‡

¡

2

+

¡

( )‡ +

¡

% ( ).

= ! ( ï ) sin  ƒñ + " ( ï )  . ¨ 
= Œ" ( ) sin ¨ + The substitution ¨ = Œ + ( ) leads to an equation of the form 1.7.4.11: I
 ( ).

ñóò

= ! ( ï ) cos  ƒñ + " ( ï )  . ¨ 
= Œ" ( ) cos ¨ + The substitution ¨ = Œ + ( ) leads to an equation of the form 1.7.4.11: I
 ( ).

ñóò

= ! ( ï ) sin2  ƒñ + " ( ï )  . ¨ 
= Œ" ( ) sin2 ¨ + The substitution ¨ = Œ + ( ) leads to an equation of the form 1.7.4.12: I
 ( ).

ñóò

= ! ( ï ) cos2 pƒñ + " ( ï )  . ¨ 
= Œ" ( ) cos2 ¨ + The substitution ¨ = Œ + ( ) leads to an equation of the form 1.7.4.12: I
 ( ).

ñ óò

1.7.5. Equations Containing Combinations of Exponential, Logarithmic, and Trigonometric Functions 1.

ó

= – sin 2ñ + cos2 ñB! ( ï )" (  2 tan ñ ).  ¨ 
=  2   ( ) (¨ ). The substitution ¨ ( ) =  2 tan leads to a separable equation: u

ñ óò

Page 201

202 2.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

=

ó

cos ñ ) . sin ñ

H (  ó

This is an equation of the type 1.7.5.5 with  (ˆ ) =  (ˆ ) ˆ . 3.

ñ óò

=

 ]

cos ï

H ( ]

sin ï ).

This is an equation of the type 1.7.5.7 with  (ˆ ) = ˆž (ˆ ). 4.

ñóò

ó

= tan ñB! ( 

sin ñ ).

 leads to an equation of the form 1.7.2.3: H 
= H! (  H ).

The substitution H = sin 5.

ñ óò

ó

= cot ñB! ( 

cos ñ ).

 leads to an equation of the form 1.7.2.3: H 
= − H! (  H ).

The substitution H = cos 6.

7.

8.

9.

10.

ñóò

= tan ï

! (  ]

cos ï ).

ñ óò

= cot ï

! ( ]

sin ï ).

ñóò

= sin 2ñB! (

ñ óò

= sin 2ñB! (

The substitution a = cos  leads to an equation of the form 1.7.2.4: a
g = −  (a  ). The substitution a = sin  leads to an equation of the form 1.7.2.4: a
g =  (a  ).

ó

tan ñ ).



The substitution H = tan  leads to an equation of the form 1.7.2.3: H 
= 2 H! (  H ).

ó

cot ñ ).

 The substitution H = cot  leads to an equation of the form 1.7.2.3: H 
= −2 H! (  H ).

ñ óò

=

ó

H (  ó

sin ñ ) . cos ñ

This is an equation of the type 1.7.5.4 with  (ˆ ) =  (ˆ ) ˆ . 11.

ñóò

=

]

sin ï

H (  ]

cos ï ).

This is an equation of the type 1.7.5.6 with  (ˆ ) = ˆž (ˆ ). 12.

tan ï ) . sin 2 ï The substitution a = tan  leads to an equation of the form 1.7.2.4: 2a
g =  (a  ).

ñóò

=

! ( ]

13.

ñ óò

=

cot ï ) . sin 2 ï The substitution a = cot  leads to an equation of the form 1.7.2.4: 2a
g = −  (a  ).

14.

ñóò

=

! ( ]

ó

 – !

(

ï

+ ln ñ ).

The substitution ‡ = Œ  + ln 15.

ñ óò

=

 ] ! ( Ė

+ ln ï ).

The substitution ‡ = Œ

leads to a separable equation: ‡
 = 

−



+ ln  leads to a separable equation: ‡
 = Π

 (‡ ) + Œ .



 (‡ ) + 1.

Page 202

1.8. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY FUNCTIONS

203

1.8. Equations of the Form c ( ì , í , í ï î ) = 0 Containing Arbitrary Functions 1.8.1. Some Equations 1.8.1-1. Arguments of arbitrary functions depend on  and . 1.

(ñó ò )2 + [ ! ( ï ) + " (ï )]ñ„ó ò + ! ( ï )" ( ï ) = 0.

The equation can be factorized: [
 +  ( )][
 + ( )] = 0, i.e., it falls into two simpler equations 
+  ( ) = 0 and 
+ ( ) = 0. Therefore, the solutions are:

 ( )  = L M

+X 2.

(ñó ò )2 + 2 !„ññó ò + "Iñ

= (" –

2

!

2

ó

) exp O –2 X

=

3.

ï ñ óò

– ñ = ! (ï

2








exp O − X









P

.



− 2  +L  ü M P  M P L exp O − X   M P    exp O − X   cosh O X  2−  +L   ü P M P M 

óò

+ ñ 2 ) (ñ

ü



M

!ý ï

D

Here,  =  ( ), = ( ). Solution:






+ X ( )  = L .

and

sin O X

if >  2 , if ≡  2 , if <  2 .

)2 + 1.

Raising the equation to the second power and applying the transformation  = & (a ) cos a , = & (a ) sin a , one arrives at the relation & 4 =  2 (& 2 )[(& g
)2 + & 2 ]. Solving it for & g
yields a separable equation:  (& 2 )& g
= A"& & 2 −  2 (& 2 ). 4.

ñ ñóò + ï

= ! (ï

2

+ ñ ) (ñ 2

ü

óò

ü

)2 + 1.

Raising the equation to the second power and applying the transformation  = & (a ) cos a , = & (a ) sin a , one arrives at the relation & 2 (& g
)2 =  2 (& 2 )[(& g
)2 + & 2 ]. Solving it for & g
yields a &ž (& 2 ) separable equation: & g
= A . & 2 −  2 (& 2 )

^_ 5.

ü

Reference: G. W. Bluman, J. D. Cole (1974, p. 100).

ï ñ óò

–ñ =

ü

ï

2

+ñ

2

!e”

ï

ü

ï

2

+ñ

2

,

ñ

ü

ï

2

+ñ

2

+ñ

2

• ü

(ñ

óò

)2 + 1.

Raising the equation to the second power and applying the transformation  = & (a ) cos a , = & (a ) sin a , one arrives at the relation & 2 =  2 (cos a , sin a )[(& g
)2 + & 2 ]. Solving it for & g
yields a separable equation:  (cos a , sin a )& g
= A"& 1 −  2 (cos a , sin a ). 6.

ññ óò

+ï =

ü

ï

2

+ñ

2

!e” ü

ï

ü

ï

2

+ñ

2

,

ü

ï

ñ 2

• ü

(ñ

óò

)2 + 1.

Raising the equation to the second power and applying the transformation  = & (a ) cos a , = & (a ) sin a , one arrives at the relation & 2 (& g
)2 =  2 (cos a , sin a )[(& g
)2 + & 2 ]. Solving it for & g
yields a separable equation: 1 −  2 (cos a , sin a ) & g
= A"&ž (cos a , sin a ).

ü

Page 203

204

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.8.1-2. Argument of arbitrary functions is "
 . 7.

ï

= ! (ñó ò ). Solution in parametric form:

 =  (a ), 8.

9.

ñ

= ! (ñó ò ). Solution in parametric form:

! (ñ óò ) + ô ï

+ õÿñ + s = 0. Solution in parametric form:

= X ªa  g
(a ) a + L .

M

a  = Xv g
(a ) M + L , a

=  (a ).

 g
(a ) a M ,  = −  − ° −  (a ).  + 'a In addition, there is a particular solution = ƒ… + ý , where ƒ and ý are determined by solving  =L −X

the system of two algebraic equations:  + ˜ƒ = 0, 10.

11.

12.

13.

14.

15.

16.

ñ = ï ñóò

 ( ƒ ) + 'ý + ° = 0.

+ ! (ñó ò ). = L3 +  ( L ). The Clairaut equation. Solution: In addition, there is a singular solution, which may be written in the parametric form as:
 = −  g
(a ), = −aª g (a ) +  (a ).

= ï ! (ñó ò ) + " (ñó ò ). The Lagrange–d’Alembert equation. For the case  (a ) = a , see equation 1.8.1.10. Having differentiated with respect to  , we arrive at a linear equation with respect to  =  (a ), where a = 
 : [a −  (a )] g
=  g
(a ) + g
(a ). See also 1.8.1.12.

ñ

ï ! (ñ óò

) + ñ" (ñ ó ò ) + % (ñ ó ò ) = 0. The Legendre transformation m [  (m ) + m (m )] n o
− (m ) n + % Inverse transformation:  = n

= 
 , n =  
 − , n o
=  (m ) = 0.


o , = mn o − n ,  = m .

leads to a linear equation:

= ï 2 ! (ñó ò ) + ï " (ñó ò ) + % (ñó ò ). Having differentiated with respect to  , we arrive at an Abel equation with respect to  =  (a ), where a = 
 : V 2  (a ) + (a ) − aÖW g
= −  g
(a ) 2 − g
(a ) − % g
(a ) (see Subsection 1.3.4).

ñ

ï

= ñ 2 ! (ñó ò ) + ñ" (ñó ò ) + % (ñó ò ). Having differentiated with respect to  , we arrive at an Abel equation with respect to = (a ), where a = 
 : V 2aª (a ) + aÝ (a ) − 1W g
= −aª g
(a ) 2 − aÝ g
(a ) − aª% g
(a ) (see Subsection 1.3.4).

= ï ÷ ! (ñ ó ò ) + ï ñ ó ò . Differentiating with respect to  and denoting a = 
, we obtain a Bernoulli equation for  =  (a ): B… (a ) g
−  g
(a ) −  2−  = 0.

ñ

( ï ñ ó ò – ñ )÷ ! (ñ ó ò ) + ñ" (ñ ó ò ) + ï % (ñ ó ò ) = 0. The Legendre transformation  = ‡ g
, = aև g
− ‡ [aÝ (a ) + % (a )]‡ g
= (a )‡ +  (a )‡  .

(
 = a ) leads to a Bernoulli equation:

Page 204

1.8. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY FUNCTIONS

205

1.8.1-3. Arguments of arbitrary functions are linear with respect to
 . 17.

ñ

= ô (ñ

óò

)2 + ! ( ï – 2 ôñ

óò

).

1 Solution: =  (L ) + ( − L )2 . In addition, there is a singular solution, which can be 4 represented in parametric form as:

 = a + 2  g
(a ), 18.

ñóò

= ! (ñó ò + ô

ï

The contact transformation leads to a linear equation: n Inverse transformation: 19.

ñóò

= ! (ñó ò + ô

ï )(ñ óò 2 + 2ôñ

ïƒ÷  ^ ]

= ! (ï

ñóò

).

The substitution

22.

23.

24.

25.

o

m




).

= 
 +  , n = 12 (
 )2 +  , n o
= 
 , where n = n (m ), = 2 (m ) n +  (m ).

=  −1 (m − n o ), = 12  −1 [2 n − ( n o )2 ], 
 = n o
.

) + " (ñó ò +

ô ï )(ñ óò 2 + 2ôñ )

.

The contact transformation m = 
+  , n = 12 ( 
)2 +  , n o
= 
, where n = n (m ), leads to a Bernoulli equation: n o
= 2  (m ) n + 2  (m ) n  . Inverse transformation:  =  −1 (m − n o
), = 12  −1 [2 n − ( n o
)2 ], 
 = n o
.

20.

21.

ô ï )(ñ óò 2 + 2ôñ

) + " (ñó ò +

=  (a ) +  [  g
(a )]2 .

ï

= ! (ñó ò )" ( ï

ñóò

¨ 
¨ ). leads to an equation of the form 1.8.1.32:   ¨ 9 =  ( 2

= ln ¨

– ñ ).

The Legendre transformation m = 
 , n =  
 − , n o
=  leads to a separable equation: n o
=  (m ) ( n ). Inverse transformation:  = n o
, = mn o
− n , 
 = m .

ï ! (ï ñ óò

– ñ )ñó ò + ï

" (ï ñ óò

ï

= ! (ï

ñóò

+ ñ )" ( ï

2

= ! (ï

ñ óò

+ñ )+ï

2

– ñ )(ñó ò )  = ô .

The modified Legendre transformation m =  
 − , n = 
 , n o
= 1  leads to a Bernoulli equation: !n o
=  (m ) n + (m )n  . Inverse transformation:  = ( n o
)−1 , = n ( n o
)−1 − m , 
 = n .

ñóò

).

The contact transformation m =  
 + , n =  2 
 , n o
=  , where n = n (m ), leads to a separable equation: n o
=  (m ) ( n ). Inverse transformation:  = n o
, = m − n ( n o
)−1 , 
= n ( n o
)−2 .

ï

ñ óò " (ï ñ óò

+ ñ ).

The contact transformation m =  
 + , n =  2 
 , n o
=  , where n = n (m ), leads to a linear equation: n o
= (m ) n +  (m ). Inverse transformation:  = n o
, = m − n ( n o
)−1 , 
= n ( n o
)−2 .

ï ñóò ! (ï ñ óò

+ñ )+

ï 3 (ñ óò )2" (ï ñ óò

+ñ )=ô .

The contact transformation m =  
 + , n =  2 
 , n o
=  , where n = n (m ), leads to a Bernoulli equation: !n o
=  (m ) n + (m ) n 2 . Inverse transformation:  = n o
, = m − n ( n o
)−1 , 
= n ( n o
)−2 .

Page 205

206 26.

FIRST-ORDER DIFFERENTIAL EQUATIONS + ï ) = ñ 2 (ñó ò 2 + 1). Setting ‡ ( ) = ! 
+  and differentiating with respect to  , we obtain

! (ñ ñóò

‡
 [  
(‡ ) − 2‡ + 2 ] = 0.

(1)

Equating the first factor to zero, after integrating we find = −( − L ) + $ . Substituting the latter into the original equation yields $ =  ( L ). As a result we obtain the solution: 2 =  ( L ) − (  − L )2 . There is also a singular solution that corresponds to equating the second factor of (1) to zero. This solution in parametric form is written as: 2

27.

28.

 = ‡ − 12  
(‡ ),

=  (‡ ) − 14 [  
(‡ )]2.

2

= ! (ññ ó ò – ï ) + ñ 2 (ñ ó ò 2 – 1)" (ññ ó ò – ï ). The contact transformation  = nIn o
− m , = n [( n o
)2 − 1]1 J 2 , 
 = n o
[( n o
)2 − 1]−1 J 2 , ú where n = n (m ), leads to the equation nCn o
=  (m ) + n 2 (m ), which is linear in = n 2.




! 2 1 J 2 2 −1 Inverse transformation: m =  −  , n = − [(  ) − 1] , n o = −  [(  ) − 1] J 2 .

ññ óò

ñóò

=

1 ! Oiñóò ï 2I

ñ ï

+

ñ " Oiñóò ï P3

P

+ Oiñó ò –

+




ï

ñ

P

.

The contact transformation m =  +   , n =  equation: mn o
= 2 (m ) n + 2mR (m ). Inverse transformation: m n o
 1  =A mn o
− n , =A m ü 2 mn 29.

ï D –1 I ! Oiñóò

–ô

ñ

ï P

+ Oiñó ò –

ñ ï

P

"3Oiñóò

–ô

For  ≠ 1, the contact transformation m = to a linear equation: ˜n o
= (m ) n +  (m Inverse transformation: 1 1  = ( n o
) 1−  , = (mn 1− 30.

2

ï D +1 I ! Oiñóò

+ô

ï

ñ

P

= " (ï

D +1 ñ óò

–ï

D¦ñ

ï

ñü

P

2

(
 )2 −

− 2n ,
o −n

2

, n o
= 2 2 
 leads to a linear


=

=õ .


 −    , n =  ).



 o − n )( n o ) 1−  ,

1−

m 2n o
. 2(mn o
− n )

 
 −  −  , n o
=  
=

1−

mn o
− !n . (1 −  ) n o


).

For  ≠ −1, the contact transformation m = 
 +    , n =   +1 
 −   , n o
=   to a separable equation:  (m ) n o
= ( n ). Inverse transformation:  1 1 mn o
+ !n 
=  = ( n o
)  +1 , = (mn o
− n )( n o
)−  +1 , .  +1 (  + 1) n o
31. 32.

 leads

ó  ^ ñ ÷

= ! (ñó ò ¦ñ ). The substitution  = ln a leads to an equation of the form 1.8.1.32: a9

ï ÷ ñ ú

+1

leads

 =  (a 
g  ).

= ! ( ï ñ ó ò ¦ñ ). We pass to a new variable ¨ ( ) =  
 , divide both sides of the equation by   + , and differentiate with respect to  . As a result we arrive at a separable equation:   ?
(¨ )¨2
= ( ¨ + B )  (¨ ). ¡ Solution in parametric form:  ?
(¨ ) ¨ + L ,   + =  (¨ ). ln | | = X M ( ¨ + B )  (¨ )

¡

In addition, there are singular solutions equation # + −  (−B ) = 0.

 J + , where #  are roots of the algebraic = #   −´

¡

Page 206

ê ( Ë , Ï , Ï ÒÍ

1.8. EQUATIONS OF THE FORM

33.

34.

35.

36.

ó

ñ óò

ó

=  ! ( ñ óò ) – ñ .  The contact transformation m =  
 , n = 
 + , n separable equation: nCn o
=  (m ). Inverse transformation:  = − ln n o
, = n −  m n

ó

ó

 ! ( ñóò

) – " (ñó ò + ñ ) = 0.  The contact transformation m =  
 separable equation: ( n ) n o
= −  (m Inverse transformation:  = − ln n

ó

ó

! ( ñ óò

) + " (  ñ ó ò )(ñ ó ò + ñ The contact transformation linear equation: !n o
= (m Inverse transformation:

ó

ó


o =  − , where n = n (m ), leads to a


o ,  = mn o .

 = 
 + , n o
=  − , where n


o ,

) – " (ñó ò + ñ ) = ï . This equation can be rewritten in the form 1.8.1.34:


o =  − , where n = n (m ), leads to a


o ,  = mn o .

! (  ñóò

where

 (‡ ) = − ln  (‡ ),

sin ï – ñ cos ï ) = ñ„ó ò ! (ñóò cos ï + ñ sin ï ). The contact transformation


cos  + sin  £ (
 )2 + 2 , 2 leads to the homogeneous equation: n o
=  ( n m ).

H ( ïƒ÷

+1

1 = £ 
(  )2 +

,

H (ï 2 ñ óò

n o
=
( 
sin  − cos  ) 

n =

ñóò , ï ñ óò

Solution:  (− L 1 B , L 39.

( ) = − ln 4 ( ).

ñ (ñ óò

m

38.

= n (m ), leads to a

= n − mn o
, 
 = mn o
.

) = ô – .  m =  
 , n = 
 + , n ) n +  (m ).  = − ln n o
, = n −  m n

    (  
) − 4 ( 
+ ) = 0,

37.

, n ).

207

) = 0 CONTAINING ARBITRARY FUNCTIONS

+ ùñ ) = 0. = L 1  −  + L 2 . Here, the constants L 2 B ) = 0.

1

and L

2

are related by the constraint

+ 2 ï ñ , ï 3 ñ ó ò + ï 2 ñ ) = 0. Solution: = L 1  −1 + L 2  −2 . Here, the constants L 1 and L 2 are related by the constraint  ( L 1 , − L 2 ) = 0. The singular solution can be represented in parametric form as:

 (‡ ,  ) = 0,





(‡ ,  ) + "



(‡ ,  ) = 0,

‡ =  2 a + 2 ,

where

 =  3a + 

2

.

The subscripts ‡ and  denote the respective partial derivatives, and a is the parameter. 40.

H ( ïƒ÷

+ þ ïƒ÷ ñ , ïƒú +1 ñó ò + ù ïƒú ñ ) = 0. Solution: = L 1  −  + L 2  − + . Here, the constants L   L 1 ( − B ), L 2 (B − )  = 0. +1

ñóò

¡

41.

42.

and L

2

are related by the constraint

¡

ó H (  ñ óò , ñ óò

+ ñ ) = 0.  Solution: = L 1 − + L  (− L 1 , L 2 ) = 0.

ó H (  ^ ñ óò

1

ó

ó

2.

Here, the constants L

1

and L

2

are related by the constraint

ó

+ J ^ ñ ,  _ ñó ò + q _ ñ ) = 0.   Solution: = L 1  − 9 + L 2  −: . Here, the constants L   L 1 (ý − ƒ ), L 2 ( ƒ − ý )  = 0.

1

and L

2

are related by the constraint

Page 207

208 43.

FIRST-ORDER DIFFERENTIAL EQUATIONS

H (ñ óò

cosh ï – ñ sinh ï ,

ñ óò

Solution: = L 1 sinh  + L constraint  ( L 1 , − L 2 ) = 0. 44.

H (ï ñ óò , ñ

–ï

ñ óò

Solution: = L  ( L 1 , L 2 ) = 0. 45.

HwO

ñ óò ñ

, ln ñ –

Solution:  ( L 2 , ln L 46.

HwO

ï ñ óò ñ

1)

H (ñ óò

ln  + L

ñóò P

ï ñ óò ñ

ñ óò

,

ñ

–

r

r òó ñ óò P

Solution: = L  ( L 1 , L 2 ) = 0. 49.

óò ñ

2

are related by the

1

and L

2

are related by the constraint

2

 ). Here, the constants L

1

and L

2

are related by the constraint

ln ï

= 0.

P

and L

1

2

cos  . Here, the constants L

r

= 0,

r (ï

=

 ( L 1 , − L 2 ) = 0.

and L

1

2

are related by the constraint

).

Y ( ) + L 2 . Here, the constants L

–

are related by the constraint

2

sin ï – ñ cos ï ) = 0.

ñ óò

ñ óò r òóñ – r ñ óò s r , r óò – óò óò – r óò P s s = L Y ( s ) + L s j ( ). Solution: 1 2 HwOr s

and L

1

k 2 . Here, the constants L

= L 1 1 ) = 0.

cos ï + ñ sin ï ,

HwO r òó

1

Here, the constants L

2.

Solution: = L 1 sin  + L  ( L 1 , − L 2 ) = 0. 48.

cosh  . Here, the constants L

= 0.

= L 1 exp( L = 0.

, ln ñ –

Solution:  ( L 2 , ln L 47.

ñ

2

ln ï ) = 0. 1

ï

sinh ï – ñ cosh ï ) = 0.

r

= 0,

=

and L

1

r (ï

),

Here, the constants L

1

are related by the constraint

2

=

s

and L

s 2

( ï ). are related by the constraint

The singular solution can be represented in parametric form as:

 (‡ ,  ) = 0,

j·  (‡ ,  ) + Ys (‡ ,  ) = 0,

where ‡ =



j Y j h




− aÖj , 
− j·Y


 =

Y Y j h




− a›Y . 
− j·Y


The subscripts ‡ and  denote the respective partial derivatives, and a is the parameter. 50.

H r ó

+

r ñ óò , r ]

– ï (r

r ñ óò )  ] ºLt , Y  = ºLt

ó

Here, Y = Y ( , ), Y  =

º

+

º

= 0.



(Y  + Y

. Differentiating with respect to  , we obtain




)
 ( 



− "



) = 0,

where   = º Ž  and  = º Ž are partial derivatives of function  (‡ ,  ). Equating the first º  solution: factor to zero,º we find the

Y ( , ) = L3 + # ,

where

It remains to be checked whether the equation  which of them satisfy the original equation.



 ( L , # ) = 0.

− "



= 0 possesses any solutions and

Page 208

1.8. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

209

) = 0 CONTAINING ARBITRARY FUNCTIONS

1.8.1-4. Arguments of arbitrary functions are nonlinear with respect to
 . 51. 52.

= ! (ï ñ óò 2) + 2ï ñ óò . Solution: [ −  ( L )]2 = 4 L3 .

ñ

= 2 ô (ñ„ó ò )3 + ! ( ï – 3 ôñó ò 2 ). This is a special case of equation 1.8.1.72 with B = 3.  − L 3J 2 Solution: =  ( L ) + 2 rO . In addition, there is the following singular solution 3 P written in parametric form:

ñ

 = a + 3  [  g
(a )]2 ,

53.

54.

55.

56.




=  (a ) + 2 [  g (a )]3 .

= ! ( ôñó ò 2 – õ ï ) + (2 ôñó ò 3 – 3 õÿñ )" ( ôñó ò 2 – õ ï ), õ ≠ 0. 

2 The contact transformation m =  (  ) − ' , n = 2  (  )3 − 3  , n o
= 3
 leads to a linear equation: n o
= 3 (m ) n + 3  (m ). 1 −1 Inverse transformation:  = 19  −1 [  ( n o
)2 − 9m ], = 81  [2  ( n o
)3 − 27 n ], 
 = 13 n o
.

ñóò

ñóò

= ! (ñó ò + ô

ï )"

( 21 ñó ò 2 + ôñ ).

The contact transformation m = 
 +  , leads to a separable equation: n o
=  (m ) Inverse transformation:  =  −1 (m − n

ñóò (ñ óò

– ñ ) = ! (ñó ò 2 – ñ 2 ).

The contact transformation m = (
 )2 − (separable) equation: 2  (m ) n o
= n .

ó

ñ óò

= ! (ñ ó ò 2 – ñ 2 )( ô + õ The contact transformation

–

ó

2

n = 12 (
 )2 +  , n o
= 
 , where n = n (m ), ( n ).



o ), = 12  −1 [2 n − ( n o )2 ],  = n o .

  , n =  (
 − ), n o
= 12  ( 
)−1 leads to a linear

).

    n = 
(   +   − ) − (   −   − ), leads to a separable equation: 2  (m ) n o
= 1. m

57.

58.

59.

60.

ñóò

= ( 
)2 −

ó

2

,

ó

ó

=  ! (ñó ò 2 – ñ 2 )" (  ñó ò –  ñ ). The contact transformation m = (
 )2 − separable equation: 2  (m ) ( n ) n o
= 1.

2

, n

  n o
= 12 (   +   − )( 
)−1



=  (
 − ), n o
=

1 2

  (
 )−1 leads to a

= ! (ñó ò 2 – ñ 2 ) cosh ï . The contact transformation m = ( 
)2 − 2 , n = 
cosh  − sinh  , n o
= leads to a separable equation: 2  (m ) n o
= 1.

ñóò ñóò

= ! (ñó ò 2 – ñ

2

1 2

cosh  ( 
)−1

1 2

sinh  ( 
)−1

) sinh ï .

The contact transformation m = (
 )2 − 2 , n = 
 sinh  − cosh  , n o
= leads to a separable equation: 2  (m ) n o
= 1.

= ! (ñó ò 2 – ñ 2 )( ô cosh ï + õ sinh ï ). The contact transformation

ñóò

m = ( 
)2 − 2 ,

n = 
(  cosh  +  sinh  )− (  sinh  +  cosh  ),

leads to a separable equation: 2  (m ) n o
= 1.

 cosh  +  sinh  n o
= 2 


Page 209

210 61.

62.

63.

64. 65.

FIRST-ORDER DIFFERENTIAL EQUATIONS = ! (ñó ò 2 – ñ 2 )" (ñó ò cosh ï – ñ sinh ï ) cosh ï . The contact transformation m = (
 )2 − 2 , n = 
 cosh  − sinh  , n o
= leads to a separable equation: 2  (m ) ( n ) n o
= 1.

ñóò

= ! (ñó ò 2 – ñ 2 )" (ñó ò sinh ï – ñ cosh ï ) sinh ï . The contact transformation m = (
 )2 − 2 , n = 
 sinh  − cosh  , n o
= leads to a separable equation: 2  (m ) ( n ) n o
= 1.

ñóò

) + ôñó ò 2 cosh ï – ôññó ò sinh ï = cosh ï . The contact transformation m = (
 )2 − 2 , n = 
 cosh  − sinh  , n o
= leads to a linear equation: 2 n o
= !  n +  (m ).

ñóò ! (ñ óò 2 – ñ

68.

1 2

cosh  (
 )−1

= ! (ñó ò 2 + ñ 2 )( ô cos ï + õ sin ï ). The contact transformation

ñóò

n o
= 12 (  cos  +  sin  )( 
)−1

= ! (ñó ò 2 + ñ 2 )" (ñó ò cos ï + ñ sin ï ) cos ï . The contact transformation m = (
 )2 + 2 , n = 
 cos  + sin  , n o
= to a separable equation: 2  (m ) ( n ) n o
= 1.

ñóò

) + ôñó ò 2 cos ï + ôññó ò sin ï = cos ï . The contact transformation m = (
 )2 + 2 , n = 
 cos  + sin  , n o
= to a linear equation: 2 n o
= !n +  (m ).

ñóò ! (ñ óò 2 + ñ

= ! (ñó ò 2 + ñ 2 )" (ñó ò sin ï – ñ cos ï ) sin ï . The contact transformation m = (
 )2 + 2 , n = 
 sin  − cos  , n o
= to a separable equation: 2  (m ) (n ) n o
= 1.

ñóò

ï 2 ñ óò

1 2

cos  (
 )−1 leads

1 2

cos  (
 )−1 leads

1 2

sin  (
 )−1 leads

2

ñ ï ï P " ( 2 ñ óò 2 – ñ

=I ! Oiñóò +

2

).

The contact transformation m = 
 +   , n =  2 (
 )2 − equation: n o
= 2  (m ) ( n ). Inverse transformation:

 =A

71.

sinh  (
 )−1

= ! (ñó ò 2 + ñ 2 ) cos ï . This is a special case of equation 1.8.1.65 with  = 1 and  = 0.

69.

70.

1 2

ñóò

n = 
(  cos  +  sin  )+ (  sin  −  cos  ), leads to a separable equation: 2  (m ) n o
= 1.

67.

cosh  (
 )−1

2

m = ( 
)2 + 2 ,

66.

1 2

m

1

ü

mn o
− n ,

=A

mn o
− 2 n , 2 mn o
− n ü

2

, n o
= 2 2 
 leads to a separable


=

m 2n o
. 2(mn o
− n )

= ! ( ôñó ò 2 – õ ï )" (2 ôñó ò 3 – 3 õÿñ ), õ ≠ 0. 
2 The contact transformation m =  (  ) − ' , n = 2  (
 )3 −3  , n o
= 3
 leads to a separable equation: n o
= 3  (m ) ( n ). 1 −1 Inverse transformation:  = 19  −1 [  ( n o
)2 − 9m ], = 81  [2  ( n o
)3 − 27 n ], 
 = 13 n o
.

ñóò

ñ

= ! (ï

ñóò ÷

Solution:

)+

ù

ù

–1

ï ñóò

.

 −1 B…L =  (L  ) +   . B −1

Page 210

1.8. EQUATIONS OF THE FORM

72.

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY FUNCTIONS

211

= ô (ù – 1)(ñ„ó ò )÷ + !  ï – ôù (ñó ò )÷ –1  . Differentiating with respect to  , we obtain a factorized equation:

ñ

V 1 − B (B − 1)( 
) 

 B ( 
) where a =  − 

−2



 W V 
−  g
(a )W = 0,

(1)

−1

. Equate the first factor to zero and integrate the obtained equation. Substituting the expression obtained into the original equation, we find the solution:

 −L  B P

=  ( L ) +  (B − 1) O



−1

.

Equating the second factor in (1) to zero, we have another solution that can be written in parametric form as:

 = a + B [  g
(a )] 

73.

−1

=  (a ) +  (B − 1)[  g
(a )]  .

,

= !  ô (ñó ò )  – õ ï  "  ôƒü (ñó ò )  +1 – õ ( ü + 1)ñ  . The contact transformation (  ≠ 0, , ≠ −1)

ñóò

=  ( 
)  − ' ,

m

n = , ( 
) 

+1

n o
= ( , + 1) 


−  ( , + 1) ,

leads to a separable equation: n o
= ( , + 1)  (m ) ( n ). Inverse transformation:

 (n o
)  m − ,   ( , + 1) 

 = 74.

=

, ( n o
)   ( , + 1) 

+1 +2

−

n

 ( , + 1)

= !u ô (ñ ó ò )  – õ ï  +  ôƒü (ñ ó ò )  +1 – õ ( ü + 1)ñ  * "  ô (ñ The contact transformation (  ≠ 0, , ≠ −1)

óò )

ñ óò


=  (  )  − ' ,

m

n = , ( 
) 

+1

–

−  ( , + 1) ,

leads to a linear equation: n o
= ( , + 1) (m ) n + ( , + 1)  (m ). Inverse transformation:

 (n o
)  m − ,  ( , + 1)  

 = 75.

76.

H  ññóò

+ Solution:

ó

H (  ñóò

ô ï ,ñ

–

2

78.

, ( n o
)   ( , + 1) 

+1 +2

−

n

 ( , + 1)

ñ óò 2 + ô

 = 0. = −  + 2 L 1  + L 2 . Here, the constants L ü

2

ó ñ , ñ óò 2 – ñ 

Solution: = L 1 + L  (−2 L 2 , −4 L 1 L 2 ) = 0.

77.

=

1

and L

  L 1 , ü L 12 + L 2  = 0 if   L 1 , − ü L 12 + L 2  = 0 if

n o
.


=

,

õï

, +1

.

n o
= ( , + 1) 


n o
.


=

,

, +1

are related by the constraint

2

> 0, < 0.

2

) = 0. − . Here, the constants L 2

1

and L

cosh ï – ñ sinh ï , ñ ó ò 2 – ñ 2 ) = 0. Solution: = L 1 sinh  + L 2 cosh  . Here, the constants L constraint  ( L 1 , L 12 − L 22 ) = 0.

2

are related by the constraint

H (ñ óò

cos ï + ñ sin ï , ñ ó ò 2 + ñ 2 ) = 0. Solution: = L 1 sin  + L 2 cos  . Here, the constants L  ( L 1 , L 12 + L 22 ) = 0.

H (ñ óò

1

and L

1

and L

2

are related by the constraint

2

are related by the

Page 211

212

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.8.2. Some Transformations 1.

2.

ï

= ! (ñ , ñó ò ). Substituting a = 
and differentiating both sides of the equation with respect to  , we obtain an equation with respect to = (a ): – – [1 − aª ( , a )] g
= aªg ( , a ), where g = º g ,  = º .   º º If = (a ) is the solution of the latter equation, the solution of the original equation can be represented in parametric form as:  =  ( (a ), a ), = (a ).

= ! ( ï , ñó ò ). Differentiating with respect to  and setting a = "
 , we obtain an equation with respect to  =  (a ): – – [a −   ( , a )] g
= g ( , a ), where g = º g ,   = º  . º º If  =  (a ) is the solution of the latter equation, the solution of the original equation can be represented in parametric form as:  =  (a ), =  ( (a ), a ).

ñ

ïƒ÷ ñ ú

3.

=I ! O À ï

Set H =  

ñ

s

,

and ¨

ï ñ óò ñ =

P

 


. . Divide both sides of the equation by   +

and differentiate

with respect to  . As a result we arrive at the following equation with respect to ¨ = ¨ ( H ): ‘ H ( ° ¨ + , )(  +  ? ¨ ‘
) = ( ¨ + B )  , where  =  ( H , ¨ ), ¡ which is usually simpler than the original equation, since it is readily solved for the derivative. If ¨ = ¨ ( H ) is the solution of the Àequation obtained, the solution of the original equation is written in parametric form as:   = H ,   + =  ( H , ¨ ( H )). 4. 5. 6.

7.

8.

]

ïƒ÷  ^ ]

= ! ( ïƒú The substitution

ó  ^ ñ ÷

ó

 _ ] , ï ñ óò

). = ln ‡ leads to an equation of the form 1.8.2.3:   ‡ 9 =    + ‡ : , ‡
 ‡  .

= ! (  _ ñ ú , ñ ó ò ¦ñ ). The substitution  = ln a leads to an equation of the form 1.8.2.3: a 9

! (ï , ï ñ óò

 =   a : + , a g
  .

– ñ , ñó ò ) = 0. The Legendre transformation  = ‡ g
, = aև g
− ‡ (
 = a ), where ‡ = ‡ (a ), leads to the equation  (‡ g
, ‡ , a ) = 0. Inverse transformation: a = 
, ‡ =  
− , ‡ g
=  .

(ñ ó ò )2 = ƒñ + ! ( ï ). For Œ ≠ 0, the transformation Œ ¨ = 2 £ Œ +  ( ) leads to an Abel equation of the second kind, ¨*¨ 
= ¨ + Y ( ), where Y = 2 Œ −2  
( ), which is outlined in Subsection 1.3.1 for specific functions Y .

= ï ñó ò + ô ï 2 + ! (ñó ò ), ô ≠ 0. Differentiating the equation with respect to  and changing to new variables a = 
and ¨ = −2  , we arrive at an Abel equation of the second kind, ¨*¨ g
= ¨ + Y (a ), where Y = −2  g
(a ), which is outlined in Subsection 1.3.1 for specific functions Y .

ñ

For information about contact transformations, see Subsection 0.1.8.

Page 212

Chapter 2

Second-Order Differential Equations 2.1. Linear Equations 2.1.1. Representation of the General Solution Through a Particular Solution 1 b . A homogeneous linear equation of the second order has the general form

 2 ( ) 

 +  1 ( ) 
+  0 ( ) = 0. Let 0 = 0 ( ) be a nontrivial particular solution ( solution of equation (1) can be found from the formula: =

0

OL

1

+L

2

X

 −Ž

2 0

 , M P

0

(1)

1 0) of this equation. Then the general

where

 =X 



1 2

M

 .

(2)

For specific equations described below in 2.1.2–2.1.9, often only particular solutions are given, while the general solutions can be obtained with formula (2) (see also Paragraph 0.2.1-1, Item 3 b ). x3y'z|{}~/€ Only homogeneous equations are considered in Subsections 2.1.2 through 2.1.8; the solutions of the corresponding nonhomogeneous equations can be obtained using relations (7) and (8) of Subsection 0.2.1. 2 b . Suppose a particular solution of a homogeneous linear equation is obtained in the closed form = [  ( )]  , with this formula valid for  ( ) ≥ 0. If the equation makes sense in a range of  where  ( ) < 0, then the function = |  ( )|  will be a particular solution of the equation in that range.

3 b . Suppose = L 1  ( )[ ( )]  + L 2  ( )[ ( )]  is the general solution of the homogeneous linear equation with  ≠  , where  and  are free parameters. Then the function = L 1  ( )[ ( )]  + L 2  ( )[ ( )]  ln ( ) will be the general solution of this equation with  =  .

2.1.2. Equations Containing Power Functions 2.1.2-1. Equations of the form 

 +  ( ) = 0. 1.

2.

ñóò ò ó

+ ôñ = 0. Equation of free oscillations.   L 1 sinh( £ | | ) + L 2 cosh( £ | | ) if  < 0, Solution: = L 1 + L 2 if  = 0, L 1 sin( £  ) + L 2 cos( £  ) if  > 0. – ( ô ï + õ )ñ = 0, ô ≠ 0. −2 J 3 The substitution ˆ =  (  +  ) leads to the Airy equation:

ñóò ò ó

ªŠ

Š − ˆ

= 0,

(1)

213

Page 213

214

SECOND-ORDER DIFFERENTIAL EQUATIONS which often arises in various applications. The solution of equation (1) can be written as: =L

1

Ai(ˆ ) + L

2

Bi(ˆ ),

where Ai( ˆ ) and Bi(ˆ ) are the Airy functions of the first and second kind, respectively. The Airy functions admit the following integral representation: 1

Ai(ˆ ) =

X ÷ cos  13 a 3 + ˆa  a , 0 M



Bi(ˆ ) =



1

X ÷ exp  − 31 a 3 + ˆa  + sin  13 a 3 + ˆa  S a . 0 Q M

The Airy functions can be expressed in terms of the Bessel functions and the modified Bessel functions of order 1 3 by the relations:

ˆ V® Bi(ˆ ) = Å 13 ˆ V ®

Ai(ˆ ) =

1 3

H ) − ® 1 J 3 ( H )W , Ai(−ˆ ) = 13 ˆ V ø ü 1 V Å ø −1 J 3 ( H ) + ® 1 J 3 ( H )W , Bi(− ˆ ) = 3ˆ −1 3 (

J

ü

−1 3 (

J

H ) + ø 1 J 3 ( H )W ,

−1 3 (

H ) − ø 1 J 3 ( H )W ,

J

where H = 23 ˆ 3 J 2 . For large values of ˆ , the leading terms of the asymptotic expansions of the Airy functions are: 1 −1 J 4 1 −1 J 4 Ai(ˆ ) = £ ˆ exp(− H ), Ai(−ˆ ) = £ ˆ sin OH +  , 4P 2   1 −1 J 4 1 −1 J 4 Bi(ˆ ) = £ ˆ exp( H ), Bi(−ˆ ) = £ ˆ cos OH +  . 4P





The Airy equation (1) is a special case of equation 2.1.2.7 with  = B = 1. 3.

ñ óò ò ó

– (ô

2

ï

2

+ ô )ñ = 0.

Particular solution: 4.

ñóò ò ó

ï

– (ô

2

0

= exp 

1 2

 2  .

+ õ )ñ = 0.

The Weber equation (two canonical forms of the equation correspond to  = A

‘

1 4 ).

1 b . The transformation H =  2 £  , ‡ =  J 2 leads to the degenerate hypergeometric equation 1  1 2.1.2.70: H´‡ ‘'

‘ + O − H ‡ ‘
− O £ + 1 ‡ = 0. 2 P 4 P  2 b . For  = ,

2

> 0,  = −(2B + 1) , , where B = 1, 2,  , there is a solution of the form: = exp  − 12 ,! 2  5   £ ,·  ,

, > 0,



where 5  ( H ) = (−1)  exp  H 2  M  exp  − H 2  is the Hermite polynomial of order B . H See also Subsection S.2.10.M

^_ 5.

ñ óò ò ó

References: H. Bateman and A. Erd´elyi (1953, Vol. 2), M. Abramowitz and I. A. Stegun (1964).

+

ô 3ï

(2 – ô

ï )ñ

Particular solution: 6.

ñóò ò ó

– (ô

ï

2

+

õï

= 0. 0

= exp  − 12  2 

2

+   .

+ ö )ñ = 0.

The substitution ˆ =  +



2

leads to an equation of the form 2.1.2.4:

ªŠ

Š − Oˆ 2 + ( − 

2

4 ·P

= 0.

Page 214

215

2.1. LINEAR EQUATIONS

7.

– ô ï ÷ ñ = 0. 1 b . For B = −2, this is the Euler equation 2.1.2.123, while for B = −4, this is the equation 2.1.2.211 (in both cases the solution is expressed in terms of elementary function).

ñ óò ò ó

2 b . Assume 2 (B + 2) = 2

=









1−2

 (

1−2

(

+ 1, where

¡

& c )+

+1

)− + ¬L

&c

¬L

1

1

¡

exp ”

is an integer. Then the solution is:

£ 



£ 

exp ”

 &

 &



•

+L

• +L

2

exp ” −

exp ” −

2

£ 

£ 





 &

 & •

•

®

≥ 0,

if

®

if

¡ ¡

< 0,

B +2

1 = . 2 2 +1 ¡ M 3 b . For any B , the solution is expressed in terms of the Bessel functions and modified Bessel functions of the first or second kind (see 2.1.2.126 and 2.1.2.127):

where c



=



where 8.

ñ óò ò ó



= M ,



=





L

1

10.

ø

1 2

”

£ −



 &

&

•

+L

2

£ 2n

1 2

”

£ −



 &

• & £ £   L 1£   ® 1 ”  & + L 2£  ; 1 ”  &  •  • 2 2 & &






if  < 0, if  > 0,

= 12 (B + 2).

– ô (ô

ï 2÷

+ù

ï ÷

Particular solution: 9.

£ 

–1 0

)ñ = 0.

= exp O



B +1

 

+1

– ô ï ÷ –2 ( ô ï ÷ + ù + 1)ñ = 0. Particular solution: 0 =  exp    B  .

.

P

ñ óò ò ó

+ ( ô ï 2÷ + õ ï ÷ –1 )ñ = 0. The substitution ˆ =   +1 leads to a linear equation of the form 2.1.2.108: (B + 1) 2 ˆ 
Šª
Š + B (B + 1)
Š + ( ˆ +  ) = 0.

ñ óò ò ó

2.1.2-2. Equations of the form 

 +  ( )
 + ( ) = 0. 11.

ñóò ò ó

+ ôñó ò + õÿñ = 0. Second-order constant coefficient linear equation. In physics this equation is called an equation of damped
vibrations. V 1 1 1  exp  − 2   L 1 exp  2 Œ   + L 2 exp  − 2 Œ   W if Œ 2 =  2 − 4  > 0, Solution:

12.

=




exp  − 12   V L

Œ   +L 1 exp  − 2      L 1 + L 2  1

sin 

1 2

2

cos 

1 2

Π W

+ ôñó ò + ( õ ï + ö )ñ = 0. 1 b . Solution with  ≠ 0:

if Œ

2

= 4 − 

if 

2

= 4 .

2

> 0,

ñóò ò ó

= exp  − 12   where

ø

1 3(

J

ü

ˆ VL

ø

£ ˆ 3 J 2  + L 2 n 2£  ˆ 3J 2  W , J  1J 3  3 "

2 1 1 3 3

ˆ = +

4( −  4

2

,

H ) and n 1 J 3 ( H ) are the Bessel functions.

2 b . For  = 0, see equation 2.1.2.11.

Page 215

216 13.

14.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï

ñóò ò ó

+

ôñóò

– (õ

ñóò ò ó

+

ôñóò

+ õ (– õ

+ ö )ñ = 0.

2

The substitution = ¨ exp  12  2 £   leads to a linear equation of the form 2.1.2.108: ¨2

 + (2 £ " +  )¨2
+ (  £ " − ( + £  )¨ = 0.

ï

2

Particular solution: 15.

ñóò ò ó

+

ôñóò

+

õï

(– õ

ï

Particular solution: 16.

ñóò ò ó

+

ôñóò

17.

ñóò ò ó

+

ôñóò

ñ óò ò ó

+

ï ñ óò

ñóò ò ó

– 2ï

+ô

ï

+ô

ïƒ÷

+ 2)ñ = 0.

= exp  − 13 ' 3  .

= exp O −

0

–ô 0

ïƒ÷

+ù

ïƒ÷



ïƒ÷

+ù

= exp O −



ù

= M  ¦ exp  − 21  2  L Q 

M

ñóò

+ 2ùñ = 0,



ù

 

–1

B +1



)ñ = 0.

–1

B +1

+ (ù + 1)ñ = 0,

Solution: 19.

3

ï 2÷

+ õ (– õ

Particular solution: 18.

= exp  − 12 ' 2  .

0

Particular solution:

+ 1)ñ = 0.

0

ï 2÷

+ õ (– õ

ô ï

+

+1

.

P

)ñ = 0.

 

+1

− 

= 1, 2, 3, 1

+L

2

= 1, 2, 3,

P

.

uuu

X exp  12  2   S § . M

uuu

Solution: = exp( 2 ) M  ¦ exp(− 2 ) L 1 + L 2 X exp( 2 )  S"§ . Q M  For L 1 = (−1)  andM L 2 = 0, this solution defines the Hermite polynomials. 20.

ñóò ò ó

+

ñ óò ò ó

+

ô ï ñóò

+

ô ï ñ óò

+

õÿñ

= 0.

Solution: = L 1 `  12  −1  , 12 , − 21  2  + L 2 i  12  −1  , 12 , − 21  2  , where ` (  ,  ;  ) and i (  ,  ;  ) are the degenerate hypergeometric functions (see equation 2.1.2.70 and Subsection S.2.7). 21.

õï ñ

= 0.

 Solution: =   J  V L 1 `  12  −3  2 , 12 , − 12 ˆ 2  + L 2 i  12  −3  2 , 12 , − 21 ˆ 2  W , ˆ =  − 2  −2  , where ` (  ,  ;  ) and i (  ,  ;  ) are the degenerate hypergeometric functions (see equation −

2.1.2.70 and Subsection S.2.7).

22. 23. 24.

+ ô ï ñó ò + ( õ ï + ö )ñ = 0. This is a special case of equation 2.1.2.108 with 

ñóò ò ó

26.

=

1

= 0 and 

+ 2 ô ï ñó ò + ( õ ï 4 + ô 2 ï 2 + ö ï + ô )ñ = 0. This is a special case of equation 2.1.2.49 with B = 1 and

ñóò ò ó ñ óò ò ó

+ (ô

ï

+ õ )ñ

óò

Particular solution:

25.

2

+ ôñ = 0. 0

= exp  − 12 

+ ( ô ï + õ )ñó ò – ôñ = 0. Particular solution: 0 =  +  .

2

¡

2

= 1.

= 2.

− '  .

ñóò ò ó

+ ( ô ï + õ )ñó ò + ö ( ô ï + õ – ö )ñ = 0.  Particular solution: 0 =  − v .

ñóò ò ó

Page 216

217

2.1. LINEAR EQUATIONS

27.

ñóò ò ó

ï

Particular solution: 28. 29.

0

31.

32.

34. 35.

ñóò ò ó

37.



−

= 

õ 2 )ñ

= 0.

.

+ õ )ñó ò + ö [( ô – ö ) ï

ï

+ (ô

0

= exp 

2

õï

+

(˜  . 2

− 12

2

= 1.

+ 2( ô ï + õ )ñó ò + ( ô 2 ï 2 + 2 ôƒõ ï + ö )ñ = 0. The substitution ‡ = exp  12  2 + '  leads to a constant coefficient linear equation of the form 2.1.2.1: ‡

 + ( ( −  −  2 )‡ = 0.

+ ( ô ï + õ )ñó ò + (  ï 2 +  ï + )ñ = 0. The substitution = ‡ exp( °˜ 2 ), where ° is a root of the quadratic equation 4 ° 2 + 2 !° + ƒ = 0, leads to an equation of the form 2.1.2.108: ‡

 + [(  + 4 ° ) +  ]‡
 + [(ý + 2 ˜° ) + = + 2 ° ]‡ = 0.

ñóò ò ó ñóò ò ó

ï

+ (ô

+ õ )ñó ò + ö (– ö 0

ï 2÷

ïƒ÷

+ô

= exp O −

+1

(

B +1

õ ïƒ÷

+

 

+1

+ ô ( ï 2 – õ 2 )ñó ò – ô ( ï + õ )ñ = 0. Particular solution: 0 =  −  .

+ù

ïƒ÷

–1

)ñ = 0.

.

P

ñóò ò ó

+ ( ô ï 2 + õ )ñó ò + ö ( ô ï Particular solution: 0 = 

ñóò ò ó ñ óò ò ó

ï

+ (ô

2

óò

+ 2 õ )ñ

2 −

+ õ – ö )ñ = 0. .

ï

2

= 

−

+ ( ôƒõ 0

v

ï

–ô





õ 2 )ñ

+

= 0.

.

+ (2 ï 2 + ô )ñó ò + ( ï 4 + ô ï 2 + 2 ï + õ )ñ = 0. The substitution ‡ = exp  13  3  leads to a constant coefficient linear equation of the form 2.1.2.11: ‡ 

 + ‡
 + '‡ = 0.

ñóò ò ó

+ ( ô ï 2 + õ ï )ñ ó ò + (  ï 2 +  ï + )ñ = 0. 1 b . This is a special case of equation 2.1.2.146 with B = 1.

ñ óò ò ó

38.

+ ( ôƒõ ï 2 + õ ï + 2 ô )ñó ò + ô 2 ( õ ï 2 + 1)ñ = 0.  Particular solution: 0 = (  + 1)  −  .

39.

ñóò ò ó

0

=  exp  − 31 

3

− 12 '  2  .

ñóò ò ó

ï

+ (ô

2

+

õï

+ ö )ñó ò + ï ( ôƒõ

Particular solution:

ñóò ò ó

ï

+ (ô

2

+

õï

ñóò ò ó

+ (ô

ï

3

0

− 13

= exp 

+ ö )ñó ò + ( ôƒõ

Particular solution: 41.

2

+ 1]ñ = 0.

2 b . Let ƒ = 0, ý = 3  , = = 2  . Particular solution:

40.

= 0 and 

ñóò ò ó

Particular solution: 36.



+ ( ô ï + õ )ñó ò + ( ö ï + ý )ñ = 0. This is a special case of equation 2.1.2.108 with 

Particular solution: 33.

–ô +

ñóò ò ó

Particular solution: 30.

ï

+ 2 õ )ñó ò + ( ôƒõ

+ (ô

0

= exp 

ï

3

= 

−

+ 2 õ )ñó ò + ( ôƒõ

Particular solution:

0





ï

− 12 –ô

ï

2

+ 3

 3

+

'

2

ï

2

õ^ö

+ 2 ô )ñ = 0.

− (˜  .

ô„ö ï

2

+ õ )ñ = 0.

− ˜(   .

+

õ 2 )ñ

= 0.

.

Page 217

218 42.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

+ (ô

ï

3

õ ï )ñ óò

+

Particular solution: 43.

ñ óò ò ó

+ ( ôƒõ

ï

3

õï

+

ñóò ò ó

+

ô ƒï ÷ ñ óò

óò

+ 2 ô )ñ

2

+ õ )ñ = 0.

2

4

− 12 '  2  .

3

+ 1)ñ = 0.

=  exp  − 14 

0

ï

+ ô 2(õ

−  . 0 = (  + 1) 

Particular solution: 44.

ï

+ 2(2 ô

= 0.

This equation is encountered in the theory of diffusion boundary layer. =L

Solution: 45.

46.

ô ƒï ÷ ñ óò

ñóò ò ó

+

ñóò ò ó

+ 2ô

+

+L

1

õ ïƒ÷ –1 ñ

48.

ïƒ÷ ñóò

ï 2÷

+ ô (ô

ñóò ò ó

+

ô ƒï ÷ ñ óò

+ (õ

ñóò ò ó

+

ô ƒï ÷ ñ óò

– õ (ô

ï 2÷

+ö

ïƒ÷ +ú

ñóò ò ó

ïƒ÷ ñóò

+ 2ô

+ (ô

2

ñóò ò ó

+ (ô

ïƒ÷

ñ óò ò ó

+ (ô

ï ÷

−1

óò

+ 2 õ )ñ

+ ( ôƒõ

ïƒ÷

+

ñóò ò ó

+ ( ôƒõ

ïƒ÷

+ 2õ

ñóò ò ó

+

ïƒ÷ [ô ï

2

)ñ = 0.

   B +1

+1

.

P

)ñ = 0.

ñ óò ò ó

+ (ô

ï ÷

+

ïƒ÷

 

 +

+1

−

0

2

0



–ô



ï ÷

–1

ï )ñ óò

0

–1

ïƒú

+ö

–1

)ñ = 0.

leads to a linear equation of the form 2.1.2.10:

P

ï ÷

+

ïƒ÷

+ ô ( ôƒõ

õ^ö ]„ñ óò

= +( . = .

.

P

õ 2 )ñ

= 0.

.

 = (  + 2)  −  .

– (ô

)ñ = 0.

.

ï ÷

–ô

–1

– ö )ñ = 0.

+ 2 ô )ñó ò + ô 2 ( õ

–1

+1

ôù ïƒ÷

−  . 0 = (  + 1) 

õ ï ú )ñ ó ò

Particular solution:

õ

+

−

= 

+1

B +1

ïƒ÷

+ ( ôƒõ



ïƒú

+þ

+



)¨ = 0.

–1

¡

õ ï 2ú

+ ( ô„ö + õ ) ï +

Particular solution: 55.

õ ï 2ú

exp O

0

õ ïƒ÷

Particular solution: 54.

+

v 0 = 

Particular solution: 53.

–1

+

+ õ )ñó ò + ö ( ô

Particular solution:

ñóò ò ó

ï 2÷

=

Particular solution:

52.

ïƒ÷

= exp O

0

¨ 

 + ( ' 2 + + (˜ +

51.

–1

+1

The substitution ˆ =   +1 leads to a linear equation of the form 2.1.2.108: (B + 1) 2 ˆ Šª

Š + (B + 1)( ˆ + B ) Š
+ ( 'ˆ + ( ) = 0.

The substitution ¨

50.

ïƒ÷

+ù

=  exp O −

0

Particular solution: 49.

= 0.

For B = −1, we obtain the Euler equation 2.1.2.123. For B ≠ −1, the substitution H =   ‘'‘ ‘ leads to an equation of the form 2.1.2.108: (B + 1) 2 H 

+ (B + 1)(!H + B )
+  = 0.

Particular solution: 47.

  +1  . B + 1 P3M

X exp O −

2

–1

+

+ 1)ñ = 0.

ïƒ÷

– ïƒ÷ ( ô

õï ú

–1

ï

+

õ ïƒ÷

–1

–ô

2

ï )ñ

= 0.

+ õ )ñ = 0.

)ñ = 0.

Page 218

219

2.1. LINEAR EQUATIONS

56. 57.

ñóò ò ó

+ (ô

ïƒ÷

+

õ ïƒú )ñ óò

+ ( ôù

ñ óò ò ó

+ (ô

ï ÷

+

õ ï ú )ñ ó ò

+ [ ô (ù + 1) ï

ñóò ò ó

ïƒ÷

+ (ô

ñóò ò ó

õ ïƒú )ñ óò

+

ïƒ÷

+ (ô

ñ óò ò ó

−

õ ïƒú )ñ óò

+

ï ÷

+ (ô

+ ö (ô

v 0 = 

Particular solution: 60.

=  exp O −

0

Particular solution: 59.

–1

õÿþ ïƒú

+

)ñ = 0.

–1

Integrating yields first-order linear equation: "
 + (   + ' + ) = L .

Particular solution: 58.

ïƒ÷

óò

+ ö )ñ

Particular solution:



 

õ ïƒú

+1

ú

–1

 +

+1

+ 1) ï

+ õ (þ



−

+1

¡

]ñ = 0. .

P

– ö )ñ = 0.

.

ïƒú +÷

+ ( ôƒõ

= exp O −

0

–1

B +1 +

=  exp O −

0

õï ú

+

+ [ ôƒõ

ïƒ÷

÷

 ¡

+ õ (þ +1

ï ú +÷ 

B +1

 

 + + +1

+ 1) ïƒú +1

–1

ïƒ÷

–ô

–1

]ñ = 0.

.

P

õ^ö ï ú − (˜

+ ôù

P

ï ÷

–1

)ñ = 0.

.

2.1.2-3. Equations of the form (  +  )

 +  ( )
 + ( ) = 0. 61.

62.

ï ñóò ò ó

+

1 2

ñóò

=

ï ñóò ò ó

+

ôñóò

= 0.

cos £ 4  + L 2 sin £ 4  if  > 0, L 1 cosh £ 4|  | + L 2 sinh £ 4|  | if   < 0.

L

™

Solution: +

ôñ

+

1

õÿñ

= 0.

1 b . The solution is expressed in terms of Bessel functions: 1− 2

=

 V L

1

Á± 2 £ '  + L 2 n Á± 2 £ '  W ,

ø

@ = |1 −  |.

where

2 b . For  = 12 (2B + 1), where B = 0, 1,  , the solution is:



=

63.

ï ñóò ò ó

+

ôñóò

+




  if  > 0, M  cos £ 4  + L 2 M  sin £ 4     M  M L 1 M  cosh 4|  | + L 2 M  sinh 4|  | if   < 0.  ü  ü M M L




õï ñ

1

= 0.

1 b . The solution is expressed in terms of Bessel functions: 1− 2

=

 V L

1

Á £ "    + L 2 n Á  £    W ,

ø

where

@ = 12 |1 −  |.

2 b . For  = 2B , where B = 1, 2,  , the solution is: =

64.

ï ñóò ò ó

+

ôñóò







L

1

O

1



 3P

cos   £   + L

2

O



1

M

 3P

sin   £  

M1   L 1O cosh   £ −   + L 2 O sinh   £ −   M M  3P  3  P M M

+ (õ

ï

1 M

M

+ ö )ñ = 0.

This is a special case of equation 2.1.2.108 with 

2

= 1 and 

1

=

2

if  > 0, if  < 0.

= 0.

Page 219

220 65.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñóò ò ó

+ ùñó ò + õ ï 1–2÷ ñ = 0. For B = 1, this is the Euler equation 2.1.2.123. For B ≠ 1, the solution is: =

66. 67.

70.

1

sin ”

L

1

exp ”

B −1 £ −

1−

 

B −1



1−

+L

•  •

+ (1 – 3ù )ñ„ó ò – ô 2 ù 2 ï 2÷ –1 ñ = 0. Solution: = L 1 (   + 1) exp(−   ) + L

2

cos ” exp ”

£ 

 B −1 −£ −

1−

if  > 0,

•

1−



+L

2

2 (−

  + 1) exp(  ).

B −1





if  < 0.

•

+ ôñó ò + õ ïƒ÷ ñ = 0. If B = −1 and  = 0, we have the Euler equation 2.1.2.123. If B ≠ −1 and  ≠ 0, the solution is expressed in terms of Bessel functions:

ï ñóò ò ó

ï ñ óò ò ó

1− 2



¬L

1

+ ôñ ó ò + õ ï ÷

Particular solution: 69.






£ 

L

ï ñóò ò ó

= 68.







ø

ÁhO

2£    B +1

(– õ ï ÷ 0

+1

+

ô

P

+L

2

n Á…O

2£    B +1

+1 2

P

® ,

where

@ =

|1 −  | . B +1

+ ù )ñ = 0.

= exp O −

ï ñóò ò ó

+1 2

+ ô ï ñó ò + ôñ = 0.  Particular solution: 0 =   −  .

   B +1

+1

.

P

ï ñóò ò ó

+ ( õ – ï )ñó ò – ôñ = 0. The degenerate hypergeometric equation. 1 b . If  ≠ 0, −1, −2, −3,  , Kummer’s series is a particular solution:

½ ( )    ` ( ,  ;  ) = 1 + ÷ ,  =1 (  )  , ! where (  )  =  (  + 1)  (  + , − 1), (  )0 = 1. If  >  > 0, this solution can be written in terms of a definite integral:

1  ( ) ` ( ,  ;  ) = w X  g a  −1 (1 − a )  −  ( ) (  −  ) 0 w w ‘ −g −1 where ( H ) = X ÷  a a is the gamma function. w M 0 If  is not an integer, then the general solution has the form:

=L

1

` ( ,  ;  ) + L 2 

1−

 ` (

−1

M

a,

−  + 1, 2 −  ;  ).

Table 14 gives some special cases where ` is expressed in terms of simpler functions. The function ` possesses the properties:

 ` (  ,  ;  ) =  ` (  −  ,  ; − );

 ( ) M  ` (  ,  ;  ) =  ` (  + B ,  + B ;  ).  ( ) M

The following asymptotic relations hold:

` ( ,  ;  ) = w ` ( ,  ;  ) =

w

w

( )   − 1 S    1 + [­O ( ) Q | | P ( ) 1 w (− )−  1 + [­O S ( −  ) | | P Q

if ED

+F ,

if ED

−F .

Page 220

221

2.1. LINEAR EQUATIONS

TABLE 14 Special cases of Kummer’s function ` (  ,  ; H )



H 







1

2

2



 +1

−

1 2

3 2

−B

1 2



−B

3 2



`

Conventional notation

 

1 

 sinh 



Incomplete gamma function

 −  = (  ,  ) £

2

−

 2

= ( ,  ) = X

2 2

2

0

−



ga

−1

M

a

Error function  2 erf  = £ X exp(−a 2 ) a

erf 

1 − B ! O− 5 2  ( ) (2B )! 2 P B ! 1 − O− 5 2  +1 ( ) (2B + 1)! 2 P

2





M

0

Hermite polynomials

2  2 5  ( ) = (−1)   M   −  ,  M  B = 0, 1, 2, 3, Laguerre polynomials

−B

@ +

1 2

B +1





2@ + 1

2

2B + 2

     −9 M    −   + 9  , B !  ƒ =  M − 1, (  )  =  (  + 1)  (  + B − 1)

0 ( 9 ) (

B ! ( −1) 0  ( ) ( ) 

w

(1 + @ ) 

2

OB +

w



 O

−

Á

2P

3    O 2P 2P



®Á (

− − 12

®

) + 12 (

)=

Modified Bessel functions ®Á ( )

 )

2 b . The following function is a solution of the degenerate hypergeometric equation:

i ( ,  ;  ) =

w

(1 −  ) w ` ( ,  ;  ) + (  −  + 1)

w

(  − 1)  ( )

w

1−

 ` (

−  + 1, 2 −  ;  ).

Calculate the limit as ‹D„B (B is an integer) to obtain

i ( , B ;  ) = ½

(−1)  −1 ™ ` ( , B +1;  ) ln  B ! ( − B )

w

+ ÷

Ÿ =0

+

Ÿ

( )Ÿ V j (  + & ) − j (1 + & ) − j (1 + B + & )W  (B + 1) Ÿ &!

 (B − 1)! ½ ( ) Ÿ

w

−1 =0

¢

Ÿ ( − B )Ÿ  − , (1 − B ) Ÿ & !

where B = 0, 1, 2,  (the last sum is omitted for B = 0), j ( H ) = [ln ( H )]
‘ is the logarithmic w derivative of the gamma function:

j (1) = −= ,

j ( B ) = −= +

½ 

−1

,

−1

,

= = 0.5772  is the Euler constant.

=1

Page 221

222

SECOND-ORDER DIFFERENTIAL EQUATIONS If  is a negative number, then the function i can be expressed in terms of the one with positive second argument using the relation

 i (

1−

i ( ,  ;  ) = 

−  + 1, 2 −  ;  ),

which holds for any value of  . 3 b . For  ≠ 0, −1, −2, −3,  , the general solution of the degenerate hypergeometric equation can be written in the form: =L

1

` (  ,  ;  ) + L 2 i (  ,  ;  ),

while for  = 0, −1, −2, −3,  , it can be represented as: 1−

=

 V L 1 ` (

−  + 1, 2 −  ;  ) + L

2

i (  −  + 1, 2 −  ;  )W .

The functions ` and i are described in Subsection S.2.7 in more detail; see also the books by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 1). 71.

ï ñóò ò ó

ï

+ õ )ñó ò + ö [( ô – ö ) ï + õ ]ñ = 0.

+ (ô

Particular solution: 72.

ï ñ óò ò ó

ï

+ (2 ô

ï ñóò ò ó

óò

+ õ )ñ =

Solution: 73.

0

+ ô (ô

  −  (L −

™

−

=

ï

+ õ )ñ = 0.

1−  ) if  ≠ 1, ln |  |) if  = 1. 2

+L 1+L

+ [( ô + õ ) ï + ù + þ

Here, B and

.

1

(L



v



2

]ñ„ó ò + ( ôƒõ

ï

ôù

+

are positive integers;  ≠  or B ≠

¡

 + = L 1  M +  M

ï ñóò ò ó

−1

74.

+ ( ô ï + õ )ñó ò + ( ö ï + ý )ñ = 0. This is a special case of equation 2.1.2.108.

75.

ï ñóò ò ó

ï

Particular solution: 76.

ï ñóò ò ó

ï

– (2 ô

ï ñóò ò ó

+ (ô

0

 =  L Q

ï

78.

ï ñóò ò ó

– (2 ô

ï

Solution:

2

79.

ï ñóò ò ó

+ ( ôƒõ

ï

0

ï ñ óò ò ó

öï

3

ô 2ï

+

(– ö

ï

2

= exp 

+ 1)ñó ò +

+ ô )ñ = 0.

õ ï 3ñ

+

− 12

ô ï

2

+

(˜  . 2

cos 

õ

2

+

õ

= 0.

– 5)ñó ò + 2 ô 2 ( õ – 2) ï

+ ( ô ï 2 + õ ï )ñ Particular solution:

0

óò

= ( 

1 2

2

 2£   S .

+ 1) = 0.

= L 1 exp V 12   + £  2 −    2 W + L

Particular solution: 80.

ï

£   +L 1 1 sin  2 

Particular solution:

.

+ ô )ñ = 0.

2

+ õ )ñó ò +

= 0.

= exp  − 12 ' 2  .

+ 1)ñ„ó ò + ( õ

Solution: 77.

õ ï 2(õ ï

+ 1)ñó ò –

õÿþ )ñ

¡  −1 V  −  (  −  ) W + L 2  −   M V  −+  (  −  ) W .  −1  M

−1

−

Solution:

– (ô

+

3

ñ

2

exp V

1 2

  − £  2 −    2W .

= 0. 2

+ 1) exp(−  ).

– [ ô„ö ï 2 + ( ô + v . 0 =  

õ^ö

+ ö 2 )ï +

õ

+ 2 ö ]ñ = 0.

Page 222

223

2.1. LINEAR EQUATIONS

81. 82. 83.

ï ñóò ò ó

+ ( ô ï 2 + õ ï + 2)ñó ò + õÿñ = 0. Particular solution: 0 =  +   .

ï ñóò ò ó

+ ( ô ï 2 + õ ï + ö )ñó ò + (2 ô ï + õ )ñ = 0. Integrating, we obtain a first-order linear equation:  "
 + ( 

ï ñóò ò ó

ï

+ (ô

2

+

õï

Particular solution: 84.

ï

+ ö )ñó ò + ( ö – 1)(ô 0

ï ñóò ò ó

=

v.

1−

2

+ ' + ( − 1) = L .

+ õ )ñ = 0.

+ ( ô ï 2 + õ ï + ö )ñó ò + (
ï 2 + ( ï + = )ñ = 0. 1 b . Let # = , , $ = , (  − , ), L  = (, , where , is an arbitrary number. Particular solution: 0 =  −  . 2 b . Let # =  (  + , ), $ =  ( ( + 1) − , (  + , ), L = − (, . Particular solution: 0 = exp  − 12  2 + ! ,  . 3 b . Let # =  (  + , ), $ = 2  − , − , 2 , L =  ( ( − 1) + , ( ( − 2). Particular solution: 0 =  1− v exp  − 12  2 + ,!  . 4 b . Let # = − , , $ =  ( ( − 1) − , (   + , ), L Particular solution: 0 =  1− v   .

85.

86.

ï ñóò ò ó

+ ( ô ï 2 + õ ï + 2)ñó ò + ( ö ï 2 + ý ï + õ )ñ = 0. ¨ 

 + (  +  )2 ¨ 
+ The substitution ¨ =  leads to a linear equation of the form 2.1.2.108: I ¨ ( (˜ + −  ) = 0.

ï ñóò ò ó

M

ï

+ (ô

3

+ õ )ñó ò + ô ( õ – 1) ï

Particular solution: 87.

ï ñóò ò ó

+ ï (ô

ï

2

0

89. 90.

0

ï ñóò ò ó ï ñóò ò ó

+ ( ôƒõ ï 3 + õ ï Particular solution:

ï ñóò ò ó

+ (ô

ï

3

+

õï

2

ï ñ óò ò ó

+

ô ï ÷ ñ óò

ï ñ óò ò ó

+ (ô

ï ÷

+ö

2

0

+ ( ôƒõ

+ 2)ñ

Particular solution: 93.

ï ñóò ò ó

+ 0

Particular solution: 92.

ï

2

ñ

= 0.

+ õ )ñ = 0.

=  exp  − 13 

3

− '  .

+ ( ô ï 3 + õ ï 2 + 2)ñó ò + õ ï ñ = 0. Particular solution: 0 =  +   .

Particular solution: 91.

=

2

.

1−

+ õ )ñó ò + (3 ô

Particular solution: 88.

=  ( ( − 1) + , ( ( − 2).

– 1)ñó ò + ô 2 õ  = (  + 1)  −  .

ï

=

ï ÷

0

óò

ô ï

0

+ ý )ñó ò + ( ý – 1)(ô 1−

–ô

= 

+ô =

ï 3ñ

x

ï

2

+

õï

+ ö )ñ = 0.

.

ï ÷ −





ï ÷ –1 ñ −1

= 0.

–1

–

õ 2ï

+ 2 õ )ñ = 0.

. = 0.

.

+ ( ïƒ÷ + 1 – ù )ñó ò + õ ï 2÷ –1 ñ = 0. 1 b . For  ≠ 14 , the solution has the form: = L 1 exp  ý 1    + L are roots of the quadratic equation: B 2 ý 2 + B/ý +  = 0. 2 b . For  = 14 , the solution has the form:

= (L

1

+L

2

2 exp

  ) exp  − 12 B

 ý 2    . Here, ý −1

1

and ý

2

  .

Page 223

224 94.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñóò ò ó

+ (ô

ïƒ÷

Particular solution: 95.

ï ñóò ò ó

+ (ô

ïƒ÷

0

ï ñóò ò ó

+ (ô

ïƒ÷

0

ï ñ óò ò ó

+ (ô

ï ÷

ï ñóò ò ó

+ ( ôƒõ

óò

+ õ )ñ

ï ñóò ò ó

+ (ô

ïƒ÷

ïƒ÷

1−

=

–1

.

õ

+

ï ÷

+ ö (ô

ï

–ö

−v  . 0 = 

ñ

= 0.

–1

ñ

= 0.

+ õ )ñ = 0.

– 3ù + 1)ñ„ó ò + ô

2

– ù )ï

ù (õ

÷ ñ

2 –1

   + 1) exp(−  ). 0 = (

Particular solution: 99.

exp(−   B ).



   B ). 0 = exp(− 

Particular solution: 98.

=

= 0.

+ õ )ñó ò + ô ( õ + ù – 1) ïw÷

Particular solution: 97.

1−

+ õ )ñó ò + ô ( õ – 1) ïƒ÷

Particular solution: 96.

ôù ïƒ÷ –1 ñ

+ õ )ñó ò +

ï 2÷

+ õ )ñó ò + ( ö

–1

+

ý ïƒ÷

–1

= 0.

)ñ = 0.

This is a special case of equation 2.1.2.146 with = = 0. 100.

ï ñóò ò ó

+ (ô

ïƒ÷

õ ïƒ÷

+

–1

ï ñ óò ò ó

+ (ô

ï ÷

õ ï )ñ ó ò

+

103.

ï ñóò ò ó

+ ( ôƒõ

ïƒ÷

õ ïƒ÷

+

ï

+ô

–1

+

ôù ï ÷

– 1)ñó ò + ô

Particular solution:

ï ñóò ò ó

+ ö )ñó ò + ( ö – 1)(ô

ïƒ÷

õ ïƒú

+

Particular solution: 104.

ï ÷

+ ( ôƒõ

. 0 = (  + 1) 

+ (ô

ï ñóò ò ó

+ ( ôƒõ

ïƒ÷ +ú

0

+ ôù

ï

1

ï

=

−

1−

ïƒ÷

+ ô 0 )ñó ò ò ó + ( õ

v.

+

+ ö )ñó ò +

Particular solution: 106. ( ô

–1

õ ïƒú

2

– õ )ñ = 0.

õ ƒï ÷ ñ

ïw÷

–1

+

= 0.

õ ïƒú

+ 1 – 2ù )ñó ò + ô

   + 1) exp(−  ). 0 = (

Particular solution: 105. ( ï + ô )ñ„ó ò ò ó + ( õ

= 0.

   B ). 0 =  exp(− 

Particular solution: 102.

õ ïƒ÷ –2 ñ

0 =  +  .

Particular solution: 101.

+ 2)ñó ò +

0

1

ï

õÿñ

õ 0 )ñ óò

2

)ñ = 0.

õÿù ï 2÷ +ú

–1

ñ

= 0.

= 0.

' + ( − 1  .  + ?M P

= exp O − X +

–1

–þ

õ 1ñ

= 0.

= 1, 2, 3,  , a polynomial of order in  is a particular solution of the equation, ¡ ½+ 1  which can be represented as: 0 = O− ,ž + ®´ −+ −1 [( 1  +  0)c 2 +  0 c ] .   + , where

If

¡

 Á



+1

=0

with @ ≠ −1. c = M , ´®  Á =  @ +1 M À ï ï 2

107. ( ô

+ õ )ñ„ó ò ò ó + s( ö

Particular solution:

 1P

+ ý )ñó ò – s [( ô + ö ) ï + 0

=



õ

+ ý ]ñ = 0.

.

Page 224

225

2.1. LINEAR EQUATIONS

TABLE 15 Solutions of equation 2.1.2.108 for different values of the determining parameters

  −“ =   ¨ ( H ), where H =

Solution:

,

Constraints

 2 ≠ 0, 2  1 ≠ 4 0 

2

 2 = 0,  1≠0  2 ≠ 0,  = 4 0  2 1

Œ

£ c − 2 2  0 −  1 −

2

 2 =  1 = 0,  0≠0

−



2



2

−

1



2

2 2 , + 



2 1

1

−



2

y

2

2 2 , + 



−

2

Notation: c

1



1



2 2

2 1 − 4 0 2,

=

Œ

Parameters

 = $ ( , ) (2  2 , +  1 ),  = (  2  1 −  1  2 )  −2 2

 = $ ( , ) (2  1), ý = −  (2  )

( ,  ; H )

  , 12 ; ý…H 2 

y

1

: H ÁJ2 / Á ý £ H 

2

 21 − 4  0  4 0  2

1

2



−

1

1

¨ “

 

@ = 1 − (2  2 , +  1 )  −1 2 , ý = 2 $ (, ) ü 1J 2 2  ý = O 0 3 2P

: H 1 J 2 1 J 3  ý…H 3 J 2  see also 2.1.2.12

$ (, ) =  2 , 2 +  1 , + 

2

0

108. ( ô 2 ï + õ 2 )ñ ó ò ò ó + ( ô 1 ï + õ 1 )ñ ó ò + ( ô 0 ï + õ 0 )ñ = 0. Let the function (  ,  ;  ) be an arbitrary solution of the degenerate hypergeometric equation :  

 + (  −  )
 y −  = 0 (see 2.1.2.70), and the function Á ( ) be an arbitrary solution of 


the Bessel equation  2  +   + ( 2 − @ 2 ) = 0 (see 2.1.2.126). The results of solving the original equation are presented in Table 15. 109. ( ï + )ñ

óò ò ó

+ (ô

ï ÷

õï ú

+

Particular solution:

0

+ ö )ñ

óò

= exp O − X

+ ( ôù ï ÷ –1 + õÿþ ï ú –1 )ñ = 0.   + ' + + ( − 1  .  +Œ M P

2.1.2-4. Equations of the form  2 

 +  ( )
 + ( ) = 0. 110.

ï 2 ñ óò ò ó

+ ôñ = 0. g This is a special case of equation 2.1.2.123. The substitution  =  leads to a constant


coefficient linear equation: g g − g +  = 0.

111.

ï 2 ñ óò ò ó

+ ( ô ï + õ )ñ = 0. This is a special case of equation 2.1.2.132.

112.

ï 2 ñ óò ò ó

+ [ô

Solution: 113.

ï 2 ñ óò ò ó ï 2 ñ óò ò ó

– ù (ù + 1)]ñ = 0, ù = 0, 1, 2, uuu L cos    + L 2 sin    +1 = ( 3 c )  O 1 , where c 2  −1

– (ô

P



– [ô 2 ï

Solution: 114.

ï

2 2

2

+ ù (ù + 1)]ñ = 0,

ù

= 0, 1, 2,

uuu

= M .

M



  L 1   + L 2  −  +1 3   = ( c ) O , where c = M .  2  −1 P  M

ï

2 2

+ 2 ôƒõ

Particular solution:

ï

0

+

õ

2

– õ )ñ = 0.

 =    .

Page 225

226 115.

116.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó

+ ( ô ï 2 + õ ï + ö )ñ = 0. The substitution =   ‡ , where Œ is a root of the quadratic equation Œ 2 − Œ + ( = 0, leads to an equation of the form 2.1.2.108: ‡

 + 2 Œ ‡
 + (  +  )‡ = 0. For  = − 14 ,  = , , and ( = 14 − 2 , the original equation is referred to as Whittaker’s ¡ equation.

ï 2 ñ óò ò ó

– 

ô ï

3



5 16

+

Particular solution: 117.

ï 2 ñ óò ò ó

– [ô

ï

2 4

ñ

= 0. 0

J exp  2 £ r 3J 2  . 3  

−1 4

=

+ ô (2 õ – 1)ï

0 =  

exp  − 21  2  .

−

Particular solution:

+ õ ( õ + 1)]ñ = 0.

2

118.

+ ( ô ï ÷ + õ )ñ = 0. This is a special case of equation 2.1.2.132.

119.

ï 2 ñ óò ò ó

ï 2 ñ óò ò ó

– [ô

ï ÷

2 2

+ ô (2 õ + ù – 1) ïw÷ + õ ( õ – 1)]ñ = 0. exp(   B ).



0

=

õ ï 2÷

+

Particular solution: 120.

+ ( ô ï 2÷ + õ ï ÷ + ö )ñ = 0. This is a special case of equation 2.1.2.146.

121.

ï 2 ñ óò ò ó

ï 2 ñ óò ò ó

+ 

ô ï 3÷

+

1 4

1 4

–

ù

2



ñ

The transformation ˆ =   +  , ¨ ¨2Šª

Š + ( B )−2 ˆ ¨ = 0.

= 0.

122.

ï 2 ñ óò ò ó

+ Vô

ï 2÷ ( õ ï ÷

+ ö )ú

1 4

+

–

1 4

The transformation ˆ = '  + ( , ¨ ¨2Šª

Š +  ( 'B )−2 ˆ +2¨ = 0. 123.



= 

ù

2

W

ñ

−1 2

leads to an equation of the form 2.1.2.7:

= 0.



= 

−1 2

leads to an equation of the form 2.1.2.7:

ï 2 ñ óò ò ó

+ ô ï ñó ò + õÿñ = 0. The Euler equation. Solution:

=











| |

1− 2

| |

1− 2

| |

1− 2





ÝL 1 | |  + L 2 | |−  

if (1 −  )2 > 4  ,

(L

if (1 −  )2 = 4  ,

 V L

1

+L

1

sin(“ ln | |) + L

2

ln | |) 2

cos(“ ln | |)W

if (1 −  )2 < 4  ,

where “ = 12 |(1 −  )2 − 4  |1 J 2 . 124.

125.

ï 2 ñ óò ò ó

+ ï ñó ò + V ï 2 –  ù + 12  Wñ = 0, ù = 0, 1, 2, uuu This is a special case of equation 2.1.2.126.  1 sin  cos  Solution: =   +1 J 2 O OL 1 +L 2 S . M

ï 2 ñ óò ò ó

2

Q 

3P

M





P

+ ï ñó ò – V ï 2 +  ù + 12  Wñ = 0, ù = 0, 1, 2, This is a special case of equation 2.1.2.127.   1  −  +1 J 2 Solution: = O OL 1 + L 2 S . M

Q 

M

2

3P



uuu

åP

Page 226

227

2.1. LINEAR EQUATIONS

126.

ï 2 ñ óò ò ó

+ ï ñó ò + ( ï 2 – b The Bessel equation.

2

)ñ = 0.

1 b . Let @ be an arbitrary noninteger. Then the general solution is given by: =L

ø

1

n Á ( ), Á ( ) + L 2

(1)

where ø Á ( ) and n Á ( ) are the Bessel functions of the first and second kind:

Á ½ (−1)  ( 2) +2  ø Á ( ) cos  @ − ø − Á ( ) . Áø ( ) = ÷ , n Á ( ) = , ! ( @ + , + 1) sin @  =0 w  : Solution (1) is denoted by = Á ( ) which is referred to as the cylindrical function.

(2)

The cylindrical functions possess the following properties:

:

2@

:

Á ( ) =  [ Á

:

 ) + Á +1 ( )], : : M [ − Á Á ( )] = − − Á Á 

−1 (

: : M [ Á Á ( )] =  Á Á −1 ( ), +1 (  ).  M ø Á ( ) and n Á ( ) can be expressed M in terms of definite integrals (with  > 0): The functions

 ø 

Á ( ) = X

n Á ( ) = X

z

0

9

9

cos( sin − @

z

9

9

sin( sin − @

0

) )

M M

9

− sin @ X ÷ exp(− sinh a − @!a ) a ,  0 M

9

− X ÷ ( 0

Á˜g + − ˜Á g cos @ ) −   

sinh



g a. M

2 b . In the case @ = B + 12 , where B = 0, 1, 2,  , the Bessel functions are expressed in terms of elementary functions:

ø 

+ 12 (

 )=



2  

+ 12

 sin  2  O− M , ø −  − 1 ( ) =  2  3P   M n  + 12 ( ) = (−1)  +1 ø −  − 12 ( ). 1

+ 12

1

O 

M

M

 cos  ,  3P

3 b . Let @ = B be an arbitrary integer. The following relations hold:

n −  ( ) = (−1)  n  ( ). The solution is given by formula (1) in which the function ø ( ) is obtained by substituting @ = B into formula (2), while n  ( ) is found by taking the limit as @*D B and for nonnegative B

ø

−

 ( ) = (−1)  ø  ( ),

becomes

2

n  ( ) = −



ø

1 ½



where j (1) = −z , j (B ) = −z +



( ) ln



2

÷ (−1)  O 7 

=0 −1 =1

,

−1

−



2P



 1 ½ 

−1

 +2

=0

(B − , − 1)! 2  O , ! 2P

−2



 j ( , + 1) + j (B + , + 1) , , ! (B + , )!

, z = 0.5772  is the Euler constant, j ( ) = [ln ( )]


is the logarithmic derivative of the gamma function.

w

For nonnegative integer B and large  , we can write

 ø 2  ( ) = (−1)  (cos  + sin  ) + [ ( −2 ),  £  ø 2  +1 ( ) = (−1)  +1 (cos  − sin  ) + [ ( −2 ).  The function ! ø  ( ) can be expressed in terms of a definite integral: 1 z B = 0, 1, 2,  ø  ( ) = X 0 cos( sin a − B/a ) M a ;  The Bessel functions are described in Subsection S.2.5 in more detail; see also the books £

by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 2).

Page 227

228 127.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó

+ ï ñó ò – ( ï 2 + b 2 )ñ = 0. The modified Bessel equation. It can be reduced to equation 2.1.2.126 by means of the substitution  = ™  ¯ (™ 2 = −1). Solution: = L 1 ®Á ( ) + L 2 ; Á ( ), where ®Á ( ) and ; Á ( ) are the modified Bessel functions of the first and second kind:

®Á

½

( ) = ÷



=0

Á ( 2)2  + , , ! ( @ + , + 1)

; Á (

w

)=

 ) − ®Á ( 2 ® − Á (sin  @

)

.

The modified Bessel function ®Á ( ) can be expressed in terms of the Bessel function:

Á (  z © J 2 ), ™ 2 = −1. The case @ = B + 12 , where B = 0, 1, 2,  , is given in 2.1.2.125. If @ = B is a nonnegative integer, we have  −1   2 + −  (B − − 1)! 1½  +1 (  ) = (−1) ® (  ) ln + (−1) + O ¡ ;|  2 2+ 2P ! =0 ¡ ½   +2 + j (B + + 1) + j ( + 1) 1 + (−1)  ÷ O ¡ ¡ , 2 ! (B + )! + =0 2 P ¡ ¡ where j ( H ) is the logarithmic derivative of the gamma function (see 2.1.2.126, Item 3 b ); for B = 0, the first sum is omitted. As þD + F , the leading terms of the asymptotic expansion are:  £   ®Á ( ) ¬ £ , ; Á ( ) ¬ £   − . 2  2

®Á

( ) = 

−

z

Á© J

2

ø



The modified Bessel functions are described in Subsection S.2.6 in more detail; see also the books by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 2). 128. 129.

ï 2 ñ óò ò ó

+ 2ï Solution: 

ï 2 ñ óò ò ó

– 2ô

Solution: 130.

ï 2 ñ óò ò ó

– 2ô

Solution: 131.

132.

– ( ô 2 ï 2 + 2)ñ = 0.   = L 1 (  − 1)   + L 2 (  + 1)  −  .

ñóò 2

ï ñóò =

™

+ [ õ 2 ï 2 + ô ( ô + 1)]ñ = 0. | |  ( L 1 sin ' + L 2 cos ' ) if  ≠ 0, L 1 | |  + L 2 | |  +1 if  = 0.

ï ñ óò =

™

+ [– õ

ï

2 2

+ ô ( ô + 1)]ñ = 0.

  | |  ( L 1   + L 2  −  ) if  ≠ 0, L 1 | |  + L 2 | |  +1 if  = 0.

+ ï ñó ò + ( ô ï 2 + õ ï + ö )ñ = 0. The substitution =   ‡ , where , is a root of the quadratic equation , 2 + ( Œ − 1) , + ( = 0, leads to an equation of the form 2.1.2.108: ‡

 + ( Œ + 2 , )‡
 + (  +  )‡ = 0.

ï 2 ñ óò ò ó

+ ô ï ñ ó ò + ( õ ï ÷ + ö )ñ = 0, ù ≠ 0. The case  = 0 corresponds to the Euler equation 2.1.2.123. For  ≠ 0, the solution is:   1−  2 2 =  2 L 1 ø Á O £ " 2 + L 2 n Á O £  2 S ,

ï 2 ñ óò ò ó

Q

B

P

B

P

where @ =  1 (1 −  )2 − 4 ( ; ø Á ( H ) and n Á ( H ) are the Bessel functions of the first and second ü kind.

Page 228

229

2.1. LINEAR EQUATIONS

133.

134.

135.

136. 137.

ï 2 ñ óò ò ó

+ ô ï ñó ò + ïƒ÷ ( õ ïƒ÷ + ö )ñ = 0. The substitution ˆ =   leads to an equation of the form 2.1.2.108: B ( 'ˆ + ( ) = 0.

+ ( ô ï + õ )ñó ò + ö^ñ = 0. ‘ The transformation  = H −1 , = H   ¨ , where , is a root of the quadratic equation , 2 + (1 −  ) , + ( = 0, leads to an equation of the form 2.1.2.108:

H ¨ '‘

‘ + [(2 −  )H + 2 , + 2 −  ]¨ ‘
+ [(1 −  )H + 2 , + 2 −  − , ]¨ = 0.

ï 2 ñ óò ò ó

+ ô ï 2 ñó ò + ( õ ï 2 + ö ï + ý )ñ = 0. The substitution = ¨ exp  − 12   leads to a linear equation of the form 2.1.2.115:

1 2 ¨2 2 2  + [( 4  +  ) + (˜ + ]¨ = 0. 

M

ï 2 ñ óò ò ó

+ (ô ï

2

+ õ )ñó ò + ö [( ô – ö ) ï  Particular solution: 0 =  − v .

ï 2 ñ óò ò ó

+ (ô

ï

2

+

õ ï )ñ óò

ï 2 ñ óò ò ó

+ (ô

ï

2

+

140. 141.

2

=

õ ï )ñ óò

−

   −

.

+ [ü (ô – ü )ï

− −   .  0 = 

ô 2 ï 2 ñ óò ò ó

+ õ ]ñ = 0.

– õÿñ = 0.

0

Particular solution: 139.

ˆ 
Šª
Š + B (B − 1 +  ) Š
+

ï 2 ñ óò ò ó

Particular solution: 138.

2

2

+ ( ôù +

– 2 üIù )ï + ù ( õ – ù – 1)]ñ = 0.

õü

+ ( ô 1 ï 2 + õ 1 ï )ñ ó ò + ( ô 0 ï 2 + õ 0 ï + ö 0 )ñ = 0. The substitution =   ¨ , where , is a root of the quadratic equation  2 , 2 +(  1 −  2 ) , + ( 0 = 0, leads to an equation of the form 2.1.2.108:  2  ¨2

 +(  1  +2  2 , +  1 )¨2
+(  0  +  1 , +  0 )¨ = 0.

ï 2 ñ óò ò ó

+ [ ô ï 2 + ( ôƒõ – 1) ï + õ ]ñ„ó ò + ô 2 õ  Particular solution: 0 = (  + 1)  −  .

ï ñ

= 0.

ï 2 ñ óò ò ó

– 2 ï ( ï 2 – ô )ñó ò + {2ù ï 2 + [(–1)÷ – 1] ô }ñ = 0. For B = 0, 1, 2,  , particular solutions are polynomials: 0

.

=  ( ), where

.

0(

 ) = 1,

.

1(

 )= ,

.

2(

 ) = 2 2 − 1 − 2  ,

.

3(

 ) = 2 3 − (3 + 2  ) , 

The polynomials contain only even powers of  for even B and only odd powers of  for odd B . 142.

ï 2 ñ óò ò ó

+ ï ( ô ï 2 + õ ï + ö )ñó ò + (
ï 3 + ( ï 2 + = ï + F )ñ = 0. 1 b . The substitution =   ¨ , where , is a root of the quadratic equation , leads to an equation of the form 2.1.2.84 (see also 2.1.2.80–2.1.2.83):

 ¨ 

 + ( 

2

+ ' + ( + 2 , )¨ 
+ [ #*

2

2

+( ( −1) , + c = 0,

+ ( $ + , ) + L + , ] = 0.

2 b . LetÀ ° and & be arbitrary parameters. For # = & , $ = !° + '& − & 2 , L = ˜° + (˜& − 2&° , c = ° ( ( − ° − 1), a particular solution is: − −Ÿ  À .  0 =  For # =  (  − & ), $ =  ( ( − ° + 1) + & (  − & ), L = ˜° + (˜& − 2&° , c = ° ( ( − ° − 1), a particular solution is: 0 =  − exp(− 21  2 − & ). 143.

ï 2 ñ óò ò ó

+

ô ƒï ÷ ñ óò

– ( ôƒõ

Particular solution:

ïƒ÷

+

ô„ö ïƒ÷

v  0 =   

–1

+

õ 2ï

2

+ 2 õ^ö

ï

+

ö

2

– ö )ñ = 0.

.

Page 229

230 144.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó

+

ô ƒï ÷ ñ óò

Particular solution: 145.

ï 2 ñ óò ò ó

+ ï (ô

ï ÷

147.

148.

óò

+ õ )ñ

õ 2 ï 4ú

ï ÷

+ õ (ô

+2

2



+1

¡

ïƒ÷

+ ôþ

 2+

+1

–1

–þ

2

–þ

)ñ = 0.

.

P

– 1)ñ = 0.

.

−

=

0

ï 2 ñ óò ò ó

–

=  − + exp O −

0

Particular solution:

146.

ïƒ÷ +2ú

+ ( ôƒõ

+ ï ( ô ïƒ÷ + õ )ñó ò + (  ï 2÷ +  ïƒ÷ + )ñ = 0. The transformation H =   , ¨ = H −  , where , is a root of the quadratic equation B 2 , 2 + B (  − 1) , + = = 0, leads to a linear equation of the form 2.1.2.108: B 2 H ¨2‘'

‘ + [B±!H + 2 ,!B 2 + B (B − 1 +  )]¨3‘
+ ( ƒhH + ,!B± + ý )¨ = 0.

ï 2 ñ óò ò ó

+ ï (2 ô ïƒ÷ + õ )ñó ò + [ ô 2 ï 2÷ + ô ( õ + ù – 1)ïw÷ +  ï 2ú +  ïƒú + ]ñ = 0. The substitution ¨ = exp(   B ) leads to a linear equation of the form 2.1.2.146:  2 ¨2

 + ' ¨2
+ ( ƒ… 2 + + ý± + + = )¨ = 0.

ï 2 ñ óò ò ó

+ (ô

ïƒ÷

+2

õï

+

+ ö )ñó ò + ( ôù

2

Particular solution:

0

= exp O −

2.1.2-5. Equations of the form ( 

2



B +1

ïƒ÷

+1

 

+

ô„ö ïƒ÷

+1

− '

P

+

õ^ö )ñ

= 0.

.

+ ' + ( )

 +  ( )
 + ( ) = 0.

149. (1 – ï 2 )ñó ò ò ó + ù (ù – 1)ñ = 0, ù = 0, 1, 2, uuu This equation is encountered in hydrodynamics when describing axially symmetric Stokes flows. 1 b . For B ≥ 2, the solution is given by: =L

1

 ( ) + L 2 ‚ {  ( ),

y

where  ( ) and {‚ ( ) are the Gegenbauer functions which can be expressed in terms of the y functions of the first and second kind (see 2.1.2.153) as follows: Legendre

 ( ) =

y

.



.

 ) −  ( ) , 2B − 1

−2 (

2 b . For B = 0 and B = 1, the solution is:

150. ( ï

2

– ô 2 )ñ

óò ò ó

+

õÿñ óò

– 6ñ = 0.

Particular solution:

0

=L



{‚ ( 1

)=

+L

2

 .

Æ 

 ) − Æ  ( ) . 2B − 1

−2 (



= (4 −  )| +  | 2   | −  | 2   . 2 +

2 −

151. ( ï 2 – 1)ñó ò ò ó + ï ñó ò + ôñ = 0. 1 b . For  = , 2 > 0, the solution is: =

L

cos( , arccosh | |) + L 2 sin( , arccosh | |) if | | > 1, L 1 exp( , arccos  ) + L 2 exp(− , arccos  ) if | | < 1,

™

1

where arccosh  = ln   + £  2 b . For  = − ,

2

2

−1 .

< 0, the solution is: =

™

L

exp( , arccosh | |) + L 2 exp(− , arccosh | |) if | | > 1, L 1 cos( , arccos  ) + L 2 sin( , arccos  ) if | | < 1. 1

3 b . For  = −B 2 , where B is a nonnegative integer, particular solutions are the Chebyshev { polynomials:  ( ) = cos(B arccos  ).

Page 230

231

2.1. LINEAR EQUATIONS

152. (1 – ï 2 )ñó ò ò ó – ï ñó ò + ù 2 ñ = 0, ù = 0, 1, 2, uuu This is a special case of equation 2.1.2.151 with  = −B

=

[ ´J B ½

2 +

{ where

2]

(−1) +

¡

=0

. Particular solution:

(−1)  £ 1− 2   12  

{

=  ( ) = cos(B arccos  ) =

0

2

(B − − 1)! ¡ (2 )  ! (B − 2 )!

−2

2

M

 M  V (1 −  2 )  

− 21

W

+ ,

¡

 ( ) is the Chebyshev polynomial of the first kind, (  )  =  (  + 1)  ( + B − 1), and [  ] stands for the integer part of a number  .

153. (1 – ï 2 )ñó ò ò ó – 2 ï ñó ò + ù (ù + 1)ñ = 0, The Legendre equation. The solution is given by: =L

.

ù 1

.

= 0, 1, 2,

uuu

 ( ) + L 2 Æ  ( ),

where the Legendre polynomials  ( ) and the Legendre functions of the second kind Æ  ( ) are given by the formulas:

.

 ( ) =

1 B ! 2

. The functions .

0(

 ) = 1,

.

M

 M  ( 

2

½  1. 1. 1+ Æ  ( ) = (  ) ln − +  2 1− + =1 ¡

− 1)  ,

.

−1 (

.

 )  − + ( ).

 =  ( ) can be conveniently calculated by the recurrence relations: 1(

 )= ,

.

2(

.

1 2

 ) = (3 2 −1),  ,



+1(

 )=

.

2B +1  B +1

B .  ( )− B +1 

−1(

 ).

Three leading functions Æ  = Æ  ( ) are:

Æ

0(

 )=

1 1+ ln , 2 1−

Æ

1(

 )=

. zeros of the polynomial



2

ln

1+ − 1, 1−

Æ

2(

 )=

3

2

−1 1+ 3 ln −  . 4 1− 2

All B  ( ) are real and lie on the interval −1 <  < 1; the . functions  ( ) form an orthogonal system on the closed interval −1 ≤  ≤ 1, with the following relations taking place:

X

.

1 −1

 (

. )

a

+ ( )  = M

0

2 2B + 1

if B ≠ if B =

,

¡

¡

.

154. (1 – ï 2 )ñó ò ò ó – 2 ï ñó ò + b ( b + 1)ñ = 0. The Legendre equation; @ is an arbitrary number. The case @ = B where B is a nonnegative integer is considered in 2.1.2.153. The substitution H =  2 leads to the hypergeometric equation. Therefore, with | | < 1 the solution can be written as: =L

1

@ 1+@ 1 O − , , ;  2

2

2

P

+L

2

"O

1−@ @ 3 , 1+ , ;  , 2 2 2 P

where  ( ƒ , ý , = ;  ) is the hypergeometric series (see 2.1.2.171). The Legendre equation is discussed in the books by Abramowitz & Stegun (1964), Bateman & Erd´elyi (1953, Vol. 1), and Kamke (1977) in more detail (see also Subsection S.2.9).

Page 231

232

SECOND-ORDER DIFFERENTIAL EQUATIONS

155. (1 – ï 2 )ñó ò ò ó – 3 ï ñó ò + ù (ù + 2)ñ = 0, Particular solution: 0

= ’  ( ) =

½ ´J

[

=

+

2]

sin[(B + 1) arccos  ]

£ 1−

(B − )! (2 )  ¡ ! (B − 2 )!

(−1) +

¡

=0

2

ù = −2

= 1, 2, 3,

uuu

(−1)  (B + 1) 1 £ 1− 2 +1  1  2  +1

2

M

 M  V (1 −  2 )  

+ 21

W

+ ,

¡

where ’  ( ) is the Chebyshev polynomial of the second kind, (  )  =  (  + 1)  (  + B − 1), and [  ] stands for the integer part of a number  .

156. ( ï

2

– 1)ñ

óò ò ó

+ 2(ù + 1)ï



= M  

Solution: 2.1.2.154. 157. ( ï

2

M

ñ óò

– ( b + ù + 1)( b – ù )ñ = 0,

ù

uuu

= 1, 2, 3,

Á ( ), where Á ( ) is the general solution of the Legendre equation

– 1)ñó ò ò ó – 2(ù – 1)ï

ñ„óò 

– ( b – ù + 1)( b + ù )ñ = 0,

Solution: = | 2 − 1|  M   M equation 2.1.2.154. 158. ( ï 2 – 1)ñó ò ò ó + (2 ô + 1)ï 1 b . Particular solution: 0

uuu

= 1, 2, 3,

Á ( ), where Á ( ) is the general solution of the Legendre

ñ„óò

– õ (2 ô + õ )ñ = 0.

=

(2  +  )   2  +  , −  ,  + 12 ; (  + 1) (2 )

w

ù

w

w

1 2

− 12   ,

(1)

where  ( ƒ , ý , = ; H ) is the hypergeometric function (see equation 2.1.2.171 and Subsection S.2.8). 2 b . For  = B , where B = 0, 1,  , the right-hand side of (1) defines the Gegenbauer polynomials,

½ (2  + B ) (  + , ) (2  + B + , )( − 1)  w w L  (  ) ( ) = w   2  + B , −B ,  + 12 ; 12 − 12   = . (B + 1) (2  ) , ! (B − , )! 2  (  ) (2  + 2 , ) =0  w w w w

159. (1 – ï 2 )ñ ó ò ò ó + (2 ô – 3)ï Particular solution: 2  0 (  ) = (1 −  )

ñ óò

+ (ù + 1)(ù + 2 ô – 1)ñ = 0,

J L (  ) ( ) = (1 −  2 )  

−1 2

ù

= 0, 1, 2,

uuu

 (  + , ) (2  + B + , )( − 1)  J ½ w w  ,  =0 , ! (B − , )! 2 w (  ) w (2  + 2 , )

−1 2

where L  (  ) ( ) are the Gegenbauer polynomials.

160. (1 – ï 2 )ñó ò ò ó + V  –  – (  +  + 2) ï WNñó ò + ù (ù + Particular solution:



+  + 1)ñ = 0,

ù

= 0, 1, 2,

uuu

 . (−1)   ) =  9 ,: (  ) =  (1 −  )− 9 (1 +  )−: M  (1 −  ) 9 +  (1 +  ): +  S 2 B !  Q M  ½ + + + + = 2−  L  + L   +−: ( − 1)  − ( + 1) , 9 + =0 . where  9 ,: ( ) are the Jacobi polynomials and L  are binomial coefficients.  0(

Page 232

233

2.1. LINEAR EQUATIONS

161. (1 – ï

2

)ñó ò ò ó + V  –  + (  +  – 2) ï WUñó ò + (ù + 1)(ù +  +  )ñ = 0,

Particular solution: 0 ( ) = (1 −  ) 9 (1 +  ):

.

mials (see 2.1.2.160).

ï

162. ( ô

2

+ õ )ñ

óò ò ó

ô ï ñ óò

+

The substitution H = X 163. ( ï

2

2

– ô 2 )ñó ò ò ó + 2 õ =L

Solution: 165. ( ï

2

+

ô 2)ñ óò ò ó

ï

166. ( ô

2

+ õ )ñ

óò ò ó

1



leads to a constant coefficient linear equation: 
‘'
‘ + (

+

2

= |

0

= 0.

+  |1−  .

2

ï ñóò

+ õ ( õ – 1)ñ = 0.

ï ñóò

+ õ ( õ – 1)ñ = 0.

 −  |1−  + L 2 | +  |1−  .

+ 2õ =L

Solution:

1|

uuu

+ 2( õ – 1)ñ = 0.

Particular solution: 164. ( ï

= 0, 1, 2,

= 0.

£ M

ï ñóò

+ ô )ñó ò ò ó + 2 õ

ö^ñ

+

ù

. ,: ,:  9 ( ), where  9 ( ) are the Jacobi polyno-



2

+  2

1− 2



sin Y + L

ï ñ óò

+ (2ù + 1)ô

+

ö^ñ

2



2

1− 2

+  2

ù

= 0,



cos Y , where Y = (1 −  ) arctan(   ).

uuu

= 1, 2, 3,

This equation can be obtained by B -fold differentiation of an equation of the form 2.1.2.162: (  2 +  )‡

 + ‡
 + ( ( − B 2 )‡ = 0. = ‡ (  ) . Solution: 167. (1 – ï

2

–ï

óò ò ó

)ñ

ñ óò

ï

+ (2 ô

2

+ õ )ñ = 0.

This is an algebraic form of the Mathieu equation. The substitution  = cos H leads to the ‘'‘ Mathieu equation 2.1.6.29: 

+ (  +  +  cos 2 H ) = 0. 168. (1 – ï

2

óò ò ó

)ñ

+ (ô

ï

óò

+ õ )ñ

ö^ñ

+

= 0.

1 b . The substitution 2 H = 1 +  leads to the hypergeometric equation 2.1.2.171: H (1 − H )/
‘'
‘ + ‘ [ !H + 12 (  −  )]
+ ( = 0. 2 b . For  = −2

¡

− 3,  = 0, and ( = Œ , the Gegenbauer functions are solutions of the equation.

3 b . In the special case  = − ƒ − ý − 2,  = ý − ƒ , and ( = B (B + ƒ + ý + 1), solutions of the equation are the Jacobi polynomials:

½ 9 : ( ) = 2−  L  + + L   +−: + ( − 1)  − + ( + 1) + , 9 + =0

.

( , )

where L  are binomial coefficients (see Paragraph S.2.1-2). 169. ( ô

ï



2

+ õ )ñó ò ò ó + ( ö

ï

Particular solution: 170. ( ô

ï

2

+ õ )ñ

óò ò ó

2

+ ý )ñó ò + [( ö – ô ) ï 0

0

+

ý

– õ ]ñ = 0.

=  .

+ [ (ö + ô )ï

Particular solution:



−

2

2

+ ( ö – ô ) ï + 2 õ ]ñ −



óò

+

2

(ö

ï

2

+ õ )ñ = 0.

= ( Π + 1)   .

Page 233

234 171.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï (ï

– 1)ñ ó ò ò ó + [(  +  + 1) ï – ]ñ ó ò + ñ = 0. The Gaussian hypergeometric equation. For = ≠ 0, −1, −2, −3,  , a solution can be expressed in terms of the hypergeometric series:

½ ( ƒ )  (ý )     (ƒ , ý , = ;  ) = 1 + ÷ , ( ƒ )  = ƒ ( ƒ + 1)  ( ƒ + , − 1), , !  =1 (= )  which, a fortiori, is convergent for | | < 1. For = > ý > 0, this solution can be expressed in terms of a definite integral:  (ƒ , ý , = ;  ) =

(= ) X ( ý ) (= − ý )

w

w

where (ý ) is the gamma function.

w

w

1 0

a:

−1

(1 − a );

:

− −1

(1 − aÖ )− 9

a, M

If = is not an integer, the general solution of the hypergeometric equation has the form: =L

1

 (ƒ , ý , = ;  ) + L 2 

1−

;  ( ƒ − = + 1, ý − = + 1, 2 − = ;  ).

In the degenerate cases = = 0, −1, −2, −3,  , a particular solution of the hypergeometric equation corresponds to L 1 = 0 and L 2 = 1. If = is a positive integer, another particular solution corresponds to L 1 = 1 and L 2 = 0. In both these cases, the general solution can be constructed by means of the formula given in 2.1.1. Table 16 presents some special cases where  is expressed in terms of elementary functions. Table 17 gives the general solutions of the hypergeometric equation for some values of the determining parameters. The function  possesses the following properties:

 ( ƒ , ý , = ;  ) =  (ý , ƒ , = ;  ),  ( ƒ , ý , = ;  ) = (1 −  ) ; − 9 −:  (= − ƒ , = − ý , = ;  ),   ( ƒ , ý , = ;  ) = (1 −  )− 9  ( ƒ , = − ý , = ; ),  −1  ( ƒ )  (ý )   ( ƒ + B , ý + B , = + B ;  ). M   (ƒ , ý , = ;  ) =  (= )  M

The hypergeometric functions are discussed in the books by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 1) in more detail; see also Subsection S.2.8. 172.

ï (ï

+ ô )ñ„ó ò ò ó + ( õ ï + ö )ñó ò + ýIñ = 0. The substitution  = − !H leads to the hypergeometric equation 2.1.2.171: H (1 − H )
‘'
‘ + [( (  ) − ˜H ]
‘ − = 0.

M

173. 2 ï ( ï – 1)ñ„ó ò ò ó + (2 ï – 1)ñ„ó ò + ( ô ï + õ )ñ = 0. The substitution  = cos2 ˆ leads to the Mathieu equation 2.1.6.29: 
Šª
Š −(  +2  +  cos 2ˆ ) = 0. 174. ( ï

2

ï

+ 2ô

Solution:

+ õ )ñó ò ò ó + ( ï + ô )ñó ò – þ = L 1  + + £ 

2

ñ

= 0.

+ 2  +  

2

+

+L

2

 + +£ 

2

+ 2  +  

−

+

.

175. ( ô ï 2 + õ ï + ö )ñó ò ò ó + ( ý ï + ü )ñó ò + ( ý – 2 ô )ñ = 0. Integrating yields a first-order linear equation: (  2 + ' + ( )
 + [( − 2  ) + , −  ] = L . 176.

(ô ï

2

+õ ï

+ ö )ñ ó ò ò ó + ( ü ï

Particular solution:

0

+ ý )ñ óò – = ,! + .

üIñ

M

= 0.

M

Page 234

235

2.1. LINEAR EQUATIONS

TABLE 16 Some special cases in which the hypergeometric function  ( ƒ , ý , = ; H ) is expressed in terms of elementary functions

=

ƒ

H

ý =

−B

ý

−B



−B −

ý

 ½ 

 ¡

ý ƒ

ƒ + 12

1 2



2

ƒ

ƒ + 12

3 2



2

ƒ

−ƒ

1 2

−

2

ƒ

1− ƒ

1 2

−

2

ƒ

ƒ − 12

2ƒ − 1



ƒ

−ƒ

1 2

sin 

ƒ

1− ƒ

3 2

sin2 

ƒ

2− ƒ

3 2

sin2 

ƒ

1− ƒ

1 2

sin2 

ƒ

ƒ + 12

1 2

− tan2 

ƒ

ƒ +1 ƒ

ƒ + 12 ƒ

ƒ + 12 1 2

=0



=0

(−B )  (ý )    , (−B − )  , !

½ 

ƒ

1 2



(−B )  (ý )    , (= )  , !

¡

 ý



2ƒ + 1



where B = 1, 2, 3, 

(1 −  )− 9

V (1 +  )−2 9 + (1 −  )−2 9 W

(1 +  )1−2 9 − (1 −  )1−2 9 2 (1 − 2 ƒ )

V  £ 1 +  2 +   29 +  £ 1 +  2 −   29 W 2 −1 2 −1  £ 1+ 2 +  9 +  £ 1+ 2 −  9 2£ 1 +  2

1 2

22 9

ƒ

1 2

1 2

where B = 1, 2, 3, 

 1+ £ 1− 

−2

2−2

cos(2 ƒ… ) sin[(2 ƒ − 1) ( ƒ − 1) sin(2 sin[(2 ƒ − 2) ( ƒ − 1) sin(2 cos[(2 ƒ − 1) cos 

] ) ] ) ]

cos2 9  cos(2 ƒ… ) (1 +  )(1 −  )− 9

−1

−2  12 + 12 £ 1 −   9 −1 J 2 1  1−   2 + 12 £ 1 −  



2ƒ

1 2

3 2



1

3 2

−

2

1

1

2

−

1 2

1

3 2



1 2

1 2

3 2

−

B +1

B + +1 ¡

B + + , +2 ¡





2

 

−1

arcsin 

−1

arctan 

−1

ln( + 1)

1−2

9

1 1+ ln 2 1−

2 2

9



−1

ln   + £ 1 + 

2



  (−1) + (B + + , + 1)!  + + ¡ M  + + ™ (1 −  ) + +  M  B ! , ! (B + )! ( + , )!   ¡ −1 ¡ M M  = − ln(1 −  ), B , , , = 0, 1, 2, 3, 

¢

,

¡

Page 235

236

SECOND-ORDER DIFFERENTIAL EQUATIONS

TABLE 17 General solutions of the hypergeometric equation for some values of the determining parameters

ý

= ý

=

ƒ

ƒ + 12

2ƒ + 1

ƒ

ƒ − 12

1 2

L

ƒ

ƒ + 12

3 2

£  QL

ƒ 0

L

ƒ ý

ƒ +1

ï

+ 2õ

The substitution ˆ = X

form 2.1.2.1: Šª

Š +

ï

178. ( ô

2

+ 2õ

ï

+ ö )ñó ò ò ó + ( ô

ï

£ 

2

= 0.

M

ï

+ ö )ñó ò ò ó + 3( ô

M



ýIñ

−2

 1+ £  

1

1

9 + L 2

1−2

 1+ £   −1

| |− 9 OL

+ õ )ñó ò +

X | |−; | − 1|; −:

2

 1+ £ 1− 

| − 1|−: OL

ƒ

2

L

| |1−; | − 1|; −:

ý

ï

1

1+

1

ƒ

177. ( ô

L

= ý

1

= ( )

Solution:

OL

1+ 1+

L

9 +L

1−2

1+

L

2 2

L

−2

2

2

1−2

 1− £  

X | |;

−1



9  1 + £ 1 −   29

−2

9

1−2

| − 1|: −;

X | |− 9 | − 1|:

X | | 9

M

 1− £  

9 +L 2

−1

| − 1|−:

−1

9 S  M P

 M P

 M P

= 0. leads to a constant coefficient linear equation of the

+ 2 ' + (

+ õ )ñó ò + ýIñ = 0.

The substitution ¨ = |  2 + 2 ' + ( | leads to a linear equation of the form 2.1.2.177: ü (  2 + 2 ' + ( )¨2

 + (  +  )¨2
+ ( −  )¨ = 0. M

179. ( ô

2

ï

Let Œ

õ 2ï

+ ö 2 )ñó ò ò ó + ( õ

2

+

1

and Œ

1 b . For Œ

1

2

1

ï

+

ö 1 )ñ óò

+ö

0

ñ

= 0.

are roots of the quadratic equation  2 Œ

≠ Œ 2 , the substitution H =

H (1 − H )
‘'
‘ − ( #* + $ )
‘ − L

Œ

 −Œ 2

−Œ

1 1

2

+  2Π+ (

2

= 0.

leads to the hypergeometric equation 2.1.2.171:

= 0, where # =





1 2

, $

=

 1Œ 1 + ( 1 (0 , L = .  2(Œ 2 − Œ 1)  2 = ˆ  ‡ , where , is a root of the

2 b . For Œ 1 = Œ 2 = Œ , the transformation  = Œ + ˆ −1 , quadratic equation  2 , 2 + (  2 −  1 ) , + ( 0 = 0, leads to a linear equation of the form 2.1.2.108:  2 ˆ‡ Šª

Š − [( ( 1 + Œ" 1 )ˆ +  1 − 2  2 ( , + 1)]‡ Š
− , ( ( 1 + Œ" 1 )‡ = 0. 3 b . Let ( 0 = −  2 B (B − 1) −  1 B , where B is a positive integer. Then, among solutions there exists a polynomial of degree ≤ B .

ï

180. ( ô

2

+

õï

+ ö )ñó ò ò ó – ( ï

Particular solution: 181. ( ô

ï

2

+

õï

0

+ ö )ñó ò ò ó + ( ï

Particular solution:

0

2

–

ü 2)ñ óò

+ ( ï + ü )ñ = 0.

= −, . 3

+

ü 3 )ñ óò

– (ï

2

–

ü ï

+

ü 2 )ñ

= 0.

= +, .

Page 236

237

2.1. LINEAR EQUATIONS

2.1.2-6. Equations of the form (  3  182. 183.

184. 185. 186.

188.

ï 3 ñ óò ò ó

+ ( ô ï 2 + õ )ñó ò + ö ï ñ = 0. ‘'‘ ‘ The substitution  = 1 H leads to an equation of the form 2.1.2.139: H 2 

+ H (2 −  − ˜H )
+ ( = 0.

ï 3 ñ óò ò ó

+ ( ô ï 2 + õ ï )ñ Particular solution:

óò

+ õÿñ = 0. =  −2+  .

0

ï 3 ñ óò ò ó

+ ( ô ï 2 + õ ï )ñó ò + ö^ñ = 0. The substitution  = 1 H leads to an equation of the form 2.1.2.108: H
‘'
‘ +(2−  − ˜H ) ‘
+ (

= 0.

+ ( ô ï 2 + õ ï )ñó ò + ( ö ï + ý )ñ = 0. 1 b . The substitution =   ‡ , where , = −  , leads to a linear equation of the form M 2.1.2.134:  2 ‡

 + [(  + 2 , ) +  ]‡
 + [ , (  + , − 1) + ( ]‡ = 0.

ï 3 ñ óò ò ó

=  (  − 2), a particular solution is:

M

ï 3 ñ óò ò ó

+ ( ô ï 3 – ï 2 + ôƒõ ï + õ )ñ ó ò + ô 2 õ  Particular solution: 0 = (  + 1)  −  .

ï 3 ñ óò ò ó

+ ï ( ô ïƒ÷ + õ )ñó ò – ( ô ïƒ÷ – ôƒõ Particular solution: 0 =  exp(   ).

ï (ô ï

+ õ )ñó ò ò ó + 2( ô Particular solution:

190.

ï (ï

ï

2

+ õ )ñó ò – 2 ô =  +   .

2 0

2 + ô )ñó ò ò ó + ( õ ï

ï ñ

ïƒ÷

–1

ï ñ

0

 = J .

= 0. + õ )ñ = 0.

= 0.

+ ö )ñó ò + s ï ñ = 0. ‘'‘ The substitution !H = − 2 leads to the hypergeometric equation 2.1.2.171: H (1 − H )"

+ ‘ 
1 V 1 −1 − (1 +  ) HžW − 4 ° = 0. 2 1 + (

ï 2(ô ï

191.

+ õ )ñó ò ò ó + [ ö Particular solution:

192.

ï 2(ô ï ï 2(ô ï

ï

L 1

=

+ (2 õ + ô ) ï +

 ). 0 = exp( Œ

2

ï

+ 2 õ )ñ„ó ò + 2( ô

+ L 2 .  +  2

=







õ ]„ñ óò ï

)ï

2

– õ (ù + þ

L 1 | |  + L 2 | | + +1  +  | |  ( L 1 + L 2 ln | |)  + 

) ï ]ñó ò + [ ôþ

(ù – 1)ï +

õÿù (þ

+ 1)]ñ = 0.

≠ B − 1,

if if

+ ( ö – 2 ô )ñ = 0.

+ 3 õ )ñ = 0.

3

+ õ )ñó ò ò ó + [ ô (2 – ù – þ

Solution:

ï 2(ï

2

+ õ )ñó ò ò ó – 2 ï ( ô

Solution:

194.

+  1  +  0 )

 +  ( )
 + ( ) = 0.

2

+ ( ô ï + õ )ñ = 0. This is a special case of equation 2.1.2.132 with B = −1.

189.

193.

+  2

ï 3 ñ óò ò ó

2 b . If ( = 0 and 187.

3

¡

¡

= B − 1.

+ ô 2 )ñó ò ò ó + ï ( õ 1 ï + ô 1 )ñó ò + ( õ 0 ï + ô 0 )ñ = 0. The substitution =   ‡ , where , is a root of the quadratic equation  2 , 2 + , (  1 −  2 )+  0 = 0, leads to a linear equation of the form 2.1.2.172:  ( +  2 )‡

 + [(2 , +  1 ) + 2 , 2 +  1 ]‡
 + [ , 2 + , (  1 − 1) +  0 ]‡ = 0.

Page 237

238

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï

195. ( ô

3

+

õï

+ö

2

ï )ñ óò ò ó

Particular solution:

ï

196. ( ô

3

+

õï

+ö

2

ï )ñ óò ò ó

Particular solution:

ï

197. ( ô

3

+

õï

+ö

2

ï )ñ ó ò ò ó

Particular solution:

ï

198. ( ô

3

+

õï

+ö

2

ï )ñ óò ò ó

Particular solution: 199. ( ô

ï

3

+

õï

+ö

2

ï )ñ óò ò ó

Particular solution:

ï

+ ( 0

ï

+

ï

2

=

ï

2

1−

 .

= ( +

200. ( ô ï 3 + õ ï 2 + ö ï )ñó ò ò ó + (ù With the constraint

ï

+þ

ï

2

2

¡

)

2

ï

+þ

a particular solution has the form:

2

2

+

õï

The substitution ˆ = X O 2
Šª
Š + Œ

203.

204.

ï (ô ï

ï

+ ö )ñó ò ò ó + ( ô

= 0.



+ õ ï + 1)ñ ó ò ò ó + (  The substitution =  1−; ¨ 2

2

2

ï



óò

+ 2( ô

ï

+ 1)ñ = 0.

+ ( ( + , )(2 − , ).

+ ü )ñó ò + ( ü – 1)[(ù – ôƒü ) ï + þ

+ (2 ö^þ

+ (2  + 4 (

– 1) ï – ö ]ñ„ó ò + (–2þ

–õ

ï

= 0.

+ 1)ñ = 0.

− 1)( + ( ).

¡

ü ]ñ

+ ü )ñó ò + [–2( ô + ù ) ï + 1]ñ = 0.

0

+ 1)[

¡

= (2  + B )

201. ( ô ï 3 + ï 2 + õ )ñó ò ò ó + ô 2 ï ( ï 2 – õ )ñó ò – ô  Particular solution: 0 = (  + 2)  −  .

ï

ï

+ 2 ö )ñó ò – ( 

2(2  + B )( ( + , ) + (2  + 2

202. 2 ï ( ô

.

– ( õ + 1) ï + ü ]ñ

– ô )ï

+ [(þ 0

ï

= ( , +  − 1)

+ (ù 0

−1

+ 2 õ –  )ñ = 0. Œ (ƒ + (  − ý )(2  − ý ) =… ƒ  + 2(ý −  ) + , where Œ = .  ƒ − 2

+ [–2 ô 0

+ 2 ö )ñó ò + ( – 2 õ )ñ = 0.

= 2  − ƒ + (2  − ý )

+ ( 0

ï

+

2

– ö )ñó ò +

J

3

1 2

+ ' + ( P

2

¡

+ 1 + 2 , (  + B )] = 0, + (2  + 2

õï ñ

= 0.

ï 2ñ

= 0.

M

¡

+ 1)( − , ).

 leads to a constant coefficient linear equation:

+  ï + )ñ ó ò + (ù ï + þ )ñ = 0. leads to an equation of the same form:

2

+ ' + 1)¨ 

 + [( ƒ + 2  − 2 ´= ) 2 + (ý + 2  − 2 = ) + 2 − = ]¨ 
+ , [B + (1 − = )( ƒ − ´= )] + + (1 − = )(ý − = ) . ¨ = 0.

 ( 

2

ï (ï

– 1)( ï – ô )ñ ó ò ò ó + , ( 

+  + 1) ï

2

– [  +  + 1 + ô ( + S ) – S ] ï

¡

+ ô/

. ñ óò

+ ( 

ï – )ñ

= 0.

Heun’s equation. 1 b . For |  | ≥ 1 and = ≠ 0, −1, −2, −3,  , a solution can be represented as the power series:

½  ( ,  ; ƒ , ý , = , > ,  ) = ÷ (    ,  =0 where the coefficients are determined by the recurrence formulas:

( 0 = 1,

´=/( 1 =  ,

 (B + 1)(= + B )( 

+1

=  (= + > + B − 1) + ƒ + ý − > + B +

Q

B



SB±( 

− V (B − 1)(B − 2) + (B − 1)( ƒ + ý + 1) + ƒ…ý W ( 

−1 .

A fortiori, the series is convergent for | | ≤ 1.

Page 238

239

2.1. LINEAR EQUATIONS

TABLE 18 Some transformations preserving the form of Heun’s equation No 1* 2 3 4 5 6 7 8 9 10 11 12

Parameters of transformed equation for ¨ = ¨ (ˆ )

New variables

  ƒ ˆ = , ¨ = ý ƒ ˆ =1− , ¨ = ƒ…ý −  ý 1−  1 ƒ −= +1 ý −=  ˆ =  , ¨ = | |; −1   2 ƒ −= +1 ý −= 1 ˆ = , ¨ = | |; −1    1 ¨ 1 3 ƒ ƒ −= ˆ = , = | | 9    1  ˆ = ,¨ = ƒ ý    1  ˆ =1− , ¨ = ƒ ý 1−     ˆ = , ¨ = | | 9  ƒ −= +1 ƒ +=   −1 ¨ 1  ƒ ƒ −= ˆ = , = | | 9 1−     ( − 1) ¨  ƒ ˆ = , = | | 9 ƒ −=  −1  (  − 1)    ˆ = , ¨ = | − 1| 9 ƒ ƒ −>  −1  −1  (  − 1) ¨ 1  ƒ ˆ = , = | − 1| 9 1− ƒ −>  ( − 1)  Notation:  1 =  + ( ƒ − = + 1)(ý − = + 1) − ƒ…ý + > (= − 1),  3 =  −1 + ƒ (ƒ − = + 1) + ƒh −1 (> − ý ) − !>

= > = > +1

2−=

>

+1

2−=

ƒ +ý −= −> +1

+1 ƒ −ý +1

>

=

ƒ +ý −= −> +1

ƒ +ý −= −> +1

=

+1

>

ƒ −ý +1

+1

>

ƒ +ý −= −> +1

−1 ƒ −ý +1 ƒ +ý −= −> +1

+1

=

ƒ −ý +1

+1

=

ƒ +ý −= −> +1

 2 =  1 + !> (1 − = .

* This row corresponds to the original equation, while the others refer to the transformed equation for



=

),

 (~ ).

2 b . If = is not an integer, the general solution of Heun’s equation can be presented as follows: =L

1

 (  ,  ; ƒ , ý , = , > ,  ) + L 2 | |1−;  (  ,  1 ; ƒ − = + 1, ý − = + 1, 2 − = , > ,  ),

where  1 =  + ( ƒ − = + 1)(ý − = + 1) − ƒ…ý + > (= − 1). Table 18 lists some transformations preserving the form of Heun’s equation. (Whenever at least one of the indicated equations is integrable by quadrature with some values of parameters, all the other equations are also integrable for those values of the parameters.)

^_

References: H. Bateman and A. Erd´elyi (1955, Vol. 3), E. Kamke (1977), S. Yu. Slavyanov, W. Lay, and A. Seeger (1955), A. Ronveaux (1995).

205. ( ô ï 3 + õ ï 2 + ö ï + ý )ñ ó ò ò ó – ( ï 2 – Particular solution: 0 =  − Œ . 206. 2( ô

ï

3

+

õï

2

+

öï

+ ý )ñó ò ò ó + (3 ô

The substitution ˆ = X 2
Šª
Š + Œ

= 0.

ï

£ 

3

2

M

+ '

2

 2

)ñ

óò

+ 2õ

+ ( ï + )ñ = 0.

ï

+ (˜ +

+ ö )ñó ò +

Ė

= 0.

leads to a constant coefficient linear equation:

M

Page 239

240

SECOND-ORDER DIFFERENTIAL EQUATIONS

207. 2(ô ï 3 + õ ï 2 + ö ï + ý )ñó ò ò ó + 3(3 ô ï 2 + 2 õ ï + ö )ñó ò + (6 ô ï + 2 õ + )ñ = 0. This equation is obtained by differentiating the equation 2.1.2.206.

208. ( ô ï 3 + õ ï 2 + ö ï + ý )ñó ò ò ó + [  ï Particular solution: 0 =  + = .

2

+ ( | +  ) ï + C ]ñ„ó ò – ( 

209. ( ô ï 3 + õ ï 2 + ö ï + ý )ñó ò ò ó + ( ï 3 + Particular solution: 0 =  + Œ .

210. 2 ï ( ô

ï

2

+

õï

+ [ ô (2 – ü ) ï

óò ò ó

+ ö )ñ

3

2

)ñó ò – ( ï

2

–

ï

2

+

+ õ (1 – ü ) ï – ö³ü ]ñ

óò

ï

+  )ñ = 0.

)ñ = 0. +

ï  +1 ñ

= 0.

The substitution ˆ = X  J 2 (  2 + ' + ( )−1 J 2  leads to a constant coefficient linear equation: M 2
Šª
Š + Œ = 0.

2.1.2-7. Equations of the form (  4  211.

212.

ï 4 ñ óò ò ó

– (ô + õ )ï =

™

ï 4 ñ óò ò ó

2

ñóò

+  1  +  0 )

 +  ( )
 + ( ) = 0.

2

leads to a constant coefficient linear equation of the

+ ( ô ï 2 + õ ï + ö )ñ = 0. The transformation H = 1  , ‡ =   H 2 ‡ ‘'

‘ + ( (H 2 + ˜H +  )‡ = 0.

leads to a linear equation of the form 2.1.2.115:

+ [( ô + õ ) ï + ôƒõ ]ñ = 0.

  L 1   − J + L 2   − J  ( L 1  + L 2 ) −  J

if  ≠  , if  =  .

+ 2 ï 2 ( ï + ô )ñó ò + õÿñ = 0. The substitution H = 1  leads to a constant coefficient linear equation:
‘'
‘ − 2  
‘ + 

ï 4 ñ óò ò ó

+

ô ƒï ÷ ñ óò

– (ô

Particular solution:

218.

+  2

ï 4 ñ óò ò ó

215.

217.

3

+ ôñ = 0. The transformation H = 1  , ‡ =   form 2.1.2.1: ‡ ‘'

‘ + ‡ = 0.

Solution:

216.

+  3

ï 4 ñ óò ò ó

213.

214.

4

ï 2(ï

ïƒ÷

–1

+

ôƒõ ïƒ÷

− J  . 0 =   

– ô )2 ñ ó ò ò ó + õÿñ = 0. Solution: = L 1 | | + | −  |1− + + L ( − 1)  2 = −  .

¡

¡

ï 2(ï

– ô )2 ñ Solution:

óò ò ó

+

õÿñ

=

ö ï 2(ï

2|

–2

+

õ 2 )ñ

= 0.

= 0.

 |1− + | −  | + , where ¡

is a root of the quadratic equation

– ô )2 .

= | | + | −  |1− + OL

(

X | |1− + | −  | +   (2 − 1) M P ¡ ( + + 1− + 1− + + | | | −  | OL 2 −  , X | | | −  |  (2 − 1) M P ¡ where is a root of the quadratic equation ( − 1)  2 = −  . ¡ ¡ ¡ 1

+

ô ï 2 (ï – 1)2 ñóò ò ó + ( õ ï 2 + ö ï + ý )ñ = 0. Let  and  be roots of the quadratic equations D ( − 1) + = 0, ³ ( M

− 1) +  + ( + = 0. M ¨ The substitution =  K ( − 1) & leads to the hypergeometric equation of the form 2.1.2.171:  ( − 1)¨2

 + 2  [( +  ) −  ]¨3
+ (2 D¿ − ( − 2 )¨ = 0. M

Page 240

241

2.1. LINEAR EQUATIONS

219.

220.

ï 2(ï

2

ï

+ ô )ñó ò ò ó + ( õ

+ ö )ï

2

ñóò

ýIñ

+

= 0.

The substitution ˆ =  leads to a linear equation of the form 2.1.2.194: 4ˆ 2 (ˆ +  )
Šª
Š + 2ˆ [(  + 1)ˆ +  + ( ] Š
+ = 0. (ï

2

+ 1)

2

ñóò ò ó

2

M

ôñ

+

= 0.

The Halm equation. Solution: 2

– 1)2 ñó ò ò ó +

ôñ

= 221. ( ï

2

+1 VL 2+1 VL £  2 + 1 (L

£ 



cos(ý arctan  ) + L 2 sin(ý arctan  )W 1 cosh( ý arctan  ) + L 2 sinh( ý arctan  )W 1 + L 2 arctan  )

Solution: =







if  + 1 = ý 2 > 0, if  + 1 = −ý 2 < 0, if  = −1.

1

£ 

= 0.

| 2 − 1| V L 1 cos ý ln | H |  + L 2 sin ý ln | H |  W ( + 1) L 1 | H |(2: −1) J 2 + L 2 | H |−(2: +1) J 2  | 2 − 1|  L 1 + L 2 ln | H | 

ü ü

if  − 1 = 4ý 2 > 0, if  − 1 = −4ý 2 < 0, if  = 1,

where H = ( + 1) ( − 1). 222. ( ï

2

Aô 2 )2 ñó ò ò ó + õ

ñ

2

= 0.

This is the equation of bending of a double-walled compressed bar with a parabolic crosssection. 1 b . For the upper sign (constricted bar), the solution is as follows: =£ 

2

+

2

(L

1

cos ‡ + L

2

sin ‡ ),

where ‡ =

2

ü 1 + (   ) arctan(  ).

2 b . For the lower sign (bar with salients), the solution is given by: =£  223. 4( ï

2

+ 1)2 ñ

óò ò ó

2

−

2

(L

ï

2

+ (ô

cos ‡ + L

1

™

=

224. ( ô

ï

2

+ õ )2 ñ

1 2

+ 2ô

ï (ô ï

The substitution ˆ = X 225. ( ï

2

( (

2 2

£ |  − 1| ln   +

óò ò ó

– 1)2 ñó ò ò ó + 2 ï ( ï



2

sin ‡ ),

where ‡ =

£  2− 2

2

ln

 + ; | | <  .  −

+ ô – 3)ñ = 0.

Solution:

where ˆ =

2

2

M2



+ 1)1 J 4 ( L + 1)1 J 4 ( L |

2

+ õ )ñ

óò

ü

1 1

cos ˆ + L 2 sin ˆ ) if  > 1, cosh ˆ + L 2 sinh ˆ ) if  < 1,

+ 1|  . + ö^ñ = 0.

leads to a constant coefficient linear equation: 
Šª
Š + (

+

– 1)ñó ò – [ b ( b + 1)( ï

2

– 1) + ù

2

= 0.

]ñ = 0.

Here, @ is an arbitrary number and B is a nonnegative integer. This is a special case of equation 2.1.2.226. If B = 0, this equation coincides with the Legendre equation 2.1.2.154. Denote its general solution by Á ( ). If B = 1, 2, 3,  , the general solution of the original equation is given by the formula:

= |

2

− 1| ´J

2

M

 M  

Á ( ).

Page 241

242

SECOND-ORDER DIFFERENTIAL EQUATIONS

226. (1 – ï 2 )2 ñó ò ò ó – 2 ï (1 – ï 2 )ñó ò + [ b ( b + 1)(1 – ï 2 ) – 2 ]ñ = 0. The Legendre equation, @ and “ are arbitrary parameters. The transformation  = 1 − 2ˆ , = | 2 − 1| žJ 2 ¨ leads to the hypergeometric equation 2.1.2.171: ˆ (ˆ − 1)¨ Šª

Š + (“ + 1)(1 − 2ˆ )¨ Š
+ ( @ − “ )( @ + “ + 1)¨ = 0 with parameters ƒ = “ − @ , ý = “ + @ + 1, = = “ + 1. In particular, the original equation is integrable by quadrature if @ = “ or @ = −“ − 1. In Subsection S.2.9, the Legendre equation are discussed in more detail. See also the books by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 1). 227.

ô (ï

– 1)2 ñó ò ò ó + õ ï ( ï 2 – 1)ñó ò + ( ö ï 2 + ý ï +  )ñ = 0. The transformation ˆ = 12 ( + 1), ¨ = | + 1|−K | − 1|− & , where  and  are parameters that are determined by solving the second-order algebraic system 2

4 ³ (  − 1) + 2 U + ( + +  = 0,

( −  )[2 ( +  − 1) +  ] = ,

M

leads to the hypergeometric equation 2.1.2.171 with respect to ¨ = ¨ (ˆ ).

ï

228. ( ô

2

ï

+ õ )2 ñó ò ò ó + (2 ô

The substitution ˆ = X



ï

2

+ õ )2 ñó ò ò ó + ( ô

ï

2

Particular solution: 230. ( ï

2

+ ô )2 ñ

óò ò ó

+

Particular solution:

231. ( ï

2

+ ô )2 ñó ò ò ó +

0

õ ï ÷ (ï õ ïƒ÷ (ï

Particular solution:

M2

+ õ )ñó ò +

2



= 0.

+ ý )ñó ò + 2( õ^ö – ô„ý ) ï (˜ 2 + = exp O − X M  .  2 + 3M P

+ õ )( ö

2

2

2

óò

+ ô )ñ

£  0 = 0

ï

üIñ

leads to a constant coefficient linear equation of the form

+ = 0.

2.1.2.11: Šª

Š + ( 
Š + , 229. ( ô

ï

+ ö )( ô

2

– (õ

= (

ï ÷

+ .

+ ô )ñó ò – þ 2

M

[õ

+ +  ) J 2.

+1

+ ô )ñ = 0.

ïƒ÷

+1

+ (þ

– 1) ï

ñ

= 0.

2

+ ô ]ñ = 0.

232. ( ï – ô )2 ( ï – õ )2 ñó ò ò ó – ö^ñ = 0, ô ≠õ.  − The transformation ˆ = ln û , = ( −  )‰ leads to a constant coefficient linear equation:  −  ûû û


û (  −  )2 (‰ ŠªŠ − ‰ Š ) − (˜‰ = 0.û Therefore, the solution is as follows: =L where Œ

2

1|

 −  |(1+  ) J 2 | −  |(1−  ) J 2 + L 2 | −  |(1−  ) J 2 | −  |(1+  ) J 2 ,

= 4 ( (  −  )−2 + 1 ≠ 0.

233. ( ï – ô )2 ( ï – õ )2 ñó ò ò ó + ( ï – ô )( ï – õ )(2 ï + )ñ„ó ò + “ñ = 0. Let , 1 and , 2 be roots of the quadratic equation (  −  )2 , 2 + (  −  )(  +  + Œ ) , + “ = 0. Solution: L |H |  1 + L 2 |H |  2 if , 1 ≠ , 2 , = ™ 1 | H |  L 1 + L 2 ln | H |  if , 1 = , 2 = , , where H = ( −  ) ( −  ).

234. ( ô

ï

2

+

õï

+ ö )2 ñó ò ò ó +


ñ

= 0.



, ¨ = leads to a constant coefficient + ' + ( |  2 + ' + ( | ü  ( − 1  2 )¨ = 0. linear equation of the form 2.1.2.1: ¨uŠª

Š + ( # + ! 4 The transformation ˆ = X



2

M

Page 242

243

2.1. LINEAR EQUATIONS

235. ( ï

2

– 1)2 ñó ò ò ó + 2 ï ( ï

– 1)ñó ò + [( ï

2

– 1)( ô

2

ï

2 2

– )–þ

2

]ñ = 0.

Equation for prolate spheroidal wave functions, = 0, 1,  It arises when separating ¡ variables in the wave equation written in the system of prolate spheroidal coordinates. 1 b . In applications, one usually looks for eigenvalues Œ = Œ +  and eigenfunctions = +  ( ) that assume finite values at  = A 1. The following functions are solutions of the eigenvalue problem:

8

½ (1) +  ( ,  ) = ÷

8

½ (2) +  ( ,  ) = ÷

Ÿ

Ÿ =−

.

+  ( ). + Ÿ +

M =0,1

+

+  ( ) Æ + Ÿ + M

Ÿ ( +

)

Ÿ (

(prolate angular functions of the first kind), ) (prolate angular functions of the second kind),

÷

where  + ( ) and Æ + ( ) are the associated Legendre Ù . functions of the first and second kind. . For −1 ≤  ≤ 1, we have  + ( ) = (1 −  2 ) + J 2 x x  Ù  ( ). The summation is performed over either even or odd values of & , depending on whether |B − | is even or odd, respectively.

¡

2 b . The following recurrence relations for the coefficients  = +  (  ) hold: M M

ƒ   M where

3 b . For eD

+2

+ (ý  − Œ +  )  + = 

M 

M

−2

= 0,

 2 (2 + , + 1)(2 + , + 2) ƒ  = ¡ ¡ , (2 + 2 , + 3)(2 + 2 , + 5) ¡ ¡ 2( + , )( + , + 1) − 2 2 − 1 ý  = ( + , )( + , + 1) +  2 ¡ ¡ ¡ , ¡ ¡ (2 + 2 , − 1)(2 + 2 , + 3) ¡ ¡  2 , ( , − 1) =  = . (2 + 2 , − 3)(2 + 2 , − 1) ¡ ¡

0, the eigenvalues are defined by: 1

− 1)(2

(2

+ 1)

Œ +  = B (B + 1) + 1 − ¡ ¡ Sž 2 + [   4  . 2Q (2B − 1)(2B + 3) 4 b . For eDGF , we have:

Œ +  = ³ + ¡

^_ 236. ( ï

2

− 18 ( 

2

+ 5) −

1 64

 ( 2 + 11 − 32¡

2

)

−1

+[  

−2

,



= 2(B −

¡

) + 1.

References: H. Bateman and A. Erd´elyi (1955, Vol. 3), M. Abramowitz and I. A. Stegun (1964). 2

+ 1)2 ñó ò ò ó + 2 ï ( ï

2

+ 1)ñó ò + [( ï

2

+ 1)( ô

ï

2 2

– )+þ

2

]ñ = 0.

¶

Equation ¶ of oblate spheroidal wave functions, ¡ = 0, 1,  The transformations  = A"™  ,  = T ™  lead to equation 2.1.2.235. See the books by Bateman & Erd´elyi (1955, Vol. 3) and Abramowitz & Stegun (1964) for more information on this equation. 237. ( ô

ï

2

+

õï

+ ö )2 ñ

óò ò ó

+ (2 ô

ï

+ ü )(ô

ï

2

+

õï

+ ö )ñ

óò

+ þxñ = 0.

 leads to a constant coefficient linear equation of the M 2 + ' + (    = 0. form 2.1.2.11: ªŠ

Š + ( , −  )
Š + ¡

The substitution ˆ = X

Page 243

244

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.1.2-8. Other equations. 238.

239.

ï 6 ñ óò ò ó

–ï

5

ñóò

ôñ

+

= 0.

The transformation ˆ =  −2 , ¨ =  form 2.1.2.1: 4¨3Šª

Š +  ¨ = 0.

ï 6 ñ óò ò ó

+ (3 ï

2

+ ô )ï

3

The substitution ˆ =  240.

ñóò ò ó + ñ óò where

½

÷

73 

õ÷

3

(1 – 

õ÷ ï

=1

ñ óò

õÿñ

+

−2

leads to a constant coefficient linear equation of the

= 0.

−2

leads to a constant coefficient linear equation: 4
Šª
Š − 2  
Š + 

÷

–

–ô

÷

÷

)

(õ 1 ï

–

ñ

– ô 1 )( õ 2 ï

R R  ÷  ÷ ÷ ï ÷ –1 – ô 3 ) ÷ =1 õ ÷ –ô ÷ ½

– ô 2 )( õ 3 ï

( ƒ  + ý  ) = 1, |   | + |   | > 0, ¢  =    

=1

+1

− 

= 0.

+1

3

  ≠ 0,  

+3

=  ,  

+3

= 0,

=  .

Riemann’s equation. Denote this equation by:



™



1



1





2 2

ƒ û ý

3



3

ƒ

1

û

1

û û

2

ý

2

ƒ ý



3 3

û û

¢

û

= 0.

(1)

û

For  1 =  2 = 0,  3 =  3 = 1, ƒ 1 = ƒ 3 = 0, ƒ 2 = ƒ , ý 1 = 1 − = , ý 2 = ý , and ý equation (1) transforms into the hypergeometric equation 2.1.2.171. À The transformation

− $çL

#

#

1

$

2

$

1

$

2

#

ƒ û ý

3 3

û û û

1 1

+ &ǃ +& ý

2 2

where #  = #2  + $ç  , $  = Lu  + c‚  . In (2), assume & = − ƒ 1 , ° = − ƒ 3 , # =  1 ¢ to obtain the hypergeometric equation (3).

ï ÷ ñ óò ò ó

+ õ )÷

ï

+ ö (ô

–4

ñ

242.

ïƒ÷ ñóò ò ó

 , ¨  + 

ˆ  ¨ = 0.

−2 −

+

ô ï ñóò

– (õ

2

Particular solution: 243.

ïƒ÷ ñóò ò ó

+ (ô

ï

ï ÷ ñ óò ò ó

+ (ô

ï ÷

–1

+

0

ï ÷ ñ óò ò ó

+ (2 ï

+ 2õ = 

$ = − 1 ¢

+° ˆ û +° û ¨

û

¢

= 0,

(3)

û

3,

L = − 2 ¢

2,

and c

=  2 ¢

2





ïƒ÷

–1

=

+

 + 

ôƒõ ï

leads to an equation of the form 2.1.2.7:

+ ô )ñ = 0.

.

÷

–1

+ô

Particular solution:

=  +  .

õ ï )ñ ó ò

Particular solution: 245.

3,

3 3

+ õ )ñó ò – ôñ = 0.

Particular solution: 244.

ïƒ÷ 0

−& −° ƒ −& −Ç ° ý

= 0.

The transformation ˆ =

¨2Šª

Š + (

(2)

≠ 0, brings the original equation into an equation of similar form:

™

241.

== −ƒ −ý ,

À Ÿ ¨ = | 1  −  1| | 3 −  3| , |  2  −  2 |Ÿ +

#  +$ * ˆ = , L3 + c where #2c

3

ï

0

=

2 0

+

+ ( ô – 1) õÿñ = 0.

1−

.

õ ï )ñ ó ò

+ = +  .

õÿñ

= 0.

Page 244

245

2.1. LINEAR EQUATIONS

246. 247. 248.

ï ÷ ñ óò ò ó

+ ( ô ï ÷ + õ )ñ Particular solution:

ï ÷ ñ óò ò ó ï ÷ ñ óò ò ó

+ (ô

0

÷

+ (ô ï ÷ – ï Particular solution:

ï ÷ +ú

ï ÷

+ õ )ñ

óò ò ó

+ ôƒõ ï + õ )ñ ó ò + −  . 0 = (  + 1) 

óò

+ 1)ñ 0

ï ÷

+ (ö

+ õ ]ñ = 0.

ô 2õ ï ñ

–1

Particular solution: 249. ( ô

÷

+ ö [( ô – ö ) ï  =  −v .

óò

+

= exp O − + ý )ñ

Particular solution:

0

ô ï ú

−

óò 

=  .

ï ÷

(1 + þ



¡

+1

 +

–1

+1

P

= 0. )ñ = 0.

.

+ [( ö – ô ) ï

÷

+ý –

õ ]ñ

= 0.

250. ( ô ï ÷ + õ ï + ö )ñ ó ò ò ó = ôù (ù – 1) ï ÷ –2 ñ . Particular solution: 0 =   + ' + ( . 251.

ï (ïƒ÷

+ 1)ñó ò ò ó + [( ô – õ ) ïƒ÷ + ô – ù ]ñó ò + õ (1 – ô ) ïƒ÷

Particular solution: 252.

ï ( ï 2÷

+ ô )ñó ò ò ó + ( ï

254.

÷

õ 2 ï 2÷ –1 ñ

+ ô – ôù )ñó ò –

J = L 1    + £  2 +    + L

Solution: 253.

2

 + 1)  J  . 0 = (

–1

ñ

= 0.

= 0.

 + £  2  +   −  J  . 2 

ï 2 ( ô 2 ï 2÷

– 1)ñ ó ò ò ó + ï [ ô 2 (ù + 1) ï 2÷ + ù – 1]ñ ó ò – b ( b + 1) ô 2 ù 2 ï 2÷ ñ = 0. Solution: = Á (   ), where Á ( ) is the general solution of the Legendre equation 2.1.2.154.

ï 2 (ô ïƒ÷

– 1)ñó ò ò ó + ï ( ô' ïƒ÷ + )ñó ò + ( ô/ ïƒ÷ + s)ñ = 0. Find the roots # 1 , # 2 and $ 1 , $ 2 of the quadratic equations 2

#

− (  + 1) # − ° = 0,

2

$

− ( − 1) $ + & = 0

and define parameters ( , ƒ , ý , and = by the relations

= = 1 + ( # 1 − # 2 )B −1 . Then the solution of the original equation has the form =  v ‡ (   ), where ‡ = ‡ ( H ) is the general solution of the hypergeometric equation 2.1.2.171: H ( H − 1)‡ ‘'

‘ + [( ƒ + ý + 1) H − = ]‡ ‘
+ ƒ…ý±‡ = 0. ( = # 1,

255. ( ïƒ÷ + ô )2 ñó ò ò ó –

ƒ = (#

õ ïƒ÷

Particular solution: 256. ( ô

ïƒ÷

+ õ )2 ñó ò ò ó + ( ô

Particular solution: 257. ( ï

÷

+ ô )2 ñ

óò ò ó

+

ïƒ÷

0

ïƒ÷

0

+ õ )2 ñó ò ò ó +

1)

B

−1

ý = (#

,

+$

2)

B

−1

,

+ ý )ñó ò + ù ( õ^ö – ô„ý ) ïƒ÷ (˜  + = exp O − X M  .   + 3M P

ïƒ÷

+ õ )( ö

+ ô )ñ

óò

–ï

÷

–2

 +  )1 J . 0 = (

0

1

= |  +  |  J  .

ö ïƒú (ô ïƒ÷

Particular solution:

+$

[( õ – 1) ïƒ÷ + ô (ù – 1)]ñ = 0.

õ ï ú (ï ÷

Particular solution: 258. ( ô

–2

1

+ õ )ñó ò + ( ö

= exp O − X



(õ

ï ú

ïƒú

+1

– ôù

–1

ñ

= 0.

+ ôù – ô )ñ = 0.

ïƒ÷

–1

– 1)ñ = 0.

. M   +  P

Page 245

246 259.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 2 (ô ïƒ÷

+ õ )2 ñó ò ò ó + (ù + 1) ï ( ô 2 ï 2÷ – õ 2 )ñó ò + ö^ñ = 0. 1    The substitution ˆ = ln O leads to a constant coefficient linear equation of the B… 
  +  P

form 2.1.2.11: ŠªŠ −  (B + 2) Š + ( = 0.

260. ( ô

ïƒ÷

+1

ïƒ÷ + ö )ñ óò ò ó

+õ

ïƒ÷ +  ïƒ÷

+( 

Particular solution:

0

–1

+ )ñó ò +[ù (  – ô – ôù ) ïw÷

= exp X

( B +  − ƒ )  + ( 'B − ý )    +1 + '  + (

–1

Q

−1

+(ù –1)( – õÿù ) ïw÷

−=

261. ( ô ïƒ÷ + õ ïƒú + ö )ñó ò ò ó + ( – ï )ñó ò + ñ = 0. Particular solution: 0 =  − Œ .

M

–2

]ñ = 0.

 S .

262. ( ô ïƒ÷ + õ ïƒú + ö )ñó ò ò ó + ( 2 – ï 2 )ñó ò + ( ï + )ñ = 0. Particular solution: 0 =  − Œ .

ï ÷

263. 2( ô

+

õï ú

óò ò ó

+ ö )ñ

264. ( ô

ïƒ÷

M

= 0.

+ õ )ú

+1

ñóò ò ó

265.

ïƒ÷

+ (ô

Particular solution:

–1

+

õÿþ ï ú

 £   +M ' + + (

The substitution ˆ = X 2
Šª
Š +

ï ÷

+ ( ôù

0

+ õ )ñó ò – ôùþ

)ñ

óò

+ ýIñ = 0.

leads to a constant coefficient linear equation:

ïƒ÷ –1 ñ

= 0.

S . M (   +  ) +

= exp − X

Q

ï g ÷ (ï )ñ óò ò ó + [2 g ÷ (ï . ( ) and Æ Here,



–1

) + ( ô ï 2 + õ ï ) i ÷ –2 ( ï )]ñ ó ò + õi ÷ –2 ( ï )ñ = 0.   −2 ( ) are arbitrary polynomials of degrees B and B − 2, respectively. Particular solution: 0 =  +   .

2.1.3. Equations Containing Exponential Functions 2.1.3-1. Equations with exponential functions. 1.

2.

3.

+ ô Solution: functions.

óñ

ñóò ò ó ñóò ò ó

= 0, =L

1

ø

≠ 0. 0 ( H ) + L 2 n 0 ( H ), where H = 2 Œ

ó

+ ( ô – õ )ñ = 0.  Solution: = L 1 ø 22  2£   J 2  + L  Bessel functions.

ñóò ò ó

ó

+ ô ( 

– ô

2

Particular solution:

ñóò ò ó



ó )ñ

ó

0

= 0.



= exp O −  

ó

Œ



n 2

P

2 2

£    J 2 ;

ø

0(

H ) and n 0 ( H ) are the Bessel

  2 £   J 2  , where

ø

Á ( H ) and n Á ( H ) are the

.

4.

– [ ô 2  2 + ô (2 õ + 1) + õ 2 ]ñ = 0.  Particular solution: 0 = exp(   + '  ).

5.

ñóò ò ó

ó



−1

ó

– ( ô 2  + õ + ö )ñ = 0.  The transformation H = ž , ¨ = H −  , where , = £ ( Œ , leads to an equation of the form ' ‘ ‘ 2 ¨

+ Œ 2 (2 , + 1)¨2‘
− ( !H +  )¨ = 0. 2.1.2.108: Œ 2 H

Page 246

247

2.1. LINEAR EQUATIONS

6.

7.

ñ óò ò ó

+ ( ô

ó 

4

+

õ 3 ó

+ ö

2

+ [ ô

2

ó ( õ  ó



+ ö )÷ –

The transformation ˆ =  ž ¨2Šª

Š +  ( Œ )−2 ˆ  ¨ = 0. 8.

9.

10.

1 4

–

2

)ñ = 0.

 The transformation ˆ = ž , ¨ =   J ¨2ªŠ

Š + Œ −2 ( ˆ 2 + 'ˆ + ( )¨ = 0.

ñóò ò ó



ó 

ôñóò

+

õ 2D ó ñ

= 0.

óò

+

õ 2D ó ñ

= 0.

ñóò ò ó

+

ñ óò ò ó

– ôñ

ñóò ò ó

+



1 4

2

2

leads to a linear equation of the form 2.1.2.6:

]ñ = 0.

 =  J

+(, ¨

2

leads to an equation of the form 2.1.2.7:

  The transformation ˆ =   , ‡ =   leads to a constant coefficient linear equation of the

−2 form 2.1.2.1: ‡ ŠªŠ + ˜ ‡ = 0.  The substitution ˆ =   leads to a constant coefficient linear equation of the form 2.1.2.1: 
Šª
Š + ˜ −2 = 0.

ôñóò

ó

+ ( õ

+ ö )ñ = 0.

  Solution: =   J 2 V L 1 ø Á/ 2 Œ −1 £    J 2  + L 2 n Á± 2 Œ ø Á (H ) and n Á (H ) are the Bessel functions. −

11.

ñ óò ò ó

–ñ

óò

+ 

ô 3 ó

õ 2 ó

+

The substitution H =  H 2 
‘'
‘ +  !H 3  + ˜H 2  + 12.

ñóò ò ó

– ñó ò + V ô

ñ óò ò ó

+ 2 ô

2



1 4

–



2

ñ

£     J 2  W , where @ = Œ

−1

£  2 − 4( ;

= 0.

leads to a second-order linear equation of the form 2.1.2.121: − 14 Œ 2  = 0.

1 4

ó ( õ  ó



1 4

+

−1

+ ö )÷ +



1 4

–

1 4

2

Wñ = 0.

leads to a second-order linear equation of the form 2.1.2.122: The substitution H =  H 2 
‘'
‘ + V !H 2  ( ˜H  + ( )  + 14 − 14 Œ 2 W = 0. 13.

ó

 ñ óò

ñóò ò ó

+ ( ô + õ ) 

ó ñóò

Œ

ñ óò ò ó

+

ô  ó ñ óò

–

ñ óò ò ó

ó ñ óò

+ 2 ü

¨2

 + (   2 

= exp O −  

+ ( ô

17.

ñóò ò ó

+  

– ( ô + 2 õD

ó )ñ óò

Particular solution:

2

0

0



+L

1

Œ

+

= exp O 2



ó

+

 “

exp O

=

“

+ ( )¨ = 0. + 0

õ 2  2D ó ñ

= exp O







.

P

õ ó

+ )ñ = 0.

   . P

õ  ó ,

 ).

+ )ñ = 0.



õ ó (ô   ó

The substitution ¨



(L

ô ó ( õ  ó

Particular solution: 16.

P

+ )ñ = 0.

+

Particular solution: 15.





= exp O −  

Solution: 14.

ô  ó (ô   ó

+

ü 2  2 ó

+

  

P

+

ü C ó

+ ö )ñ = 0.

leads to a linear equation of the form 2.1.3.5:

= 0.

   . P

Page 247

248 18.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

+ ( ô

2

ó 

óò

+ )ñ

Particular solution: 19.

ñóò ò ó

ó

ñóò ò ó

ó

– ô 0

ó

 ñ

2



=  

= 0.



+ Œ  − .

ó

+ ( ô – )ñó ò + õ 2  ñ = 0.  The substitution ˆ = ž leads to a constant coefficient linear equation: Œ 2 
Šª
Š + Œ 
Š + 

ó

20.

+ ( ô + õ )ñó ò + ö ( ô +  Particular solution: 0 =  − v .

21.

ñ óò ò ó

õ 2 ó )ñ óò

+ (ô +

+ (ô –

Particular solution: 22.

ñ óò ò ó + (ôƒõ ó

õ

+

0

=  

0

ó



−

2

ó )ñ



+   .

(õ – )

2



= (  

– ö )ñ = 0.

– õ

– 3 )ñó ò + ô

Particular solution:

ñóò ò ó



õ

= 0. 2

ó

 ñ

= 0.



+ 1) exp(−    ).

ó

ó

23.

+ (2 ô – )ñó ò + ( ô 2  2  + ö  )ñ = 0. This is a special case of equation 2.1.3.28 with  = , = 0.

24.

ñ óò ò ó

ó

+ (2 ô

+ õ )ñ



óò

+ [ô

 

2 2

= 0.

ó

+ ô (õ + )



ó

+ ö ]ñ = 0.

  The substitution ¨ = exp O   leads to a constant coefficient linear equation of the form Œ P 3 ¨

2 ¨
¨ 2.1.2.11:  +   + ( = 0.

ó

ñóò ò ó

ó

ó

25.

+ ( ô + 2 õ – )ñó ò + ( ö 2   The transformation ˆ = ž  Œ , ¨ ¨2Šª

Š +  ¨2Š
+ ( ¨ = 0.

26.

ñ óò ò ó

+ ( ô + õ )ñ ó ò + [ ö ( ô – ö )  2 + ( ôƒü +  Particular solution: 0 = exp(− (  − ,! ).

27.

ñóò ò ó

+ ( ô + õ )ñó ò + ( q 2  + J + )ñ = 0.  The substitution ˆ =  leads to an equation of the form 2.1.2.146: ˆ 2 
Šª
Š + ( ˆ  +  + 1)ˆ 
Š + ( ƒ…ˆ 2  + ý±ˆ  + = ) = 0.

28.

ñóò ò ó

ó

ó

30.

ó

+ (2 ô

– )ñó ò + ( ô

 

2 2

õ^ö

+ ö – 2 ö³ü ) 

ó

+ ü ( õ – ü )]ñ = 0.

ó

ó

õ 2 ó

+



+

ö  ó

+ ü )ñ = 0.

 Œ  leads to a linear equation of the form 2.1.3.5:   − Œ ¨2

 +    2   + (    + , − 1 Œ 2  ¨ = 0. 2 P 4

ñ óò ò ó

+ (2 ô

ó

ñ óò ò ó + (ô ó

+

õ

exp O

=

óò

M

+

ñ óò ò ó

ó

– )ñ

õ  ó )ñ óò ó

0

+ (ô

ó

ó

ó

ó

+ ôƒõ  + ö 2 + ý + ü )ñ = 0.    −Œ The substitution ¨ = exp O   +  leads to an equation of the form 2.1.3.5: Œ ¨2

 + V (  2   +    + , − 1 (  − Œ )2 W ¨ = 20. P 4



Particular solution: 31.

ó

ó

The substitution ¨

29.

+ ôƒõ + õ 2 – õ )ñ = 0.  =   leads to a constant coefficient linear equation:

+ ô

 

2 2

ó ( õ   ó 

= exp O −  

Œ

ó



P

+ )ñ = 0. .

+   ( ô 2 + õ )ñ ó ò + [   ( õ – ô   Particular solution: 0 =    +   −  .

2



ó

) – ]ñ = 0.

Page 248

249

2.1. LINEAR EQUATIONS

32.

ñ óò ò ó

+ ( ô

ó 

+

õ ó

Particular solution: 33.

ñóò ò ó

ó

+ ( ô

+

õ  ó

ñóò ò ó

+ ( ô

ó

+ 2 õ



+ ( ô



= exp O −  

0

Œ

+ ö )ñó ò + [ ôƒõ

ó 



+

−

“

 



( + )

õ C ó )ñ

ó

– )ñó ò + [ ôƒõ

 

= 0.

   − (˜

ó

+ ô„ö

  − (˜ . 0 = exp O −  “ P 

Particular solution: 34.

óò

+ ö )ñ

( + )

ó

.

ó

P

+

õ C  ó ]ñ

+

õ 2  2 ó

ö 2 ó

+

= 0.

+ õ ( – )



ó ]ñ

= 0.

1 b . If Œ = 0, the equation transforms into 2.1.3.24, and if “ = 0, into 2.1.3.25. 1    2 b . For Œ “ ≠ 0, the transformation ˆ =   , ¨ = exp O   leads to a constant coefficient linear equation: ¨3Šª

Š +  ¨2Š
+ ( ¨ = 0. 35.

ñóò ò ó

+ [ ôƒõ

 

( + )

ó

+

ô ó

Particular solution: 36.

ñóò ò ó

ô

+

v 0 =  −

ó

( ô

+ õ )ñó ò ò ó – ô

2

Particular solution: 38.

(ô

ó

 

2 2

2( ô

ó



+ õ )ñ

=L

óò ò ó

1



  

+



ô 2 õ  (2 + )ó ñ

= 0.

ïƒ÷

) – ö ]ñ = 0.

= 0.



=  

0

– 2 ]ñó ò +

P

.

+ õ )ñó ò ò ó – õƒñó ò – ô

Solution: 39.

ó

 ñ

“

+ 1) exp(−    ).

+ ö [ ô exp( õ

Particular solution: 37.



= (  

0

ïƒ÷ )ñ óò

exp( õ

õ  ó

+

Œ

+ .

ü  óñ

2

2 2 2

= 0.

  + £  2  2 +   + L

ô  ó ñ óò

+



ö^ñ

2

  



+ £  2  2



+ 

−



.

= 0.

The substitution ˆ = X (    +  )−1 J 2  leads to a constant coefficient linear equation of the M form 2.1.2.1: 2 Šª

Š + ( = 0.

40.

ó

( ô

+ õ )ñ



óò ò ó

ó

+ ( ö



( ô

42.

ó (

ó

+ õ )ñ



óò ò ó

+ ( ö

óò

+ ü [( ö – ôƒü ) 

+ ý )ñ

óò

+ (ùJ

 . 0 =  −

Particular solution: 41.

+ ý )ñ

ó





ó

+þ



ý

–õ

ü ]ñ

= 0.



For the case  = 0, see equation 2.1.3.27. For  ≠ 0, the transformation ˆ =  ! , ¨ = ˆ −  , where , is a root of the quadratic equation Œ 2 , 2 + Œ", + = 0, leads to an equation of the form ¡ M 2.1.2.172: Œ 2 ˆ (ˆ +  )¨2Šª

Š + Œ [(2 , Œ + Œ + ( )ˆ +  (2 , Œ + Œ + )]¨3Š
+( , 2 Œ 2 + (, Œ + B )¨ = 0.

ó + ü )ñó ò ò ó + ( ô

ó + õ 

ó

ó ( ˆõ

( ô

+ õ )2 ñó ò ò ó + ö

Particular solution: 44.

+

)ñ = 0.

+ ö ) ñ óò + (ô ó

Integrating yields a first-order linear equation: (  43.

ó

( ô

ó

0

0



ó

ó )ñ



M

–

ó )ñ

= 0.

   + , )
 + (   +    −  + ( ) = L .

= 0.

+ õ )ñó ò + ö

= (  



( +  )  , where , = − . Œ

= (  

+ õ )2 ñó ò ò ó +  ( ô

Particular solution:

– ö

ó + õ C 

ó (

+

ˆõ

– ö

( +  )  , where , = − . Œ

ó )ñ

= 0.

Page 249

250 45.

SECOND-ORDER DIFFERENTIAL EQUATIONS ( ô

ó 

óò ò ó

+ õ )2 ñ

+ ( ô

The substitution ˆ = X

2.1.2.11: Šª

Š + ( 
Š +

46.

ó ( ô 

+õ )

2

ñ óò ò ó

4( ô

ó

+ õ )÷

ñóò ò ó

+ ö )( ô

M

  



¡

0

+ [ ü

+ þxñ = 0.

ó ( ü  ó

óò

+ ö



+ ý )÷

–4

–

– ö

( +  )  , where , = − . Œ

ó (ö  ó



2

óò

+ õ )ñ

+ õ )ñ



= (  

ó 

leads to a constant coefficient linear equation of the form

+ = 0.

ü ó (ô   ó

+

Particular solution: 47.

ó 



2

ó

( ô



ó

+

ˆõ )ñ

= 0.

+ õ )÷ ]ñ = 0.

   +  ¨   J 2 The transformation ˆ = , = leads to an equation of the form 2.1.2.7:   (  +  + ¨42Šª

Š + , ( ¢eŒ )−2 ˆ −  ¨ = 0,(  where M ¢ =  − ˜( . M M

2.1.3-2. Equations with power and exponential functions. 48.

ñóò ò ó

+

ô ó ñóò

ïƒ÷  ó

+ õ (ô

Particular solution: 49.

ñóò ò ó

ó ñóò

+ 2 ô

0

+ (ô

The substitution ¨

¨2

 + ( ' 2  + (˜ 

+ ( ô ï + õ )  Particular solution:

51.

ñóò ò ó

ñóò ò ó

+ (ô

ï  ó

53.

ó



55.

+ ( ô ï + õ ) exp( Particular solution:

ñ óò ò ó + ô ïƒ÷

56.

ï ñóò ò ó

– (2 ô ï

2

+1





0

– ( ô = .

ó )ñ óò

.

= exp O −

ïw÷ )ñ óò

ó



B +1

 

0

–ô = .

3

ôù ïƒ÷

+

0

+1

exp(2

57.

ï ñ óò ò ó

+

ô ï   ó ñ óò

Particular solution:

+

ï 2 )ñ

2

ô  ó 0

ïƒ÷

–1

)ñ = 0.

(1 +

ï )ñ 

=  exp O −  

Œ

2

õ 2 )ñ

= 0.

= 0. –1

)ñ = 0.

.

ïƒú )ñ

Solution: = exp    L 1 ø  ( H ) + L Q 2 n Á ( H ) are the Bessel functions.  1 2

P

ïƒ÷ )ñ

exp( õ

–1

+

õ  ó )ñ

+

– ô exp( =  +  .

ïƒ÷

ó

– ô

ïƒ÷  ó

+ ( ôƒõ

ï

+ö

leads to a linear equation of the form 2.1.2.10:

P

ï  ó

= 

+ 1)ñó ò + 4 õ

)ñ = 0.

.

P

õ ï 2÷

+



–1

ó

−

exp( õ ïƒú ) ñ óò

Particular solution:

 

0

0

ñóò ò ó

Œ

ó

 

– ô ñ = 0. 0 =  +  .

ñ óò ò ó + (ô ïƒ÷ + õ ó )ñ óò

Particular solution: 54.

exp O

+ 2 õ )ñó ò + ( ôƒõ

+ ï ( ô + õ Particular solution:

ñóò ò ó

B +1



ïƒ÷

+ù



+ ô

)¨ = 0.

Particular solution: 52.

ó

 

2 2

ó ñóò

50.

= exp O −

=

−1

õ ï 2÷

–

= 0. = 0.

= 0.

n  ( H )S , where H = Œ 2 

−1

£  exp ݌  2  ; ø Á ( H ) and

= 0.



P

.

Page 250

251

2.1. LINEAR EQUATIONS

58.

ï ñ óò ò ó

+

ô ï   ó ñ óò

– [ô ( õ

Particular solution: 59.

ï ñóò ò ó

+ (ô

ï  ó

ï ñóò ò ó

+ [ô ( õ

ï

62. 63.

65. 66.

ó

+ 1) -

.

ó

1−

+ ( ô ï ÷ + õ  Particular solution:

ï ñ óò ò ó ï ñóò ò ó

ó + (ô ï   

õï

– 1]ñó ò + ôƒõ −

= ( ' + 1) 

0

70.

0

ó

+ [( ô ïƒ÷ + 1)  Particular solution:

73.

ï  ó ñ

0

= 0.

.

õ ó ñ

= 0.

ó

ó

+ [ ô ( õ ïƒ÷ – 1)  =  exp(− '  B ).

ó

õÿù ïƒ÷

+

+ ôù ïƒ÷ + 1 – 2ù ]ñó ò + = (   + 1) exp(−   ).

ó + õ )ñ ó ò

–1

]ñ = 0.

ô 2 ù ï 2÷ –1   ó ñ

= 0.

ó

+ ( ô  + õ C )ñ = 0.   Integrating, we obtain a first-order linear equation:  "
 + (   +    − 1) = L .

ï ñóò ò ó

+ [ô

ïƒ÷

exp( õ

ó ( ï + ô )ñ ó ò ò ó + ( õ  4ï

2

ñóò ò ó

+ [ô

ï 2÷

ïƒú

) + ö ]ñó ò + ô ( ö – 1) ïƒ÷

0

1−

=

óò

+ ö )ñ 0

exp( õ

v.

õ  ó ñ

+

ïƒ÷

= 0.

1 − ( −    +

= exp O X

)+1–ù

2



–1

exp( õ

ïƒú )ñ

= 0.

 . M P

]ñ = 0.

= 



−1 2

leads to a linear equation of the form 2.1.3.1:

ó

ï 2 ñ óò ò ó

+ 2 ô ï ñó ò + [( õ 2  2 } – b 2 ) ö 2 ï 2 + ô ( ô – 1)]ñ = 0.   Solution: =  −  V L 1 ø Á (   v )+ L 2 n Á (   v )W , where ø Á ( H ) and n Á ( H ) are the Bessel functions.

ï 2 ñ óò ò ó

+

ô ï   ó ñ óò

ï 2 ñ óò ò ó

+ ï ( ô

ó

ï 4 ñ óò ò ó

ó

+ õ ( ô 0

=



−

ó

.

– 1)ñ = 0.

ï

− −v  . 0 =   

+ (  2 ) – b 2 )ñ = 0.  Solution: =  V L 1 ø Á (  1 J ) + L

ï 4 ñ óò ò ó

õ

–

+ 2 õ )ñó ò + [ ô ( ö

Particular solution: 72.

2

+ õ ïƒ÷ ) ñ óò

Particular solution: 71.

+ 2)]ñ = 0.

+ ô ï ÷ –1 ( õ  + ù – 1)ñ = 0. = exp(−   B ).

The transformation ˆ = '  , ¨ ¨42Šª

Š +  ( 'B )−2  Š ¨ = 0. 69.





ï

= 0.

ó )ñ ó ò

ï ñóò ò ó

ï ñ óò ò ó + (ô  ó

óñ

.

+ [( ô ï 2 + õ ï )  + 2]ñó ò + Particular solution: 0 =  +   .

Particular solution: 68.

+

0

ï ñóò ò ó

Particular solution:

67.

=

0

Particular solution: 64.



+ õ (õ

+ õ )ñó ò + ô ( õ – 1) 

Particular solution:

61.



ó 

+ 1)

= 

0

Particular solution: 60.

ï

2

+ õ ) 

ó

–ö

2

ï

2

+ õ ( õ – 1)]ñ = 0.

 n Á (  1 J )W , where ø Á ( H ) and n Á ( H ) are the Bessel functions.

+ [ ô exp(2 # ï ) + õ exp( # ï ) + ö ]ñ = 0. The transformation ˆ = 1  , ¨ =   leads to a linear equation of the form 2.1.3.5: ¨2Šª

Š + (   2  Š +   Š + ( )¨ = 0.

Page 251

252 74. 75.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ó

ï 4 ñ óò ò ó

ó

+ ô ï 2   ñ óò + [ô ( õ – ï )  – Particular solution: 0 =  exp(   ).

(ï

õ ó (ï

+ ô )2 ñó ò ò ó +

2

ïƒ÷

(ô

( ô



ïƒ÷

+ õ )2 ñó ò ò ó + ö ( ô

ó

õï

+

0

óò ò ó

+ ö )ñ

79.

0

= exp O − X 2

0

ó

  ñ 

=  

+ ö ]ñó ò ò ó – ö

Particular solution:

ó ñóò

+ õ ) 

– ô

Particular solution:

ó [( ô ï + õ )  

– ïƒ÷

–2

 +  )1 J . 0 = (

Particular solution: 78.

ó ñóò

( ïƒ÷ + ô )2 ñó ò ò ó + õ ( ïƒ÷ + ô )  Particular solution:

77.

+ .

2

2 −

=(  

ñ

= 0.

ï  ó

+ ô )ñó ò – ( õ

£  0 =

Particular solution:

76.

2

õ 2 ]ñ

+ ô )ñ = 0.

ï  ó

(õ

ó

+ ( ö

ïƒ÷

– ôù



ôù

+

– ô )ñ = 0. – 1)ñ = 0.

–1

. M   +  P = 0.

+ ' + ( . = 0.



+  +  .

2.1.4. Equations Containing Hyperbolic Functions 2.1.4-1. Equations with hyperbolic sine. 1.

ñóò ò ó

+ ( ô sinh2 ï + õ )ñ = 0.

ñ óò ò ó

+

Applying the formula sinh2  = 12 cosh 2 − 12 , we obtain the modified Mathieu equation 2.1.4.9: 

 +   − 12  + 12  cosh 2  = 0. 2.

ô

ï )ñ ó ò

sinh(

Particular solution: 3.

ñóò ò ó

+ [ ô sinh ÷ (

ï

5.

0 =  

) – õ ]ñ = 0.

.

) + ö ]ñó ò + ôƒõ sinh ÷ ( 0

=

−





.

+ ( ô ï + õ ) sinh ÷ ( ï )ñó ò – ô sinh ÷ ( Particular solution: 0 =  +  .

ñóò ò ó

+ ( ô sinh ÷

ï ñ óò ò ó

ï

Particular solution: 6.



−

Particular solution: 4.

ï

+ õ [ ô sinh(

sinh2 ( ô

ï )ñ óò ò ó

–

õÿñ

+

õï ú

0

+1

óò

)ñ

= exp O −

= 0.

¡

+



õ ï ú (ô

+1

 +

ï )ñ

= 0.

ï )ñ

= 0.

sinh÷ +1

P

ï

+þ

)ñ = 0.

.

H

The substitution  = A ln £ ( H > 0) leads to a linear equation of the form 2.1.2.190: H 2+1 ' ‘ ‘ ‘ 


2 2 −2 H ( H + 1) + (3 H + 1) − 4  ˜H = 0. 7.

ñóò ò ó

sinh2 ï – [ ô

Solution: 8.

sinh2 ï + ù (ù – 1)]ñ = 0,   1 = sinh   O M (L 1   + L sinh   P

ô

2

M

2

≠ 0;

  −  ).

ù

= 1, 2, 3,

uuu

[ ô sinh( ï ) + õ ï + ö ]ñ„ó ò ò ó – ô 2 sinh( ï )ñ = 0. Particular solution: 0 =  sinh( Œ  ) + ' + ( .

Page 252

253

2.1. LINEAR EQUATIONS

2.1.4-2. Equations with hyperbolic cosine. 9.

– ( ô – 2 cosh 2 ï )ñ = 0. The modified Mathieu equation. The substitution  = ™Ýˆ leads to the Mathieu equation 2.1.6.29: Šª

Š + (  − 2  cos 2ˆ ) = 0.

ñóò ò ó

For eigenvalues  =   (  ) and  =   (  ), the corresponding solutions of the modified Mathieu equation are:

½

# 22  ++K cosh[(2 , +  ) ], K  =0 ½ Se2  + ( ,  ) = −™ se2  + (™Ý ,  ) = ÷ $ 22  ++K sinh[(2 , +  ) ], K K K  =0 where  can be either 0 or 1, and the coefficients # 22  ++K and $ 22  ++K are specified in 2.1.6.29. K K The modified Mathieu equation is discussed in the books by Abramowitz & Stegun (1964), Ce2 

+

( ,  ) = ce2 

K

+

K (™Ý ,  ) = ÷

Bateman & Erd´elyi (1955, vol. 3), and McLachlan (1947) in more detail. 10.

ñóò ò ó

+ ( ô cosh2 ï + õ )ñ = 0.

ñ óò ò ó

+

Applying the formula cosh 2 = 2 cosh2  − 1, we obtain the modified Mathieu equation 2.1.4.9: 

 +  12  +  + 12  cosh 2  = 0. 11.

ô

cosh(

ï )ñ ó ò

+ õ [ ô cosh(

Particular solution: 12.

ñóò ò ó

+ [ ô cosh÷ (

ï

0

14. 15.

0

18.

+ ô ïƒ÷ coshú ( Particular solution:

ï )ñ óò

cosh÷ (

ï )ñ óò

ñóò ò ó

ï ñóò ò ó

+

ô ï

ï ñóò ò ó

+ [ô

ï

cosh÷ (

ï )ñ

0

ï

0

.

0

ïƒ÷

–ô = .

– [ô ( õ





= 

= 0.

ï )ñ

ï

–1

coshú (

= 0.

ï )ñ

+ 1) cosh ÷ (

= 0.

ï

) + õ (õ

) + õ ]ñó ò + ô ( õ – 1) cosh ÷ ( =

1−

ï 2 ñ óò ò ó

+

ô ï

cosh÷ (

ï )ñ óò

.

+ õ [ ô cosh÷ (

−

ï

+ 2)]ñ = 0.

.

+ [( ô ï 2 + õ ï ) cosh÷ ( ï ) + 2]ñó ò + õ cosh÷ ( Particular solution: 0 =  +   .

ï ñóò ò ó

Particular solution: 19.

.

+ ( ô ï + õ ) cosh ÷ ( ï )ñó ò – ô cosh÷ ( Particular solution: 0 =  +  .

Particular solution: 17.





) – õ ]ñ = 0.

ñóò ò ó

Particular solution: 16.

=

−



ï

) + ö ]ñó ò + ôƒõ cos÷ (

Particular solution: 13.

=

−



.

0

=

0

=  − .

ï

)–

õ

ï )ñ ï )ñ

= 0.

= 0.

– 1]ñ = 0.

( ô cosh ï + õ )ñ„ó ò ò ó + ( ö cosh ï + ý )ñ„ó ò + [( ö – ô ) cosh ï + ý – õ ]ñ = 0. Particular solution:



Page 253

254

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï )ñ óò ò ó

20.

cosh2 ( ô

õÿñ

21.

[ ô cosh( ï ) + õ ï + ö ]ñ ó ò ò ó – ô 2 cosh( ï )ñ = 0. Particular solution: 0 =  cosh( Œ  ) + ' + ( .

–

= 0. H 1 The substitution  = ln (0 < H < 1) leads to the hypergeometric equation 2.1.2.171: 2  1 − H ' ‘ ‘ ‘


H ( H − 1) + (2 H − 1) +  −2  = 0.

2.1.4-3. Equations with hyperbolic tangent.

ï

22.

ñóò ò ó

+ [ ô tanh(

) + õ ]ñ = 0. 1 − tanh( Œ  ) ¨ The transformation H = , = H − J  , where , is a root of the quadratic 1 + tanh( Œ  ) equation 4 , 2 +  −  = 0, leads to a linear equation of the form 2.1.2.172: 4 Œ 2 H ( H + 1)¨2‘'

‘ + ‘ 4 Œ (2 , + Œ )( H + 1)¨3
+ (4 , 2 +  +  )¨ = 0.

23.

ñóò ò ó

– 4ô

24.

tanh2 (3 ô

ï )ñ

Particular solution:

0

ñ óò ò ó

2

ñóò ò ó

= sinh(3  )[cosh(3  )]−1 J 3 .

ï )]ñ

+ [ ô – ô ( ô + ) tanh2 (

Particular solution: 25.

= 0.

+ [3 ô –

2

0

= 0.

= [cosh( Œ  )]  J  . −

– ô ( ô + ) tanh2 (

Particular solution:

0

ï )]ñ

= sinh( Œ  )[cosh( Œ  )]−  J  .

26.

+ ôñó ò – [ + ô tanh( ï )]ñ = 0. Particular solution: 0 = cosh( Œ  ).

27.

ñóò ò ó

ñóò ò ó

+ 2 tanh ï

Solution: 28.

ñóò ò ó

+

ô

ñ„óò

30. 31.

32.

+ ôñ = 0.

cosh  =

tanh(

ï )„ñ óò

Particular solution: 29.

= 0.

™

L

cos( ' ) + L 2 sin( ' ) if  − 1 =  2 > 0, L 1 cosh( ' ) + L 2 sinh( ' ) if  − 1 = −  2 < 0. 1

+ õ [ ô tanh( 0

=

−





ï

) – õ ]ñ = 0.

.

+ 2 tanh ï ñ„ó ò + ( ô ï 2 + õ ï + ö )ñ = 0. The substitution ¨ = cosh  leads to a second-order linear equation of the form 2.1.2.6: ¨2

 + (  2 + ' + ( − 1)¨ = 0.

ñóò ò ó

+ 2 tanh ï ñ„ó ò + ( ô ïƒ÷ + 1)ñ = 0. ¨ 

 +   ¨ = 0. The substitution ¨ = cosh  leads to a linear equation of the form 2.1.2.7: I

ñóò ò ó

+ 2 tanh ï ñ„ó ò + ( ô ï 2÷ + õ ïƒ÷ –1 + 1)ñ = 0. The substitution ¨ = cosh  leads to a second-order linear equation of the form 2.1.2.10: ¨2

 + (  2  + '  −1 )¨ = 0.

ñóò ò ó

+ (2 tanh ï + ô )ñ„ó ò + ( ô tanh ï + õ )ñ = 0. ¨ 

 +  2 ¨ 
+ The substitution ¨ = cosh  leads to a constant coefficient linear equation: I ¨ (  − 1) = 0.

ñóò ò ó

Page 254

255

2.1. LINEAR EQUATIONS

33.

+ ô tanh÷ ( ï )ñó ò – [ + ô tanh÷ Particular solution: 0 = cosh( Œ  ).

34.

ñóò ò ó

ñóò ò ó

+ [ ô tanh ÷ (

ï

) + ö ]ñó ò +

Particular solution: 35. 36. 37.

0

40.

+ ô ï ÷ tanhú ( Particular solution:

ï )ñ ó ò

tanh÷ (

ï )ñ óò

ñ óò ò ó

ï ñóò ò ó

+

ô ï

0

ï ñ óò ò ó

ï

+ [ô

tanh ÷ (

ï

+ ( ô tanh÷

ï ñ óò ò ó

ï

ï 2 ñ óò ò ó

+

ô ï

+





ï )ñ

=

–1

tanhú (

= 0.

ï )ñ

+ 1) tanh÷ (

ï

= 0.

ï

) + õ (õ

õï ú

+ ô ( õ – 1) tanh÷ (

óò

.

+1

)ñ

óò

= exp O −

ï )ñ ó ò

¡



+1

 +

.

0

=

0

=  − .

tanh÷

õ ï ú (ô

+

+ õ [ ô tanh ÷ (

−

ï

+ 2)]ñ = 0.

.

) + õ ]ñ

0

tanh÷ (

ï )ñ

+1

ï

P

ï

= 0.

ï )ñ

= 0.

+þ

)ñ = 0.

. ) – õ – 1]ñ = 0.

( ô tanh ï + õ )ñ„ó ò ò ó + ( ö tanh ï + ý )ñ„ó ò + [( ö – ô ) tanh ï + ý – õ ]ñ = 0. Particular solution:

43.

ï ÷

– [ô ( õ

1−

= 0.

.

–ô = . = 

= 0.

+ [( ô ï 2 + õ ï ) tanh÷ ( ï ) + 2]ñó ò + õ tanh÷ ( Particular solution: 0 =  +   .

ï ñóò ò ó

Particular solution: 42.

0

0

Particular solution: 41.



ï )ñ

+ ( ô ï + õ ) tanh ÷ ( ï )ñó ò – ô tanh÷ ( Particular solution: 0 =  +  .

Particular solution: 39.



tan÷ (

ôƒõ

ï )]ñ

(

ñóò ò ó

Particular solution: 38.

=

−

+1

) + õ ]ñ„ó ò ò ó + [ ö tanh( ï ) + ý ]ñ„ó ò + [ù tanh( ï ) + þ ]ñ = 0. 1 + tanh( Œ  ) ¨ The transformation H = , = H − J  , where , is a root of the quadratic equation 1 − tanh( Œ  ) 4(  −  ) , 2 + 2( ( − ) , + B − = 0, leads to a linear equation of the form 2.1.2.172:

[ ô tanh(

ï



¡

M

4 Œ 2 H [(  +  ) H +  −  ]¨ ‘'

‘ + 2 Œ {[2(2 , + Œ )(  +  ) + ( + ] H + 2(2 , + Œ )(  −  ) + − ( }¨ ‘
M M + [4(  +  ) , 2 + 2( ( + ) , + B + ]¨ = 0.

M

¡

2.1.4-4. Equations with hyperbolic cotangent. 44.

45.

ñóò ò ó

+ [ ô coth(

ï

) + õ ]ñ = 0. 1 − tanh( Œ The transformation H = 1 + tanh( Œ equation 4 , 2 +  −  = 0, leads to 4 Œ (2 , + Œ )( H − 1)¨3‘
+ (4 , 2 +  +  )¨ coth2 (3 ô

ï )ñ

Particular solution:

0

ñóò ò ó

– 4ô

2

 ) ¨ ,  )

= H −  J  , where , is a root of the quadratic

‘'‘

an equation of the form 2.1.2.172: 4 Œ 2 H ( H − 1)¨2

+ = 0.

= 0.

= cosh(3  )[sinh(3  )]−1 J 3 .

Page 255

256 46.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

Particular solution: 47. 48.

ñóò ò ó

ï )]ñ

+ [ ô – ô ( ô + ) coth2 ( + [3 ô –

= [sinh( Œ  )]  J  .

0

– ô ( ô + ) coth2 (

2

0

ñ óò ò ó

+ õ [ ô coth(

ô

coth(

ï )ñ ó ò

Particular solution:

= 0.

=

0





−

.

ï

) – õ ]ñ = 0.

coth÷ (

ï )ñ

= 0.

50.

+ ( ô ï + õ ) coth ÷ ( ï )ñ ó ò – ô coth÷ ( Particular solution: 0 =  +  .

ï )ñ

= 0.

51.

ñóò ò ó

49.

ñóò ò ó

+ [ ô coth÷ (

ï )]ñ

= cosh( Œ  )[sinh( Œ  )]−  J  .

Particular solution: +

= 0.

−

ï

) + ö ]ñó ò +

Particular solution:





.

ñ óò ò ó

+ 2ù coth ï

Solution: 52.

=

0

−

ôƒõ

ñóò

+ (ù

1 = O sinh 



M M

ï )]„ñ óò ò ó

– ô 2 )ñ = 0,

2

3P

ù

uuu

= 1, 2, 3,

  ( L 1   + L 2  −  ).

[ ô + õ coth( + [ ö + ý coth( ï )]ñ„ó ò + [ù + þ coth( ï )]ñ = 0. Multiply this equation by tanh( Œ  ) to obtain equation 2.1.4.43.

2.1.4-5. Equations containing combinations of hyperbolic functions. 53.

ñóò ò ó

ï

– ô [ ô cosh2 ( õ

) + õ sinh( õ

Particular solution: 54.

ñ óò ò ó

ï

– ô [ ô sinh2 ( õ

ñóò ò ó

Q 

) + õ cosh( õ

Particular solution: 55.

= exp

0



= exp

0



Q 

ï )]ñ

= 0.

sinh( ' )S .

ï )]ñ

= 0.

cosh( ' )S .

+ ( ô cosh2 ï + õ sinh2 ï + ö )ñ = 0.

Apply the formulas 2 sinh2  = cosh(2 ) − 1 and 2 cosh2  = cosh(2 ) + 1 to obtain an  −  + equation of the form 2.1.4.9: 

 + +( + cosh(2 )S = 0. Q 2 2 56.

+ ô sinh( ï )ñ„ó ò – [ + cosh( Particular solution: 0 = sinh( Œ  ).

ñóò ò ó

57.

+ ô cosh( ï )ñ Particular solution:

58.

ñóò ò ó

ñ óò ò ó

–

tanh(

Solution: 59.

ñóò ò ó

– tanh ï

Solution:

=L

ñóò

óò

Р[ + sinh( 0 = cosh( Π ).

ï )„ñ óò 1

–ô

exp

+

ô

2

= £ sinh 

Q Œ

2



ï )]ñ

= 0.

ï )]ñ

= 0.

ï )ñ

cosh2 (

sinh( Π )S + L

= 0. 2

coth2 ï (sinh ï )2ú

Q

L

1

ø

2

1

n Á ( H ) are the Bessel functions.

+

O ¡



exp −



Q Œ

–2

ñ

+ sinh 

sinh( Π )S .

P

= 0. +L

2

n

2

1

+

O ¡



+

sinh 

P

S , where ø Á ( H ) and

Page 256

257

2.1. LINEAR EQUATIONS

60.

61.

sinh ÷ (

ï )ñ óò ò ó

+ [ ô cosh÷

¨2Šª

Š + Œ

−2 −

cosh ÷ (

ï )ñ ó ò ò ó

¨2Šª

Š + Œ

−2 −  ¨ ˆ = 0.

–4

ï

(

The transformation ˆ = tanh( Œ  ), ¨

=

ˆ  ¨ = 0.

+ [ ô sinh÷

–4

ï

(

sinh ÷ (

2

)–

cosh÷ (

The transformation ˆ = coth( Œ  ), ¨

=

= 0.

leads to an equation of the form 2.1.2.7:

cosh( Π )

2

)–

ï )]ñ ï )]ñ

= 0.

leads to an equation of the form 2.1.2.7:

sinh( Π )

2.1.5. Equations Containing Logarithmic Functions 2.1.5-1. Equations of the form  ( )

 + ( ) = 0. 1.

ñóò ò ó

– (ô

2

ï

2

ln2 ï + ô ln ï + ô )ñ = 0.

Particular solution: 2.

ñóò ò ó

– (ô

2

ï 2÷

0

ln2 ï + ôù

Particular solution: 3.

0

7.

8.

ï ñ óò ò ó

– [ô

ï 2 ñ óò ò ó

10.

ln ï + ô

=  −Ž  (

2

ï

ln2÷ ( õ

ï

) + ôù ln÷ 0

+1)

–1

ïƒ÷

ï )]ñ

(õ

= 0.

Q

M

+ ( ô ln2 ï + õ ln ï + ö )ñ = 0.

The transformation ˆ = ln  , ¨ ¨2Šª

Š + ( ˆ 2 + 'ˆ + ( − 1 )¨ = 0. 4

ï 2 ñ óò ò ó

=

+ V ô ( õ ln ï + ö )÷ + 41 UW ñ = 0.

The transformation ˆ =  ln  + ( , ¨ ¨2Šª

Š +  −2 ˆ  ¨ = 0.

ï

2

ln( ô

ï )ñ óò ò ó

ï (ô ï

ln ï

J

= 



J

−1 2

−1 2

= 

leads to an equation of the form 2.1.2.2:

J

leads to an equation of the form 2.1.2.6:

−1 2

leads to an equation of the form 2.1.2.7:

+ ñ = 0. =L

1

ln(  ) + L

+õ ï

+ ö )ñ Particular solution:

ï 2(ô

)ñ = 0.

 +1 Ž , where  =   . (B + 1)2

+ õ )ñ = 0. The transformation ˆ =  ln  +  − 14 , ¨ ¨2Šª

Š +  −2 ˆ ¨ = 0.

ï 2 ñ óò ò ó

–1

= exp lX ln  ( ' )  S .

+ ( ô ln ï

Solution: 9.

–1

– ( ô 2 ï ln2 ï + ô )ñ = 0.   Particular solution: 0 =  −    .

Particular solution:

6.

ïƒ÷

ï ñ óò ò ó

4.

5.

2 2 =  − J 4   J 2.

ln ï + õ ï + ö )ñ Particular solution:

óò ò ó 0

óò ò ó 0

2

ln(  ) X [ln(  )]−2  .

M

– ôñ = 0. =  ln  + ' + ( . + ôñ = 0. =  ln  + '  + ( .

Page 257

258

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.1.5-2. Equations of the form  ( )

 + ( )
 + % ( ) = 0. 11. 12. 13. 14. 15.

+ ô ln ÷ ( õ ï )ñó ò + ö [ ô ln ÷ ( õ  Particular solution: 0 =  − v .

17. 18. 19.

+ [ ô ln÷ ( õ ï ) + ö ]ñó ò + ô„ö ln ÷ ( õ  Particular solution: 0 =  − v . + ( ô ï + õ ) ln ÷ ( ö Particular solution:

ñóò ò ó

0

+ ô ïƒ÷ lnú ( õ ï )ñó ò – ô ïƒ÷ Particular solution: 0 =  .

ï ñóò ò ó

+

ô ï

ln ï

22. 23.

0

ï ñóò ò ó

25. 26.

= 0.

ï )ñ

= 0.

ln ú ( õ

–1

ï )ñ

= 0.

 =  

1−

 .

+ ( ô ï ln ï + õ )ñ„ó ò + ( ôƒõ ln ï + ô )ñ = 0.   Particular solution: 0 =    −  . + (2 ô ï ln ï + 1)ñ„ó ò + ( ô   Solution: =    − (L 1 + L

ï ñóò ò ó

ï

ln2 ï + ô ln ï + ô )ñ = 0. 2 ln  ). 2

ï ñóò ò ó

+ ln ï ( ô ï + õ )ñ„ó ò + ô ( õ ln2 ï + 1)ñ = 0.   Particular solution: 0 =    −  .

ï ñóò ò ó

+

ln÷ ( õ

ô ï

ï ñ óò ò ó

+

ln÷

ô ï

+ ôù ln÷

ï )ñ óò

0

ï ñ óò

–1

Q

÷

+ ( ô ï ln ÷ ï + 1)ñó ò – ô ln÷ Particular solution: 0 = ln  .

ï ñóò ò ó

ï

ln ÷

ï

–1

ñ

M

–1

–1

=

1−

.

ï )ñ

 . M P

ï ñ

= 0.

ï ñ

= 0.

+ [( ô ï 2 + õ ï ) ln÷ ( ö ï ) + 2]ñó ò + õ ln ÷ ( ö Particular solution: 0 =  +   .

ï ñóò ò ó

+ ( ô ïƒ÷ + õ ln ú Particular solution:

ï ñóò ò ó

+ ô ïƒ÷ –1 ( õ lnú = exp(−   B ).

ï )ñ óò

0

= 0.

= 0.

+ õ )ñó ò + ô ( õ – 1) ln ÷ 0

= 0.

=  exp O − lX ln 

+ ( ô ï ÷ ln ï + 1)ñ ó ò – ô ï Particular solution: 0 = ln  .

+ (ô

ï )ñ

+ ôù ln÷

ï

ï ñ óò ò ó

ï ñóò ò ó

(õ

= exp − lX ln  ( ' )  S .

+ ( ô ln÷ 0

Particular solution: 24.

ï )ñ

+ ô (ln ï + 1)ñ = 0.

ñóò

Particular solution: 21.

– ô ln÷ ( ö =  +  .

ï )ñ óò

ñóò ò ó

Particular solution: 20.

) – ö ]ñ = 0.

ñóò ò ó

Particular solution:

16.

ï

ñóò ò ó

ï

ï )ñ

+ ù – 1)ñ = 0.

+ ( ô ïƒ÷ + õ ï lnú ï )ñó ò + [ õ ( ô ïƒ÷ – 1) lnú Particular solution: 0 =  exp(−   B ).

ï ñóò ò ó

= 0.

ï

+ ôù

ïƒ÷

–1

]ñ = 0.

Page 258

259

2.1. LINEAR EQUATIONS

ï 2 ñ óò ò ó

27.

+

=

Solution: where 28.

29.

30.

31.

32.

ø

+ ô ln÷ ( õ

ï ñóò

ü

ln( ' ) ¬L

+

ln ( ' ) + L

2

•

+ ï (2 ô ln ï + 1)ñ„ó ò + ( ï

n

2

1

+

”

£  ¡

+

ln ( ' ) ® ,

•

¡

=

ï 2 ñ óò ò ó

+ ï (2 ln ï + ô + 1)ñ„ó ò + (ln2 ï +

ï 2 ñ óò ò ó

+ ï (2 ln ï + ô )ñ„ó ò + [ln2 ï + ( ô – 1) ln ï +

2

+ô

2

1 (B + 2), 2

−1

) = 0.

ln2 ï + õ )ñ = 0.

¨ Š
+ ¨ ªŠ

Š +  2 The substitution = ¨ exp(− 21  ln2  ) leads to the Bessel equation 2.1.2.126:  2 2 ¨ ( 2 +  −  ) = 0.

ô

ln ï + õ )ñ = 0.

The transformation ˆ = ln  , ¨ = exp( 21 ln2  ) leads to a constant coefficient linear equation: ¨2Šª

Š +  ¨2Š
+ (  − 1)¨ = 0.

õ ïw÷

+ ö ]ñ = 0.

The substitution ¨ = exp( 12 ln2  ) leads to a linear equation of the form 2.1.2.132:  2 ¨2

 +  ¨2
+ ( '  + ( − 1)¨ = 0. + ô ï ln÷ ( õ ï )ñó ò + ö [ ô ln÷ ( õ Particular solution: 0 =  − v .

ï

ï 2 ñ óò ò ó ï 2 ñ óò ò ó

+ ï (ô

ïƒ÷

+ õ ln ï )ñó ò + õ ( ô

ï (ï

ïƒ÷

+ ô )ñ„ó ò ò ó + ï ( õ ln ï + ö )ñ„ó ò +

+ ô ï 2 ln ÷ ( õ Particular solution:

ï 4 ñ óò ò ó

) – ö – 1]ñ = 0. ln ï – ln ï + 1)ñ = 0.

= exp  − 12  ln2   .

0

Particular solution:

õÿñ

= 0.  ln  + ( − 1  .  + M P

= exp O − X

0

0

+ [ ô ( ö – ï ) ln ÷ ( õ =  exp( (  ).

0

=  − .

ï )ñ ó ò

ï

) – ö 2 ]ñ = 0.

( ô ln ï + õ )ñ„ó ò ò ó + ( ö ln ï + ý )ñ„ó ò + [( ö – ô ) ln ï + ý – õ ]ñ = 0. Particular solution:

ï

ln ï

37.

ñóò ò ó

– ùñó ò – ô

Solution:

ï

ln( ô

38.

ï )„ñ óò ò ó

Solution:

ï

ln2 ï

ñóò ò ó

2



ï

(ln ï )2÷

+1

ñ

= 0.

= L 1   Ž + L 2  −  Ž , where  = X ln  – [ù ln( ô

ï

)+þ

]ñ„ó ò –

õ 2 ï 2÷

+1

ln2ú

+1

M (ô

 .

ï )ñ

= 0.

+ = L 1   Ž + L 2  −  Ž , where  = X   ln (  )  . M + (ô

ï

+ 1) ln ï

The substitution ˆ = X 40.

£ 

”

1

ï 2 ñ óò ò ó

34.

39.

ø

2+ ¡ functions. Á ( H ) and n Á ( H ) are the Bessel

Particular solution:

36.

1

= 0.

+ ï ñ ó ò + ( ô ln2÷ ï + õ ln ÷ –1 ï )ñ = 0. The substitution ˆ = ln  leads to an equation of the form 2.1.2.10:
Šª
Š + ( ˆ 2  + 'ˆ 

ï 2 ñ óò ò ó

33.

35.

ï )ñ

ln ÷ ( ô ï )ñó ò ò ó + ( õ 2 – ï Particular solution:

M



ln 

ñ„óò

+

õï ñ

= 0.

leads to a constant coefficient linear equation: 
Šª
Š +  
Š + 

= 0.

)ñó ò + ( ï + õ )ñ = 0. 0 =  − . 2

Page 259

260

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.1.6. Equations Containing Trigonometric Functions 2.1.6-1. Equations with sine. 1.

+ ô 2 ñ = õ sin( ï ). Equation of forced oscillations.


ñóò ò ó

=

Solution:

2.

3.

4.






6. 7.

ñóò ò ó

+

ô

sin( õ

ï )„ñ óò

ïƒ÷

+ ö [ô 0

ï

sin( õ

= exp O −

(

)–ö

 

B +1

+ ô sin ÷ ( õ ï )ñó ò + ö [ ô sin÷ ( õ  Particular solution: 0 =  − v .

ï

ñóò ò ó

ï 2÷ +1

= 0.

+ ( ô ï + õ ) sin ÷ ( ö ï )ñó ò – ô sin ÷ ( ö Particular solution: 0 =  +  .

ï )ñ

= 0.

ñóò ò ó

+

ô ïƒ÷

sin ú ( õ

ï )ñ óò

Particular solution:

0

+ ö [ô

sinú ( õ

–1

ïƒ÷ +

= exp O −

ï ñóò ò ó

(

ï )ñ

sin ú ( õ

, +1

 

+1

= 0.

ï

)–ö

+ (ô

ïƒ÷

+1

+ õ sinú'ï )ñó ò +

Particular solution:

0

= exp O −

ô ïƒ÷ ( õ

   B +1

ï )ñ

ï ñóò ò ó

+ ô ïƒ÷ sinú ( õ Particular solution:

ï ) ñ óò 0

– [ô (ö  =  v .

ï

+

ü ï –1 ]ñ

= 0.

= 0.

sinú'ï + ù )ñ = 0.

+1

P

.

+ ( ô ï ÷ + õ ï sin ú ï )ñ ó ò + [ õ ( ô ï ÷ – 1) sin ú Particular solution: 0 =  exp(−   B ).

ï ñ óò ò ó

ï 2

.

P

+ [( ô ï 2 + õ ï ) sin÷ ( ö ï ) + 2]ñó ò + õ sin÷ ( ö Particular solution: 0 =  +   .

ï ñóò ò ó

]ñ = 0.

.

P

ï )ñ

ïƒ÷

–1

) – ö ]ñ = 0.

+ [ ô sin ÷ ( õ ï ) + ö ]ñ ó ò + ô„ö sin÷ ( õ  Particular solution: 0 =  − v .

ñóò ò ó

ïƒ÷

+ù

ñ óò ò ó

ñóò ò ó

13.

, we obtain the Mathieu equation 2.1.6.29:



+ ( ô sin2 ï + õ )ñ = 0. Applying the formula 2 sin2  = 1 − cos 2 , we obtain the Mathieu equation 2.1.6.29: 

 + ( 1  +  − 1  cos 2 ) = 0. 2 2

9.

12.

1 2

2

ñóò ò ó

+ ô ïƒ÷ sin ú ( õ ï )ñó ò – ô Particular solution: 0 =  .

11.

cos(  ) +

+ [ ô sin( ï ) + õ ]ñ = 0. Applying the substitution Œ  = 2ˆ + 
Šª
Š + (4 Œ −2 cos 2ˆ + 4 Œ −2 ) = 0.

8.

10.

2

ñóò ò ó

Particular solution: 5.

1

sin(  ) + L



sin( Œ  ) if  ≠ Œ ,  −Œ 2  L 1 sin(  ) + L 2 cos(  ) −  cos(  ) if  = Œ . 2

L

+ 1)ïw÷

–1

sinú ( õ

ï ï

+

ôù ï ÷

)+ö

2

ï

–1

]ñ = 0.

+ 2 ö ]ñ = 0.

Page 260

261

2.1. LINEAR EQUATIONS

14.

ï ñóò ò ó

+ [ô

sin ú ( õ

ïƒ÷

Particular solution: 15.

ï 2 ñ óò ò ó

+ ï ( ô sin÷

0

ï

17.

ï 2 ñ óò ò ó

+ ï ( ô sin÷

0

ï

19.

0

ï 2 ñ óò ò ó

ï )ñ óò

+

sin ú ( õ

ô ïƒ÷

ï 4 ñ óò ò ó

ï

+ [ ô sin( #

−

−

ï 4 ñ óò ò ó

+

ô ï

2

sin ÷ ( õ

ïƒ÷

v. 0 = 

0

sin(2 ï ) ñ„ó ò ò ó – ñó ò + 2 ô =L

sin2 ï

ñóò ò ó

1

–1

−

) + õ ]ñ = 0.

ï ) ñ óò

ï

.

+ ö [ô

sinú ( õ

=  

ñ

sin( ‡ ) + L

2

= 0.

ï

) – ö – 1]ñ = 0.

leads to a linear equation of the form 2.1.6.2:

+ [ ô ( ö – ï ) sin ÷ ( õ

sin2 ï

ï )ñ

– 1)ñ = 0.

ï

) – ö 2 ]ñ = 0.

=  exp( (  ).

2

sin ú ( õ

– õ )ñ = 0.

.

The transformation ˆ = 1  , ¨ ¨2Šª

Š + [  sin( Œ ˆ ) +  ]¨ = 0.

Solution: 21.

=

Particular solution:

Particular solution: 20.

=

ï

+ õ )ñó ò + õ ( ô sin÷

Particular solution: 18.

=

–1

v.

1−

+ 1)ñó ò + õ ( ô sin÷

Particular solution: 16.

) + ö ]ñó ò + ô ( ö – 1) ïƒ÷

ï

= 0. cos( ‡ ), where ‡ = X

£ tan  M

 .

+ ôñ = 0.

This is a special case of equation 2.1.6.23. 22.

sin2 ï

ñóò ò ó

– [ ô sin2 ï + ù (ù – 1)]ñ = 0, 1 = sin  ‹  O sin 

Solution: 23.

sin2 ï

ñóò ò ó

M M



3P

ù

= 1, 2, 3,

 2  2 Ý L 1   + L 2  −   .

uuu

+ ( ô sin2 ï + õ )ñ = 0.

Set  = 2ˆ . Applying the trigonometric formulas sin 2ˆ = 2 sin ˆ cos ˆ and  =  (sin 2 ˆ + cos2 ˆ )2 and dividing both sides of the equation by sin 2  , we arrive at an equation of the form 2.1.6.131: Šª

Š + (  tan2 ˆ +  cot2 ˆ + 4  + 2  ) = 0. 24.

sin2 ï

ñóò ò ó

– {[( ô

õ

2 2

Particular solution: 25.

ï

[ ô sin(

)+õ

ï

0

sin ÷ ( ô

ï ) ñ óò ò ó

+ (ï

Particular solution: 27.

( ô sin ÷

ï

 =    sin   (cos  +  sin  ).

+ ö ]ñ„ó ò ò ó + ô

Particular solution: 26.

– ( ô + 1)2 ] sin2 ï + ô ( ô + 1) õ sin 2 ï + ô ( ô – 1)}ñ = 0.

0 2

sin(

ï )ñ

= 0.

=  sin( Π ) + ' + ( .

– õ 2 )ñó ò – ( ï + õ )ñ = 0. 0

= − .

+ õ )ñó ò ò ó + ( ö sin ÷

Particular solution:

2

0

ï −



+ ý )ñó ò + [( ö – ô ) sin ÷

ï

+ý –

õ ]ñ

= 0.

=  .

Page 261

262

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 19 The general solution of the Mathieu equation 2.1.6.29 expressed in terms of auxiliary periodical functions Y 1 ( ) and Y 2 ( )

Constraint 1( 1(

)>1

 L 1  2  Y 1 ( ) + L 2 

−2

Y

2(

) < −1

 L 1  2 ~ Y 1 ( ) + L 2 

−2

~Y

2(





General solution = ( )

| 1 ( )| < 1



Y

Period of 1 and Y 2

“ is a real number

 )



 )

2

L 1 Y 1 ( ) + L 2 "Y 2 ( )



“ =  + 12 ™ , ™ 2 = −1,  is the real part of “





“ = ™›@ is a pure imaginary number, cos(2 @ ) = 1 ( )



“ =0

( L 1 cos @! + L 2 sin @! ) Y 1 ( ) + + ( L 1 cos @! − L 2 sin @! ) Y 2 ( )

1( ) = A 1

Index





2.1.6-2. Equations with cosine. 28.

+ ô 2 ñ = õ cos( ï ). Equation of forced oscillations.


ñóò ò ó

Solution: 29.

=






 cos( Œ  ) if  ≠ Œ ,  2−Œ 2  L 1 sin(  ) + L 2 cos(  ) +  sin(  ) if  = Œ . 2 L

1

sin(  ) + L

2

cos(  ) +

+ ( ô – 2 cos 2 ï )ñ = 0. The Mathieu equation.

ñóò ò ó

1 b . Given numbers  and  , there exists a general solution ( ) and a characteristic index “ such that ( + ) =  2 zf ( ).



For small values of  , an approximate value of “ can be found from the equation: cosh( “ ) = 1 + 2 sin2 



1 2



£   +

  (1 −  ) £ 2



sin  £   + [ (  4 ).



1 (  ) is the solution of the Mathieu equation satisfying the initial conditions 
(0) = 0, the characteristic index can be determined from the relation: 1

If

cosh(2 “ ) =



1(

= 1 and

1 (0)

).



The solution 1 ( ), and hence “ , can be determined with any degree of accuracy by means of numerical or approximate methods. The general solution differs depending on the value of 1 ( ) and can be expressed in  terms of two auxiliary periodical functions Y 1 ( ) and Y 2 ( ) (see Table 19). 2 b . In applications, of major interest are periodical solutions of the Mathieu equation that exist for certain values of the parameters  and  (those values of  are referred to as eigenvalues). The most important solutions are listed in Table 20. The Mathieu functions possess the following properties: ce2  ( , −  ) = (−1)  ce2  O −  ,  , P 2 se2  ( , −  ) = (−1) 

−1

se2  O −  ,  , 2 P

 , −  ) = (−1)  se2 

+1

O −  ,



 , −  ) = (−1)  ce2 

+1

O −  ,



ce2 

+1 (

se2 

+1 (

2 2

,

P

P

.

Page 262

263

2.1. LINEAR EQUATIONS

TABLE 20 Periodical solutions of the Mathieu equation ce  = ce  ( ,  ) and se  = se  ( ,  ) (for odd B , the functions ce  and se  are 2 -periodical, and for even B , they are -periodical); certain eigenvalues   =   (  ) and  =   (  ) correspond to each value of the parameter  ; B = 0, 1, 2,  Recurrence relations for coefficients

Mathieu functions

7

# 22 =  2 # 20 ;  # =( 2 −4)# 22 −2# 20 ; # 22+ +2 =( 2 −4¡ 2)# 22+ − # 22 + −2, ≥2 ¡ 2  +1 # 3 =( 2 +1−1−  )# 21 +1; # 22+ +1+3 =[ 2 +1−(2¡ +1)2] 2  +1 ≥1 × # 22 + +1 +1 − # 2 + −1, ¡ $ 42 =(  2 −4)$ 22 ; $ 22+  +2 =(  2 −4¡ 2)$ 22+  − $ 22+  −2, ≥2 ¡ $ 32 +1 =(  2 +1−1−  )$ 12 +1; $ 22+  +1+3 =[  2 +1−(2¡ +1)2] 2  +1 ≥1 × $ 22+  +1 +1 − $ 2 + −1, ¡

+1(

7

se2  ( ,  )= ÷

+

se2 

2 2

7  ,  )= ÷ # 22 + + =0

+1(

+1 +1 cos

V (2 +1) W ¡

$ 22+  sin(2  ), ¡ =0

se0 =0

7  ,  )= ÷ $ 22+  + =0

+1 +1 sin

(#

2 4

ce2  ( ,  )= ÷ # + cos(2  ) ¡ + =0

ce2 

Normalization conditions

V (2 +1) W ¡

 )2 + 7  # 2   + + ÷ =0 2 2 if B =0 =¦ 1 if B ≥1 2 0

7

+÷

=0

# 22 +

2

+1 2 +1 =1



7

2 2 ÷+ $ 2 +  =1 =0

7

$ 22+  +÷ =0

+1 2 +1 =1



Selecting a sufficiently large and omitting the term with the maximum number in the ¡ recurrence relations (indicated in Table 20), we can obtain approximate relations for the eigenvalues   (or   ) with respect to parameter  . Then, equating the determinant of the corresponding homogeneous linear system of equations for coefficients # + (or $ +  ) to zero, we obtain an algebraic equation for finding   (  ) (or   (  )). For fixed real  ≠ 0, the eigenvalues   and   are all real and different, while: if if



> 0 then < 0 then





0



0

< <

< 1 < 

1

1 1

< <

2 2

< <

2 2

< ))) ; < 3<

3

<

4

< )))

The eigenvalues possess the following properties:

 2  (−  ) =  2  (  ),

 2  (−  ) =  2  (  ),

 2

+1 (−

 ) =  2

+1 (

 ).

The solution of the Mathieu equation corresponding to eigenvalue   (or   ) has B zeros on the interval 0 ≤  < (  is a real number).  Listed below are two leading terms of asymptotic expansions of the Mathieu functions ce  ( ,  ) and se  ( ,  ), as well as of the corresponding eigenvalues   (  ) and   (  ), as ·D 0: 1  ce0 ( ,  ) = £ O 1 − cos 2 , 2 P 2 ce1 ( ,  ) = cos 

 −

8

 2 + 7 4 ; 2

128

 1 ( ) = 1 +  ;

cos 4 5 2 ,  2 ( ) = 4 + ; 4 3 P 12  cos(B + 2) − cos(B − 2) S ,  ( ) = B ce  ( ,  ) = cos B/ +  4Q B +1 B −1 ce2 ( ,  ) = cos 2 +



cos 3 ,

 0( ) = −

O 1−

2

+

2(B

2 2

− 1)

(B ≥ 3);

Page 263

264

SECOND-ORDER DIFFERENTIAL EQUATIONS



se1 ( ,  ) = sin  −

se2 ( ,  ) = sin 2 −

 1 ( ) = 1 −  ;

sin 3 ,

8



12

12 sin(B + 2) sin(B − 2) − S , 4Q B +1 B −1

 −

se  ( ,  ) = sin B/

 2;

 2 ( ) = 4 −

sin 4 ,

  ( ) = B

2

+

2(B

2 2

− 1)

(B ≥ 3).

The Mathieu functions are discussed in the books by McLachlan (1947), Whittaker & Watson (1952), Bateman & Erd´elyi (1955, vol. 3), and Abramowitz & Stegun (1964) in more detail. 30.

31. 32. 33.

+ ( ô cos2 ï + õ )ñ = 0. Applying the formula 2 cos2  = 1 + cos 2 , we obtain the Mathieu equation 2.1.6.29: 

 + ( 12  +  + 12  cos 2 ) = 0.

ñóò ò ó

+ ô cos÷ ( õ ï )ñó ò + ö [ ô cos÷ ( õ  Particular solution: 0 =  − v .

+ [ ô cos÷ ( õ ï ) + ö ]ñó ò + ô„ö cos÷ ( õ  Particular solution: 0 =  − v .

ï )ñ

= 0.

+ ( ô ï + õ ) cos÷ ( ö ï )ñ ó ò – ô cos÷ ( ö Particular solution: 0 =  +  .

ï )ñ

= 0.

ñ óò ò ó

+ ô ïƒ÷ cosú ( õ ï )ñó ò – ô Particular solution: 0 =  .

35.

ñ óò ò ó

38.

39.

+

ô ï ÷

cosú ( õ

ï )ñ ó ò 0

41.

–1

cosú ( õ

ï ÷ +

= exp O −

+ ô ï cos÷ ( õ ï ) ñó ò – [ ô ( ö  Particular solution: 0 =   v .

ï

(

ï )ñ

cosú ( õ

= 0.

, +1

 

+1

ï

+ (ô

ïƒ÷

+1

+ 1) cos÷ ( õ

+ õ cosú'ï )ñó ò +

ô ïƒ÷ ( õ 

cosú

 

+1

Particular solution:

0

= exp O −

ï ñóò ò ó

ï

) + ö ]ñó ò + ô ( ö – 1) ïƒ÷

+ [ô

ïƒ÷

cosú ( õ

ï ñóò ò ó

0

=

1−

B +1

v.

ï

)+ö

P

ï

+ ï ( ô cos÷

ï 2 ñ óò ò ó

+ ï ( ô cos÷

Particular solution:

+ 1)ñó ò + õ ( ô cos÷

ï

ï

0

=

−

=

−

.

+ õ )ñó ò + õ ( ô cos÷ 0

.

ï

2

ï

ï )ñ

ü ï  –1 ]ñ

= 0.

+ 2 ö ]ñ = 0. = 0.

+ ù )ñ = 0.

. –1

cosú ( õ

+ ( ô ïƒ÷ + õ ï cosú ï )ñó ò + [ õ ( ô ïƒ÷ – 1) cosú Particular solution: 0 =  exp(−   B ).

ï 2 ñ óò ò ó

+

.

P

+ [( ô ï 2 + õ ï ) cos÷ ( ö ï ) + 2]ñó ò + õ cos÷ ( ö Particular solution: 0 =  +   .

ï ñóò ò ó

ï 2

)–ö

ï ñóò ò ó

Particular solution: 42.

+ ö [ô

ï ñóò ò ó

Particular solution:

40.

ïƒ÷

ñóò ò ó

Particular solution:

37.

) – ö ]ñ = 0.

ñóò ò ó

34.

36.

ï

ñóò ò ó

ï

ï )ñ +

= 0.

ôù ïƒ÷

–1

]ñ = 0.

ï

– õ )ñ = 0. – 1)ñ = 0.

Page 264

265

2.1. LINEAR EQUATIONS

43.

ï 2 ñ óò ò ó

ô ïƒ÷

+

cosú ( õ

ï )ñ óò

v. 0 = 

Particular solution: 44.

ï 4 ñ óò ò ó

ô ï

+

2

cos÷ ( õ

cos2 ï

ñ óò ò ó

Solution: 46.

cos2 ï

ñóò ò ó ï

[ ô cos( cos÷ ( ô

ï

ï

) – ö – 1]ñ = 0. ) – ö 2 ]ñ = 0.

M

= 1, 2, 3,

2  −  2

uuu

.

+ ( ô cos2 ï + õ )ñ = 0.

)+

ï ) ñ óò ò ó

( ô cos÷

ï

– [ ô cos2 ï + ù (ù – 1)]ñ = 0, ù 2   1 = cos   O M Ý L 1   + L cos   P

õï

1 2

leads to a linear equation of the form 2.1.6.23: sin 2 ˆ



+ ö ]ñ„ó ò ò ó + ô

+ (ï

0 2

–

Particular solution: 49.

cosú ( õ

–1

0 =  exp( (  ).

Particular solution: 48.

ïƒ÷

+ [ ô ( ö – ï ) cos÷ ( õ

The substitution  = ˆ + (  sin2 ˆ +  ) = 0. 47.

−

ï ) ñ óò

Particular solution: 45.

+ ö [ô

0

0

cos(

ï )ñ

= 0.

=  cos( Π ) + ' + ( .

– ( ï + õ )ñ = 0.

õ 2 )ñ óò

= − .

+ õ )ñó ò ò ó + ( ö cos÷

Particular solution:

2

ªŠ

Š +



−

ï

+ ý )ñó ò + [( ö – ô ) cos÷

ï

+

ý

–

õ ]ñ

= 0.

=  .

2.1.6-3. Equations with tangent. 50.

ñóò ò ó

Particular solution: 51.

52.

0

= 0.

= [cos( Œ  )]  J  .

ñóò ò ó

+ ( ô tan2 ï + õ )ñ = 0.

ñóò ò ó

+ ( ô – ) tan(

The transformation H = sin2  , ‡ = cos   , where , is a root of the quadratic equation , 2 + , +  = 0, leads to the hypergeometric equation 2.1.2.171: H ( H − 1)‡ ‘'

‘ + [(1 − , )H − 12 ]‡ ‘
− 1 4 ( , +  ) ‡ = 0.

ï )„ñ óò

Particular solution: 53.

ï )]ñ

+ ô [ + ( – ô ) tan2 (

ñ óò ò ó

tan ï

ô

+

ñ óò

+

0

õÿñ

+ ôƒñ = 0.

= [cos( Œ  )]  J  . = 0.

1 b . The substitution ˆ = sin  leads to a linear equation of the form 2.1.2.168: (ˆ (1 −  )ˆ 
Š −  = 0.

2

− 1)
Šª
Š +

2 b . Solution for  = −2: cos  =

™

sin( ,! ) + L 2 cos( ,! ) if  + 1 = , 2 > 0, L 1 sinh( ,! ) + L 2 cosh( ,! ) if  + 1 = − , 2 < 0.

L

1

3 b . Solution for  = 2 and  = 3: 54.

ñóò ò ó

+

ô

tan ï

ñóò

=L

1

+ ( õ tan2 ï + ö )ñ = 0.

cos3  + L

2

sin  (1 + 2 cos2  ).

This is a special case of equation 2.1.6.131.

Page 265

266 55.

56.

57.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

– 2 tan(

ï ) „ñ óò

+ (ô

ï

ñ óò ò ó

– 2 tan(

ï ) ñ óò

+ (ô

ï 2÷

ñ óò ò ó

+

tan÷ ( õ

ï ) ñ óò

+ ö [ ô tan÷ ( õ

ô

ñóò ò ó

tan÷ (

ô

+

ñóò ò ó

+

tan÷

ô

ï ñóò

+ ( ô tan÷

ñ óò ò ó

+ [ ô tan÷ ( õ

ï

ñóò ò ó

+ tan ï ( ô tan÷

ñóò ò ó

+ (ô

ï

ñóò ò ó

+

tanú ( õ

Particular solution: 64.

ñ óò ò ó

+

ô ï ÷

tanú ( õ

Particular solution: 65.

66.

ï ñóò ò ó

–2

ï ñóò ò ó

+

ï

ï

)ñ = 0.

ï

+ 0

– ô tan÷

– 1)ñó ò + ( ôƒõ tan÷

ïƒ÷

–ô

= .

ï ) ñ óò 0

– ô tan÷ ( ö

=  +  .

ï ) ñ óò 0

ï )ñ

= sin  cos   .

+ ö [ô

ï ÷ + (

= exp O −

tan÷ ( õ

ï ) ñ óò

– [ô (ö

, +1

ï

ï )ñ

+ 4)ñ = 0.

ï

+2

ï )ñ

+1

) + ]ñ = 0.

– ô tan÷

ï

+ 2 õ + 2)ñ = 0.

= 0.

tanú ( õ

 

ï

= 0.

tanú ( õ

–1

ï

–1

õ

ï ) ñ óò 0

) + ( – õ ) tan2 (

+ ô„ö tan÷ ( õ

+ (ô

= 0.

ï

)–ö

ï 2

+

ü ï  –1 ]ñ

= 0.

.

P

+ õ )ñ = 0.

The substitution ¨ = cos( Œ  ) leads to a second-order linear equation of the form 2.1.2.64:  ¨2

 + [(  + Œ 2 ) +  ]¨ = 0.

ô ï

ï ñóò ò ó

+ [( ô

ï

2

+

õï

ï ñóò ò ó

+ (ô

ïƒ÷

+1

ï ñóò ò ó

+ (ô

ïƒ÷

+

v . 0 =  

) tan÷ ( ö 0

õï

Particular solution:

0

tanú

ï

ï

+ 1) tan ÷ ( õ

ï

)+

) + 2]ñó ò + õ tan÷ ( ö

= +  .

+ õ tanú

Particular solution: 69.

(

óò

ï ) „ñ óò

Particular solution: 68.

+1

ï

+1

tan(

Particular solution: 67.

2

–

) – ö ]ñ = 0.

−v  . 0 = 

+ õ ) tan ÷ ( ö

ô ïƒ÷

ï

= sin  cos  .

0

) + ö ]ñ

Particular solution: 63.

–1

J 0 = [cos( Π )]   .

Particular solution: 62.

õï ÷

+ õ [ ô tan÷

ï ) ñ óò

Particular solution: 61.

+

−v  . 0 = 

Particular solution: 60.

+ ö )ñ = 0.

The substitution ¨ = cos( Œ  ) leads to a second-order linear equation of the form 2.1.2.10: ¨2

 + (  2  + '  −1 )¨ = 0.

Particular solution: 59.

õï

+

The substitution ¨ = cos( Œ  ) leads to a second-order linear equation of the form 2.1.2.6: ¨ 

 + (  2 + ' + ( + Œ 2 )¨ = 0.

Particular solution: 58.

2

ï )ñ óò

= exp O −

ï )ñ óò

+

ô ïƒ÷ ( õ 

 

+ [õ (ô

ïƒ÷

B +1

tanú +1

   B ). 0 =  exp(− 

P

ï

ö 2ï

ï )ñ

+ 2 ö ]ñ = 0. = 0.

+ ù )ñ = 0.

.

– 1) tanú'ï + ôù

ïƒ÷

–1

]ñ = 0.

Page 266

267

2.1. LINEAR EQUATIONS

70.

ï ñóò ò ó

+ [ô

ïƒ÷

tanú ( õ

Particular solution: 71.

72. 73.

74.

0

ï 2 ñ óò ò ó

77. 78. 79.

1−

=

–1

v.

tanú ( õ

ï )ñ

= 0.

ï 2 ñ óò ò ó

+ ï (1 – 2 ï tan ï )ñ„ó ò – ( ï tan ï + b 2 )ñ = 0. Solution: cos  = L 1 ø Á ( ) + L 2 n Á ( ), where ø Á ( ) and n Á ( ) are the Bessel functions.

ï 2 ñ óò ò ó

– ï (2 ï tan ï + ü )ñ„ó ò + ( ô ï 2 + õ ï + ö + ü ï tan ï )ñ = 0. The substitution ¨ = cos  leads to a second-order linear equation of the form 2.1.2.131:  2 ¨2

 − ,! ¨2
+ [(  + 1) 2 + ' + ( ]¨ = 0.

ï 2 ñ óò ò ó

+ ï ( ô tan÷

ï 2 ñ óò ò ó

+ ï ( ô tan÷

ï

óò

+ 1)ñ 0

ï

−

=

óò

+ õ )ñ

Particular solution:

76.

) + ö ]ñó ò + ô ( ö – 1) ïƒ÷

– 2 ï 2 tan( ï ) ñó ò + ( ô ï 2 + õ ï + ö )ñ = 0. The substitution ¨ = cos( Œ  ) leads to a second-order linear equation of the form 2.1.2.115:  2 ¨2

 + [(  + Œ 2 ) 2 + ' + ( ]¨ = 0.

Particular solution: 75.

ï

0

−

=

+ ô ï ÷ tanú ( õ ï ) ñ Particular solution: 0 = 

ï 2 ñ óò ò ó

+ ô ï 2 tan÷ ( õ Particular solution:

ï 4 ñ óò ò ó

óò

.

+ õ ( ô tan÷

.

+ ö [ô −v .

ï ÷

–1

ï

ï

– õ )ñ = 0. – 1)ñ = 0.

tanú ( õ

+ [ ô ( ö – ï ) tan÷ ( õ =  exp( (  ).

ï ) ñ óò 0

+ õ ( ô tan÷

( ô tan ÷ ï + õ )ñ ó ò ò ó + ( ö ï + ý )ñ ó ò – ö^ñ = 0. Particular solution: 0 = (˜ + . ( ô tan ÷

ï

+ õ )ñ

óò ò ó

+ ( ö tan÷

Particular solution:

0

−



ï

M

=  .

+ ý )ñ

óò

ï

ï

) – ö – 1]ñ = 0. ) – ö 2 ]ñ = 0.

+ [( ö – ô ) tan÷

ï

+

ý

–

õ ]ñ

= 0.

2.1.6-4. Equations with cotangent. 80.

ñ óò ò ó

Particular solution:

81. 82.

83.

0

= 0.

= [sin( Œ  )]  J  .

+ ( ô cot2 ï + õ )ñ = 0. The substitution  = ˆ + z 2 leads to an equation of the form 2.1.6.51: 
Šª
Š + (  tan2 ˆ +  ) = 0.

ñóò ò ó

+ cot ï ñó ò + b ( b + 1)ñ = 0. The substitution ˆ = cos  leads to the Legendre equation 2.1.2.154: (ˆ @ ( @ + 1) = 0.

ñóò ò ó ñ óò ò ó

+ 2 ô cot( ô

ï ) ñ óò

Particular solution: 84.

ï )]ñ

+ ô [ + ( – ô ) cot2 (

ñóò ò ó

+ ( – ô ) cot(

Particular solution:

+ (õ 0

− 1)
Šª
Š + 2ˆ 
Š −

– ô 2 )ñ = 0. cos( ' ) . = sin(  )

ï ) „ñ óò 0

2

2

+

ôƒñ

= 0.

= [sin( Œ  )]  J  .

Page 267

268 85.

86.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ô cot( ï ) ñ ó ò + õÿñ = 0. The substitution H = Œ  + z 2 leads to a second-order linear equation of the form 2.1.6.53: 
‘'
‘ − Œ −1 tan H 
‘ +  Œ −2 = 0.

ñ óò ò ó ñóò ò ó

ô

+

ï ) ñ óò

cot(

J 0 = [sin( Π )]   .

Particular solution: 87.

ñóò ò ó

cot ï

ô

+

ï )]ñ

+ õ [ + ( – ô – õ ) cot2 (

= 0.

+ ( õ cot2 ï + ö )ñ = 0.

ñóò

This is a special case of equation 2.1.6.131. 88.

89.

90.

ñóò ò ó

ï ) „ñ óò

+ 2 cot(

ñóò ò ó

+

ï ) ñ óò

cot÷ ( õ

ô

ñóò ò ó

+ [ ô cot÷ ( õ

ï

−v  . 0 =  −v  . 0 = 

= 0.

ï )ñ

= 0.

+ ( ô ï + õ ) cot÷ ( ö ï ) ñó ò – ô cot÷ ( ö Particular solution: 0 =  +  .

ñ óò ò ó

+

ô ï ÷

cotú ( õ

ï ) ñ óò

Particular solution:

0

ï ÷

+ ö [ô

cotú ( õ

–1

ï ÷ +

= exp O −

ï ñ óò ò ó

(

ï )ñ

cotú ( õ

, +1

 

+1

= 0.

ï

)–ö

ï 2

+

ü ï  –1 ]ñ

= 0.

.

P

+ 2 ï cot( ï ) ñ ó ò + ( ô ï + õ )ñ = 0. The substitution ¨ = sin( Œ  ) leads to a second-order linear equation of the form 2.1.2.64:  ¨2

 + [(  + Œ 2 ) +  ]¨ = 0.

ï ñóò ò ó

+

ô ï

ï ) ñ óò

– [ô (ö

+ õ cotú

ï )ñ óò

cot÷ ( õ

v . 0 =  

Particular solution:

ï ñóò ò ó

+ (ô

ïƒ÷

+1

Particular solution:

ï ñóò ò ó

+ (ô

ïƒ÷

+

õï

0

ï ñóò ò ó

+ [ô

ïƒ÷

cotú ( õ

Particular solution:

ï

+ 1) cot÷ ( õ

+ô

= exp O −

ïƒ÷ ( õ

   B +1

cotú'ï )ñó ò + [ õ ( ô

ïƒ÷

+1

P

0

=

1−

v.

)+

ö 2ï

+ 2 ö ]ñ = 0.

.

– 1) cotú

) + ö ]ñó ò + ô ( ö – 1) ïƒ÷

ï

ï

cotú'ï + ù )ñ = 0.

   B ). 0 =  exp(− 

Particular solution: 99.

ï )ñ

ñóò ò ó

ñ óò ò ó

98.

) – ö ]ñ = 0.

) + ö ]ñó ò + ô„ö cot÷ ( õ

94.

97.

ï

+ ö [ ô cot÷ ( õ

+ ô ï ÷ cotú ( õ ï ) ñ ó ò – ô Particular solution: 0 =  .

96.

+ ö )ñ = 0.

+ 2 cot( ï ) ñ ó ò + ( ô ï 2÷ + õ ï ÷ –1 – 2 )ñ = 0. The substitution ¨ = sin( Œ  ) leads to a second-order linear equation of the form 2.1.2.10: ¨2

 + (  2  + '  −1 )¨ = 0.

93.

95.

õï

+

ñ óò ò ó

Particular solution: 92.

2

The substitution ¨ = sin( Œ  ) leads to a second-order linear equation of the form 2.1.2.6: ¨2

 + (  2 + ' + ( + Œ 2 )¨ = 0.

Particular solution: 91.

ï

+ (ô

–1

ï

cotú ( õ

+

ôù ïƒ÷

ï )ñ

–1

]ñ = 0.

= 0.

Page 268

269

2.1. LINEAR EQUATIONS

100.

101.

102.

ï 2 ñ óò ò ó

+2

ï 2 ñ óò ò ó

+ ï (2 ï cot ï + ü )ñ„ó ò + ( ô

ï 2 ñ óò ò ó

+ ï ( ô cot÷

ï

2

cot(

ï ) ñ óò

ï

+ ö )ñ = 0.

ï

2

+

õï

+ 1)ñó ò + õ ( ô cot÷

ï 2 ñ óò ò ó

+ ï ( ô cot÷

0

ï

+ö +

ï 2 ñ óò ò ó

+

ô ï ÷

ï 4 ñ óò ò ó

+

ô ï

2

0

cotú ( õ cot÷ ( õ

ï

=

−

=

ï ) ñ óò

. .

+ ö [ô

v. 0 =  −

ï ) ñ óò

Particular solution: 106. ( ô cot÷

−

ï

+ õ )ñó ò + õ ( ô cot÷

Particular solution: 105.

õï

+

ü ï

cot ï )ñ = 0.

The substitution ¨ = sin  leads to a second-order linear equation of the form 2.1.2.131:  2 ¨2

 + ,! ¨2
+ [(  + 1) 2 + ' + ( ]¨ = 0.

Particular solution: 104.

2

The substitution ¨ = sin( Œ  ) leads to a second-order linear equation of the form 2.1.2.115:  2 ¨2

 + [(  + Œ 2 ) 2 + ' + ( ]¨ = 0.

Particular solution: 103.

ï

+ (ô

0

0

–1

– 1)ñ = 0.

cotú ( õ

+ [ ô ( ö – ï ) cot÷ ( õ

=  exp( (  ).

+ õ )ñó ò ò ó + ( ö cot÷

Particular solution:

ï ÷

ï

– õ )ñ = 0.

−



ï

ï

ï

) – ö – 1]ñ = 0. ) – ö 2 ]ñ = 0.

+ ý )ñó ò + [( ö – ô ) cot÷

ï

+

ý

–

õ ]ñ

= 0.

=  .

2.1.6-5. Equations containing combinations of trigonometric functions. 107.

ñ óò ò ó

– ô [ ô sin2 ( õ

ï

) + õ cos( õ

Particular solution: 108.

ñóò ò ó

– ô [ ô cos2 ( õ

ï

0

109.

ñ óò ò ó

+ ( ô sin ï + õ )ñ

0

ñóò ò ó

+ ( ô sin ÷

ï

0

ñ óò ò ó

+ ( ô cos ï + õ )ñ

Particular solution: 112.

ñóò ò ó

+ ( ô cos÷

ï

ñ„óò ò ó

= exp −



Q 

= 0. sin( ' )S .

= exp(  cos  ).

+ cos ï

0

ñóò

+1

ï

+ cos ï )ñ = 0.

= exp(  cos  ).

+ ô ( õ cos ï – sin ï )ñ = 0.

óò 0

= exp(−  sin  ).

+ õ cos ï )ñó ò + õ ( ô cos÷

Particular solution: 113. sin ï

ï )]ñ

cos( ' )S .

+ õ sin ï )ñó ò + õ ( ô sin ÷

Particular solution: 111.

Q 

= 0.

+ ô ( õ sin ï + cos ï )ñ = 0.

óò

Particular solution: 110.



= exp −

) + õ sin( õ

Particular solution:

ï )]ñ

0

= exp(−  sin  ).

+1

ï

– sin ï )ñ = 0.

+ b ( b + 1) sin ï

ñ = 0. The substitution ˆ = cos  leads to the Legendre equation 2.1.2.154: (1 − ˆ 2 )
Šª
Š − 2ˆ 
Š + @ ( @ + 1) = 0.

Page 269

270

SECOND-ORDER DIFFERENTIAL EQUATIONS

114. sin ï

+ (2ù + 1) cos ï

ñ óò ò ó

ñ óò

+ ( b – ù )( b + ù + 1) sin ï

ñ

= 0.

Here, @ is an arbitrary number and B is a positive integer. The substitution ˆ = cos  leads to an equation of the form 2.1.2.156: (ˆ 2 − 1)
Šª
Š + 2(B + 1)ˆ Š
+ (B − @ )( @ + B + 1) = 0. 115. sin2 ï

ñóò ò ó

+ sin ï cos ï

ñóò ò ó

+ sin ï ( ô cos ï + õ )ñ„ó ò + (  cos2 ï +  cos ï + )ñ = 0.

ñ„óò

+ [ b ( b + 1) sin2 ï – ù

2

]ñ = 0.

Here, @ is an arbitrary number and B is a nonnegative integer. The transformation ˆ = cos  , = ¨ sin   leads to an equation of the form 2.1.2.156: 2 (ˆ − 1)¨ Šª

Š + 2(B + 1)ˆ ¨ Š
+ (B − @ )(B + @ + 1)¨ = 0. 116. sin2 ï

Set  = 2ˆ . Applying the trigonometric formulas

sin(2ˆ ) = 2 sin ˆ cos ˆ , cos(2ˆ ) = cos2 ˆ − sin2 ˆ ,  =  (sin2 ˆ + cos2 ˆ ), ý = ý (sin2 ˆ + cos2 ˆ ), = = = (sin2 ˆ + cos2 ˆ )2 , and dividing all the terms by sin2  , we arrive at an equation of the form 2.1.6.131:

Šª

Š + [(  −  ) tan ˆ + (  +  ) cot ˆ ] Š
+ [( ƒ − ý + = ) tan2 ˆ + ( ƒ + ý + = ) cot2 ˆ + 2= − 2 ƒ ] = 0. 117. cos2 ï

ñóò ò ó

+ ô sin(2 ï ) ñ„ó ò + [ õ cos(2 ï ) + ö ]ñ = 0.

Dividing the equation by cos2  and applying the formulas sin(2 ) = 2 sin  cos  ,

cos(2 ) = cos2  − sin2  ,

( = ( (sin2  + cos2  ),

we obtain an equation of the form 2.1.6.131: "

 + 2  tan  
 + [( ( −  ) tan2  +  + ( ] = 0. 118. cos2 ( ô

ï ) ñ óò ò ó

+ (ù – 1) ô sin(2 ô

Particular solution: 119. cos2 ï

ñóò ò ó

0

ï ) ñ óò

+ ù,ô 2 [(ù – 1) sin2 ( ô

= cos  (  ).

ï

) + cos2 ( ô

ï )]ñ

= 0.

+ cos ï ( ô sin ï + õ )ñ„ó ò + (  sin2 ï +  sin ï + )ñ = 0.

The substitution  = ˆ + z 2 leads to a second-order linear equation of the form 2.1.6.116: sin2 ˆ 
Šª
Š − sin ˆ (  cos ˆ +  ) Š
+ ( ƒ cos2 ˆ + ý cos ˆ + = ) = 0.

120. sin ï cos2 ï

ñóò ò ó

+ cos ï ( ô sin2 ï + õ )ñó ò + ö sin ï

ñ

= 0.

1 b . Dividing the equation by sin  cos  and assuming  =  (sin2  +cos2  ), ( = ( (sin2  +cos2  ), we obtain equation 2.1.6.131: 

 + [(  +  ) tan  +  cot  ] 
+ ( (tan2  + 1) = 0. 2

2 b . Particular solutions: 0 0 0

121. sin ï cos2 ï

ñóò ò ó

= cos   = tan1−   = sin1−   cos 

 

+ −1

if ( =  (  + 1), if ( = (  + 2)(  − 1), if ( = 2(  +  − 1).

+ cos ï ( ô sin2 ï – 1)ñó ò + õ sin3 ï

ñ

= 0.

= L 1 (cos  )  1 + L 2 (cos  )  2 , where , 1 and , 2 are roots of the quadratic equation

Solution: , 2 − , +  = 0. 122. sin2 ï cos2 ï

ñóò ò ó

+ ( ô sin2 ï + õ cos2 ï + ö sin2 ï cos2 ï )ñ = 0.

Dividing the equation by sin2  cos2  and assuming  =  (sin2  + cos2  ),  =  (sin2  + cos2  ), we obtain equation 2.1.6.131: 

 + (  tan2  +  cot2  +  +  + ( ) = 0.

Page 270

271

2.1. LINEAR EQUATIONS

123. sin ÷ (

ï ) ñ óò ò ó

sin÷ (

2

+[

ï

ô

)+

The transformation ˆ = tan( Œ  ), ¨

¨2Šª

Š + Œ

cos÷

–4

=

ˆ  ¨ = 0.

(

ï )]ñ

= 0. leads to an equation of the form 2.1.2.7:

cos( Π )

−2 −

124. cos÷ ( ï ) ñó ò ò ó + [ 2 cos÷ ( ï ) + ô sin ÷ –4 ( ï )]ñ = 0. The substitution Œ  = z 2 − Œ ˆ leads to an equation of the form 2.1.6.123. 125.

ñóò ò ó

+ ù tan ï

Solution: 126.

ñóò ò ó

+ tan ï

ñóò

ô

+

2

=L

1

sin( ‡ ) + L

ñóò

+

ô

L Q

1

ø

ñóò ò ó

+ tan ï



128.

ñ óò ò ó

129.

ñóò ò ó

=L

– ù cot ï

130.

131.

ï ) ñ óò 1

ñóò =L

Solution:

–2

ñ

= 0.

ñ

sin 

P

+L

2

n

1 2



O B



 . M

sin  

P

S , where ø Á ( H ) and n Á ( H )

= 0.

L 1 | sin  |  + L 2 | sin  |1−  if  ≠ 12 , £ | sin  | ( L 1 + L 2 ln | sin  |) if  = 1 . 2

– 2 ô cot(2ô

Solution:

B



– 1) cot2 ï

ñóò a – ô (ô =

Solution:

= 0.

cos( ‡ ), where ‡ = X cos  

O

1 2

are the Bessel functions. 127.

2

ñ

cos2 ï (sin ï )2÷

2

= £ sin 

Solution:

(cos ï )2÷

exp +

1

–

ô

Q 

2



õ

2

sin2 (2 ô

ï )ñ

sin2 (  )S + L

(sin ï )2÷

sin( ‡ ) + L

2

ñ

= 0. 2

exp −

Q 



sin2 (  )S .

= 0.

cos( ‡ ), where ‡ = X sin  

 . M

– 2 cot(2 ï ) ñ„ó ò + ô tan2 ï ñ = 0. The substitution ˆ = cos  leads to the Euler equation 2.1.2.123: ˆ

ñóò ò ó

Šª

Š − ˆ Š
+ 

2

= 0.

+ ( ô tan ï + õ cot ï )ñ„ó ò + (  tan2 ï +  cot2 ï + )ñ = 0. The transformation ˆ = sin2  , = ¨ sin   cos +  , where B and are roots of the quadratic ¡ equations 2 B 2 + (  − 1)B + ý = 0, − (  + 1) + ƒ = 0,

ñóò ò ó

¡

¡

leads to the hypergeometric equation 2.1.2.171: 4ˆ (ˆ − 1)¨ Šª

Š + 2[(2B + 2 132. sin(2 ï ) ñ Solution:

óò ò ó

– 2ùñ =L

1

óò

¡

+ 2ô

+ 2 +  −  )ˆ − 2B −  − 1]¨ Š
+ (2B 2

sin2 ï (tan ï )2÷

sin( ‡ ) + L

2

–1

ñ

+B +

¡

¡

+

¡

− B − = )¨ = 0.

= 0.

cos( ‡ ), where ‡ = X tan  

M

 .

2.1.7. Equations Containing Inverse Trigonometric Functions 2.1.7-1. Equations with arcsine. 1.

+ ( ô ï + õ + ö arcsin ï )ñ ó ò + [ ö ( ô ï + õ ) arcsin ï + ô ]ñ = 0. Particular solution: 0 = exp(− 21  2 − ' ).

ñ óò ò ó

Page 271

272

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.

+ õ (arcsin ï ) ÷ Particular solution:

ñ óò

3.

ñóò ò ó

ñóò

ñ óò ò ó

+ õ (arcsin ï ) ÷

Particular solution: 4. 5. 6.

9.

+ ô ïƒ÷ (arcsin ï ) ú ñó ò – ô Particular solution: 0 =  .

12. 13.

ï ñ óò ò ó

+

ô ï

arcsin ï

ñ óò

ï ñóò ò ó

+ [ô ( õ ï

16. 17.

 +

+1





.

ï 2ú

õï

ñ

−





ñ

+ [ô

ôƒõ 2 ï

ï ñ óò ò ó

+ (ô

ï

ï ÷

+1

=

0

1−

.

+ õ arcsin ï )ñ

óò

= exp O −

0

ï ñóò ò ó

+

+ 2)]ñ = 0.

arcsin ï

ñ

ñ

= 0.

ô ï ÷ (õ

   B +1

ñ

= 0.

arcsin ï + ù )ñ = 0.

+1

.

P

+ ( ô ïƒ÷ + õ ï arcsin ï )ñ„ó ò + [ õ ( ô ïƒ÷ – 1) arcsin ï + ôù Particular solution: 0 =  exp(−   B ).

ï 2 ñ óò ò ó

+ õ ï arcsin ï Particular solution:

ï 2 ñ óò ò ó

ñóò 0

ï

(ô

2

+ õ )ñ

óò ò ó

ï

+ ö (ô

2 0

+ ô ï 2 arcsin ï Particular solution:

( ô ï 2 + õ )2 ñ ó ò ò ó + ( ö Particular solution: 2

+ ô )2 ñó ò ò ó + õ ( ï

(ô

ï

2

+ õ )2 ñó ò ò ó + ö ( ô

Particular solution:

óò

ï

+ [õ (ô

−1 −   .  0 = 

ï 4 ñ óò ò ó

(ï

+ ô ( õ arcsin ï – ô – 1)ñ = 0. =  − .

+ ï ( õ arcsin ï + 2)ñ

+ 1) arcsin ï – ô

+ õ )(arcsin ï ) ÷ = 

2

= 0.

.

(arcsin ï ) ÷ + õ ]ñó ò + ô ( õ – 1)(arcsin ï ) ÷

ï

]ñ = 0.

= 0.

+ [( ô ï 2 + õ ï ) arcsin ï + 2]ñ„ó ò + õ arcsin ï Particular solution: 0 =  +   .

ï ñóò ò ó

–1

= 0.

(arcsin ï ) ú

– 1]ñ„ó ò +

ïƒú

+þ

.

P

+ 1) arcsin ï + õ ( õ

= ( ' + 1) 

0

ï ñóò ò ó

Particular solution: 18.

= 

+ 1) arcsin ï +

Particular solution: 15.

+1

–1

ï

– [ô ( õ

0

Particular solution: 14.

¡

ïƒ÷

ñóò ò ó

Particular solution: 11.

= exp O −



+ ( ô ï + õ )(arcsin ï ) ÷ ñó ò – ô (arcsin ï ) ÷ Particular solution: 0 =  +  .

Particular solution: 10.

(arcsin ï )÷ – ô

ñóò ò ó

Particular solution: 8.

ïƒú

+ ô [õ 0

Particular solution: 7.

+ ö [ õ (arcsin ï ) ÷ – ö ]ñ = 0. −v  . 0 = 

ñ óò

+ .

– 2ô [ö

ï

2

ïw÷

ï 2 ]ñ

–1

]ñ = 0.

= 0.

(arcsin ï ) ÷ + 1]ñ = 0.

+ [ ô ( õ – ï ) arcsin ï – õ 2 ]ñ = 0. =  exp(   ).

ñóò 0

+ ý )(arcsin ï ) ÷ 0 = (˜ + .

ï

M

+ ô )(arcsin ï ) ÷

2

ï

0

=£ 

2 0

2

ñ óò

– ö (arcsin ï ) ÷

ñóò

– [õ

+ .

+ õ )(arcsin ï ) ÷

= exp O − X



M2



ñóò +  P

ï

ñ

= 0.

(arcsin ï ) ÷ + ô ]ñ = 0.

+ [ ö (arcsin ï ) ÷ – 2 ô

ï

– 1]ñ = 0.

.

Page 272

273

2.1. LINEAR EQUATIONS

2.1.7-2. Equations with arccosine. 19.

+ ( ô ï + õ + ö arccos ï )ñ„ó ò + [ ö ( ô ï + õ ) arccos ï + ô ]ñ = 0. Particular solution: 0 = exp(− 21  2 − ' ).

ñóò ò ó

20.

+ õ (arccos ï ) ÷ Particular solution:

ñ óò

21.

ñóò ò ó

ñóò

ñ óò ò ó

+ õ (arccos ï ) ÷

Particular solution: 22. 23. 24. 25.

27.

30. 31. 32.

+ ô ïƒ÷ (arccos ï )ú ñó ò – ô Particular solution: 0 =  .

34.

¡

ïƒ÷

ñóò ò ó

ï ñóò ò ó

arccos ï

ñóò

Particular solution:

0

ï ñ óò ò ó

+

ô ï

ï

+ [ô ( õ

ï

– [ô ( õ

= 



+1

 +

+1



0

ï ñóò ò ó

–1

ï 2ú

.

ï

ñ

(arccos ï )ú

= ( ' + 1) 

óò

– 1]ñ

−





+

+ [ô

ï

ñ

ï ñ óò ò ó

+ (ô

ï ÷

+1

0

=

.

+ õ arccos ï )ñ 0

óò

= exp O −

ï ñóò ò ó

+ô

ôƒõ 2 ï

ï

+ 2)]ñ = 0.

arccos ï

ñ

ï ÷ (õ

   B +1

+ õ ï arccos ï Particular solution:

0

Particular solution:

−1 −   .  0 = 

ï 2 ñ óò ò ó

(ô

ï

2

ñ

= 0.

ñ

= 0.

arccos ï + ù )ñ = 0.

+1

P

.

+ ô ( õ arccos ï – ô – 1)ñ = 0. =  − .

ï

2 0

ï 4 ñ óò ò ó

ïw÷

–1

]ñ = 0.

ñ óò

+ ï ( õ arccos ï + 2)ñ„ó ò + [ õ ( ô

+ õ )ñó ò ò ó + ö ( ô

= 0.

.

+ ( ô ïƒ÷ + õ ï arccos ï )ñó ò + [ õ ( ô ïƒ÷ – 1) arccos ï + ôù Particular solution: 0 =  exp(−   B ).

ï 2 ñ óò ò ó

]ñ = 0.

= 0.

(arccos ï ) ÷ + õ ]ñó ò + ô ( õ – 1)(arccos ï ) ÷ 1−

–1

= 0.

+ [( ô ï 2 + õ ï ) arccos ï + 2]ñ„ó ò + õ arccos ï Particular solution: 0 =  +   .

ï ñóò ò ó

ïƒú

+þ

.

P

+ 1) arccos ï + õ ( õ

+ 1) arccos ï + õ

Particular solution: 33.

= exp O −



+ ( ô ï + õ )(arccos ï ) ÷ ñ ó ò – ô (arccos ï ) ÷ Particular solution: 0 =  +  .

Particular solution: 29.

(arccos ï )÷ – ô

ñ óò ò ó

Particular solution: 28.

ïƒú

+ ô [õ 0

Particular solution: 26.

+ ö [ õ (arccos ï ) ÷ – ö ]ñ = 0. −v  . 0 = 

ï

+ õ )(arccos ï ) ÷ = 

2

+ .

+ 1) arccos ï – ô

ñóò

– 2ô [ö

ï

ï ]ñ

2 2

= 0.

(arccos ï ) ÷ + 1]ñ = 0.

+ ô ï 2 arccos ï ñó ò + [ ô ( õ – ï ) arccos ï – õ 2 ]ñ = 0. Particular solution: 0 =  exp(   ).

( ô ï 2 + õ )2 ñó ò ò ó + ( ö Particular solution:

ï

+ ý )(arccos ï ) ÷ 0 = (˜ + .

ñóò

– ö (arccos ï ) ÷

ñ

= 0.

M

Page 273

274 35.

SECOND-ORDER DIFFERENTIAL EQUATIONS (ï

2

+ ô )2 ñó ò ò ó + õ ( ï

+ ô )(arccos ï ) ÷

2

Particular solution: 36.

(ô

ï

2

óò ò ó

+ õ )2 ñ

=£ 

0

+ ö (ô

ï

2

Particular solution:

0

ñóò

+ .

2

+ õ )(arccos ï ) ÷

= exp O − X

M2





(arccos ï ) ÷ + ô ]ñ = 0.

ï

– [õ

+ [ ö (arccos ï ) ÷ – 2 ô

ñ óò

ï

– 1]ñ = 0.

.

+  P

2.1.7-3. Equations with arctangent. 37. 38. 39.

+ ( ô ï + õ + ö arctan ï )ñ ó ò + [ ö ( ô ï + õ ) arctan ï + ô ]ñ = 0. Particular solution: 0 = exp(− 21  2 − ' ).

ñ óò ò ó

+ õ (arctan ï ) ÷ Particular solution:

ñ óò

+ õ (arctan ï ) ÷

ñóò

ñ óò ò ó ñóò ò ó

0

Particular solution: 40.

ñóò ò ó

+ (ô

ï

ñóò ò ó

+

0

(arctan ï ) ú

Particular solution: 42.

ï ñóò ò ó

+

ô ï

0

arctan ï

ñóò

Particular solution: 43.

ï ñóò ò ó

ï

+ [ô ( õ

0

44.

ï ñóò ò ó

ï

+ [( ô

2

+

0

õï

ï ñóò ò ó

+ [ô

ï

ï ñóò ò ó

+ (ô

ï ñóò ò ó

+ (ô

ïƒ÷

+1

ï ñóò ò ó

+

ô ï

ïƒ÷

+

õï

ï ñóò ò ó

ïƒ÷

= .

ï

– [ô ( õ = 





–1

ñ

(arctan ï )ú

ñ

= 0.

ï

+ 2)]ñ = 0.

.

õï

= ( ' + 1) 

1−

+ õ arctan÷

Particular solution:

ï

.

0

=

0

= exp O −

ôƒõ 2 ï

– 1]ñ„ó ò +

−





arctan ï

ô ïƒ÷ ( õ

ï ñóò

   B +1

0

+ ô ( õ arctan÷

 =  − .

= 0.

ñ

= 0.

+1

ïƒ÷

ï

ñ

= 0.

arctan ï + ù )ñ = 0.

P

. – 1) arctan ï + ôù

+ õ )ñó ò – ô (arctan÷ = .

ñ

.

   B ). 0 =  exp(−  0

]ñ = 0.

= 0.

+ 1) arctan ï + õ ( õ

arctan ï )ñ„ó ò + [ õ ( ô

(arctan÷

Particular solution: 49.

–ô

+ õ arctan ï )ñ„ó ò +

Particular solution: 48.

ñóò

–1

.

P

– ô (arctan ï ) ÷

=  +  .

ïƒú

+þ

(arctan ï ) ÷ + õ ]ñó ò + ô ( õ – 1)(arctan ï ) ÷

Particular solution: 47.

¡

+1

0 =  +  .

Particular solution: 46.

+1

 +

ï 2ú

) arctan ï + 2]ñ„ó ò + õ arctan ï

Particular solution: 45.



ñóò

+ 1) arctan ï +

Particular solution:

(arctan ï ) ÷ – ô

= exp O −

+ õ )(arctan ï ) ÷

ô ïƒ÷

ïƒú

+ ô [õ 0

Particular solution: 41.

+ ö [ õ (arctan ï ) ÷ – ö ]ñ = 0.  =  −v .

–ô

ï

ïw÷

–1

]ñ = 0.

+ õ )ñ = 0.

ï )ñ

= 0.

Page 274

275

2.1. LINEAR EQUATIONS

50.

ï ñ óò ò ó

+ ô (arctan÷

Particular solution: 51.

ï ñóò ò ó

+ õ arctan÷

ï

0

ï ñóò

Particular solution: 52. 53.

0

ï 2 ñ óò ò ó

+ õ ï arctan ï Particular solution:

ï 2 ñ óò ò ó

55.

0

58.

ï 2 ñ óò ò ó

+ ô ï (arctan÷ Particular solution:

ï

61.

arctan÷

− 12

= exp 

  .

óò

+ [õ (ô

0

+ ô (arctan÷ ï

0

64.

ï 2 ñ óò ò ó

+ 1)ñ = 0.

2

õ ï 2 )ñ óò

=

−





ï

–ô

ï 2 )ñ

= 0.

+ ôƒõ arctan÷

ï ñ

= 0.

ï

(ï

2

+ 1)ñó ò ò ó – [ ô 2 ( ï

2

ï

(ô

2

+ õ )ñ

óò ò ó

2 0

ï 4 ñ óò ò ó

ï 2 ]ñ

= 0.

.

ï ñ

= 0.

+ 1)(arctan ï )2 + ô ]ñ = 0.

0

ï

+ ö (ô

2

+ õ )ñ = 0.

+ ï [( ô ï + õ ) arctan÷ ï + 2]ñó ò + õ arctan÷ Particular solution: 0 =  +   . = (

+ 1)−  J 2 exp(  arctan  ).

2

+ õ )(arctan ï ) ÷ = 

2

ñ óò

+ .

– 2ô [ö

ï

(arctan ï ) ÷ + 1]ñ = 0.

+ ô ï 2 arctan ï ñó ò + [ ô ( õ – ï ) arctan ï – õ 2 ]ñ = 0. Particular solution: 0 =  exp(   ).

(ï

2

+ 1)2 ñ

óò ò ó

+ [ ô (arctan ï )2 + õ arctan ï + ö ]ñ = 0.

The transformation ˆ = arctan  , ¨

(ï

2

= £ 

The transformation ˆ = arctan  , ¨

( ô ï 2 + õ )2 ñ ó ò ò ó + ( ö Particular solution: (ï

2

(ô

ï

2

+1

2

+1

leads to an equation of the form 2.1.2.6:

+ 1)2 ñó ò ò ó + [ õ (arctan ï ) ÷ – 1]ñ = 0.

+ ô )2 ñó ò ò ó + õ ( ï

Particular solution: 65.

= 0.

+ 1) arctan ï – ô

+ ô ( õ arctan÷  =  − .

+ 0

ï

+ õ )ñó ò – ô (arctan÷ = .

ï 2 ñ óò ò ó

¨2Šª

Š + 'ˆ  ¨ = 0.

63.

ï

–ô

2

¨2Šª

Š + ( ˆ 2 + 'ˆ + ( + 1)¨ = 0.

62.

ï

ï ñ

+ ô ( õ arctan ï – ô – 1)ñ = 0. =  − .

+ õ arctan÷ ï ñ ó ò

Particular solution: 60.

ï (õ

+ô

ï 2 ñ óò ò ó

Particular solution: 59.

.

arctan÷

ôƒõ

−1 −   .  0 = 

Particular solution: 57.



−

+ ï ( õ arctan ï + 2)ñ

Particular solution: 56.

=

+



ñóò

Particular solution: 54.

õ ï )ñ ó ò

+

2

+ õ )2 ñó ò ò ó + ö ( ô

Particular solution:

= £ 

+ ý )(arctan ï ) ÷ 0 = (˜ + .

ï

M

+ ô )(arctan ï ) ÷

2

ï

0

=£ 

2 0

2

ñ óò

– ö (arctan ï ) ÷

ñóò

– [õ

+ .

+ õ )(arctan ï ) ÷

= exp O − X



leads to an equation of the form 2.1.2.7:

M2



ñóò +  P

ï

ñ

= 0.

(arctan ï ) ÷ + ô ]ñ = 0.

+ [ ö (arctan ï ) ÷ – 2 ô

ï

– 1]ñ = 0.

.

Page 275

276

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.1.7-4. Equations with arccotangent. 66.

ñóò ò ó

ï

+ (ô

+ ö arccot ï )ñ„ó ò + [ ö ( ô

õ

+

Particular solution: 67.

+ õ (arccot ï ) ÷

ñóò ò ó

ñóò

ñ óò ò ó

Particular solution: 70.

ñóò ò ó

+

ô ïƒ÷

71.

72.

ï ñóò ò ó

(arccot ï )ú

Particular solution:

0

ï

+ [ô ( õ

ï ñóò ò ó

ï

+ [( ô

ï ñ óò ò ó

2

+

õï

ï ñóò ò ó

ïƒ÷

+ (ô

+1

ï ñ óò ò ó

ï ÷

+ (ô

+

ï 2 ñ óò ò ó

+

ï 2 ñ óò ò ó (ô

ï

2

õï

ï 4 ñ óò ò ó





õï

1−

=

ï

+

ô ï

2

0

ï

– 1]ñ„ó ò + ôƒõ −





= 0.

ï

óò

ïƒ÷ ( õ

   B +1 + [õ (ô

+ 2)]ñ = 0.

2

ï

arccot ï

ñ

= 0.

ñ

= 0.

+1

ï ÷

ñ

= 0.

arccot ï + ù )ñ = 0.

P

. – 1) arccot ï + ôù

ï ÷

–1

]ñ = 0.

+ ô ( õ arccot ï – ô – 1)ñ = 0.

=  − .

ï

2

+ õ )(arccot ï ) ÷

0

= 

ñóò 0

ï

.

−1 −   .  0 = 

arccot ï

Particular solution:

ñ

+ ô ( õ – 1)(arccot ï ) ÷

óò

= exp O −

ñóò

]ñ = 0.

= 0.

+ 1) arccot ï + õ ( õ

+ ï ( õ arccot ï + 2)ñ„ó ò + [ õ ( ô + õ )ñó ò ò ó + ö ( ô

ñ

(arccot ï )ú

   B ). 0 =  exp(− 

arccot ï

–1

.

.

arccot ï )ñ

Particular solution: 80.

ïƒ÷

= ( ' + 1) 

0

Particular solution: 79.

–ô

–1

ïƒú

+þ

.

P

– ô (arccot ï )÷

+ õ arccot ï )ñó ò + ô

Particular solution: 78.

= 

0

Particular solution: 77.

+1

0 =  +  .

Particular solution: 76.

¡

+1

 +

ï 2ú

) arccot ï + 2]ñ„ó ò + õ arccot ï

Particular solution: 75.



ñ óò

(arccot ï ) ÷ + õ ]ñ

ï

+ [ô

(arccot ï )÷ – ô

– [ô ( õ

0

Particular solution: 74.

ïƒú

+ 1) arccot ï + õ

Particular solution: 73.

.

= .

0

ñóò

ï ñóò ò ó

ô ï

ñóò

arccot ï

+

− ' ).

=  +  .

0

Particular solution:

v

= exp O −

0

+ õ )(arccot ï ) ÷

ï

+ (ô

−

+ ô [õ

Particular solution: 69.

=

0

+ õ (arccot ï ) ÷

ñóò ò ó

2

+ õ ) arccot ï + ô ]ñ = 0.

+ ö [ õ (arccot ï ) ÷ – ö ]ñ = 0.

ñóò

Particular solution: 68.

= exp(− 21 

0

ï

2

+ 1) arccot ï – ô

ñóò

– 2ô [ö

ï

2

ï 2 ]ñ

= 0.

(arccot ï ) ÷ + 1]ñ = 0.

+ .

+ [ ô ( õ – ï ) arccot ï – õ 2 ]ñ = 0.

=  exp(   ).

Page 276

277

2.1. LINEAR EQUATIONS

81.

(ï

2

+ 1)2 ñó ò ò ó + [ ô (arccot ï )2 + õ arccot ï + ö ]ñ = 0.

The transformation ˆ = arccot  , ¨

= £ 

¨2Šª

Š + ( ˆ 2 + 'ˆ + ( + 1)¨ = 0.

82.

(ï

2

The transformation ˆ = arccot  , ¨

84.

( ô ï 2 + õ )2 ñ ó ò ò ó + ( ö Particular solution: (ï

2

= £ 

+ ô ) ñó ò ò ó + õ ( ï 2

+ ý )(arccot ï ) ÷ 0 = (˜ + .

ï

(ô

ï

2

M

+ ô )(arccot ï ) ÷

2

Particular solution: 85.

+1

2

+1

leads to an equation of the form 2.1.2.6:

+ 1)2 ñó ò ò ó + [ õ (arccot ï ) ÷ – 1]ñ = 0.

¨ Šª

Š + 'ˆ  ¨ = 0.

83.

2

+ õ )2 ñó ò ò ó + ö ( ô

0

ï

=£ 

2

Particular solution:

0

2

ñ óò

– ö (arccot ï ) ÷

ñóò

– [õ

+ .

+ õ )(arccot ï ) ÷

= exp O − X

M2



leads to an equation of the form 2.1.2.7:



ï

ñ

= 0.

(arccot ï )÷ + ô ]ñ = 0.

+ [ ö (arccot ï ) ÷ – 2 ô

ñóò

ï

– 1]ñ = 0.

.

+  P

2.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions 1.

ñóò ò ó

+ ô ñó ò + õ [ õ + ô tan( õ Particular solution: 0 = cos( ' ).

ó

2.

ñ óò ò ó

ó

3.

ó

ó

+ ô  ñ ó ò + õ [ õ – ô  cot( õ Particular solution: 0 = sin( ' ).

ï )]ñ

= 0.

ï )]ñ

= 0.

+ ô cosh÷ ( ï ) ñó ò + õ [ õ + ô cosh ÷ ( Particular solution: 0 = cos( ' ).

4.

+ ô cosh÷ ( ï ) ñó ò + õ [ õ – ô cosh ÷ ( Particular solution: 0 = sin( ' ).

5.

ñóò ò ó

7. 8.

cosh÷ ( ü

ô

+

ï ) ñ óò

+ ô sinh ÷ ( ï ) ñ Particular solution:

ñ óò ò ó

ñ óò ò ó + ô sinh÷

( Particular solution:

ñóò ò ó

+

ô

sinh ÷ ( ü

ï ) ñ óò

+ ô tanh÷ ( ï ) ñ Particular solution:

ñ óò ò ó

+

õ ó [ô



cosh÷ ( ü



0

= exp O −  

0

+ õ [ õ + ô sinh ÷ ( = cos( ' ).

óò

ï ) ñ óò

Particular solution: 9.

ï

ñóò ò ó

Particular solution: 6.

ï

ñóò ò ó

0

Œ

0

óò 0

õ ó [ô



sinh ÷ ( ü

= exp O −  

Œ



ï )]ñ

= 0.

) cot( õ

ï )]ñ

= 0.

)–

õ ó

+ ]ñ = 0.

.

P

+ õ [ õ – ô sinh ÷ ( = sin( ' ). +

ï

) tan( õ

P

ï ï ï

) tan( õ

ï )]ñ

= 0.

) cot( õ

ï )]ñ

= 0.

) – õ

ó

+ ]ñ = 0.

.

+ õ [ õ + ô tanh÷ ( = cos( ' ).

ï

) tan( õ

ï )]ñ

= 0.

Page 277

278 10. 11.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ô tanh÷ ( ï ) ñ Particular solution:

ñ óò ò ó ñóò ò ó

tanh÷ ( ü

ô

+

0

ï ) ñ óò

Particular solution: 12. 13. 14.

õ ó [ô

+



tanh÷ ( ü

Œ

+ ô coth÷ ( ï ) ñ Particular solution:

ñ óò ò ó ñóò ò ó

coth÷ ( ü

ô

+

ï ) ñ óò

0

+

õ ó [ô



Œ

+ ô ln ÷ ( ï ) ñó ò + õ [ õ + ô ln ÷ ( Particular solution: 0 = cos( ' ).

17.

ñóò ò ó

ln ÷ ( ü

+

Particular solution:

0

ñóò ò ó

cos÷ ( ü

ô

+

ï ) ñ óò

ñóò ò ó

sin ÷ ( ü

ô

+

ï ) ñ óò

Particular solution:

ñóò ò ó

tan÷ ( ü

ô

+

ï ) ñ óò

Particular solution: 21.

ñóò ò ó

+

ô

cot÷ ( ü

ï ) ñ óò

Particular solution: 22.

õ ó [ô

ï ) ñ óò

ô

+

ï

ñóò ò ó

ó

+ õ ln ÷

+ ( ô

Particular solution:

õ ó [ô

+

Œ

õ ó [ô

Œ



cot÷ ( ü

ï )ñ óò

Œ

+



) cot( õ

ï )]ñ

= 0.



= exp O −  

Œ

ó

ï

ó

+ ]ñ = 0.

) – õ

)–

ï ï

ln÷



+ ]ñ = 0.

ó

+ ]ñ = 0.

õ ó

+ ]ñ = 0.

)–

õ ó

+ ]ñ = 0.

)–

õ ó

+ ]ñ = 0.

.

P

ô ó ( õ

ó

.

P

= exp O −  

) – õ

.

P



= 0.

= 0.

tan÷ ( ü



ï )]ñ

ï )]ñ

ï





Œ

) cot( õ

.

P

sin÷ ( ü

= exp O −  

+

0





ï

= 0.

) tan( õ

cos÷ ( ü

õ ó [ô

ï

ï )]ñ

.

P

= exp O −  

+

0

õ ó [ô

+ ]ñ = 0.

) tan( õ

) – õ





Œ

= exp O −  

0

0

ï

= exp O −  

+

0

ln÷ ( ü

õ ó

= 0.

.

P

ï

ñ óò ò ó

Particular solution:

20.



ñóò ò ó

+ ô ln ÷ ( ï ) ñ ó ò + õ [ õ – ô ln÷ ( Particular solution: 0 = sin( ' ).

19.

ï

coth÷ ( ü

= exp O −  

0

)–

ï )]ñ

.

P

+ õ [ õ – ô coth ÷ ( = sin( ' ).

óò

) cot( õ

ï

+ ô coth÷ ( ï ) ñó ò + õ [ õ + ô coth÷ ( Particular solution: 0 = cos( ' ).

16.

18.



= exp O −  

0

ï

ñóò ò ó

Particular solution: 15.

+ õ [ õ – ô tanh÷ ( = sin( ' ).

óò

ï

+ )ñ = 0.

.

P

ó

23.

+ ( ô  + õ cos ï )ñ ó ò + õ ( ô  cos ï – sin ï )ñ = 0. Particular solution: 0 = exp(−  sin  ).

24.

ñóò ò ó

ñ óò ò ó

+ ( ô

ó

+ õ cos÷

Particular solution:

ï )ñ óò

0

+

ô ó ( õ 

= exp O −  

Œ



P

cos÷

ï

+ )ñ = 0.

.

Page 278

279

2.1. LINEAR EQUATIONS

25.

ñ óò ò ó

+ ( ô

ó

+ õ cos÷



Particular solution: 26.

ñóò ò ó

ó

ñóò ò ó

ó

ñóò ò ó

ó

ñóò ò ó

ó

ñóò ò ó

ó

ñóò ò ó

ó

ñóò ò ó

+ ( ô

ó

ñóò ò ó

+ ( ô cosh÷

ñóò ò ó

+ ( ô cosh÷

ñóò ò ó

+ ( ô cosh÷

ñóò ò ó

+ ( ô cosh÷

ï

ñóò ò ó

+ ( ô cosh÷

ñóò ò ó

+ ( ô cosh÷

ï

ñóò ò ó

–1

ï (ô  ó ó

= cos 

ï )ñ óò

+1

0

+

 .

ô ó ( õ

= exp O −  

Œ

0

+ ( ô sinh ÷

ï

= sin1−   .

ï )ñ óò 0



= exp O −  

Œ



tan ï + 1)ñ = 0.

ï

0

0

0

0

+ )ñ = 0.

.

P

ó (õ

+ ô

 . M P

tan÷





ó

cot ï – 1)ñ = 0.

cot÷

P

ï

+ )ñ = 0.

.

ï

= exp(−  sin  ).

= exp O −  X cos + 

–1

+ õ sinú

= exp O −  X sin + 

cos ï – sin ï )ñ = 0. cosh÷

ï (ô

0

= cos 

+1

0

ï

–1

+ õ cos ï )ñó ò + õ ( ô sinh ÷

Particular solution:

0

= exp(−  sin  ).

cos ï – þ

sin ï )ñ = 0.

sin ï + cos ï )ñ = 0.

ï (ô

cosh÷

 . M P

 .

= sin1−   .

ï

 . M P

= exp(  cos  ).

ï )ñ óò

sin ï + ù cos ï )ñ = 0.

+ õ cot ï )ñó ò + ( õ – 1)( ô cosh÷

Particular solution: 39.

.

P

+ õ sin ÷

+ )ñ = 0.

+ õ tan ï )ñó ò + ( õ + 1)( ô cosh÷

ï

Particular solution: 38.

Œ

= exp O −  X sin  

+ õ sin ú

Particular solution: 37.

= exp O −  

ï

+ õ sin ï )ñó ò + õ ( ô cosh÷

ï

Particular solution: 36.





sin ÷

+ õ cosú'ï )ñó ò + õ cosú

ï

Particular solution: 35.

ô ó ( õ

+ õ cos ï )ñó ò + õ ( ô cosh÷

ï

Particular solution: 34.

0

+ õ cot÷

Particular solution: 33.

+

+ õ cot ï )ñó ò + ( õ – 1)(ô-

+ ( ô

Particular solution: 32.

0

sin ï + cos ï )ñ = 0.

= exp(  cos  ).

ï )ñ óò

+ õ tan÷

+ ( ô

Particular solution: 31.

ó

cos ï – ù sin ï )ñ = 0.

 . M P

+ õ tan ï )ñó ò + ( õ + 1)( ô-

+ ( ô

Particular solution: 30.

0

ï (ô   ó

–1

= exp O − /X cos  

ï )ñ óò

+ õ sin ÷

+ ( ô

Particular solution: 29.

0

+ õ sin ÷

+ ( ô

Particular solution: 28.

0

+ õ cos÷

+ õ sin ï )ñó ò + õ ( ô

+ ( ô

Particular solution: 27.

ï )ñ ó ò

ï

ï

ï

ï

sin ï + þ

cos ï )ñ = 0.

tan ï + 1)ñ = 0. cot ï – 1)ñ = 0.

cos ï – sin ï )ñ = 0.

Page 279

280 40.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

+ ( ô sinh ÷

+ õ cosú

ï

Particular solution:

0

ï )ñ óò

+ õ cosú

= exp O −  X cos + 

41.

+ ( ô sinh ÷ ï + õ sin ï )ñó ò + õ ( ô sinh ÷ Particular solution: 0 = exp(  cos  ).

42.

ñóò ò ó

ï

+ õ sin ú

Particular solution: 43.

ñóò ò ó

+ ( ô sinh ÷

ï

ñóò ò ó

+ ( ô sinh ÷

+ õ sinú

 . M P

= cos 

0

= sin1−   .

+1

 .

+ õ cot ï )ñó ò + ( õ – 1)(ô sinh ÷

ï

Particular solution: 45.

+ ( ô tanh ÷ ï + õ cos ï )ñó ò + õ ( ô tanh ÷ Particular solution: 0 = exp(−  sin  ).

46.

ñóò ò ó

+ ( ô tanh ÷

+ õ cosú

ï

0

ï )ñ óò

+ õ cosú

–1

= exp O −  X cos + 

47.

+ ( ô tanh ÷ ï + õ sin ï )ñó ò + õ ( ô tanh÷ Particular solution: 0 = exp(  cos  ).

48.

ñóò ò ó

ñóò ò ó

+ ( ô tanh ÷

+ õ sin ú

ï

Particular solution:

ñóò ò ó

+ ( ô tanh ÷

ñóò ò ó

+ ( ô tanh ÷

0

ï )ñ óò

+ õ sin ú

+

ï

0

= cos 

0

= sin1−   .

+1

52.

ñóò ò ó

+ õ cosú'ï )ñó ò + õ cosú

ï

Particular solution:

0

+

= exp O − /X cos 

53.

+ ( ô coth÷ ï + õ sin ï )ñó ò + õ ( ô coth÷ Particular solution: 0 = exp(  cos  ).

54.

ñóò ò ó

ñóò ò ó

+ ( ô coth÷

ï

+ õ sinú'ï )ñó ò + õ sin ú

Particular solution:

ñ óò ò ó

+ ( ô coth÷

ï

0

= exp O − /X sin + 

+ õ tan ï )ñ

Particular solution:

0

= cos 

óò

+1

cos ï – sin ï )ñ = 0. tanh÷

ï (ô

ï

cos ï – þ

sin ï )ñ = 0.

sin ï + cos ï )ñ = 0.

ï (ô

ï

tanh÷

ï

–1

ï

ï

sin ï + þ

cos ï )ñ = 0.

tan ï + 1)ñ = 0. cot ï – 1)ñ = 0.

cos ï – sin ï )ñ = 0. coth÷

ï (ô

ï

cos ï – þ

sin ï )ñ = 0.

 . M P

ï

–1

sin ï + cos ï )ñ = 0.

ï (ô

 . M P

+ ( õ + 1)(ô coth÷

 .

cos ï )ñ = 0.

cot ï – 1)ñ = 0.

 .

+ ( ô coth÷ ï + õ cos ï )ñó ò + õ ( ô coth÷ Particular solution: 0 = exp(−  sin  ).

55.

ï

ñóò ò ó

+ ( ô coth÷

ï

 . M P

51.

sin ï + þ

tan ï + 1)ñ = 0.

+ õ cot ï )ñó ò + ( õ – 1)(ô tanh ÷

Particular solution:

ï

 . M P

–1

= exp O − /X sin 

ï

+ õ tan ï )ñó ò + ( õ + 1)( ô tanh ÷

ï

Particular solution: 50.

ï

ñóò ò ó

Particular solution:

49.

sinh÷

ï (ô

–1

= exp O −  X sin + 

0

sin ï )ñ = 0.

sin ï + cos ï )ñ = 0.

+ õ tan ï )ñó ò + ( õ + 1)( ô sinh ÷

ï

Particular solution: 44.

0

ï )ñ óò

cos ï – þ

ï

 . M P

ñóò ò ó

+ ( ô sinh ÷

sinh÷

ï (ô

–1

coth÷

ï

ï

sin ï + þ

cos ï )ñ = 0.

tan ï + 1)ñ = 0.

Page 280

281

2.1. LINEAR EQUATIONS

56.

ñ óò ò ó

+ ( ô coth÷

+ õ cot ï )ñ

ï

Particular solution: 57.

ñ óò ò ó

+ ( ô ln÷

ñóò ò ó

+ ( ô ln÷

ñóò ò ó

+ ( ô ln÷

ñóò ò ó

+ ( ô ln÷

ñóò ò ó

+ ( ô ln÷

ñóò ò ó

+ ( ô ln÷

ñóò ò ó

ô ó

+

64.

ñóò ò ó

ô

Particular solution: 66.

ñóò ò ó

ô

ï

Particular solution: 67.

ñ óò ò ó

cosh÷ (

ô

+

ï

ñ óò ò ó

ô

Particular solution: 69.

ñ óò ò ó

cosh÷ (

ô

+

ï

Particular solution: 70.

ñóò ò ó

ô

ï

Particular solution: 71.

ñóò ò ó

+

ô

cosh( õ

0

0

0

0

0

0

= cos( ' ).

+ õ [ õ – ô

= sin( ' ).

ï ) ñ óò

ï

0

0

ï

ï

tan ï + 1)ñ = 0.

cot ï – 1)ñ = 0.

ï )]ñ

= 0.

cos( õ

ï )]ñ

= 0. ) ln ÷ (

ï

ï ) ñ óò

– õ [õ +

ï ) ñ óò

+ õ [ õ + ô cosh ÷ (

ï ) ñ óò

– õ [ õ + ô sinh( õ

ï ) ñ óò

+ õ [ õ – ô cosh ÷ (

ï ) ñ óò

– õ [ õ + ô sinh( õ

ï ) ñ óò

– õ [õ +

ô

ï

sinh( õ

= cos( ' ).

ï

= sin( ' ).

= cosh( ' ).

sinh( õ

ï ï

ï )]ñ

) cos÷ (

= cosh( ' ).

ô

cos ï )ñ = 0.

 . M P

– õ [ õ + ô sinh( õ

= cosh( ' ).

sin ï + þ

ï

sin( õ

= cosh( ' ).

) cot÷ (

Particular solution:

ó

ln÷

ï (ô

= cosh( ' ).

) sin( õ 0

ó

sin ï )ñ = 0.

 . M P

 .

+ õ [ õ + ô

) tan ÷ (

cosh( õ

+

+1

= sin1−   .

) sin ÷ (

ï

cosh( õ

+

–1

= exp O − /X sin + 

= cos 

cos ï – þ

ï

sin ï + cos ï )ñ = 0.

ï

= exp(  cos  ).

) cos( õ

Particular solution: 68.

+

= exp O − /X cos 

) cos ÷ (

cosh( õ

+

0

) ln ÷ (

ï

cosh( õ

+

0

ï ) ñ óò

sin( õ

Particular solution: 65.

0

ï ) ñ óò

cos( õ

Particular solution:

ñ óò ò ó + ô ó

0

ln ÷

ï (ô

–1

+ õ cot ï )ñó ò + ( õ – 1)(ô ln ÷

ï

Particular solution: 63.

= exp(−  sin  ).

+ õ tan ï )ñó ò + ( õ + 1)( ô ln ÷

ï

Particular solution: 62.

0

cot ï – 1)ñ = 0.

ï

cos ï – sin ï )ñ = 0.

ï

+ õ sinú'ï )ñó ò + õ sin ú

ï

Particular solution: 61.

+ õ ( ô ln÷

+ õ sin ï )ñó ò + õ ( ô ln÷

ï

Particular solution: 60.

0

óò

+ õ cosú'ï )ñó ò + õ cosú

ï

Particular solution: 59.

= sin1−   .

+ õ cos ï )ñ

ï

Particular solution: 58.

0

+ ( õ – 1)(ô coth÷

óò

= 0.

ï )]ñ

= 0.

ï )]ñ

= 0.

) sin ÷ (

ï )]ñ

= 0.

ï

) cos( õ

ï )]ñ

= 0.

) tan ÷ (

ï )]ñ

= 0.

) cot÷ (

ï )]ñ

= 0.

ï

) sin( õ

Page 281

282 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ô sinh( õ ï ) ln ÷ ( ü ï ) ñ ó ò – õ [ õ + Particular solution: 0 = sinh( ' ).

ñ óò ò ó

ô

+ ô sinh( õ ï ) cos ÷ ( ü ï ) ñó ò – õ [ õ + ô cosh( õ Particular solution: 0 = sinh( ' ).

+ ô sinh ÷ ( ï ) cos( õ ï ) ñ„ó ò + õ [ õ + ô sinh ÷ ( Particular solution: 0 = cos( ' ). + ô sinh( õ ï ) sin ÷ ( ü ï ) ñó ò – õ [ õ + ô cosh( õ Particular solution: 0 = sinh( ' ).

+ ô sinh ÷ ( ï ) sin( õ ï ) ñ„ó ò + õ [ õ – ô sinh ÷ ( Particular solution: 0 = sin( ' ).

ñóò ò ó

+ ô sinh( õ ï ) tan ÷ ( ü ï ) ñ ó ò – õ [ õ + Particular solution: 0 = sinh( ' ).

ñ óò ò ó

+ ô sinh( õ ï ) cot÷ ( ï ) ñó ò – õ [ õ + Particular solution: 0 = sinh( ' ).

ô

ñóò ò ó

ï

cosh( õ

ô

ï

cosh( õ

+ ô tanh÷ ( ï ) cos( õ ï ) ñ ó ò + õ [ õ + Particular solution: 0 = cos( ' ).

ñ óò ò ó

= 0.

ï

) sin( õ

ï )]ñ

= 0.

) sin ÷ ( ü

ï )]ñ

= 0.

ï

) cos( õ

ï )]ñ

= 0.

) tan ÷ ( ü

ï )]ñ

= 0.

) cot÷ (

ï )]ñ

= 0.

) sin( õ

ï )]ñ

= 0.

) cos( õ

ï )]ñ

= 0.

) sin( õ

ï )]ñ

= 0.

) cos( õ

ï )]ñ

= 0.

+ ô tanh÷ ( ï ) sin( õ ï ) ñ„ó ò + õ [ õ – ô tanh ÷ ( Particular solution: 0 = sin( ' ).

ï ï

ñóò ò ó

+ ô coth÷ ( ï ) cos( õ ï ) ñ„ó ò + õ [ õ + Particular solution: 0 = cos( ' ).

ñóò ò ó

coth÷ (

ô

+ ô coth÷ ( ï ) sin( õ ï ) ñ„ó ò + õ [ õ – ô coth÷ ( Particular solution: 0 = sin( ' ).

ñóò ò ó

+ ô ln ÷ ( ï ) cos( õ ï ) ñ„ó ò + õ [ õ + Particular solution: 0 = cos( ' ).

ñóò ò ó

ô

ln÷ (

ï )]ñ

= 0.

ï

) cos( õ

ï )]ñ

= 0.

õ 2 ó

) ln÷ ( ü

ï

ñóò ò ó

+ (ô +

õ 2 ó

) ln÷ ( ü

Particular solution:

ñ óò ò ó

+ (ô +

õ 2 ó

+ (ô +

õ 2 ó

0

Particular solution:

0

−



+   .

+ [( ô – õ

2

ï ) ñ óò

+ [( ô – õ

2

 

=  

) tan÷ ( ü

+ [( ô –

ï ) ñ óò

=  

) sin ÷ ( ü

Particular solution:

ñóò ò ó

0



=  

) cos÷ ( ü

Particular solution:

ñ óò ò ó + (ô + õ 2 ó

0

ï ) ñ óò

+   .

=  



−

+   .

ï ) ñ óò 



−

+ [( ô – −



+   .

ï

ï

) sin( õ

ñ óò ò ó

ï

= 0.

ï )]ñ

tanh ÷ (

ô

85.

88.

ï

ñóò ò ó

ï )]ñ

) cos÷ ( ü

ñóò ò ó

+ ô ln ÷ ( ï ) sin( õ ï ) ñ ó ò + õ [ õ – ô ln÷ ( Particular solution: 0 = sin( ' ).

87.

ï

ñóò ò ó

84.

86.

) ln ÷ ( ü

ï

cosh( õ



ó 

ó

õ 2 ó

) – ]ñ = 0.

) cos÷ ( ü ) sin ÷ ( ü ) tan÷ ( ü

ï ï

) – ]ñ = 0.

ï

) – ]ñ = 0. ) – ]ñ = 0.

Page 282

283

2.1. LINEAR EQUATIONS

89.

ñ óò ò ó

+ (ô +

õ 2 ó

) cot÷ ( ü

Particular solution: 90.

91.

92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104.

ï ) ñ óò

=  

0



õ 2 ó

+ [( ô – −



+   .

) cot÷ ( ü

ï

) – ]ñ = 0.

+ ( ô sn2 ï + õ )ñ = 0. The Lam´e equation in the form of Jacobi, sn  is the Jacobi elliptic function. See the books by Whittaker & Watson (1952), Bateman & Erd´elyi (1955, Vol. 3), and Kamke (1977) for information on this equation.

ñóò ò ó

+ [
4€ ( ï ) + ( ]ñ = 0. The Lam´e equation in the form of Weierstrass, < ( ) is the Weierstrass function. See the books by Whittaker & Watson (1952), Bateman & Erd´elyi (1955, Vol. 3), and Kamke (1977) for information on this equation.

ñóò ò ó

ï ñóò ò ó

+ ( ô ï ln ï + Particular solution:

õ ó )ñ óò

ï ñ óò ò ó + (1 – ô ï  ó

0

+ ô ( õ  −   =  .

ln ï )ñó ò + ô Particular solution: 0 = ln  .

óñ

+ (1 – ô ï cosh ÷ ï ln ï )ñó ò + Particular solution: 0 = ln  .

ô

ó

ln ï + 1)ñ = 0.

= 0.

+ ( ô ï ln ï + õ cosh ÷ ï )ñó ò + ô ( õ cosh÷   Particular solution: 0 =    −  .

ï ñóò ò ó ï ñóò ò ó

cosh÷

ï

ï ñ

+ ( ô ï ln ï + õ sinh ÷ ï )ñó ò + ô ( õ sinh ÷   Particular solution: 0 =    −  .

ï ñóò ò ó

+ (1 – ô ï sinh ÷ Particular solution:

ï ñ óò ò ó

ï

0

ln ï )ñ = ln  .

+ ô sinh ÷

óò

= 0. ln ï + 1)ñ = 0.

ï

ï ñ

= 0.

+ ( ô ï ln ï + õ tanh ÷ ï )ñó ò + ô ( õ tanh ÷   Particular solution: 0 =    −  .

ï ñóò ò ó

+ (1 – ô ï tanh÷ ï ln ï )ñ Particular solution: 0 = ln  .

ï ñ óò ò ó

óò

+ ô tanh÷

ï

ï ñ

+ ( ô ï ln ï + õ coth÷ ï )ñó ò + ô ( õ coth÷   Particular solution: 0 =    −  .

ï ñóò ò ó

+ (1 – ô ï coth÷ ï ln ï )ñó ò + Particular solution: 0 = ln  .

ï ñóò ò ó

ô

coth÷

+ ( ô ï ln ï + õ cos ÷ ï )ñó ò + ô ( õ cos÷   Particular solution: 0 =    −  . + (1 – ô ï cos÷ Particular solution:

ï

0

ln ï )ñó ò + ô cos÷ = ln  .

+ ( ô ï ln ï + õ sin ÷ ï )ñ ó ò + ô ( õ sin ÷   Particular solution: 0 =    −  .

ï ñ óò ò ó

ï

ï ñ

ï ñóò ò ó ï ñóò ò ó

ln ï + 1)ñ = 0.

ï

ï ñ ï

ln ï + 1)ñ = 0. = 0. ln ï + 1)ñ = 0. = 0.

ln ï + 1)ñ = 0. = 0. ln ï + 1)ñ = 0.

Page 283

284 105.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñ óò ò ó

sin ÷

ï

+ (1 – ô

ï

Particular solution:

ln ï )ñ

106.

ï ñóò ò ó

0

ln ï + õ tan ÷

ï

+ (ô

ï ñóò ò ó

tan÷

ï

+ (1 – ô

ï

Particular solution: 108.

ï ñóò ò ó

+ (ô

ï

0

ï ñóò ò ó

+ (1 – ô

ï

cot÷

Particular solution: 110.

ï 2 ñ óò ò ó

+ ï ( ô ln ï +

Particular solution: 111.

ï 2 ñ óò ò ó ï 2 ñ óò ò ó ï 2 ñ óò ò ó ï 2 ñ óò ò ó ï 2 ñ óò ò ó ï 2 ñ óò ò ó ï 2 ñ óò ò ó ï 2 ñ óò ò ó

ñóò ò ó

0

0

0

0

0

0

ñóò ò ó

0

0

+ ô ( õ

0

ó

ï

ln ï + 1)ñ = 0.

ï ñ

= 0.

ln ï – ln ï + 1)ñ = 0.

ï )ñ óò

+ ô ( õ cosh÷

ï )ñ óò

+ ô ( õ sinh ÷

ï )ñ ó ò

+ ô ( õ tanh ÷

ï )ñ óò

+ ô ( õ coth÷

ï

= exp(− 21  ln2  ). = exp(− 21  ln2  ).

ï )ñ óò

+ ô ( õ cos÷

ï )ñ óò

+ ô ( õ sin ÷

ï )ñ ó ò

+ ô ( õ tan÷

ï )ñ óò

+ ô ( õ cot÷

ó )ñ óò

+ ô ( õ

= exp(− 21  ln2  ).

ï

ln ï – ln ï + 1)ñ = 0.

= exp(− 21  ln2  ). = cot   = cot  

1 2

1 2

 .

ï )ñ óò  .

ó

ï ï

ln ï – ln ï + 1)ñ = 0. ln ï – ln ï + 1)ñ = 0.

ln ï – ln ï + 1)ñ = 0.

ï

ln ï – ln ï + 1)ñ = 0.

= exp(− 21  ln2  ). = exp(− 21  ln2  ).

ln ï – ln ï + 1)ñ = 0.

ï

= exp(− 21  ln2  ).

+ sin ï ( ô + õ cosh ÷

Particular solution:



= 0.

= exp(− 21  ln2  ).

+ sin ï ( ô + õ-

Particular solution: 120. sin2 ï

0

cot÷

ô

ln ï + 1)ñ = 0.

= exp(− 21  ln2  ).

+ ï ( ô ln ï + õ cot ÷

Particular solution: 119. sin2 ï

0

+ ï ( ô ln ï + õ tan ÷

Particular solution: 118.

õ  ó )ñ óò

+ ï ( ô ln ï + õ sin ÷

Particular solution: 117.

= ln  .

+ ï ( ô ln ï + õ cos ÷

Particular solution: 116.

0

+ ï ( ô ln ï + õ coth ÷

Particular solution: 115.

ln ï )ñó ò +

ï

ï ñ

+ ô ( õ cot÷

   −  . 0 = 

+ ï ( ô ln ï + õ tanh ÷

Particular solution: 114.

ï )ñ óò

+ ï ( ô ln ï + õ sinh ÷

Particular solution: 113.

= ln  .

+ ï ( ô ln ï + õ cosh ÷

Particular solution: 112.

ï

= 0.

+ ô ( õ tan÷

ln ï )ñó ò + ô tan÷

ln ï + õ cot÷

Particular solution: 109.

ï )ñ óò

ï ñ

   −  . 0 = 

Particular solution: 107.

= ln  .

+ ô sin÷

óò

ï ï

ln ï – ln ï + 1)ñ = 0. ln ï – ln ï + 1)ñ = 0.

– cos ï )ñ = 0.

+ ô ( õ cosh÷

ï

– cos ï )ñ = 0.

Page 284

285

2.1. LINEAR EQUATIONS

121. sin2 ï

+ sin ï ( ô + õ sinh ÷

ñóò ò ó

Particular solution: 122. sin2 ï

Particular solution:

Particular solution:

Particular solution:

= cot  

0

0

0

0

Particular solution:

0

ï )ñ óò

+ ô ( õ coth÷

õ ó )ñ óò

= cot   = cot   = cot   = cot   = cot  

+ cos ï ( ô + õ ln ÷

ñ óò ò ó

+ ô ( õ tanh ÷

1 2

1 2

+ cos ï ( ô + õ coth÷

ñóò ò ó

Particular solution: 130. cos2 ï

0

ï )ñ óò

 .  .  .

+ ô ( õ ln÷

ï )ñ óò

+ cos ï ( ô + õ tanh÷

ñóò ò ó

Particular solution: 129. cos2 ï

0

1 2

+ ô ( õ sinh ÷

1 2

+ cos ï ( ô + õ sinh ÷

ñ óò ò ó

Particular solution: 128. cos2 ï

= cot  

+ cos ï ( ô + õ cosh ÷

ñóò ò ó

Particular solution: 127. cos2 ï

0

+ cos ï ( ô +

ñóò ò ó

Particular solution: 126. cos2 ï

= cot  

+ sin ï ( ô + õ ln ÷

ñóò ò ó

125. cos2 ï

0

+ sin ï ( ô + õ coth ÷

ñóò ò ó

124. sin2 ï

= cot  

+ sin ï ( ô + õ tanh ÷

ñóò ò ó

123. sin2 ï

0

ï )ñ óò

= cot  

 .

+ ô ( õ

1 2

 +

1 2

 +

1 2

 +

1 2

 +

1 2

 +

1 2

 +

 

+ ô ( õ sinh ÷



+ sin ï )ñ = 0.

ï

+ sin ï )ñ = 0.

+ ô ( õ tanh÷

ï

.

1 4



+ ô ( õ coth÷

ï )ñ óò



.



.

1 4

+ ô ( õ ln ÷ 1 4

ï

.

1 4

ï )ñ óò

ï )ñ ó ò

– cos ï )ñ = 0.

.

1 4

ï )ñ ó ò

ï

+ ô ( õ cosh÷

ï )ñ óò

– cos ï )ñ = 0.

+ sin ï )ñ = 0.

.

1 4

ï

– cos ï )ñ = 0.

ï

ó

– cos ï )ñ = 0.

ï

ï

ï

+ sin ï )ñ = 0. + sin ï )ñ = 0.

+ sin ï )ñ = 0.

2.1.9. Equations with Arbitrary Functions ]

Notation:  =  ( ) and = ( ) are arbitrary functions;  ,  , ( , , B , , , , Œ , ƒ , ý , and = are ¡ M arbitrary parameters. 2.1.9-1. Equations containing arbitrary functions (but not containing their derivatives). 1.

ñóò ò ó

+

ôñ

=! .

Equation of forced oscillations without friction. Solution:

=

where 

0









 X   (ˆ ) sin[ , ( − ˆ )] ˆ M 0  −1 L 1 cosh( ,! ) + L 2 sinh( ,! ) + , X   (ˆ ) sinh[ , ( − ˆ )] ˆ M 0  L 1  + L 2 + X  ( − ˆ )  (ˆ ) ˆ M 0






L




1

cos( ,! ) + L

2

sin( ,! ) + ,

−1

if  = ,

2

if  = − ,

> 0, 2

< 0,

if  = 0,

is an arbitrary number.

Page 285

286

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.

ñ óò ò ó

+ ôñ ó ò + õÿñ = ! . Equation of forced oscillations with friction. The substitution = ¨ exp(− 21  ) leads to an equation of the form 2.1.9.1: ¨u

 + (  − 14  2 )¨ =  exp( 12  ).

3.

ñóò ò ó

+

!„ñóò

=" . =L

Solution:

1

+ X  − Ž O L

2

4.

+ !„ñ ó ò + ô ( ! – ô )ñ = 0.  Particular solution: 0 =  −  .

5.

ñóò ò ó

+

+ ô ( ïƒ÷

!„ñóò

–ô

!

ï 2÷

+ ï !„ñó ò – !„ñ = 0. Particular solution: 0 =  .

7.

ñ óò ò ó

+ (! +

ô ï ÷

ï ñóò ò ó

+ ï !„ñó ò – [( ô Particular solution:

9.

ï ñóò ò ó

+ (ï

!

0

ï

16.

+ [( ô

ï ÷

 

+1

= exp O −

+ õ )! +



B +1

+ 1) ! + ô ( ô  . 0 =  

0

ï ñóò ò ó

ï

 

.

P

ôù ï ÷

+1

− '

–1

]ñ = 0.

.

P

+ 2)]ñ = 0.

=

1−

.

+ [( ô ï + 1) ! + ô ï – 1]ñ„ó ò + ô 2 ï  Particular solution: 0 = (  + 1)  −  .

!„ñ

= 0.

ï ñóò ò ó

+ [( ô ï 2 + õ ï ) ! + 2]ñó ò + õ!„ñ = 0. Particular solution: 0 =  +   .

ï ñ óò ò ó

+ (! + ô

ï ÷

+1

Particular solution:

15.

B +1

)ñ = 0.

+ ô )ñó ò + ( ô – 1) !„ñ = 0.

Particular solution:

14.

óò

+ õ )ñ

8.

13.

–1



ñóò ò ó

Particular solution:

12.

ïƒ÷

+ù

= exp O −

0

6.

11.

  , where  = X¯  . M u P M M

ñ óò ò ó

Particular solution:

10.

+X  Ž

ï ñóò ò ó

)ñ

óò 0

+ô

ï ÷ (!

= exp O −

+ ù )ñ = 0.

   B +1

+1

P

+ ( ï ! + ô ïƒ÷ )ñó ò + [( ô ïƒ÷ – 1) ! + ôù Particular solution: 0 =  exp(−   B ).

ï ñóò ò ó

.

ïƒ÷

+ [( ô ïƒ÷ + 1) ! + ôù ïƒ÷ + 1 – 2ù ]ñó ò + ô Particular solution: 0 = (   + 1) exp(−   ).

–1

2

]ñ = 0.

ù ï 2÷ –1 „! ñ

= 0.

ï 2 ñ óò ò ó

+  ï ñ ó ò + ñ = ! . g The nonhomogeneous Euler equation. The substitution  =  leads to an equation of the g


form 2.1.9.2: g g + ( ƒ − 1) g + ý =  (  ).

ï 2 ñ óò ò ó

+ ï ñó ò + ( ï 2 – b 2 )ñ = ! . The nonhomogeneous Bessel equation. The general solution is expressed in terms of Bessel functions: = L 1 ø Á ( ) + L 2 n Á ( ) + 12 n Á ( ) X  ø Á ( )  ( )  − 12 ø Á ( ) X "n Á ( )  ( )  .



M



M

Page 286

287

2.1. LINEAR EQUATIONS

17. 18.

ï 2 ñ óò ò ó

+ ï !„ñó ò + ô ( ! – ô – 1)ñ = 0. Particular solution: 0 =  −  .

ï 2 ñ óò ò ó

+ ï ( ! + 2 ô )ñ

óò

Particular solution: 19.

ï 2 ñ óò ò ó

+

ï !„ñóò

21. 22. 23.

26. 27.

+ ù )! – ô

ï 2(ô ï

+2



 

!„ñ

–ù

– ù ]ñ = 0.

2

2 +1

2B + 1

P

.

= 0.

+ õ )ñ ó ò ò ó + ï ( ô ï 2 + õ ) !„ñ ó ò – [( ô Particular solution: 0 =  +   .

(ï

2

(ï

2

+ ô )2 ñó ò ò ó + ( ï

+ ô ) !„ñó ò – ( ï

2

+ ô )2 ñó ò ò ó + ( ï

£  0 =

2

+ .

[ï

ïƒ÷

( ô ï ÷ + õ ï )ñ ó ò ò ó + ( ô Particular solution:

0

= (

+ õ ) !„ñó ò – ôù  + . 0 = 

ï ÷ 0

ïƒ÷

+ õ )2 ñó ò ò ó + ( ô

0

30.

ñóò ò ó

+ ô

ó (!

Particular solution:

ñóò ò ó

+ (! +

ô ó )ñ óò

Particular solution:

ñóò ò ó

+ (ô +

– ô

ó

õ^ö ]ñ óò

!

+

ô ó ( !

0

= exp O −  

Œ

+ ô ]ñ = 0.

+ ù – 1)ñ = 0.

!

ï ÷

–1

+ õ ) ! + ôù (ù – 1) ï

÷

–2

]ñ = 0.

+ ôù – ô )ñ = 0.

ïƒ÷ ï

–1

– 1)ñ = 0.

+ õ )ñ = 0.

+ [( ô –



.

P

+ )ñ = 0.

 Œ

=  





= exp O −  

0

– (ô

2

+ )ñ = 0.

0

õ 2 ó ) „! ñóò

Particular solution:

(ï

(ï

 M .   +  P

= exp O − X

+ [ ô ï 2 + ( ô„ö + õ ) ï + Particular solution: 0 =  + ( .

!„ñóò

–2

+ õ ) !„ñó ò + ( ! – ôù

! ( ï )ñ óò ò ó +

–2

 +  )1 J . 0 = (

ïƒ÷

ïƒ÷

+ õ ï ) !„ñ ó ò – [( ôù =   + ' .

( ïƒ÷ + ô )2 ñó ò ò ó + ( ïƒ÷ + ô ) !„ñó ò – ïƒ÷

– 1) ï

+ (þ

!

+  )+ J 2.

2

– õ ) ! + 2 õ ]ñ = 0.

2

+ ô )ñ = 0.

!

+ ô ) !„ñó ò – þ

2

( ô ïƒ÷ + õ )ñó ò ò ó + ( ô Particular solution:

(ô

ï

2

29.

32.

ï 4÷

2

+ ô ( ô – 1)]ñ = 0.

2

+ ï 2 !„ñó ò + [( – ï ) ! – 2 ]ñ = 0. Particular solution: 0 =  exp( Œ  ).

Particular solution:

31.

ï

ï 4 ñ óò ò ó

Particular solution: 28.

+1

( ô ï 2 + õ ï + ö )ñó ò ò ó + ( ï + ü ) !„ñó ò – Particular solution: 0 =  + , .

Particular solution:

25.

  − . 0 = 

 exp O − 0 = 

Particular solution:

24.

2



−

Particular solution: 20.

+ ô )! – õ

−

ï 2÷

+ [( ô

ï

+ [( õ



P

.

õ 2 ó ) ! −



+   .

– ]ñ = 0.

Page 287

288 33.

SECOND-ORDER DIFFERENTIAL EQUATIONS ( ô



ó

óò ò ó

+ õ )2 ñ

+ ( ô

Particular solution: 34.

ñóò ò ó ñóò ò ó

– ô [ô +

Particular solution: 36.

ï ñóò ò ó

+ (1 –

Particular solution: 37.

ï ñóò ò ó

+ (! + ô

ï

ï 2 ñ óò ò ó ï 2 ñ óò ò ó ñóò ò ó

sin( ô

!

ñóò ò ó

+

cos( ô

!

ï )„ñ óò

ñóò ò ó ñóò ò ó

ï )ñ óò

ñóò ò ó ñ óò ò ó

cos÷

ô

+ (! +

sin2 ï

ñóò ò ó

48.

= 0.

= 0.

1 4

– (ln ï +

ô

+ 2) !hWUñ = 0.

= exp(− 21  ln2  ).

!

cos( ô

ï )]ñ

= 0.

!

sin( ô

ï )]ñ

= 0.

0

= cos(  ).

0

= exp(  cos  ).

0

= exp(−  sin  ).

ô

sin ÷

0

ï )ñ ó ò 0

cos÷

ô

+

–1

ï (!

= exp O − lX cos   +

sin ÷

ô

–1

ï (!

cos ï – ù sin ï )ñ = 0.

 . M P

sin ï + ù cos ï )ñ = 0.

= exp O −  X sin  

 . M P

+ sin ï ( ! + ô )ñ„ó ò + ô ( ! – cos ï )ñ = 0.

Particular solution: 47.

= 0.

= sin(  ).

ï )ñ óò

Particular solution: 46.

ˆõ )ñ

cos ï )ñ„ó ò + ô ( ! cos ï – sin ï )ñ = 0.

ô

+ (! +

+ (! +

+

sin ï )ñ„ó ò + ô ( ! sin ï + cos ï )ñ = 0.

ô

+ (! +

Particular solution: 45.

ó

+ ô ( ! ln ï – ln ï + 1)ñ = 0.

óò

+ ô [ô +

Particular solution: 44.

ï )]ñ

  =    − .

0

Particular solution: 43.

sinh( ô

0

+ ô [ô –

Particular solution: 42.

= 0.

= ln  .

0

Particular solution: 41.

ï )]ñ

0

+ ï ( ! + ô ln ï )ñ

+

cosh( ô

!



 .

£  (ln  +  ). 0 =

Particular solution: 40.

!„ñ

+ 2 ï (ln ï + ô ) !„ñ„ó ò + V

Particular solution: 39.

!

– ö

ln ï )ñ„ó ò + ô ( ! ln ï + 1)ñ = 0.

Particular solution: 38.

+ 

ó (! 

= cosh(  ).

0

ln ï )ñ„ó ò +

! ï



+ ö −v

= sinh( ).

0

ï )„ñ óò

cosh( ô

!

+

óò

+ õ ) !„ñ

– ô [ô +

Particular solution: 35.

ó

=   

0

ï )„ñ óò

sinh( ô

!

+



0

= cot  

1 2

 .

cos2 ï ñ ó ò ò ó + cos ï ( ô + ! )ñ ó ò + ô ( ! + sin ï )ñ = 0. Particular solution: 0 = cot   12  + 14  .

ñóò ò ó

+

!„ñóò

+ô [ +

Particular solution:

!

tan(

ï



) + ( – ô ) tan2 (

J 0 = [cos( Œ  )]  .

ï )]ñ

= 0.

Page 288

289

2.1. LINEAR EQUATIONS

49.

ñ óò ò ó

tan ï )ñ

ô

+ (! +

Particular solution: 50.

ñóò ò ó

0

ñóò ò ó

+

!„ñóò

0

+ô [ –

!

Particular solution: 52.

ñóò ò ó

= cos 

+1

 .

+ tan ï ( ! + ô – 1)ñ„ó ò + [( ô tan2 ï – 1) ! + 2 ô + 2]ñ = 0.

Particular solution: 51.

+ ( ô + 1)( ! tan ï + 1)ñ = 0.

óò

+ (! +

ô

= sin  cos   .

ï

cot(

) + ( – ô ) cot2 (

J 0 = [sin( Œ  )]  .

ï )]ñ

= 0.

cot ï )ñ„ó ò + ( ô – 1)( ! cot ï – 1)ñ = 0.

Particular solution:

0

= sin1−   .

2.1.9-2. Equations containing arbitrary functions and their derivatives. 53.

ñóò ò ó

– (!

2

+

!„ó ò )ñ

= 0.

Particular solution: 54.

ñóò ò ó

+

!„ñóò

0

– [ ô ( ô + 1) !

ñóò ò ó

+ 2 !„ñó ò + ( !

2

= (L

2

Solution: 56.

57.

ñ óò ò ó

!„ó ò )ñ

= 0.

2

+

! ó ò )ñ

= 0.

Particular solution:

0

= exp O X

ñ óò ò ó

2

+"

0

= exp O − X

+

!„ñ óò

+ ( !" – "

ñ óò ò ó

+ 2 !„ñ

óò

+ (!

ñóò ò ó

+ 2 !„ñó ò + ( !

¨2

 +  ( 2  + ' 

2

+

2

+

ñóò ò ó

−1

ñóò ò ó

ó ò )ñ

 . M P

= 0.

 . M P

+ ô )ñ = 0.

=

exp O X¯

 M P

ï 2÷

õ ïƒ÷

!„ó ò

+ô

+

exp OX¯

=

2

+ ô! +

0

!„ó ò

!„ó ò )ñ

–1

)ñ = 0.

+ õ )ñ = 0.

exp OX¯

+ ( ! + " )ñó ò + ( !" +

leads to a constant coefficient linear equation:

 leads to a linear equation of the form 2.1.2.10: M P

)¨ = 0.

+ (2 ! + ô )ñó ò + ( !

Particular solution:



! óò

The substitution ¨ = ¨ 

 +  ¨ 
+  ¨ = 0. 61.

= exp O X

– ô (!

The substitution ¨

60.

= 0.

 . M P

+

The substitution ¨ ¨2

 +  ¨ = 0. 59.

ô!„ó ò ]ñ

 + L 1 ) exp O − X 

0

Particular solution: 58.

+

 . M P  . M P

óò

+ (1 – ô ) !„ñ

2





Particular solution: 55.

= exp O X

 M P

leads to a constant coefficient linear equation:

= 0.

= exp O − X



 . M P

Page 289

290 62.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñ óò ò ó

+

ï !„ñ óò

+ (! + ï

Particular solution: 63.

ï ñ óò ò ó

+ (ï

+ ( ô! + ï

óò

Particular solution: 64.

( ï + ô )ñ

óò ò ó

0

66.

ï 2 ñ óò ò ó

= exp O − Xv

 . M P

!„ó ò

Particular solution:

0

ï 2 ñ óò ò ó

! óò

+ 2ï

+ (ï

!„ñ óò

+

67.

ï 2 ñ óò ò ó

68.

ï 2 ñ óò ò ó

1− −  .  + M P

– ô ( ô + 1)]ñ = 0. +1

!

+

2

–

óò

+ (!

2

+

+ ï (2 ! + ô )ñó ò + [ ! =

2

−1

2

+ [ï

!„ñóò

2

The transformation ¨

 ¨2

 + (  2

70.

ï 2 ñ óò ò ó

2

The substitution ¨

 . M P

õï

ö ï )ñ

+

2

– ô )ñ = 0.

¨ 

 +  leads to the Bessel equation 2.1.2.126:  2 2 M P

!ƒó ò

+

( !„ó ò +

+ ô ) + õ ]ñ = 0.



!

2

exp O X

+ õ )ñó ò + [ ! exp O XÈ

= 0.

 leads to an equation of the form 2.1.2.115: M P

−1

=

ô ïƒ÷





exp O X 

+  )¨ = 0.

+ ï (2 ! +



+

+ ( ô – 1) ! + ï

 ¨ 

 +  ¨ 
+ ( '  + ( )¨ = 0. + 2ï

2

+ï

2

ï 2 ñ óò ò ó

−1

ï ! óò

−  )¨ = 0.

The substitution ¨

69.

ï

+ô

!

exp OXÈ

=

−1

exp O − XÈ

= ¨ exp O − XÈ

The substitution 2

= 0.

2 + ' + ( )¨ = 0.

+ ï (2 ! + 1)ñ

¨ 
+ (  2

ô!

= 

The transformation ¨

 2 ¨2

 + ( 

! óò ñ

+

+ [ï

+

ï !„ñóò

= 0.

= exp O X

0

 . M P

! ó ò )ñ

óò

+ ( ! + õ )ñ

Particular solution: 65.

= 0.

=  exp O − Xv

0

+ ô )ñ

!

! ó ò )ñ

õ ïƒ÷

+ ö ]ñ = 0.

 leads to a linear equation of the form 2.1.2.132: M P



 leads to a linear equation of the form 2.1.2.115: M P

2

+ (ô

ïƒ÷

+

– 1) ! + ï

õ

!„ó ò

+

ï 2÷

+

ïƒ÷

+ ]ñ = 0.

−1

  leads to a linear equation of the form 2.1.2.146: M P 2 ¨

¨ 2
 2    + (  +  )  + ( ƒ… + ý±  + = )¨ = 0. =

2

71.

2 !„ñó ò ò ó +

!„óò ñóò

ôñ

+

= 0.

The substitution ˆ = Xv 72.

!„ñóò ò ó

–

!„ó ò ñóò

Solution:

– ô! =L

1





3

ñ

J

−1 2

= 0. +L

2



−



M

 leads to a constant coefficient linear equation: 2
Šª
Š + 

, where ‡ = £  X

 M

= 0.

 .

Page 290

291

2.1. LINEAR EQUATIONS

73.

!„ñóò ò ó

– ô!„ó ò

ñóò =L

Solution: 74.

!„ñ óò ò ó

óò

– (!

õ! 2D +1 ñ = 0.    + L  − , where

– 1

2

+ ô! =L

Solution:

2

!„ñóò ò ó

+

exp O Œ

1

quadratic equation Π75.

óò

)ñ

2

76.

!„ñóò ò ó

 +L M P

õ! 2 (ô

= 0.

= exp O −  X 

 . M P

+

– ( !„ó ò + 2 ô! )ñó ò + ( ô!„ó ò +

! 2 ñ óò ò ó ! 2 ñ óò ò ó

óò

ô 2!

2

õ 2 ! 3 )ñ

–

  exp O/Xv 0 = 

+ ! ( !„ó ò + ô )ñó ò +

+ ! (!

exp O Œ

2

õ! )ñ

õÿñ

−1

The substitution ˆ = Xv 78.

X¯

!"Iñóò ò ó

ñ óò ò ó

+

=L

Solution: 80.

"

óò

+ 2 !„ñ

1

+ (!

 2

The substitution ¨



+L

2

! óò

1

and Œ

2

are roots of the

 . M P

 leads to a constant coefficient linear equation: 
Šª
Š +  
Š +  = 0.

óò

+ ( !"

õ!"Ióò )ñ óò +

 , where ΠM P

= 0.

M

+ 2" + ô ) ñ

– ( ô!„ó ò

X¯

= 0.

óò

+"

2

+ ô" + õ )ñ = 0.

The transformation ˆ = Xv −1  , ‡ = exp O Xv M equation: ‡ Šª

Š + ‡ Š
+ ' ‡ = 0. 79.

 . M

− Œ +  = 0.

0

Particular solution: 77.

= 0.

1

– ( !„ó ò + ô! )ñó ò –

Particular solution:

õ! 3 ñ

‡ = £ /Xv 

#! 2D +1 " 2 +1 ñ

–

 −

−1



 leads to a constant coefficient linear M P

= 0.

, where ‡ = £ Œ3Xv 

+ ô

2

ó 

+

õ  ó

  M

.

+ ö )ñ = 0.

= exp O X¯  leads to a linear equation of the form 2.1.3.5: M P ¨2

 + (   2   +    + ( )¨ = 0.

81.

82.

ñ óò ò ó

! ó ò ñ óò

2 2

Solution:

=L

1

ñóò ò ó

–ô

2 2

=L

1

–

!„óò ñóò

Solution: 83.

!„ñóò ò ó

–

!„ó ò ò ó¿ñ

4 ! 2 ñ ó ò ò ó – [2 ! Solution:

=











= 0.

exp O Xv

–



+L

M P –



M P

+L

2

cos O lX 

2

–

exp O −  XZ

 . M P –

 . M P

= 0. 1

! óò òó

 + L 23  X¯ – (!

−2

M

ó ò )2 + ô ] ñ

1

ü

 exp O

1 2

£  X 

 . = 0.

 + L 2  exp O − 21 £  X  M P ü −1 1 L 1  cos O 2 |  | Xv X   + L 2  sin O 12 |  | v ü ü ü M P ü L 1  + L 2  X  −1  ü ü M L






= 0.

sin O lX 

  ñ

=L

Solution: 84.

  ñ

+ô

–

−1

−1

 M P −1  M P

if  > 0, if  < 0, if  = 0.

Page 291

292 85.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

! óò òó ñóò ! óò

–

2

¥¬L

ø

=

+ ô 2 ( !„ó ò )

Solution:

ü

1

! 2÷

86.

ñóò ò ó

+ ”

! ! óò !

2

=L

Solution: 87.

ñ óò ò ó

! óò òó ! óò

– ¬



ñ óò ò ó

+ ¬

=

Solution: 89.

ñóò ò ó

+

! óò ñóò !

ñóò ò ó

! óò !

–” 2

ñóò ò ó

" óò ò ó " óò

–”

ü

+

–

 ü

1

3 ” 4

–

ø

1

= %  [ L

 

)+ L

1

2

õ 2 ( ! ó ò )2 ñ ! 2+ô

2 2

+L

P



n

O

1 2



B



  ® , where ø + ( H ) and n + ( H ) are the P

= 0.

+L

2

 +

+ ¬ (ô

2

–

õ

2

ü

)”



2

+ 

! óò ! •

2

−

.

+ (!

ó ò )2 ® ñ

= 0.

n  (  )], where ø  (  ) and n  (  ) are the Bessel functions. 1 –ô 4

+ ”

2

•

”

! óò ! •

2

+ (!

ó ò )2 ®¦ñ

= 0.

ø

 (  )+ L 2 n  (  )], where ø  (  ) and n  (  ) are the Bessel functions.

–

1 2

2

1

ø

! óò òó !

–

" óò ò ó " óò •

3 ” 4

2

+

1 2

" óò ò ó ò ó " óò

+”

" óò " •

–

1 –ô 4

2

” •

" óò " •

2

" óò " •

2

+("Ió ò )2 ®¦ñ = 0.

 ( )+ L 2 n  ( )], where ø  ( ) and n  ( ) are the Bessel functions.

" óò ñóò " •

+¬

! óò ! óò ” 2 ! !

+

" óò ò ó " óò

–

! óò òó !

–ô

2

”

+ ("Ió ò )2 ®¦ñ = 0.

 ( ) + L 2 n  ( )], where ø  ( ) and n  ( ) are the Bessel functions.

+(2 ô –1)

Solution:



+ 

! óò òó ! óò •

 
[ L

" óò ò ó " óò

= 0.

! óò ® ñ óò !

! óò ! •

3 ” 4

B –

2

ø  (

  
[L

=  [L

Solution: 91.

ü

+¬ =

Solution: 90.

 +

=   [L

! óò òó ò ó ! óò

1 2

1

! óò òó ñóò ! óò •

+ (2 ô – 1)

Solution: 88.

ô

+

–

O

1 2

Bessel functions.

ñ

–2

" óò " 1

ø

+2

% óò ñóò % •

% óò " óò ò ó ” % " óò

+¬

+(2 ô –1)

" óò "

+2

% óò % •

–

% ò òó ó %

+("Ió ò )2 ®¦ñ = 0.

 ( ) + L 2 n  ( )], where ø  ( ) and n  ( ) are the Bessel functions.

2.1.10. Some Transformations ]

Notation:  , , and % are arbitrary composite functions of their arguments, which are written in parentheses following the name of a function (the argument is a function of  ). 1.

2.

3.

ñóò ò ó

+

ï

–4

!

(1 

ï )ñ

= 0.

¨ ªŠ

Š +  (ˆ )¨ = 0. The transformation ˆ = 1  , ¨ =   leads to the equation p

ô ï öï

+õ ñ = 0. +ý P  +  ¨ The transformation ˆ = , = (˜ + ˜(  + M M where ¢ =  − ˜( .

ñóò ò ó

+ (ö

ï 2 ñ óò ò ó

ï

+ Vï

+ ý )– 4 !IO

M

2

÷ ! (ô ï ÷

+õ )+

1 4

–

1 4

ù

The transformation ˆ =   +  , ¨ = 

2

W 

ñ −1 2

¨ ªŠ

Š + ¢ leads to a simpler equation: u

−2

 (ˆ )¨ = 0,

= 0.

¨ ªŠ

Š +( B )−2  (ˆ )¨ = 0. leads to a simpler equation: u

Page 292

293

2.1. LINEAR EQUATIONS

4.

5.

ï 2 ñ óò ò ó

+ ï ( ï ! + ô )ñó ò + ( ï " + õ )ñ = 0, ! = ! (ï ), " = " (ï ). The substitution =   ¨ , where , is a root of the quadratic equation , leads to the equation  ¨3

 + (  +  + 2 , )¨3
+ ( + ,  )¨ = 0.

ï g ÷ (ï )ñ óò ò ó

i ÷ (ï )ñ óò

+

g ÷ (ï

where   6.

ï (ï

−1 (

÷

– 1) g

.

ú

)=

=0

¨

 ( )   + [ Æ  ( ) + 2 , −1

 ) = ,!

–1 (

)=

ï )ñ = 0, 7÷ ô ú ï ú , i ÷ (ï –1 (

ú

7÷ =0

õú ï ú

,

h ÷

+ (  − 1) , +  = 0,

–1 (

ï

)=

ú

÷ 7 –1 =0

öú ï ú

.

=   ¨ , where , = 1 −  0  0 , leads to an equation of the same form:

The substitution



h ÷

+

2

ï )ñ óò ò ó

.

¨
 ( )]  + [ ² 

.

−1 (

 )+ 

−1 (

 )]¨ = 0,

[ Æ  ( ) + ( , − 1)  ( )].

i ÷ (ï )ñ óò

ï )ñ = 0, ÷ 7 –1 g ÷ –1 (ï ) = ô ú ï ú , i ÷ (ï ú =0

+

+

h ÷

–1 (

)=

ú

7÷

õú ï ú

=0

,

h ÷

–1 (

ï

÷ 7 –1 ï ú ö ú ú

)=

=0

.

 , ¨ = | − 1|−  , where , is a root of the quadratic equation  −1   −1 , 2 + (   −   −1 ) , + (  −1 = 0, leads to an equation of the same form: . . . ˆ (ˆ − 1) ‚  −1 (ˆ )¨ Šª

Š + [2(1 − , )ˆ‚  −1 (ˆ ) − Æ ‚  (ˆ )]¨ Š
+ [ , ( , − 1) ‚  −1 (ˆ ) +   −1 (ˆ )]¨ = 0,

The transformation ˆ =

where

.‚ 

²‚  7.

ñ óò ò ó

−1 (

ˆ )=

−1 ( ˆ ) =

+ V

2

ó

+

+

½

−1

½

−1

 + ˆ + (ˆ − 1)  − +

Æ ‚  (ˆ ) =

,

=0

=0

( + ˆ + (ˆ − 1)  − +

 ! (ô  

ó

+õ )–

The transformation ˆ =  ž

ó

−1

 

,



ó

+

 + ˆ + (ˆ − 1)  − + , =0

−1 ( ˆ ) =

WUñ = 0.  +  , ¨ =  J

1 4

½ ²‚ 

−1 (

.

ˆ ) + ,ƒÆ ‚  (ˆ ) + , ( , − 1) ‚  ˆ −1

−1 (

ˆ )

2

¨ ªŠ

Š + ( Œ )−2  (ˆ )¨ = 0. leads to the equation 3

+ ! (  )ñó ò + " (  )ñ = 0.  The substitution H = ž  leads to the equation Œ 2 H 2 
‘'
‘ + ŒH [  ( H ) + Œ ] ‘
+ ( H ) = 0.

9.

ñóò ò ó

11.

+ V–

2

+ sinh– 4 (

ï )!

 coth(

ï )  Wñ

The transformation ˆ = coth( Œ  ), ¨ =

ñ óò ò ó

+ V–

+ ¬–

2

1 4

+ cosh– 4 (

2

+

 2

ó ( ö 

ï )!

= 0.

sinh( Π )

leads to the equation ¨3Šª

Š + Œ

 tanh( ï )  WUñ = 0. ¨ ªŠ

Š + Œ The transformation ˆ = tanh( Œ  ), ¨ = leads to the equation 3 cosh( Œ  )

ñóò ò ó

ó

+ý )

! O 4

ô ö



ó 

ó

+õ ® +ý P

ñ

12.

−2

 (ˆ )¨ = 0.

−2

 (ˆ )¨ = 0.

= 0.

     + ¨   J2 , =   (  + (  + M ( ¢eŒ )−2  (ˆ )¨ = 0, where ¢ =  M − ˜( . M The transformation ˆ =

.

2

8.

10.

ñóò ò ó

−1

¨ ªŠ

Š + leads to a simpler equation: u

+ (2 ! tanh ï + " )ñ„ó ò + (" tanh ï + % )ñ = 0, ! = ! (ï ), " = " ( ï ), % = % (ï ). The substitution ‡ = cosh  leads to a simpler equation: "‡

 + !‡
 + ( % −  )‡ = 0.

!„ñóò ò ó

Page 293

294 13.

14.

15.

SECOND-ORDER DIFFERENTIAL EQUATIONS

!„ñ óò ò ó

+ (2 ! coth ï + " )ñ

+ (" coth ï + % )ñ = 0,

ï 2 ñ óò ò ó

+ V ! ( ô ln ï + õ ) + 14 W ñ = 0.

óò

The transformation ˆ =  ln  +  , ¨ =  (ï

2

– 1)2 ñó ò ò ó + I ! O ln

ô ï ï

–ô +1 P

[  (ˆ ) − 1]¨ = 0.

17.

ï 2!

(ln ï )ñó ò ò ó + ï

ñóò ò ó

+[

"

ñ

20.

21.

22.

= % ( ï ).

−2

 (ˆ )¨ = 0.

ü

|

2

leads to a simpler equation: 4¨uŠª

Š +

− 1|

(ln ï )ñó ò + % (ln ï )ñ = 0.

The substitution ˆ = ln  leads to the equation  (ˆ )"
Šª
Š + [ (ˆ ) −  (ˆ )] Š
+ % (ˆ ) = 0. 2

ï )!

+ sin– 4 (

(cot(

ï

))]ñ = 0.

+ V

ñóò ò ó

2

ï )!

+ cos– 4 (

 tan(

sin( Π )

ï )  Wñ

¨ ªŠ

Š + Œ leads to a simpler equation: u

−2

 (ˆ )¨ = 0.

leads to a simpler equation: ¨uŠª

Š + Œ

−2

 (ˆ )¨ = 0.

= 0.

cos( Π )

sin( ï + ô ) ®¦ñ = 0. sin4 ( +õ ) sin( ï + õ ) P sin( Œ  +  ) ¨ , = leads to a simpler equation: ¨uŠª

Š + The transformation ˆ = sin( Œ  +  ) sin( Œ  +  ) [ Œ sin(  −  )]−2  (ˆ )¨ = 0.

ñ óò ò ó

+ ¬n

2

1

!IO

ï

+

!„ñóò ò ó

+ (" – 2 ! tan ï )ñ„ó ò + ( % – " tan ï )ñ = 0,

!„ñ óò ò ó

+ (" + 2 ! cot ï )ñ

!

= ! ( ï ),

"

= " ( ï ),

%

= % ( ï ).

The substitution ‡ = cos  leads to a simpler equation: "‡

 + !‡
 + (  + % )‡ = 0.

óò

+ ( % + " cot ï )ñ = 0,

!

= ! ( ï ),

"

= " ( ï ),

%

= % ( ï ).

The substitution ‡ = sin  leads to a simpler equation: "‡

 + !‡
 + (  + % )‡ = 0. (ï

2

+ 1)2 ñ

óò ò ó

+ ! (arctan ï + õ )ñ = 0.

[  (ˆ ) + 1]¨ = 0. (ï

2

[  (ˆ ) + 1]¨ = 0.

ñóò ò ó

= £ 

2

+1

= £ 

2

+1

+ 1)2 ñó ò ò ó + ! (arccot ï + õ )ñ = 0.

The transformation ˆ = arccot  +  , ¨

24.

%

¨ ªŠ

Š +  J leads to a simpler equation: u

The transformation ˆ = arctan  +  , ¨

23.

= " ( ï ),

−1 2

 −  ¨ , =  +1

The transformation ˆ = tan( Œ  ), ¨ = 19.

"

= 0.

The transformation ˆ = cot( Œ  ), ¨ = 18.

= ! ( ï ),

The substitution ‡ = sinh  leads to a simpler equation: "‡

 + !‡
 + ( % −  )‡ = 0.

The transformation ˆ = ln

16.

!

¨ ªŠ

Š + leads to a simpler equation: u

¨ ªŠ

Š + leads to a simpler equation: u

+ ! ( ï )ñ = 0.

The transformation  = Y ( H ), = ¨ | Y ‘
| leads to an equation of the same form: ¨u‘'

‘ + ‘'‘ ' ‘ ' ‘ ‘


ü 1Y 3 Y

2
‘ ` ( H )¨ = 0, where ` (H ) = − O + ( Y )2  ( Y ). ‘ ‘ 2 Y
4 Y
P

Page 294

2.2. AUTONOMOUS EQUATIONS

Ï Ò˜Í Í Ò

=

ê ( Ï , Ï ÒÍ

295

)

2.2. Autonomous Equations í ïwî²î ï = c ( í , í ï î ) Preliminary remarks. Equations of this type often arise in different areas of mechanics, applied mathematics, physics, and chemical engineering science. 1 b . The substitution ¨ ( ) = 
 leads to a first-order equation:

¨
=¨ 

−1

 ( , ¨ ).

(1)

2 b . The solution of the original autonomous equation can be represented in implicit form:

 =X where ¨ = ¨ ( , L

1)

¨ ( M , L ) + L 2, 1

(2)

is the solution of the first-order equation (1).

3 b . The solution of the original autonomous equation can be written in parametric form:


(U , L ) 1 ¨ * (U , L ) M U + L 2 , 1

 =X

= (U , L

1 ),

(3)

where = (U , L 1 ), ¨ = ¨ (U , L 1 ) is a parametric form of the solution of the first-order equation (1). Formula (2) is a special case of formula (3) with = U . 4 b . For the special cases 

=  (
 ), see equations 2.9.1.1 and 2.9.4.35.

=  ( ) and 

2.2.1. Equations of the Form

â$äãÑã ä – âäã = å (â )

Preliminary remarks. Equations of this type arise in the theory of combustion and the theory of chemical reactors. 1 b . The substitution ¨ ( ) = 
 leads to the Abel equation ¨*¨3
− ¨ =  ( ), which is considered in  Subsection 1.3.1 for some specific functions  . 2 b . The solution of the original autonomous equation can be written in the parametric form (3), where = (U , L 1 ), ¨ = ¨ (U , L 1 ) is a parametric form of the solution to an Abel equation of the second kind ¨*¨
− ¨ =  ( ).



1.

ñóò ò ó

– ñó ò = –

2(þ

+ 1)

ñEA þ

(þ + 3)2 2ô Solution in parametric form:

 = ¡ =L 2.

ñóò ò ó

– ñó ò = A 2 ô

ñ

2 –1

+1 2

ñ ú

+3 ln O L −1

¡ 2 1

U O L

1− 1

þ

,

1− 1

≠ A 1,

þ

≠ –3.

U M + +1 + L 2 , £ 1 AC P U 2 U + −1 M + +1 + L 2 . £ 1 AI P U

+ ¡ −1 X +3 ¡

+ ¡ −1 X +3 ¡

.

Solution in parametric form:

 = − ln L Q

1

X exp(A"U 2 ) U + L 2 S , M

= L

1

exp(AU 2 ) L

Q

1

X exp(AU 2 ) U + L 2 S M

−1

.

Page 295

296 3.

SECOND-ORDER DIFFERENTIAL EQUATIONS – ñó ò = – 29 ñ + 16 ô 3 ) 2 ñ –1 ) 2 . 9 Solution in parametric form:

ñóò ò ó

 = −3 ln , L

exp(−U ) V exp(3U ) + L

1

9 – ñó ò = – 100 ñ‚A 1009 ô 8 ) 3 ñ Solution in parametric form:

)

ñóò ò ó

–5 3

 = − 54 ln V A (U 3

=  (U 5.

3 – ñó ò = – 16 ñ – 643 ô 8 ) 3 ñ Solution in parametric form:

ñóò ò ó

 =L ]

1

sin  £ 3 U  W. ,

V 2 exp(3U ) − L 2 sin  £ 3 U  + £ 3 L 2 cos  £ 3 U  W 2 . V exp(3U ) + L 2 sin  £ 3 U  W 2

=  exp(2U ) 4.

2

4

.

− 3U + L

)

–5 3

− 2 ln[sin U cosh(U + L

2

− 6U 1)

+ 4L

J V A (U

3 2

1

U − 3)W + L 2 ,

4

− 6U

2

+ 4L

1

U − 3)W

J

−9 8

.

.

2)

+ cos U sinh(U + L

=  [tan U + tanh(U + L

2 )],

2 )]

J .

−3 2

In the solutions of equations 6–9, the following notation is used:

for the upper sign, 1 ø Á ( U ) + n Á ( U ) L 1 ® Á (U ) + ; Á (U ) for the lower sign, where ø Á (U ) and n Á (U ) are the Bessel functions, and ®Á (U ) and ; Á (U ) are the modified Bessel

:

L

™

=

functions. 6.

– ñó ò =
ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

 = −2 XwU

−1

:

where @ = 13 , # = T 7.

−1

:
*

(U

 J

+

1 3

:

) U +L

= U

2,

M

J :

−4 3

−2

[(U

:
*

1 3

+

:

)2 AIU

2

:

2

],

1 3 2 . 3

ñóò ò ó

– ñó ò =
ñ –2 . Solution in parametric form:

 =T

2 3

XÈU

: 2V :
(U *

+

1 3

:

)2 AIU

2

:

2 −1

W

M

= 2 U 4 J

U + L 2,

3

: 2V :
(U *

+

1 3

:

)2 AIU

2

:

2 −1

W

,

where @ = 13 , # = −36  3 . 8.

– ñó ò = 2
2 –
ñ 1 ) 2 . Solution in parametric form:

ñóò ò ó

−1

 = A 2 XÈU where @ = 0, # 9.

=  1J 2.

(

:
−1 : * ) (U A

– ñó ò =
ñ –1 ) 2 + 2 ( 2 + Solution in parametric form:

ñóò ò ó

 = −2 XwU where # = (1 − @ 2 ) $

3

−1

:

(­ñ 1 ) −1

(U

:
*

2

2

:
* )M U

+L

2,

:
−2 : * ) (U A

= (

2

:
2 *),

.

:

− ) U +L

M

2,

=$

2

:

−2

(U

:
*

:

− )2 ,

.

Page 296

Ï Ò˜Í Í Ò

2.2. AUTONOMOUS EQUATIONS

]

€

In the solutions of equations 10–14, the function

U =X

M

A (4 < ü

€ („

=

<

−L

− 1)

3

ê ( Ï , Ï ÒÍ

=

297

)

) is defined in implicit form:

1.

The upper sign in this formula corresponds to the classical elliptic Weierstrass function < = < (U + L 1 , 0, 1). 10.

ñ óò ò ó

9 – ñ ó ò =
ñ 2 – 625
–1 . Solution in parametric form:

= 5  (U 2
 = 5 ln U + L 2 , 11.

<

,

where

# =A

6 −1 . 125



= 5  (U 2
6 −1 . 125

# =A

where



– ñó ò = 12ñ +
ñ –5 ) 2 . Solution in parametric form:

ü

X<

3

− 1), # = T 147  7 J 2.

A (4 <

−1

 pA 2U'< 2 

−1

2 7

M

= -<

U + L 2,

J  pA 2U'< 2 

−6 7

J

−4 7

,

– ñó ò = 63 ñ +
ñ –5 ) 3 . 4 Solution in parametric form:

ñóò ò ó

 = − 34 X ( pA 2U'< 2 )(U + 2 < )−1 U + L 2 , M 2 2J 3 where  = A (4 < 3 − 1), # = − 128 . 3  (2  ) ü

= 2 spA 2U'<

 J (U + 2 < )−9 J 8 ,

2 3 2

In the solutions of equations 15–18, the following notation is used:

U

=X

² = ü

U

+ L 1 (incomplete elliptic integral of the second kind), A (4U − 1) ü A (4U 3 − 1), ® 1 = 2UÈ®lTç² , ® 2 = U −1 (2U ²$®·Tç² 2 − 1).

M

3

– ñó ò =
ñ 1 ) 2 – 12 ñ . 49 Solution in parametric form:

ñóò ò ó

®

−1 −1

 = −7 XwU ² 16.

2

ñóò ò ó

®

15.

= 5 U

6 – ñ ó ò =
ñ 2 + 25 ñ . Solution in parametric form:

where  =

]



ñ óò ò ó

 =T

14.

6 −1 . 125

6 – ñó ò =
ñ 2 – 25 ñ . Solution in parametric form:

 = 5 ln U + L 2 , 13.

# =A

where

ñóò ò ó

 = 5 ln U + L 2 , 12.

1 2 ),

ñ óò ò ó

– ñ ó ò = 6ñ +
ñ – 4 . Solution in parametric form:

 = − 15 X U

−1

²

®

M

−1 −1 1

U + L 2,

M

U + L 2,

= 7 U

= U

®

2 −4

J

®

,

J

where

−3 5 −2 5 , 1

# =A

where

12 49 (7

 )1 J 2 .

# = T 150 5.

Page 297

298 17.

SECOND-ORDER DIFFERENTIAL EQUATIONS – ñó ò = 20ñ +
ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

1 3

 = 18.

20.

21.

22.

23.

24.

U + L 2, M

ñóò ò ó

– ñó ò = 15 ñ +
ñ –7 . 4 Solution in parametric form:

® 1  4UÈ® 12 T ® 22  −1 M U ñ óò ò ó – ñ óò =
ñ + (­ñ –1 – ( 2 ñ  = Xv²

19.

® ®

−1 −1 1 2

XZ²

−1

+L –3

2,

J

= ³®

−4 3 2 2, 1

®

= ³®

1 2 1

# = A 108 3 J 2 .

where

J  4UÈ® 2 T 1

® 22  −3 J 8 ,

# =A

where

3 8 . 4



.

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.1.5: ¨*¨
− ¨ = # + $ −1 − $ 2 −3 . 

3 – ñó ò = – 16 ñ +
ñ –1 ) 3 + (­ñ –5 ) 3 . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.1.61:

ñóò ò ó

¨*¨
− ¨ = − 3 + # 16 

J +$

−5 3

J .

J +$

−7 5

−1 3

5 – ñ ó ò = – 36 ñ +
ñ –3 ) 5 + (­ñ –7 ) 5 . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.1.62:

ñ óò ò ó

¨*¨
− ¨ = − 5 + # 36 

ñóò ò ó

J .

−3 5

– ñó ò = 94 ñ + 2
ñ 2 + 2
2 ñ 3 . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.1.14:

ñ óò ò ó – ñóò =
ñ 

–1

¨*¨
− ¨ = 

– ü/(­ñ 

+ ü/( ñ 

2 2 –1

4 9

+ 2#

2

+ 2#

2 3

.

.

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.1.6: ¨*¨
− ¨ = #  −1 − ,$  + ,$ 2 2  −1 . 

ñóò ò ó

– ñó ò = A

ü

ñ

2ô 2

2

A 8ô

2

.

Solution in parametric form:

 =T X « where « 25.

26.

27.

ñ óò ò ó

−1



−1

(

= X exp(T"U 2 ) U + L

M

2 1,

A 2«

2

) U +L

M

2,

= A !«

−1



−1

(

2

T 2«

2

),

 = 2U «ZA exp(T"U 2 ).

– ñ ó ò =
+ ( exp(–2ñ
). The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.1.8:

ñóò ò ó

¨*¨
− ¨ = # + $ exp(−2 # ). 

– ñó ò = ô 2  2 ] – ô ( õ + 1) ] + õ . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.1.73:

ñ óò ò ó

¨*¨
− ¨ =  2 Œ  2   −  ( Œ + 1)   +  . 

– ñ ó ò = ô 2  2 ] + ôƒñ ] + õ ] . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.1.74:

¨*¨
− ¨ =  2 Œ  2   + Œ   +    . 

Page 298

2.2. AUTONOMOUS EQUATIONS

Ï Ò˜Í Í Ò

=

ê ( Ï , Ï ÒÍ

299

)

â$äãÑã ä + å (â )âäã + â = 0

2.2.2. Equations of the Form 2.2.2-1. Preliminary remarks.

Equation of this form are often encountered in the theory of nonlinear oscillations, where  plays the role of time. 1 b . The transformation

2

H = − 21

+ ,

¨ = 


leads to an Abel equation:

¨*¨ ‘
= ( H )¨ + 1,

where ( H ) =  ( )  ,

=A

ü

2(  − H ),

whose special cases are outlined in Subsection 1.3.2. 2 b . For oscillatory systems with a weak nonlinearity



 + ž ( ) 
+ = 0, two leading terms of the asymptotic solution, as ID

0, are described by the formula

= # cos( + $ ), where the functions # = # (ˆ ) and $ = $ (ˆ ) depend on the slow variable ˆ =  ; they are determined from the autonomous system of first-order differential equations:

# # Š
= −

2



X

2

z

 ( # cos Y ) sin2 Y

0

M

1 $ Š
= − X

Y ,

2



2

z

 ( # cos Y ) sin Y cos Y

0

M

Y .

The right-hand sides of these equations depend only on # . The system is solved consecutively starting from the first equation. 2.2.2-2. Solvable equations and their solutions. 1.

ñ óò ò ó

+ ôññ ó ò + ñ = 0. Solution in parametric form:

 = − #X 2.

U (L

1 + 2#

2

M

U

+L

2,

=  cos( − ) −

1 32

ln |U | − 2 #*U )1 J

2

= (L

1 +2

2

#

ln |U | − 2 #*U )1 J 2 ,

where # =

1



.

ñóò ò ó

– W (1 – ñ 2 )ñó ò + ñ = 0. Van der Pol oscillator. 1 b . Solution, as ID

0:

where



2

=

9

4 1 + (4 L

−2 1

− 1) 

−

ª

,

In applications,  plays the role of time, L initial phase with = 0.

9

9

ž 3 sin[3( − )] + [ ( 2 ), =

1

1 7

ln  − ž 8 64

2

+

1 2

 +L 16

2.

is the initial oscillation amplitude, and L

2

is the

2 b . As CD + F , the periodic solution of the Van der Pol equation consists of intervals with { fast and slow oscillations and describes damping oscillations with period = (3 − 2 ln 2) + [ ( −1 J 3 ).

Page 299

300 3.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

+ ñ ( ôñ 2 + õ )ñó ò + ñ = 0. 1 b . The transformation H = − 12 ¨*¨2‘
= (−2 !H +  )¨ + 1.

2

, ¨

= 
 leads to an Abel equation of the form 1.3.2.1:

2 b . Solution in parametric form with  < 0:

: : 1: 2 4  −1 J 2 U +L , X U −1 J 3 A , 2 U −2 J 3 −1 OU
+ − S * 3 P  3 Q 3 M : : 4 1:  1J 2 9 = A , 2 U −2 J 3 −1 OU
+ − S ,  = − , −3 , * 3 P  Q 3 4

 =T

where

:

ø J U ) + L 2n ®1 1 J 3 (U ) + L 2 ;

U ) for the upper sign, ( 1 J 3 U ) for the lower sign, ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; 1 J 3 (U ) are the modified Bessel functions. 4.

5.

L

2,

™

=

1 1 3(

L

ñ óò ò ó

+ ñ ( ôñ 2 + õ )–2 ñ ó ò + ñ = 0. The transformation H = − 21 2 , ¨ ¨*¨2‘
= (−2 !H +  )−2 ¨ + 1.

= 


+ ñ ( ôñ 2 + õ )–1 ) 2 ñ ó ò + ñ = 0. 1 b . The transformation H = − 12 2 , ¨ ¨*¨ ‘
= (−2 !H +  )−1 J 2 ¨ + 1.

ñ óò ò ó

1 3(

J

leads to an Abel equation of the form 1.3.2.2:

= 
 leads to an Abel equation of the form 1.3.2.4:

2 b . Solution in parametric form:

 = − L where « 6.

ñóò ò ó

= exp O − X

– ñpO 2 ô +

ôñ

2 1

X OL

1

U 1

2

2

U

M

−

  P ·

U

+õ P

J

−1 2

U

ñóò

+ ñ = 0.

1 2

2

−

2

«

M

U

−U +



2

= OL

2,

2 1

«

2

−

 1J 2 ,  P ·

, ¨ = 
 leads to an Abel equation of the form 1.3.2.3:

¨*¨ ‘
= O# − 1 ¨ + 1, #2HrP

ñóò ò ó

+L

.

− U + ·P

The transformation H = −

7.

2

«

+ ôñ exp( ƒñ 2 )ñó ò + ñ = 0. The transformation H = − 21 2 , ¨ ¨*¨2‘
=  exp(−2 ŒH )¨ + 1.

= 


where

# = −2 .

leads to an Abel equation of the form 1.3.2.7:

8.

ñóò ò ó

+ ñ [ ô exp( ƒñ 2 ) + õ exp(– ƒñ 2 )]ñó ò + ñ = 0. The transformation H = − 21 2 , ¨ = 
 leads to an Abel equation of the form 1.3.2.8: ¨*¨2‘
= [  exp(2 ŒH ) +  exp(−2 ŒH )]¨ + 1.

9.

ñóò ò ó

+ 2 ôñ exp[ ô ( õ – ñ 2 )]ñó ò + ñ = 0. Solution in parametric form:

 = T 2 , X¥  − 4 , 2 U where  = T

1 4

,

−2

, «

…

2

− ln | ,«

… −1|  −1 J 2 U M

= X exp(T U 2 ) U + L

M

+L

2,

=   − 4, 2U

2

− ln | ,«

… −1 |  1 J 2 ,

1.

Page 300

2.2. AUTONOMOUS EQUATIONS

10. 11.

12.

Ï Ò˜Í Í Ò

=

ñóò ò ó

+
ñ cosh( ƒñ 2 )ñó ò + ñ = 0. This is a special case of equation 2.2.2.8 with  =  =

ê ( Ï , Ï ÒÍ 1 2

ñ óò ò ó

+
ñ sinh( ƒñ 2 )ñ ó ò + ñ = 0. This is a special case of equation 2.2.2.8 with  = −  =

ñóò ò ó

+ 2
ññó ò sinh2 [
( ( – ñ 2 )] + 2
ü Solution in parametric form:

 = 2 lX ( 

2

2

+ 2«

−1

)4

# . 1 2

# .

+ ñ = 0.

–1

−1

Æ

301

)

U + L 2, M

=Æ ;

# = 14 

−2

,

where

13.

ñóò ò ó

« = X exp(−U 2 ) U + L 1 , M 2 Æ = $ − 4  arcsinh[!« ü

– 2
ññó ò cosh2 [
(ñ 2 – ü Solution in parametric form:

 = 2 X (

2

 = 2U « + exp(−U 2 ), −1

(

− 2«



−1 (



)] – 2


–1

2

−1

)4

−1

Æ

− 2«

2

2 )],

4 =£ 

2

− 2«

2

+ 8«

arcsinh H = ln  H + £ H

2

2



2,

+1 .

M

+ ñ = 0.

U + L 2,

=Æ ;

# = 14 

−2

,

where

14.

15.

ñóò ò ó

« = X exp(U 2 ) U + L 1 ,  = 2U « − exp(U 2 ), 4 = £  2 + 2 « 2 − 8 « 2  2 , M 2 Æ = $ + 4  arccosh[ !« −1  −1 (  2 + 2 « 2 )], arccosh H = A ln H + £ H 2 − 1  . ü

+
ñ cos( ƒñ 2 )ñó ò + ñ = 0. The transformation H = − 12 2 , ¨ ¨*¨2‘
= # cos(2 ŒH )¨ + 1.

ñóò ò ó

+
ñ sin( ƒñ 2 )ñó ò + ñ = 0. The transformation H = − 12 2 , ¨ ¨*¨2‘
= − # sin(2 ŒH )¨ + 1.

2.2.3. Lienard Equations

= 


leads to an Abel equation of the form 1.3.2.11:

= 


leads to an Abel equation of the form 1.3.2.12:

â$äãÑã ä + å (â )âäã + ç (â ) = 0

2.2.3-1. Preliminary remarks. Equations of this form are encountered in various fields of applied mathematics, mechanics, and physics. 1 b . For  ( ) = 0, see equation 2.9.1.1.

2 b . The substitution ¨ ( ) = 
 leads to an Abel equation of the second kind:

¨*¨
+  ( )¨ + ( ) = 0, 

whose special cases are outlined in Subsection 1.3.3. 3 b . The transformation ¨ ( H ) = 
 , H = − Xv ( )

¨*¨ ‘
− ¨ = Y ( H ),

M

where

leads to an Abel equation of the second kind:

Y (H ) = ( )  ( ),

whose special cases are outlined in Subsection 1.3.1.

Page 301

302

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.2.3-2. Solvable equations and their solutions. 1.

ñ óò ò ó

+ ñ + ôñ 3 = 0. Duffing equation. This is a special case of equation 2.9.1.1 with  ( ) = − −  1 b . Solution:

 = AuXÊÖL

1

2

−

The period of oscillations with amplitude L integral of the first kind:

{

4 = £ 1 + L

L

; O 2

2

2 + 2 L

2 b . The asymptotic solution, as eD

¶

=L

¶

1

¶

cos[(1 + 38  L

¶

2 1)

2

P

J

4 −1 2



− 12 

+L

M

,

 + L 2] +

.

2.

is expressed in terms of the complete elliptic

; (¡

where

)=X

0, has the form:

¶

3

1 32

 L

¶

3 1

cos[3(1 + 38  L

¶

z´J 2

a £ 1 − M sin2 a . ¡

0

2 1)

¶  + 3 L 2 ] + [ (  2 ),

where L 1 and L 2 are arbitrary constants. The corresponding asymptotics for the period of { = 2 (1 − 3 L 2 ) + [ ( 2 ). oscillations with amplitude L is described by the formula: 8 2.

ñ óò ò ó

+ ôññ ó ò + õÿñ + ö^ñ = 0. The transformation ¨ ( H ) = 
, H = − 12  ¨*¨ ‘
− ¨ = − 2  H + ( . 2



3.

4.

7.

8.

leads to an Abel equation of the form 1.3.1.2:

ñóò ò ó

= ( ôñ + 3 õ )ñó ò + ö^ñ 3 – ôƒõÿñ 2 – 2 õ 2 ñ . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.1:

ñóò ò ó

¨*¨
= (  

+ 3  )¨ + (

3

− 

2

− 2

2

.

= (3 ôñ + õ )ñó ò – ô 2 ñ 3 – ôƒõÿñ 2 + ö^ñ . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.2: +  )¨ − 

2 3

2

− 

+( .

2ñ ó ò ò ó = (7 ôñ + 5 õ )ñ ó ò – 3 ô 2 ñ 3 – 2 ö^ñ 2 – 3 õ 2 ñ . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.3: 2¨*¨
= (7 

6.

2



¨*¨
= (3  

5.



3

+ 5  )¨ − 3 



2 3

− 2(

2

− 3

2

.

= ñ ÷ –1 [(1 + 2ù )ñ + ôù ]ñ„ó ò – ùñ 2÷ (ñ + ô ). The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.8:

ñóò ò ó

¨*¨
=  

−1

[(1 + 2B ) + B ]¨ − B

2

 ( +  ).

= ô (ñ – ù)õ )ñ ÷ –1 ñó ò + ö [ñ 2 – (2ù + 1) õÿñ + ù (ù + 1) õ 2 ]ñ 2÷ –1 . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.9:

ñóò ò ó

¨*¨
=  ( − B… )  

−1

¨ +( [

2

+ B (B + 1)  2 ] 2 

− (2B + 1) 

−1

.

= [ ô (2ù + ü )ñ  + õ ]ñ ÷ –1 ñ ó ò + (– ô 2 ùñ 2  – ôƒõÿñ  + ö )ñ 2÷ –1 . The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.10:

ñ óò ò ó

¨*¨
= [  (2B + , )  +  ]  

−1

¨ + (−  2 B

2

 −   + ( ) 2 

−1

.

Page 302

2.2. AUTONOMOUS EQUATIONS

9.

10.

ñ óò ò ó

= [ ô (2þ

+ ü )ñ

2



+ õ (2þ

– ü )]ñ

Ï Ò˜Í Í Ò

ú – –1 ñ óò

=

– (ô

ê ( Ï , Ï ÒÍ 2

þxñ 4

303

)

+

ö^ñ 2

+

õ 2 þ )ñ 2ú

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.11: ¨*¨
= [  (2 + , ) 2  +  (2 − , )] + −  −1¨ − (  2 4  + ( 2  +  2 ) 2 +  ¡ ¡ ¡ ¡

ñóò ò ó

ô]¦ñóò

=

+

õ]



–2 –1

.

 .

−2 −1

.

Solution in parametric form:

 =−

# Œ

XÈU

−1

L

1

+#

2

ln |U | − #*U 

−1

M

where # =   . 11.

ñóò ò ó

= ( ô] + õ )ñó ò +

ö 2]

U + L 2,

=

1

Œ

Œ

ln − ( L

1

Q 

+#

2

ln |U | − #*U )S ,

– ôƒõ] – õ 2 .

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.67:

¨*¨
= (    +  )¨ + (  2  −    −  2 .  12.

13.

14.

15.

16.

17.

18.

ñóò ò ó

= [ ô (2 + )-] + õ ] 

 ]Èñóò

+ (– ô

ñóò ò ó

= ( ô] + õ )ñó ò + ö [ ô

ñ óò ò ó

=

 ]

(2 ôƒñ +

ñóò ò ó

=

Df]

(2 ôñ

ñóò ò ó

= ( ô cosh ñ + õ )ñ„ó ò – ôƒõ sinh ñ + ö .

2

– ôƒõ] + ö ) 

C 2 ]

2

 ]

.

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.68: ¨*¨
= [  (2“ + Œ )   +  ]    ¨ + (−  2 “  2  −    + ( )  2   . 

 ]

+

óò

2

2 2

ôƒõ ( ƒñ

+ 1) ] +

õ 2 ƒñ

].

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.69: ¨*¨
= (    +  )¨ + ( [  2  2  +  ( Œ + 1)   +  2 Œ ]. 

ô

+ õ )ñ

–

] (ô 2 ƒñ

2

+

ôƒõÿñ

+ ö ).

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.70: ¨*¨
=   (2 Œ +  +  )¨ −  2  (  2 Œ 2 +  + ( ).  2

+ 2ñ + õ )ñó ò +

 2fD ]

(– ôñ

4

– õÿñ

2

+ ö ).

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.71: ¨*¨
=    (2  2 + 2 +  )¨ +  2   (−  4 −  2 + ( ).  The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.75:

ñóò ò ó

¨*¨
= (  cosh +  )¨ −  sinh + ( . 

= ( ô sinh ñ + õ )ñ„ó ò – ôƒõ cosh ñ + ö .

The substitution ¨ ( ) = 
leads to an Abel equation of the form 1.3.3.76:

ñóò ò ó

+

ô

sin ñ = 0.

¨*¨
= (  sinh +  )¨ −  cosh + ( . 

This is the equation of oscillations of the mathematical pendulum, where the variable  plays the role of time, and is the angle of deviation from the equilibrium state.

Page 303

304

SECOND-ORDER DIFFERENTIAL EQUATIONS 1 b . Solution:

 = AuX (2  cos + L 1 )−1 J

2

+L

M

2.

2 b . With  > 0 and the initial conditions (0) = L > 0 and "
 (0) = 0, the oscillations of the mathematical pendulum are described by

L

= sin . = sn  £ Ñ  , ¡ 2 ¡ 2 where sn = sn( H ) is the Jacobi elliptic function defined parametrically by the following relations: : ý sn( H ) = sin ý , H = X M 2 2 . 0 1− sin ý ¡ ü 3 b . The period of oscillations of the mathematical pendulum is expressed in terms of the complete elliptic integral of the second kind: sin

{

4 = £ 

; (¡

At small amplitudes, as LÃD

{

19.

),

z´J 2

)= X

0

1−

ü

ý M

2

¡

.

sin2 ý

0, the following asymptotic formula holds for the period:

2 1 = £  O 1+ L 16 

2

+ [ (L

P

4

L D

),

0.

ñóò ò ó + ô sin( ƒñ )ñ„óò + õ sin( ƒñ ) = 0. Solution in parametric form:  = − #XÈa −1   2 − Œ 2  where # =   , 

20.

; (¡

where

= #*a − #

2

ln |a | + L

J

2 −1 2



M

a + L 2,

=

1

Œ

arccos O

Π

 P

,

1.

ñóò ò ó + ô cos( ƒñ )ñóò + õ cos( ƒñ ) = 0. The substitution Œ = Œ ‡ + z 2 leads to an equation of the form 2.2.3.19: ‡

 −  sin( Œ ‡ )‡
 −  sin( Œ ‡ ) = 0.

2.2.4. Rayleigh Equations

â$äãÑã ä + å (âäã ) + ç (â ) = 0

2.2.4-1. Preliminary remarks. Some transformations. Equations of this form arise in the theory of nonlinear oscillations. 1 b . Let us discuss the special case ( ) = , which corresponds to the equation



 +  ( 
) + = 0.

(1)

Differentiating equation (1) with respect to  and substituting H ( ) =
 , we obtain the equation of

nonlinear oscillations:

H 

 + ` ( H ) H 
+ H = 0,

where

` (H ) =  ‘
( H ),

(2)

which is considered in Subsection 2.2.2. The solution of equation (1) can be written in parametric form:

 =  (U , L 1 , L 2 ), where  =  (U , L

1,

= − (H ) −

H
* , 


*

L 2 ), H = H (U , L 1 , L 2 ) is a parametric representation of the solution of equation (2).

Page 304

2.2. AUTONOMOUS EQUATIONS

2 b . The transformation

ˆ = − 21 ( 
)2 +  ,

Ï Ò˜Í Í Ò

=

ê ( Ï , Ï ÒÍ

305

)

¨ = − −  ( 
),

reduces equation (1) to an Abel equation of the second kind:

¨*¨ Š
= 5 (ˆ )¨ + 1, where 5 (ˆ ) = H −1 ` ( H ), H = A 2(  − ˆ ), (3) ü where function ` = ` ( H ) is defined above in equation (2). Specific equations of the form (3) are outlined in Subsection 1.3.2.

3 b . The equation of the special form



 +  ( 
)2 + ( ) = 0 (4) ¨ ¨ "
u ¨
2 is reduced, with the aid of the substitution ( ) = (  ) , to the first-order linear equation + 2 +  2 ( ) = 0. Therefore, the solution of equation (4) can be written in implicit form: −1 J 2  = L 2 A X V L 1  −2   − 4 ( )W , where 4 ( ) = 2  −2   XZ 2   ( ) . M M 4 b . The equation of the special form 

 +  ( 
)4 +  ( 
)2 + ( ) = 0 (5) ¨ "
3 ¨
¨ ¨ 2 2  is reduced, with the aid of the substitution ( ) = ( ) , to the Riccati equation + 2 + 2 +  2 ( ) = 0, which is outlined in Section 1.2. 5 b . For the oscillatory systems with a weak nonlinearity 

 + ž ( 
) + = 0, two leading terms of the asymptotic solution, as ID 0, are described by the formula = # cos( + $ ), where the functions # = # (ˆ ) and $ = $ (ˆ ) depend on the slow variable ˆ =  and are determined from the autonomous system of first-order differential equations: 1 2z 1 2z # Š
= X  (− # sin Y ) sin Y Y , #2$ Š
= X  (− # sin Y ) cos Y Y . 2 0 M 2 0 M   The right-hand sides of these equations depend only on # . The system is solved consecutively starting from the first equation. 2.2.4-2. Solvable equations and their solutions. 1.

2.

ñóò ò ó

+ ô (ñó ò )2 + õÿñ = 0. This equation describes small oscillations in the case where the drag force is proportional to the speed squared. −1 J 2 Solution in implicit form:  = L 2 AClX V L 1  2  −2   +  ( 21 −  )W .

ñóò ò ó

M

+ W V 31 (ñó ò )3 – ñó ò W + ñ = 0.

Van der Pol equation.

1 b . Differentiating the equation with respect to  and passing on to the new variable ¨ ( ) = /
 , we arrive at an equation of the form 2.2.2.2: ¨u

 − (1 − ¨ 2 )¨2
+ ¨ = 0. 2 b . Solution, as ID

3.

ñóò ò ó

0: 2L = £ 1−L

1 2



−

ª

2 1−L

 cos  + £ ü 1 − L 2

2 1 −

ª

sin  + [ ( 2 ).

+ ô (ñó ò )4 + õ (ñó ò )2 + ñ = 0. The transformation ˆ = − 12 (
 )2 , ¨ = − −  (
 )4 −  (
 )2 leads to an Abel equation of the form 1.3.2.1: ¨*¨3Š
= (−8 ˆ + 2  )¨ + 1.

Page 305

306 4.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

+ (ñ

óò

)2 V ô ( ñ

óò

)2 + õ W

–1

+ ñ = 0.


 )2 , ¨ = − − (
 )2 [  (
 )2 +  ]−1 leads to an equation of the The transformation ˆ = * ¨ 3 ¨
Š form 1.3.2.2: = 2  (  − 2 ˆ )−2 ¨ + 1. − 21 (

5.

ñóò ò ó

+
exp[ (ñó ò )2 ] + ( + ñ = 0. Differentiating the equation with respect to  and passing on to the new variable ¨ ( ) = /
 , we arrive at an equation of the form 2.2.2.7: ¨u

 + 2 #3Œ ¨ exp( Œ ¨ 2 )¨2
+ ¨ = 0.

6.

ñóò ò ó

+
cosh[ (ñ„ó ò )2 ] + ( + ñ = 0. Differentiating the equation with respect to  and passing on to the new variable ¨ ( ) = /
 , we arrive at an equation of the form 2.2.2.11: ¨p

 + 2 #3Œ ¨ sinh( Œ ¨ 2 )¨2
+ ¨ = 0.

7.

ñóò ò ó

+
sinh[ (ñ„ó ò )2 ] + ( + ñ = 0. Differentiating the equation with respect to  and passing on to the new variable ¨ ( ) = /
 , we arrive at an equation of the form 2.2.2.10: ¨p

 + 2 #3Œ ¨ cosh( Œ ¨ 2 )¨2
+ ¨ = 0.

8.

ñóò ò ó

+ ô (ñó ò )2 + õ sin ñ = 0. This equation describes the oscillations of the mathematical pendulum in the case where the drag force is proportional to the speed squared. −1 J 2 2 Solution in implicit form:  = L 2 AçX L 1  −2   + 2 (cos − 2  sin )S . Q 4 + 1 M

9.

ñóò ò ó

+
cos[ (ñó ò )2 ] + ( + ñ = 0. Differentiating the equation with respect to  and passing on to the new variable ¨ ( ) = /
 , we arrive at an equation of the form 2.2.2.15: ¨p

 − 2 #3Œ ¨ sin( Œ ¨ 2 )¨2
+ ¨ = 0.

10.

ñóò ò ó

+
sin[ (ñ„ó ò )2 ] + ( + ñ = 0. Differentiating the equation with respect to  and passing on to the new variable ¨ ( ) = /
 , we arrive at an equation of the form 2.2.2.14: ¨p

 + 2 #3Œ ¨ cos( Œ ¨ 2 )¨2
+ ¨ = 0.

2.3. Emden–Fowler Equation í ïwî²î ï = † ì ù í þ 2.3.1. Exact Solutions 2.3.1-1. Preliminary remarks. Classification table. In this subsection, the value of the insignificant parameter # is in many cases defined in the form of a function of two (one) auxiliary coefficients  and  ,

# = Y (  ,  ),

(1)

and the corresponding solutions are represented in parametric form,

 =  1 (U , L 1 , L 2 ,  ),

=  2 (U , L

1,

L 2 ,  ),

(2)

where U is the parameter, L 1 and L 2 are arbitrary constants, and  1 and  2 are some functions. Having fixed the auxiliary coefficient sign  > 0 (or  > 0), one should express the coefficient  in terms of both # and  with the help of (1). As a result, one obtains:

 = j ( # ,  ). Substituting this formula into (2), we find a solution of the equation under consideration (where the specific numerical value of the coefficient  can be chosen arbitrarily). The case  < 0 (or

Page 306

2.3. EMDEN–FOWLER EQUATION

Ï Ò˜Í Í Ò

=

 ËEϱ

TABLE 21 Solvable cases of the Emden–Fowler equation

307

No

B ¡

Equation

No

1 2 3 4 5

0 −3 ¡ − 21 ( + 3) ¡ arbitrary arbitrary

arbitrary arbitrary arbitrary 0 1

2.3.1.2 2.3.1.3 2.3.1.4 2.3.1.1 2.3.1.5

−

Isolated points −7 −7 −5 2 −2 −2 −5 3 −5 3

6 7 8 9 10 11 12

1 3 −1 2 −2 1 −10 3 −7 3

¡

−5 3 −5 3 −5 3 −5 3 −7 5 −7 5 −1 2 −1 2 −1 2 −1 2 −1 2 −1 2 −1 2 2 2 2

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

One-parameter families

2.3.1.15 2.3.1.16 2.3.1.22 2.3.1.28 2.3.1.27 2.3.1.10 2.3.1.8



 = #*  + B

Equation

−5 6 −1 2 1 2 −13 5 1 −7 2 −5 2 −2 −4 3 −7 6 −1 2 1 −5 −20 7 −15 7

2.3.1.23 2.3.1.24 2.3.1.7 2.3.1.9 2.3.1.14 2.3.1.13 2.3.1.12 2.3.1.6 2.3.1.26 2.3.1.17 2.3.1.18 2.3.1.25 2.3.1.11 2.3.1.19 2.3.1.21 2.3.1.20

 < 0), which may lead to another branch of the solution or to a different domain of definition of the variables  and in (2), should be considered in a similar manner. One can also use a different approach by setting one of the auxiliary coefficients (e.g.,  ) equal to A 1 in (1) and (2); then the other coefficients will be identically expressed in terms of # by means of (1). Table 21 presents all solvable Emden–Fowler equations whose solutions are outlined in Subsection 2.3.1. The one-parameter families (in the space of the parameters B and ) and isolated points ¡ are presented in a consecutive fashion. Equations are arranged in accordance with the growth of ¡ and the growth of B (for identical ). The number of the equation sought is indicated in the last ¡ column in this table.

2.3.1-2. Solvable equations and their solutions. 1.

ñóò ò ó

=


ïƒ÷

Solution:

2.

ñóò ò ó

=


ñ ú

.

=








#* 

+ L 1 + L (B + 1)(B + 2) − # ln | | + L 1  + L 2






#X ln | |  + L 1  + L M

.

Solution:  =



+2







AuXÈO

2#

+

+1

+1 ¡ A X (2 # ln | | + L

if B = −2; if B = −1.

2

J

−1 2

+L 1)

if B ≠ −1, B ≠ −2;

2

J

1

P

−1 2

+L

M

M

+L

2

2

≠ −1,

if if

¡

¡

= −1.

Page 307

308

SECOND-ORDER DIFFERENTIAL EQUATIONS Special cases.

= −1 2, the solution can be written in the parametric form:

1 b . In the case

¡  = L 13 (U

2 b . In the case

3

− 3U + L

A (4U ü

− 1)2 ,

U

 J

4 −2 3 2 . 9

# =A

where



= −4, the solution can be written in the parametric form:

¡

 = L where ² =

4 2 1(

= L

2 ),

3 b . In the case

3

5 −1 1

U

U

O 2U*X

− 1), # = T 6 

M ²

U

+L

2

U3Tç²

2 −1 , 1

= L

,

P

U

 .

−2 5

= 2, the solution can be written in the parametric form:

¡

−1 1

 = L

U ,

2 1

= L

<

−2 −1

# = A 6

;



,

the function < of the parameter U is defined in implicit form:

U =X

M

A (4 < ü

<

3

−L

− 1)

2.

The upper sign in this formula corresponds to the classical elliptic Weierstrass function < = < (U + L 2 , 0, 1). 4 b . In the case

<

= −5 2, the solution can be written in the parametric form:

¡

7 −2 1

 = L

V

A (4 < ü

3

− 1) A 2U'< 2 W ,

<

4 −2 , 1

= L

 J .

−2 7 2

# = T 3

where

The function < of the parameter U is defined in the previous case. 3.

=
ï –ú –3 ñ ú . 1 b . Solution in parametric form with

ñ óò ò ó

 = L 1+

−1

Q

X (1 AIU +

) J

+1 −1 2

M

≠ −1:

¡

−1

U + L 2S

+ 1 + +1 1− +   . 2 2 b . Solution in parametric form with

= L 1+

,

+1

U X (1 AIU + Q

) J

+1 −1 2

M

U + L 2S

−1

,

where # = A ¡

 =L

1

= −1:

¡

−1

X exp(T"U 2 ) U + L 2 S Q M

=  exp(T U 2 ) X exp(T"U 2 ) U + L

,

Q

M

2

S

−1

,

where # = T 2  2 . 4.

ú

=
ï – 2 ñ ú . 1 b . Solution in parametric form with

ñóò ò ó

+3

8 U + Q +1 ¡ 8 =  L 2 U exp XÈO U + Q +1

 = L

where # = O



 2P

+

−1 2

2 2

exp 2 X O

¡

2 2

exp 2 Xp 8 ln |U | + U

Q

where # =  2  .

+1 +1

+U

+U

2 2

+L

+L

J

−1 2 1 1

P

P

J

M

−1 2

M

U!S , U!S ,

.

2 b . Solution in parametric form with

 = L

≠ −1:

¡

2

+L

1



¡

J

= −1:

−1 2

M

U!S ,

= L

2

U exp I X  8 ln |U | + U Q

2

+L

1



J

−1 2

M

U!S ,

Page 308

2.3. EMDEN–FOWLER EQUATION

5. 6.

309

=
ï –5 ) 2 ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

−3 3 1 (

U

− 3U + L

2)

−1

= L

,

1(

U

2

− 1)2 (U

3

− 3U + L

2)

−1

,

where

# =A

=
ï ñ –5 ) 3 . Solution in parametric form:

 J  J

4 1 2 3 2 . 9

ñ óò ò ó

4

2

− 6U

+ 4L

2

U − 3),

= L

9 3 1(

U

− 3U + L

2)

J ,

3 2

# =A

where

=
ï –7 ) 3 ñ –5 ) 3 . Solution in parametric form:

 J

9 −3 8 3 . 64



ñóò ò ó

 =A 9.

 ËEϱ

=
ï ÷ ñ . For B ≠ −2, see equation 2.1.2.7. For B = −2, see equation 2.1.2.123.

 = A L 18 (U 8.

=

ñ óò ò ó

 = L 7.

Ï Ò˜Í Í Ò

U

4

− 6U

L

2

−8 1

+ 4L

2

U −3

=A

,

L 1 (U 3 − 3U + L 2 )3 J 2 , U 4 − 6U 2 + 4 L 2 U − 3

where

# =A

 J  J

9 1 3 8 3 . 64

=
ï 2 ñ –5 ) 3 . 1 b . Solution in parametric form with # < 0:

ñóò ò ó

 = L

2 1

cos U cosh(U + L

3 where # = − 16 

2 )[tan

U + tanh(U + L 2 )],

= L

3 1 [cos

U cosh(U + L 2 )]3 J 2 ,

 J .

−4 8 3

2 b . Solution in parametric form with # > 0:

 = L where # = 34  10.

2 1 [sinh

U + cos(U + L 2 )],

= L

3 1 [cosh

U − sin(U + L 2 )]3 J 2 ,

 J .

−4 8 3

=
ï –10 ) 3 ñ –5 ) 3 . 1 b . Solution in parametric form with # < 0:

ñ óò ò ó

 = L 1−2 [cos U cosh(U + L 2 )]−1 [tan U + tanh(U + L 2 )]−1 , = L 1 [cos U cosh(U + L 2 )]1 J 2 [tan U + tanh(U + L 2 )]−1 , 3 4J 3 8J 3 where # = − 16   .

2 b . Solution in parametric form with # > 0:

 = L

−2 1 [sinh

U + cos(U + L 2 )]−1 ,

= L

1 [cosh

U − sin(U + L 2 )]3 J 2 [sinh U + cos(U + L 2 )]−1 ,

where # = 34  4 J 3  8 J 3 . 11.

=
ï ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

 = L

1

= L

2 1

where # = 16

exp(−U ) V exp(3U ) + L

2

exp(−2U ) V 2 exp(3U ) − L

sin  £ 3 U  W , 2

sin  £ 3 U  + £ 3 L

2

2 cos  £ 3 U  W ,

 J .

−3 3 2

Page 309

310

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 12–14, the following notation is used:

8 8

12.

1

= exp(3U ) + L

3

=2

8

8

1( 2)


−(

*

8

sin  £ 3 U  ,

2

8


8 1) *

−

2

ñóò ò ó =
ï –7 ) 2 ñ –1 ) 2 .

8 8

= 2 exp(3U ) − L

2

13.

−1 1 –7 5

−1 1 ,

= L

4 1

exp(−2U )

8

5 1

= L

3,

=
ï –13 ) 5 ñ –7 ) 5 . Solution in parametric form:

1

ñóò ò ó

 = L ]

8

=
ï ñ ) . Solution in parametric form:

ñóò ò ó

 = L

14.

exp(U )

−4 1

exp(2U )

8

sin  £ 3 U  + £ 3 L

cos  £ 3 U  ,

2

1 2.

Solution in parametric form:

 = L

2

−1 3 ,

= L

exp(−U )

8

−1 2 1 2,

where

8 5J 2 ,

exp  − 25 U 

where

1

8 5J 2 8

exp  − 12 U 

1

8

−1 3 ,

1

# = 16( )3 J 2.



# =

where

 J

−3 12 5 5 . 1024

# =

 J 

J

3 5 12 5 5 . 1024

In the solutions of equations 15–18, the following notation is used:

 = 2UÈ® (U ) + L 2 2 U Tç² ,

®

U

(U ) = X

²

M

U

² =

,

ü

A (4U

3

− 1),

where ® (U ) is the incomplete elliptic integral of the second kind in the form of Weierstrass. 15.

=
ï ñ –7 . Solution in parametric form:

ñóò ò ó

TCU

−2

 = L 18 [4U 2 TçU 3 −5 8 where # = A 64   .

−2

 = L

16.

17.

8 1 [4

=
ï 3 ñ –7 . Solution in parametric form:

ñ óò ò ó

= L

(  ² − 1)2 ]−1 ,

 J

3 1 2 , 1

 J

5 1 2 [4 1

= L

# =A

where

U

2

TCU

−2

3 −3 8 . 64





(  ² − 1)2 ]−1 ,

=
ï –4 ) 3 ñ –1 ) 2 . Solution in parametric form: 9 3 , 1



=
ï –7 ) 6 ñ –1 ) 2 . Solution in parametric form:

4 −2 ( 1

= L

U

 ² − 1)2 ,

# =A

where

ñóò ò ó

 = L ]

(  ² − 1)2 ],

ñóò ò ó

 = L

18.

2

U

9 −3 , 1



= L

5 −2 −3 ( 1

U

 ² − 1)2 ,



€

In the solutions of equations 19–24, the function

U =X

<

where

=

€ („

J  J

4 −2 3 3 2 . 3



# =A

J  J

4 −5 6 3 2 . 3



) is defined in implicit form:

− L 2. − 1) ü The upper sign in this formula corresponds to the classical elliptic Weierstrass function < = < (U + L 2 , 0, 1). 19.

M

A (4 <

3

=
ï –5 ñ 2 . Solution in parametric form:

ñ óò ò ó

 = L 1 U

−1

,

= L

3 −1 1

U

<

,

where

# = A 6 3 

−1

.

Page 310

Ï Ò˜Í Í Ò

2.3. EMDEN–FOWLER EQUATION

20.

=
ï –15 ) 7 ñ 2 . Solution in parametric form: 7 7 , 1

7 −7 , 1

U

1

U (U 2 p < T 1),

U
U

< V A

7 2 1

(4 <

ü

3

−1

− 1) A 2U'< 2 W

3 1

= L

,

V

J  J .

−3 2 7 2

A (4 < ü

A (4 <

L

ü

3

16 1

− 1) + 2 < W

=

, 2

L

7 1

=
ï –1 ) 2 ñ –5 ) 3 . Solution in parametric form:

ñóò ò ó

16 1

VU

A (4 < ü

3

2

− 1)+2 <"W ,

V VU

= L

A (4 < ü

A (4 < ü

9 1

3

V

3

A (4 < ü

− 1) A 2U'< 2 W

−1

,

− 1) + 2 < W

3

J

where # = − 61 

−7 6 8 3

3 2

, where # = − 61 

−3 2 8 3

J

− 1) A 2U'< 2 W

J  J .

,

2

J  J .

In the solutions of equations 25–28, the following notation is used: =

ø J U ) + L 2n ®1 1 J 3 (U ) + L 2 ;

L

™

1 1 3(

L

U ) for the upper sign, ( 1 J 3 U ) for the lower sign,

1 3(

J

=
ï –1 ) 2 ñ –1 ) 2 . Solution in parametric form:

J U ) are the modified Bessel

1 3(

ñóò ò ó

:  = U 2 J 3 2 ,

=
ï –2 ñ –1 ) 2 . Solution in parametric form:

= 'U

J  U :
+ 1 :  2, 3

−2 3

J :

−2 3

−2

,

=
ï ñ –2 . Solution in parametric form:

= 'U

J :

−4 3

−2

U

ñ óò ò ó

: J V  U :
+ 1:  2I A U 2 2W , 3

−2 3

 = U

: : 2 2: 2 W  = U 2 J 3 V  U
+ 13  / A U *

−1

,

= 'U 4 J

+

1 3

= 'U 2 J

*

=
ï –2 ñ –2 . Solution in parametric form:

ñóò ò ó

:
*

3

: 2V :
 U *

# = 13 (T±  )3 J 2 .

where

*

ñóò ò ó

 = U

28.

− 1) A 2U'< 2 W

3 2

where ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; functions.

27.

3

=
ï –5 ) 6 ñ –5 ) 3 . Solution in parametric form:

:

26.

 J 

6 6 7 −1 . 49

# =A

where

ñóò ò ó

 = L

25.

 J 

6 1 7 −1 . 49

# =A

where

6 −6 2 ( 1

= L

=
ï –1 ) 2 ñ –5 ) 2 . Solution in parametric form:

 = V U

]

= L

ñóò ò ó

where # = T 3 

24.

U

=
ï –20 ) 7 ñ 2 . Solution in parametric form:

 = L 23.

311

ñ óò ò ó

 = L

22.

 ËEϱ

ñóò ò ó

 = L

21.

=

: 

3

2

:

,

:

2

where

,

# =T

# = − 29 (   )3.

where

2

+ 13  A U

2

:

2 −1

W

 J

1 3 2 . 3

,

where

# = − 29  3 .

Page 311

312

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.3.2. First Integrals (Conservation Laws) In this subsection, first integrals of the form

½  9

 9 ( , )( 
) 9 = L ,

=0

, = 2, 3, 4, 5,

where

for the Emden–Fowler equation 

 = #*  + are given. 2.3.2-1. First integrals with , = 2. 1 b . For B = 0 and arbitrary

2 b . For B = − 12 (

¡

(

≠ −1),

¡

2# ( 
)2 −

+ 3) and arbitrary

¡

(

¡

¡

¡

+1

+

+1

=L .

≠ −1),

+ +1 2#  ( 
)2 − ! 
−  − 2 + +1 = L . +1 ¡ 3 b . For B = − − 3 and arbitrary ( ≠ −1), ¡ ¡ ¡ 2# 2 
2  ( ) − 2 ! 
+ 2 −  − + −1 + +1 = L . +1 ¡ = 2, 4 b . For B = − 20 7 , ¡ 8 J 7 
2 1J 7 2 −12 J 7 3 343 ( ) −  49 −   
− 343  + 76 #* −6 J 24 #* 3 #* 36 # 5 b . For B = − 15 7 ,

7 2

=L .

−

= 2,

¡ 343 * #  6 J 7 ( 
)2 −  24

49 4

+ 1  
−

J

−1 7

#*

343 36

J

2 −9 7 3

#



J

−8 7 2

− 78 * # 

=L .

2.3.2-2. First integrals with , = 3. = − 12 ,

1 b . For B = 0,

2 b . For B = 1, 3 b . For B =

¡

( 
)3 − 6 #


+ 6#  ( 
)3 − ( 
)2 − 6 #*

= − 12 ,

¡

− 43 ,

=

¡

J

1 2

( 
)3 − 6 #*

2

J

1 2


+ 4#

− 12 ,

 ( 
)3 − ( 
)2 − 6 #*

4 b . For B = − 52 ,

 =L , J 
+

1 2

J

16 3

#

J + 3#

3 2

J + 2# 2 

J

3 2

−1 3 1 2


− 9#

3

2 2



=L .

=L .

J =L .

2 −2 3



= − 12 ,

¡

 2 ( 
)3 − 2 ( 
)2 + ( 2 − 6 #* −1 J 2 1 J 2 ) 
+ 23 #* −3 J 2 3 J 2 − 3 # 2  −2 = L ,  3 ( 
)3 − 3 2 ( 
)2 + 3( 2 − 2 #* 1 J 2 1 J 2 ) 
− 3 + 6 #* −1 J 2 3 J 2 − 6 # 2  −1 = L .

5 b . For B = − 76 ,

= − 12 ,

¡

 2 ( 
)3 − 2  ( 
)2 + (

6 b . For B = − 72 ,

¡

 3 ( 
)3 − 3 

2

= − 12 , 2

( 
)2 + 3(

2

J ) 
+ 6 #*

− 6 #* 5 J − 2 #*

J

6 1 2

J ) 
−

−1 2 1 2

3

J

J + 9#

−1 6 3 2

+ 2 #*

J

 J =L .

2 2 3

J − 2#

−3 2 3 2

2 −3



=L .

Page 312

Ï Ò˜Í Í Ò

2.3. EMDEN–FOWLER EQUATION

=

 ËEϱ

313

2.3.2-3. First integrals with , = 4. = − 53 ,

1 b . For B = 1,

¡

( 
)4 + 6 #*

J ( 
)2 − 18 # 1 J 3 
+ 9 # 2  2 −4 J 3 = L ,  ( 
)4 − ( 
)3 + 6 #* 2 −2 J 3 ( 
)2 − 27 #* 1 J 3 
+ 814 # 4 J 3 + 9 # −2 3

= − 53 ,

2 b . For B = 2,

¡

( 
)4 + 6 #*

= − 53 ,

3 b . For B = 0,

¡

 ( 
)4 − ( 
)3 + 6 #*  2 ( 
)4 − 2  ( 
)3 + ( 4 b . For B = − 12 ,

J ( 
)2 − 36 #*

J =L .

4 −4 3

J ( 
)2 − 9 # 1 J 3 
+ 9 # 2  −4 J 3 = L , 2 + 6 #* 2 −2 J 3 )( 
)2 − 18 #* 1 J 3 
+ 12 #

 ( 
)4 − ( 
)3 + 6 #* 1 J

5 b . For B = − 43 ,

1 3



−2 3

= − 53 ,

¡


+ 9# 2 

J

2 −2 3

J =L .

2 3 −4 3

J ( 
)2 + 9 #

J + 9#

4 3

J =L .

2 2 −4 3



J =L .

2 −2 3

2 −4 3

= − 53 ,

¡

 2 ( 
)4 − 2 ( 
)3 + ( 2 + 6 #* 2 J 3 −2 J 3 )( 
)2 + 6 #* −1 J 3 1 J 3 
+ 9 # 2  −2 J 3 −4 J 3 = L ,  3 ( 
)4 − 3 2 ( 
)3 + 3 ( 2 + 2 #* 2 J 3 −2 J 3 )( 
)2 − ( 3 + 3 #* 2 J 3 1 J 3 ) 
− 3 #* −1 J 3 4 J 3 + 9 # 2  1 J 3

6 b . For B = − 73 ,

 3 ( 
)4 − 3   4 ( 
)4 − 4 

J )( 
)2 − ( 3 − 15 #* −1 J 3 1 J 3 ) 
− 34 #* ( 
)3 + 6 2 ( 2 + #* −1 J 3 −2 J 3 )( 
)2
− 2 (2 3 − 3 #* −1 J 3 1 J 3 )  + 4 − 12 #*

3

7 b . For B = − 56 ,

 3 ( 
)4 − 3 

= − 53 ,

¡

( 
)3 + 3  (

2

2

+ 2 #*

J

−1 3 −2 3

= − 53 ,

¡

( 
)3 + 3  (

2

2

+ 2 #* 7 J

J )( 
)2

 4 ( 
)4 − 4 

= − 53 ,

¡


(  )3 + 6  2 (

3

9 b . For B = 1,

 ( 
)4 −3 x3y'z|{}~/€ 3

¡

2

3

+ 12 #* 7 J

2 −5 3 −4 3

J

J + 9#

2 −2 3 −4 3

−1 3 4 3

J ) 
+ 6 #* 1 J

6 1 3


+ #* −4 J 3 −2 J 3 )(  )2 − 4 ( 3 − 6 #* −4 J 3 1 J 3 ) 
+

 ( 
)4 − ( 
)3 + 23 * # 

10 b . For B = 3,

J + 9#

J

J =L ,

J

J =L .

 J

J =L .

J

J =L .

 

J + 9#

6 4 3

2 4 3 −4 3

2

= −7,

¡

J

−4 3 4 3

6 −2 3

−( 8 b . For B = − 10 3 ,

J =L .

−4 3

2 −6

( 
)2 − 13 * # 

= −7,


(  )3 +3 ( 2 + 29 #*

5 −6

−5

4


−


)(  )2 − ( 2 + #*

− 30 #* 1 12

#

5 −6

−4

J

J + 9#

+ 19 #

2 3 −12

−4 3 4 3


)  + 14 #*



4 −4

+ 19 #

2 −8 3 −4 3



=L . 2 9 −12



=L .

In the case , = 4 we omitted the first integrals of the form

ƒh

2

+ ý… + = = L ,
where function  =  ( , ,  ) is the left-hand side of the above integrals for , = 2, and ƒ , ý , and = are some constants.

Page 313

314

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.3.2-4. First integrals with , = 5. 1 b . For B = 0,

= − 23 ,

¡

2 b . For B = − 73 ,

( 
)5 − 15 #

¡

J ( 
)3 +

1 3

135 2

#


−

J

2 2 3

135 2

#

3

 =L .

= − 23 ,

 5 ( 
)5 − 5 4 ( 
)4 + 5 3 (2 − 3 #* −1 J 3 −2 J 3 )( 
)3 − 5 2 −2 J 3 2 J 3 
+ 5 ( 4 − 9 #* −1 J 3 7 J 3 + 27  ) + 15 # ( 2 #

(2 − 9 #* −1 J 3 −1 J 3 10 J 3 − 92 #* −2 J 2 2

J )( 
)2 5J 3 − 92 # 2 

−2 3 3

−1

)=L .

2.3.3. Some Formulas and Transformations 1 b . With

≠ 1, the Emden–Fowler equation has a particular solution:

¡



+2 1−

=Π

+ ,

(B + 2)(B + + 1) + ¡ S Q # ( − 1)2

where Π=

1 −1

.

¡

2 b . The transformation = ¨u a ,  = 1 a leads to the Emden–Fowler equation with the independent variable raised to a different power:

¨ g

g = #*a −  − +

−3

¨ + .

3 b . Some more complicated transformations leading to the Emden–Fowler equation are outlined in Subsection 2.5.3 (see Fig. 3). 4 b . With

¡

≠ 1 and

¡

≠ −2B − 3, the transformation

ˆ = leads to an Abel equation:

2B +

¡

¡

+ 3 +  −1

+2 −1

,

‡ = +



+2 −1

B +2 O 
+ −1 P ¡

2 −1 (B + 2)(B + + 1) ¡ 2 ˆ +# O ¡ ‡"‡ Š
− ‡ = − ˆ + , (2B + + 3) 2B + + 3 P ¡ ¡

whose special cases are given in Subsection 1.3.1. 5 b . Some more complicated transformations leading to other Abel equations are outlined in Subsection 2.5.3.

2.4. Equations of the Form í ïwî²î ï = † See Section 2.3 for the special cases #

1

= 0 and #

2

1ì

ù 1í þ

1

+†

2ì

ù 2í þ

2

= 0.

2.4.1. Classification Table

Table 22 presents all solvable equations of the form "

 = # 1   1 + 1 + # 2   2 + 2 whose solutions are outlined in Subsection 2.4.2. Two-parameter families (in the space of the parameters 1 , 2 , ¡ B 1 , and B 2 ), one-parameter families, and isolated points are presented in a consecutive ¡ fashion. Equations are arranged in accordance with the growth of 1 , the growth of 2 (for identical 1 ), ¡ ¡ ¡ the growth of B 1 (for identical 1 and 2 ), and the growth of B 2 (for identical 1 , 2 , and B 1 ). ¡ ¡ ¡ ¡ The number of the equation sought is indicated in the last column in this table.

Page 314

2.4. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

 1Ë E Ï ±

=

1

TABLE 22 Solvable equations of the form 

 = # No

1

¡

B

2

¡

1

arbitrary

arbitrary

2

arbitrary

arbitrary

−

3

arbitrary

arbitrary

− 21 (

4

arbitrary

0

5

arbitrary

0

B

1

0

−

1

¡

¡

¡

1

+ 3)

−

− 21 (

0 1

−3

6

1

arbitrary

−2

7

1

arbitrary

−2

8

1

−3

arbitrary (B 1 ≠ −2)

9

−7

−7

10

−5

11

¡

¡

1

+ #

1

2

2

+#

2

1

315

 

+

2

#

2

2

Equation

arbitrary

2.4.2.1

arbitrary

arbitrary

2.4.2.2

arbitrary

arbitrary

2.4.2.3

0

arbitrary

arbitrary

2.4.2.19

−3

arbitrary

arbitrary

2.4.2.20

arbitrary

2.4.2.4

arbitrary

2.4.2.5

−3

2

+ 3)

−2 −

 

 2Ë E Ï ±

arbitrary

2

¡

+

2

0 −3

1

1

2

−1

2( − ¡ ( ¡ 2( − ¡ (

¡

2 + 1) 2 2 + 3) 2 + 1) 2 2 + 3)

0

arbitrary

arbitrary

2.4.2.83

4

3

arbitrary

arbitrary

2.4.2.39

−5

2

0

arbitrary

arbitrary

2.4.2.16

−3

−7

0

1

arbitrary

arbitrary

2.4.2.42

12

−3

−7

0

3

arbitrary

arbitrary

2.4.2.43

13

−3

−4

0

0

arbitrary

arbitrary

2.4.2.17

14

−3

−4

0

1

arbitrary

arbitrary

2.4.2.18

15

−2

−3

−2

0

arbitrary

arbitrary

2.4.2.88

16

−2

−3

1

0

arbitrary

arbitrary

2.4.2.87

17

−2

−2

−1

−2

arbitrary

arbitrary

2.4.2.28

18

− 53 − 53 − 53 − 53 − 53 − 53 − 53 − 32 − 32 − 75 − 43 − 43 − 35 − 35 − 12

− 53 − 53 − 53 − 53 − 53 − 53 − 53

− 37 − 34 − 34 − 32

arbitrary

arbitrary

2.4.2.48

arbitrary

arbitrary

2.4.2.49

arbitrary

arbitrary

2.4.2.24

arbitrary

arbitrary

2.4.2.90

0

− 10 3 10 −3 − 37 − 34 − 32

arbitrary

arbitrary

2.4.2.89

2

0

arbitrary

arbitrary

2.4.2.47

2

1

arbitrary

arbitrary

2.4.2.46

−2

− 23

−2

arbitrary

arbitrary

2.4.2.81

−2

0

1

arbitrary

arbitrary

2.4.2.80

− 75 − 53 − 53 − 75 − 75 − 12

− 58 − 35

− 13 5 − 37

arbitrary

arbitrary

2.4.2.25

arbitrary

arbitrary

2.4.2.102

0

1

arbitrary

arbitrary

2.4.2.101

− 12 5

− 13 5

arbitrary

arbitrary

2.4.2.53

0

1

arbitrary

arbitrary

2.4.2.52

− 25

− 27

arbitrary

arbitrary

2.4.2.23

19 20 21 22 23 24 25 26 27 28 29 30 31 32

Page 315

316

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 22 (Continued) Solvable equations of the form 

 = # 1   1 +

No

1

¡

B

1

B

2

#

1

+#

2

#

  2 + 2

2

Equation

− 35

− 83

− 10 3

arbitrary

arbitrary

2.4.2.55

− 35 − 35 − 35 − 35 − 35

− 83 − 83

− 73 − 43

arbitrary

arbitrary

2.4.2.59

arbitrary

arbitrary

2.4.2.57

0

0

arbitrary

arbitrary

2.4.2.56

0

1

arbitrary

arbitrary

2.4.2.58

38

− 13 − 13 − 13 − 13 − 13

0

2

arbitrary

arbitrary

2.4.2.54

39

0

−2

−3

−2

arbitrary

arbitrary

2.4.2.108

40

0

−2

0

1

arbitrary

arbitrary

2.4.2.107

41

0

−1

−3

−2

arbitrary

arbitrary

2.4.2.22

42

0

−1

0

0

arbitrary

arbitrary

2.4.2.21

43

0

−3

− 73

arbitrary

arbitrary

2.4.2.73

44

0

0

0

arbitrary

arbitrary

2.4.2.72

−4

arbitrary

arbitrary

2.4.2.96

arbitrary

arbitrary

2.4.2.51

−3

− 52 − 72 − 52

arbitrary

arbitrary

2.4.2.45

−3

−2

arbitrary

arbitrary

2.4.2.106

−3

− 12 − 76 − 52

arbitrary

arbitrary

2.4.2.85

arbitrary

arbitrary

2.4.2.41

arbitrary

arbitrary

2.4.2.100

−2

arbitrary

arbitrary

2.4.2.79

− 12

arbitrary

arbitrary

2.4.2.78

0

arbitrary

arbitrary

2.4.2.99

− 43

arbitrary

arbitrary

2.4.2.40

0

−2

arbitrary

arbitrary

2.4.2.86

0

− 12

arbitrary

arbitrary

2.4.2.105

0

0

arbitrary

arbitrary

2.4.2.44

0

1

arbitrary

arbitrary

2.4.2.50

1

0

arbitrary

arbitrary

2.4.2.95

− 10 3

− 73

arbitrary

arbitrary

2.4.2.98

0

1

arbitrary

arbitrary

2.4.2.97

arbitrary

2.4.2.35

arbitrary

2.4.2.36

33 34 35 36 37

− 13

2

¡

1

57

0

58

0

59

0

60

0

61 62

1 3 1 3

− 32 − 32 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 21 − 35 − 35

63

1

−7

−2

−2

64

1

−7

−2

6

15 4 15 4

65

1

−4

−2

−2

6

arbitrary

2.4.2.31

66

1

−4

−2

3

6

arbitrary

2.4.2.32

45

0

46

0

47

0

48

0

49

0

50

0

51

0

52

0

53

0

54

0

55

0

56

0

−3

− 53 − 32 − 32 − 32 − 32 − 43

Page 316

Ï Ò˜Í Í Ò

2.4. EQUATIONS OF THE FORM

=

 1Ë E Ï ± 1

1

+

 2Ë E Ï ±

TABLE 22 (Continued) Solvable equations of the form 

 = # 1   1 + No 67 68

¡

1

2

¡

B

1

B

2

#

1

1

2

2

+#

2

317

  2 + #

2

2

Equation

−3

−5

0

arbitrary

arbitrary

2.4.2.84

1

−3

1

0

arbitrary

arbitrary

2.4.2.82

1

−2

−2

12

arbitrary

2.4.2.64

−2

3 2

12

arbitrary

2.4.2.65

1

70

1

− 52 − 52

71

1

−2

−2

−2

2

arbitrary

2.4.2.6

72

1

−2

−2

1

2

arbitrary

2.4.2.7

73

1

−2

−2

arbitrary

2.4.2.26

74

1

−2

−2

arbitrary

2.4.2.10

75

1

−2

−2

arbitrary

2.4.2.12

76

1

−2

−2

arbitrary

2.4.2.66

−2

arbitrary

2.4.2.27

arbitrary

2.4.2.11

arbitrary

2.4.2.13

−2

2 3 2 3 2 3 2 3

arbitrary

2.4.2.67

−2

−2

arbitrary

2.4.2.29

−2

2 5

arbitrary

2.4.2.30

−2

−2

arbitrary

2.4.2.14

−2

−2

3 − 16 9 − 100 3 4 63 4 3 − 16 9 − 100 3 4 63 4 5 − 36 5 − 36 − 92 4 − 25

arbitrary

2.4.2.8

−2

−2

20

arbitrary

2.4.2.33

−2

− 92 4 − 25

arbitrary

2.4.2.15

arbitrary

2.4.2.9

−2

− 21 − 21 − 21

20

arbitrary

2.4.2.34

69

86

1

87

1

88

1

− 53 − 53 − 53 − 53 − 53 − 53 − 53 − 53 − 75 − 75 − 12 − 12 − 12 − 12 − 12 − 12

89

1

0

−5

−3

arbitrary

arbitrary

2.4.2.77

90

1

0

1

0

arbitrary

arbitrary

2.4.2.76

1

−2

−2

arbitrary

2.4.2.37

−2

− 23

− 12 49 − 12 49

arbitrary

2.4.2.38

77

1

78

1

79

1

80

1

81

1

82

1

83

1

84

1

85

1

−2 −2

−2

92

1

1 2 1 2

93

2

0

−5

−4

arbitrary

arbitrary

2.4.2.92

94

2

0

−5

−3

arbitrary

arbitrary

2.4.2.69

− 13 7 − 12 7 − 79 − 78

arbitrary

arbitrary

2.4.2.94

arbitrary

arbitrary

2.4.2.71

arbitrary

arbitrary

2.4.2.70

arbitrary

arbitrary

2.4.2.93

91

95

2

0

96

2

0

97

2

0

98

2

0

− 20 7 − 20 7 15 −7 − 15 7

99

2

0

0

0

arbitrary

arbitrary

2.4.2.68

100

2

0

0

1

arbitrary

arbitrary

2.4.2.91

Page 317

318

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 22 (Continued) Solvable equations of the form 

 = # 1   1 + No

¡

101

1

2

B

2

¡

B

1

−3

1

#

2

−2

+#

1

2

  2 +

#

1

2

Equation

2

arbitrary

6 − 25

2.4.2.61 2.4.2.63

102

2

1

−3

−2

arbitrary

103

2

1

−2

−2

arbitrary

104

2

1

−2

−2

arbitrary

6 25 6 − 25 6 25

105

3

1

−6

−5

arbitrary

arbitrary

2.4.2.104

106

3

1

0

1

arbitrary

arbitrary

2.4.2.103

− 18 5 12 −5

− 14 5 11 −5

arbitrary

arbitrary

2.4.2.74

arbitrary

arbitrary

2.4.2.75

107

3

2

108

3

2

2.4.2.60 2.4.2.62

2.4.2. Exact Solutions 1.

=
1ñ ú 1 +
2ñ ú 2 , þ 1 b . Solution in parametric form:

ñóò ò ó

 = lX ( L where #

1

= 12 



−2 1−

+ 1(

1

¡

+

+U

1



2 b . Solution in parametric form:

where #

2.

1

= − 21 



−2 1−

+ 1(

1

¡

−U +

1

=
1 ï –ú 1 –3 ñ ú 1 +
2 ï –ú 2 –3 ñ 1 b . Solution in parametric form:

ñ óò ò ó

 = l” ù where #

1

L ü

= 12 

1

+U

1+

+ 1

+M

U 1 +1

1−

AIU +

+ 1(

1

¡

2 +1

ú

2



where # 3.

1

L ü

1

= − 21 

ú

−U 1+

+M

+ 1

U 1 +1

þ

•

1 1+ 2

=A

2



1−

+ 1(

2 +1

1

¡

ú

1 +3

 =L

2 1

=L

1

exp XÈOL

Q

U exp

2

+

1 XÈOL Q2

2

+ 1), #

=
1ï – 2 ñ ú 1 +
2ï – 2 ñ 1 b . Solution in parametric form with

ñóò ò ó

−1

+L

1 U 4 2

ú

2 +3

+

2

1 U 4

2

1

¡

+

=A

2

¡

2

1

+

+ 2(

+ 2

1 1+ 2



1

+1 2#

¡

þ

+ 2(

1

U + 1

¡

1 +1

+1

U +

1−

≠ –1.

2

L ü 2

¡

+ 2(

ü ¡

1

+U

+M

U 1 +1

AIU +

2 +1

−1

+L

2

+L

2

,

•

+ 1).

L

1

−U

2

+ 1).

+M

U 1 +1

AIU +

2 +1

−1

•

,

≠ −1:

2

+

= 'U ,

+ 1).

2

¡

1−

+ 2

. ≠ −1 and

2#

U + L 2, M

= 'U*” ù

,

•

2

= 'U ,

+ 1).

2

¡

= 'U*” ù

,

2

U + L 2, M

≠ –1,

1

−1

+L

+ 1), #

AIU +

)−1 J



,

2 b . Solution in parametric form:

 = l” ù

2 +1

2

+ 2(

1 −2 1− 2

=A

2

)−1 J



ACU +

1 +1

+ 1), #

2 +1

1 −2 1− 2

=A

2

≠ –1.

2

ACU +

1 +1

+ 1), #

 = lX ( L

þ

≠ –1,

1

¡

2#

1 +1

2

+

2

+1 2#

¡

2

U + 2

2 +1

+1

U +

J

−1 2

P

M

2 +1

U!S , J

P

−1 2

M

U!S .

Page 318

Ï Ò˜Í Í Ò

2.4. EQUATIONS OF THE FORM

2 b . Solution in parametric form with

4.

 =L

2 1

=L

1

U exp

U M + £ 1 AI U

1O X

¡

where # = A

O X

1

(

U M + £ 1 AC U

¡

ñ ú

+1

+ 1)( − 1)2 ¡  2( + 3)2

ú

2#

+

1 U 4

1

1

¡

2

+

+ 2

P

1

U +

1 +1

U +

1

+1

+

1

2

¡

 2Ë E Ï ± 2

+ 2# 1 +1

ln |U |

2

+ 2#

2

J

−1 2

P

U!S , M

ln |U |

J

−1 2

P

U!S . M

≠ –1.

U M + £ 1 AC U

= 'U O X

,

319

2

= −1:

þ

≠ –3,

+3 −1

+

+1 2#

¡

þ

,

+L

1−

–

2

+

2

+1

¡

þ + 1) ï –2 ñóò ò ó = – 2( ñ +
ï (þ + 3)2 Solution in parametric form:  =L

–2

+ 1)( − 1)2 ¡  2( + 3)2

(

1 U 4

+

2

1 XÈOL Q2

=–

where # = A

6.

Q

2(þ + 1) ï –2 ñ +
ï (þ + 3)2 Solution in parametric form:

ñóò ò ó

 =L

5.

exp XwOL

1

≠ −1 and

1

¡

 1Ë E Ï ±

=

+1

+L

2

+ P

2 −1

,

+ .

–1

ñ ú −

+L

2

1−

+ .

P

þ

,

+ +

+3 −1

≠ –3,

= L

,

1

þ

≠ –1.

U M + £ 1 AC U

U·O X

+1

+L

− 2

P

+ +

+1 −1

,

¡

= 2 ï –2 ñ +
ï –2 ñ –2 . Solution in parametric form:

ñ óò ò ó

V £ U (U + 1) − ln  £ U + £ U + 1  + L W −1 J 3 , 2 −2 J 3 V £ = ' U £ U (U + 1) − ln  £ U + U + 1  + L 2 W ,

 =L

1

where # = − 29  3 . 7.

= 2 ï –2 ñ +
ï ñ –2 . Solution in parametric form:

ñóò ò ó

V £ U (U + 1) − ln  £ U + £ U + 1  + L 2 W 1 J 3 , −1 J 3 =  L 1 U V £ U (U + 1) − ln  £ U + £ U + 1  + L 2 W ,

 =L

1

where # = − 29  3 . 8.

4 ï –2 = – 25 ñ +
ï –2 ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

 = L 1 (U 4 3J 2 where # = A 25  . 9.

3

− 3U + L

2)

4 ï –2 = – 25 ñ +
ï –1 ) 2 ñ –1 ) 2 . Solution in parametric form:

J ,

−5 3

=  (U

2

− 1)2 (U

U

2

3

− 3U + L

2)

J ,

−4 3

ñóò ò ó

 = L 1 (U 4 3J 2 . where # = A 25 

3

− 3U + L

2)

J ,

5 3

= L

1(

− 1)2 (U

3

− 3U + L

2)

J ,

1 3

Page 319

320 10.

SECOND-ORDER DIFFERENTIAL EQUATIONS 9 ï –2 = – 100 ñ +
ï –2 ñ –5 ) 3 . Solution in parametric form:

ñóò ò ó

 =L

1

V A (U

4

− 6U

J V A (U

3

− 3U + L

U

3

− 3U + L

A 1)3 J 2 (U

3

A 3U + L 2 )−3 J 4 ,

=  (U

,

2)

3 2

4

2

− 6U

+ 4L

2

U − 3)W

J

−9 8

,

9 ï –2 = – 100 ñ +
ï 2 ) 3 ñ –5 ) 3 . Solution in parametric form: 4

− 6U

2

+ 4L

2

U − 3)]5 J 4 ,

= L

1(

2)

J [A ( U

3 2

4

2

− 6U

+ 4L

2

U − 3)]1 J 8 ,

 J

9 8 3 . 100

= 34 ï –2 ñ +
ï –2 ñ –5 ) 3 . Solution in parametric form:

ñóò ò ó

3

 = L 1 (U

A 3U + L 2 )−1 J 2 ,

2

=  (U

= 34 ï –2 ñ +
ï 2 ) 3 ñ –5 ) 3 . Solution in parametric form:

# =A

where

 J

4 8 3 . 3

ñóò ò ó

 = L 1 (U

14.

J

−5 4

 J

where # = A

13.

U − 3)W

2

ñ óò ò ó

 = L 1 [A ( U

12.

+ 4L

9 8 3 . 100

where # = A 11.

2

3

A 3 U + L 2 )1 J 2 ,

= L

= – 29 ï –2 ñ +
ï –2 ñ –1 ) 2 . Solution in parametric form:

1(

A 1)3 J 2 (U

2

U

A 3U + L 2 )−1 J 4 ,

3

where

# =A

 J

4 8 3 . 3

ñóò ò ó

−3  = L 1 ( L 1  2  * + L 2  −  * sin   ) ,   = £ 3 ,!U , −2 2 = , 2 ( L 1  2  * + L 2  −  * sin   ) V 2 L 1  2  * + L 2  −  *  £ 3 cos   − sin    W ,

where # = 15.

16 9

, 3 .

= – 29 ï –2 ñ +
ï –1 ) 2 ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

3  = L 1 ( L 1  2  * + L 2  −  * sin   ) ,   = £ 3 ,!U , 2 = , 2 L 1 ( L 1  2  * + L 2  −  * sin   ) V 2 L 1  2  * + L 2  −  *  £ 3 cos   − sin    W ,

where # = 16.

16 9

, 3 .

=
1 ï 2 ñ –5 +
2 ñ –5 . Solution in parametric form:

ñóò ò ó

 = O =# 17.

#

#

2 1

J

1 2

P

tan X OL

Q

J Uh¦ cos X OL Q

1 2 2

1

1

−

− 2#

1

2# 1 # 1 1

#

2

2

U

U

−4

−4

2

−U

−U

2

J

P

−1 2

J

M

−1 2

P

M

U + L 2S ,

U + L 2" S §

−1

.

ñ óò ò ó

=
1 ñ –3 +
2 ñ –4 . 1 b . Solution in parametric form:

 =  X (L Q

1

+U

−3

ACU

J

−2 −1 2

)

M

U + L 2S ,

= 'U ,

where #

1

= T/

−2 4

 , #

2

= − 32 

−2 5

 .

Page 320

2.4. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

=

 1Ë E Ï ± 1

1

+

 2Ë E Ï ± 2

321

2

2 b . Solution in parametric form:

 =  X (L Q

18.

1

−3

−U

AIU

J

−2 −1 2

)

U + L 2S ,

M

= 'U ,

=
1 ñ –3 +
2 ï ñ –4 . 1 b . Solution in parametric form:

ñóò ò ó

−1

 =  X ( L 1 + U −3 AIU −2 )−1 J 2 U + L 2 S Q M 3 −3 5 −2 4 where # 1 = T/  , # 2 = − 2   . 2 b . Solution in parametric form:  =  X ( L 1 − U −3 AIU −2 )−1 J 2 U + L 2 S Q M 3 −3 5 −2 4 where # 1 = T/  , # 2 = 2   . 19.

= 'U

,

= 'U

=
1ñ ú +
2, þ ≠ –1. 1 b . Solution in parametric form:

Q

Q

1

= 12 



−2 1−

 =  X (L Q

1

+ ( + 1), # ¡

+U +

2

+1

ACU )

1 −2 2

=A



2 b . Solution in parametric form:

J

−1 2

M

=
1 ï –ú –3 ñ ú +
2 ï –3 , 1 b . Solution in parametric form:

ñóò ò ó

ñ óò ò ó

=




1

+


2ñ

–1

þ

. 1

.

= T 

 , #

+U

−3

AIU

−2 −1 2

X (L

1

−U

−3

AIU

−2 −1 2

) J

2

) J

= 32 

U + L 2S M

U + L 2S M

−2 5

 .

−1

−1

,

,

= 'U ,

+ 2#

+ 2#

1

2

ln | |)−1 J

2

=
1 ï –3 +
2 ï –2 ñ –1 . Solution in parametric form: 2 ln |

U |)−1 J

=
1 ï –5 ) 2 ñ –1 ) 2 +
2 ï –7 ) Solution in parametric form:

ñóò ò ó

 = where 

=L

1



1

,

=

, 

2

2

M

U +L 2S

ñ )

2 –1 2

= 'U ,

≠ –1.

ñóò ò ó

 = X ( L 1 +2 # 1 U +2 # Q

23.

1

−1 J 2  =  X ( L 1 − U + +1 ACU ) U + L 2S , Q M where # 1 = − 21  −2  1− + ( + 1), # 2 = A 12  −2  . ¡ 3 b . For the case = −1, see equation 2.4.2.21. ¡

Solution:  = X ( L 22.

X (L

U + L 2S ,

−1 −1 J 2  =  X ( L 1 + U + +1 AIU ) = 'U U + L 2S , Q M where # 1 = 12  1+ +  1− + ( + 1), # 2 = A 12  . ¡ 2 b . Solution in parametric form: −1 −1 J 2  =  X ( L 1 − U + +1 AIU ) = ' U U + L 2S , Q M where # 1 = − 21  1+ +  1− + ( + 1), # 2 = A 12  . ¡

21.

1

ñóò ò ó

where #

20.

−1

,

−2 4

where #

−1

+L

M

=U

,

Q

Q

J

X (L

1

+U +

+1

AIU )

−1 2

X (L

1

−U +

+1

AIU )

−1 2

U + L 2S

J

M

M

U + L 2S

−1

−1

,

,

2.

Q

X ( L 1 +2 # 1 U +2 #

2

ln |U |)−1 J

2

M

U + L 2S

−1

.

.

, 2L 1  2 *

 2  * + L 2  −  * sin( ÑU ) − #

+L 1

2

2  −  * ¤V £ 3 cos( ÑU ) − sin( ÑU )W. ,

# , # 2

2

= 16 , 3 ,   = , £ 3.

Page 321

322 24.

SECOND-ORDER DIFFERENTIAL EQUATIONS =
1 ï –4 ) 3 ñ –5 ) 3 +
2 ï –7 ) Solution in parametric form:

ñóò ò ó

 = ( 361 #

2

U 4 + L 1 U 3 + L 2 U + L 3 )−1 ,

where the constants L 25.

1,

=
1 ï –8 ) 5 ñ –7 ) 5 +
2 ï –13 ) Solution in parametric form: 4 1

where

8

=L

1

3

5 –7 5

#

 − #

1 2

2

U 3 +3 L 1 U 2 + L 2 )3 J 2 ( 361 # 2 U 4 + L 1 U 3 + L 2 U + L 3 )−1 ,

are related by the constraint 9 L

ñ )

ñ óò ò ó

.

= ( 91 #

L 2 , and L

 = OL

26.

ñ )

3 –5 3

1

L

=#

2

1

+#

2

L 3.

.

−1

P

= L

,

5 1

8 5J 2 4 O L 1 

8 8 8  2  * + L 2  −  * sin( £ 3 ,!U ),  = (
)2 − 2

, # *

3 ï –2 = – 16 ñ +
ï –2 ñ –5 ) 3 . 1 b . Solution in parametric form with # < 0:

*+*

− 2

# #

1 2

=−

−1

P

,

5  12 J 5 . 1024  3 , 6

ñóò ò ó

 = L 1 [cosh(U + L 2 ) cos U ]−2 [tanh(U + L 2 ) + tan U ]−2 , where # = − 3  8 J 3 .

=  [tanh(U + L

2)

+ tan U ]−3 J 2 ,

64

2 b . Solution in parametric form with # > 0:

 = L 1 [sinh U + cos(U + L 2 )]−2 , where # = 3  8 J 3 .

=  [cosh U − sin(U + L

2 )]

J [sinh U + cos(U + L 2 )]−3 J 2 ,

3 2

16

27.

3 ï –2 = – 16 ñ +
ï 2 ) 3 ñ –5 ) 3 . 1 b . Solution in parametric form with # < 0:

ñ óò ò ó

 = L 1 [cosh(U + L 2 ) cos U ]2 [tanh(U + L 2 ) + tan U ]2 , = L 1 [cosh(U + L 2 ) cos U ]2 [tanh(U + L 2 ) + tan U ]1 J 2 , 3 8J 3 where # = − 64  .

2 b . Solution in parametric form with # > 0:

 = L 1 [sinh U + cos(U + L 2 )]2 , where # = 3  8 J 3 .

= L

U − sin(U + L 2 )]3 J 2 [sinh U + cos(U + L 2 )]1 J 2 ,

1 [cosh

16

28.

=
1 ï –1 ñ –2 +
2 ï –2 ñ –2 . Solution in parametric form:

ñóò ò ó

J V (U :
+ 1 : )2 AIU 2 : 2 W − # 3

−2 3

 = ¦ L 1 U = L where

:

1

: U 2J 3 L

2

*

¦"L 1 U

#

1 2

§

−1

,

J V (U :
+ 1 : )2 AIU 2 : 2 W − # 3

−2 3

ø J U ) + L 2n ®1 1 J 3 (U ) + L 2 ;

*

#

1 2

§

−1

,

U ) for the upper sign, ( 1 J 3 U ) for the lower sign, ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; 1 J 3 (U ) are the modified Bessel functions; # 2 = − 92  −3  3 . =

™

L

1 1 3(

1 3(

J

Page 322

Ï Ò˜Í Í Ò

2.4. EQUATIONS OF THE FORM

]

=

 1Ë E Ï ± 1

1

+

 2Ë E Ï ± 2

323

2

In the solutions of equations 29 and 30, the following notation is used:

8

 2  * + L 2  −  * sin  £ 3 ,!U  , 2 * + , L 2  −  * V £ 3 cos  £ 3 ,!U  − sin  £ 3 ,!U  W , 2 = 2, L 1  8 8 2 8 8
= −2 ( ) . =L

1

8

1

3

ñóò ò ó

5 – 36

+
ï –2 ñ –7 ) 5 .

ï –2 ñ

8

J

−3 2 , 1 3

 =L

8 5J 2 8

=

J

where

5 # = − 2304 

J ,

.

8 J 8 J

where

5 # = − 2304  12 J 5 ,

−6

−5 4 , 3

5 ï –2 = – 36 ñ +
ï 2 ) 5 ñ –7 ) 5 . Solution in parametric form:

1

12 5 −6

ñ óò ò ó

30.

 =L

8 J

3 2 1 3 ,

5 2 1 4 1 1 3 ,

= L

.

In the solutions of equations 31–39, the following notation is used:

² = where 31.

*

2

= Solution in parametric form:

29.

]

1

2

ü

A (4U

3

®

− 1),

= XeU ²

−1

M

U ,



= 2UÈ® + L

1

2

U2Tç² ,



2

=U

−1

( ²u

1

− 1),

® = ® (U ) is the incomplete elliptic integral of the second kind in the form of Weierstrass. ñóò ò ó = 6ï –2 ñ +
ï –2 ñ –4 . Solution in parametric form:

 = L 1U

32.

= 6 ï –2 ñ +
ï 3 ñ –4 . Solution in parametric form:

−2 5 , 1

where

# = T 150  5 .

= L

1

J 

−2 5

U

J

−3 5 , 1

where

# = T 150  5.

J

1 3 1 ,

= 20 ï –2 ñ +
ï –1 ) 2 ñ –1 ) 2 . Solution in parametric form:

J 

−4 3 2 2, 1

= ˜

# = A 108  3 J 2 .

where

ñ óò ò ó

J

−1 3 , 1

ï –2 ñ +
ï –2 ñ –7 . = 15 4 Solution in parametric form:

= L

J 

−5 3 2 1 1 2,



# = A 108  3 J 2 .

where

ñóò ò ó

 = L 1 (4U 

36.

J

−1 5 , 1

= 20 ï –2 ñ +
ï –2 ñ –1 ) 2 . Solution in parametric form:

 = L 1

35.

J

J 

−3 5

= 'U

ñóò ò ó

 = L 1

34.

1 5 1 ,

ñóò ò ó

 = L 1 U 1J 5 

33.

J

J 

−1 5

2 1

Tç

J

2 1 4 , 2)

ï –2 ñ +
ï 6 ñ –7 . = 15 4 Solution in parametric form:

= ˜

J

1 2 1 (4

U 

2 1

Tç

J

2 −3 8 , 2)

where

# =A

3 8 4 .



ñ óò ò ó

 = L 1 (4U 

2 1

Tç

J

2 −1 4 , 2)

= L

J

1 2 1 1 (4



U 

2 1

Tç

J

2 −5 8 , 2)

where

# =A

3 8 4 .



Page 323

324

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï = – 12 49

ñóò ò ó

37.

ñ

–2

+


ï

ñ )

–2 1 2

.

Solution in parametric form:

 = L 1 ( ® + L 2 )−7 ,

ï = – 12 49

ñ óò ò ó

38.

–2

ñ

+


ï

) ñ )

–3 2 1 2

= 'U 2 ( ® + L

−4

 J

,

where

# =A

12 1 2 . 49

1

U 2 ( ® + L 2 )3 ,

where

# =A

12 1 2 . 49

,

= L

2)

.

Solution in parametric form:

ñóò ò ó

39.

=


1ï 4ñ

 = L 1 ( ® + L 2 )7 , –7

+


2ï 3ñ

–7

= L

 J

.

Solution in parametric form: 8 1 (4

 =  L Q where #

]

2 1

Tç

2 2)

−

#

1

#

2

S

−1

J

3 1 2 1 1



Q

L 18 (4U 

2 1

Tç

2 2)

−

#

#

1 2

S

−1

,

3 −3 8 . 64

=A





In the solutions of equations 40–43, the following notation is used:

² ² 40.

2

U 

1

=

2

=

ñóò ò ó

L ü

ü L

=

1

+U

−3

AIU

−2 ,

1

−U

−3

AIU

−2 ,


1ï

)

–4 3

+

« «


2ï

=X

1

= XM

2

²

) ñ

U M

²

U

)

–4 3 –1 2

1 2

+L

2,



1

=U −²

1

« 1, 5

1

= 3U 3 

2 1

+ 3(1 ACU ) «

+L

2,



2

=U −²

2

« 2, 5

2

= 3U 3 

2 2

+ 3(−1 AIU )«

2 1, 2 2.

.

Solutions in parametric form:

« 3 , = ˜  2 , where # 1 = T 29  −2 J 3  , # 2 = 13  −2 J 3  3 J 2 (−1)  ; , = 1 and , = 2.  = U

41.

ñóò ò ó

=


1ï

)

–5 3

+


2ï

) ñ

)

–7 6 –1 2

−3

.

Solutions in parametric form:

 = U 3 « −3 , where # 1 = A 29  −1 J 3  , # 2 = 13  −5 J 6  3 J 2 (−1)  42.

ñ óò ò ó

=


1ñ

–3

+


2ï ñ

–7

= 'U 3 « −3   2 , +1

; , = 1 and , = 2.

.

Solutions in parametric form:

 = U where # 43.

ñóò ò ó

=

1


1ñ

=T –3

1 −2 4 , 36

#

2

+

–7

.






2ï 3ñ

1 = − 36 

−3

5 ,

= 'U

J « 1J 2, 

−1 2

 ; , = 1 and , = 2.

−3 8

Solutions in parametric form:

 = U 3 5  −1 , where #

1

=A

1 −2 4 , 36





#

2

1 = − 36 

1J 2 = 'U 5 J 2 «  5  −1 ,

 ; , = 1 and , = 2.

−5 8

Page 324

Ï Ò˜Í Í Ò

2.4. EQUATIONS OF THE FORM

]

 1Ë E Ï ±

=

1

1

 2Ë E Ï ±

+

2

2

325

In the solutions of equations 44 and 45, the following notation is used:







 1=

 2= where , = Å



L





1

sin( ,!U ) + L

2

#

U

1

# , (L 1   * − L 2  − * ) − 1

cos( ,!U ) − L

2

#

U

1

2

#

, [L

#

cos( ,!U ) −

2

1

#

sin( ,!U )] −

2

2

#

1

if #

1

> 0,

if #

1

< 0,

if #

1

> 0,

if #

1

< 0,

# 1 |.

=
1 +
2 ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

44.

=  22 .

 =  1,

=
1 ï –3 +
2 ï –5 ) 2 ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

45.

]

1 2|

# L 1   * + L 2  − * −



−1 1 ,

 =

−1 2 1 2.

=



In the solutions of equations 46 and 47, the following notation is used:

For #

1

> 0,

{

  * + L 2  −  * + L 3 sin( ,!U ), , = ( 34 # 1 )1 J 4 , { = , ( L   * − L  −  * ) + , L cos( ,!U ). 2 1 2 3 À À For # 1 < 0, { =  * À [ L sin(°˜U ) + L cos(°˜U )] + L  − * sin(°˜U ), ° =À (− 1 # )1 J 4 , 1 1 2 3 3 1 { = °  * [( L − L ) sin(°˜U ) + (L + L ) cos(°˜U )] − °L  − * [sin( °˜U ) − cos( °˜U )]. 2 1 2 1 2 3 46.

1

=L

1

=
1 ï 2 ñ –5 ) 3 +
2 ï ñ –5 ) 3 . Solution in parametric form:

ñóò ò ó

 = where the constants L

1,

L 2 , and L 4L

1

L 1L 47.

2 3

+L

2

1 16

=

3

−

1

#

2#

2 1

=

,

{

J

3 2 2 ,

are related by the constraint

3

L

{

=
1 ï 2 ñ –5 ) 3 +
2 ñ –5 ) 3 . Solution in parametric form:

= 14 #

#

−2 1

#

−2 1

2 2

#

2 2

if #

1

> 0,

if #

1

< 0.

ñóò ò ó

 = where the constants L

1,

L 2 , and L 4L

L 1L

1

L

3 2

3

=

1,

{

J

3 2 2 ,

are related by the constraint

+L

=

{

− 14

2 3

#

= − 21 # −1 1

#

2

−1 1

#

2

if #

1

> 0,

if #

1

< 0.

Page 325

326

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 48 and 49, the following notation is used:

For #

For #

48.

2

2

> 0,

{

< 0,

{

{ À

  * + L 2  −  * + L 3 sin( ,!U ), , = ( 34 # 2 )1 J 4 ,  − * ) + , À L 3 cos( ,!U ). 2 = , (L 1  * − L 2  1

=L

1

À

À

* [ L 1 sin( °˜U ) + L 2 cos(°˜U )] + L 3  − * sin( °˜U ), ° = (− 31 # 2 )1 J 4 , { = °  * [( L − L ) sin(°˜U ) + (L + L ) cos(°˜U )] − °L  − * [sin( °˜U ) − cos( °˜U 2 1 2 1 2 3 1

=

=
1 ï –7 ) 3 ñ –5 ) 3 +
2 ï –10 ) Solution in parametric form:

ñóò ò ó

 = O where the constants L

49.

1

−

.

#

−1

2#

1

ñ )

3 –5 3

J O

3 2 2

{

1

−

#

1

{

−2 2

if #

2

> 0,

if #

2

< 0.

−1

,

.

{

−1 1 ,

{

J

3 2

= 1−1 2 , are related by the constraint

 =

1,

{

=

2# 2 P are related by the constraint 2

P

,

L 2 , and L 3 4 L 1 L 2 + L 32 = 14 # 21 # L 1 L 3 = 161 # 21 # −2 2

=
1 ï –4 ) 3 ñ –5 ) 3 +
2 ï –10 ) Solution in parametric form:

ñóò ò ó

where the constants L

]

1,

{

ñ )

3 –5 3

)].

L 2 , and L 3 4 L 1 L 2 + L 32 = − 21 # L 1 L 3 = − 14 # 1 # −1 2

1

#

−1 2

if #

2

> 0,

if #

2

< 0.

In the solutions of equations 50–53, the following notation is used:

U  1 + L 2U  2 + L 3U  3 , À ² 2 = ( L 1 + L 2 U )   * + L 3  Ÿ* , ² 3 = L 1   * +  * ( L 2 sin  ÑU + L 3 cos  ÑU ), Æ 1 = L 1 , 1U  1 + L 2 , 2U  2 + L 3 , 3U  3 , À Æ 2 = ( , L 1 + L 2 + , L 2 U )   * +  ·L 3  Ÿ* , Æ 3 = , L 1   * +  * [( °L 2 −  ·L 3 ) sin  ÑU + (°L 3 +  ·L 2 ) cos  ÑU ], 8 8 8


1 = U (Æ 1) , 2 = (Æ 2) , 3 = (Æ 3) , * * * where , 1 , , 2 , and , 3 (real numbers) or , and °"A*™  (one real and two complex numbers) are roots 8 of the cubic equation Œ 3 − 12 $ 2 Œ − 12 $ 1 = 0. The subscripts of the functions ² + , Æ + , and + ( = 1, 2, 3) are selected depending on the sign of the expression ¢ = 2 $ 23 − 27 $ 12 : ¡ ¢ > 0 subscript = 1, ¡ ¢ = 0 subscript = 2, ¡ ¢ < 0 subscript = 3. ¡ 3 2 If 2 $ 2 = 27 $ 1 (subscript 2), then , = ( 16 $ 2 )1 J 2 ,   = −2( 61 $ 2 )1 J 2 if $ 1 < 0, , = −( 61 $ 2 )1 J 2 ,   = 2( 61 $ 2 )1 J 2 if $ 1 > 0. x3y'z|{}~/€ The expressions for ² + , Æ + contain three constants L 1 , L 2 , and L 3 . One of them can be arbitrarily fixed to set it equal to any nonzero number (for example, we can set L 3 = A 1), and ²

1

=L

1

the other constants can be arbitrary.

Page 326

Ï Ò˜Í Í Ò

2.4. EQUATIONS OF THE FORM

50.

=
1 +
2 ï ñ –1 ) 2 . Solution in parametric form:

52.

 1Ë E Ï ± 1

1

+

=
1 ï –3 +
2 ï –7 ) 2 ñ –1 ) 2 . Solution in parametric form:

= Æ 2+ ,

where

#

327

2

=$

#

1

#

2,

2

=$

#

2

1.

ñóò ò ó

)

 = ² −1 + ,

+
2 ï ñ –7 ) 5 .

ñ óò ò ó =
1 ñ Solution in parametric form: –3 5

=
1 ï –12 ) 5 ñ –3 ) 5 +
2 ï Solution in parametric form:

ñóò ò ó

 =  (2 Æ 2+ − 4 ² + 5 2J 5 8J 5   $ where # 1 = 32 ]

2

1

2 = ² −1 + Æ + ,

8

where

 =  (2 Æ 2+ − 4 ² + + + $ 2 ² 2+ ), 5 −3 12 J 5 −2 where # 1 = −  −4 J 5 # 2 $ 2 , # 2 = − 32   $ 1 . 53.

 2Ë E Ï ±

ñóò ò ó

 =² + ,

51.

=

) ñ

)

–13 5 –7 5

=$

2,

=$

1.

= ˜² 5+ J 2 ,

.

8

8 −1 = ˜² 5+ J 2 (2 Æ 2+ − 4 ² + + + $ + + $ 2 ² 2+ ) , 5 3 J 5 12 J 5 −2 −2  $ 1 . 1 $ 2 , # 2 = − 32 

2

−1

² 2+ ) ,

In the solutions of equations 54 and 55, the following notation is used:

1 b . For #

{

> 0, #

≠ 0:

{ = , ( L   * − L  −  * ) +  ·L cos  ÑU , L 1   * + L 2  −  * + L 3 sin  ÑU , 2 1 2 3 2 2 1J 2 1J 2 2 1J 2 1J 2 where , = # 1 + 3 # 2 ) + # 1 ]} ,   = { 3 [( # 1 + 3 # 2 ) − # 1 ]} ; the constants L 1 , L 2 , and L 3 are related by the constraint 4 , 2 L 1 L 2 +   2 L 32 = 0. 2 b . For − # 21 < 3 # 2 < 0, # 1 > 0: { = L U  1 + L U −  1 + L U  2 + L U −  2 , { = , ( L U  1 − L U −  1 ) + , ( L U  2 − L U −  2 ), 1 1 2 3 4 2 1 1 2 2 3 4 where , 1 = { 23 [ # 1 + ( # 21 + 3 # 2 )1 J 2 ]}1 J 2 , , 2 = { 32 [ # 1 − ( # 21 + 3 # 2 )1 J 2 ]}1 J 2 ; the constants L 1 , L 2 , and L 3 are related by the constraint ( L 1 L 2 + L 3 L 4 )( # 21 + 3 # 2 )1 J 2 + ( L 1 L 2 − L 3 L 4 ) # 1 = 0. 3 b . For − # 21 < 3 # 2 < 0, # 1 < 0: { = L sin   U + L cos   U + L sin   U , { =   ( L cos   U − L sin   U ) +   L cos   U , 1 1 1 2 1 3 2 2 1 1 1 2 1 2 3 2 2 2 2 1J 2 1J 2 2 1J 2 1J 2 where   1 = {− 3 [ # 1 + ( # 1 + 3 # 2 ) ]} ,   2 = {− 3 [ # 1 − ( # 1 + 3 # 2 ) ]} ; the constants L 1 , L 2 , and L 3 are related by the constraint   12 ( L 12 + L 22 ) −   22 L 32 = 0. 4 b . For # 21 + 3 # 2 = 0, # 1 > 0: { = ( L + L U )  * + ( L + L U ) −  * , { = ( , L + L + , L U )  * − ( , L − L + , L U ) −  * , 1 1 2 3 4 2 1 2 2 3 4 4 where , = ( 32 # 1 )1 J 2 ; the constants L 1 , L 2 , and L 3 are related by the constraint L 2 L 4 + ( L 1 L 4 − L 2 L 3 )( 61 # 1 )1 J 2 = 0. 5 b . For # 21 + 3 # 2 = 0, # 1 < 0: { = ( L + L U ) sin  ÑU + L U cos  ÑU , { = ( ·L + L +  ·L U ) cos  ÑU + ( L −  ·L U ) sin  ÑU , 1 1 2 3 2 1 3 2 2 3 2 1J 2 where   = (− 3 # 1 ) ; the constants L 1 , L 2 , and L 3 are related by the constraint L 22 + L 32 + L 1 L 3 (− 32 # 1 )1 J 2 = 0. 6 b . For 3 # 2 < − # 21 : { =   * ( L sin  ÑU + L cos  ÑU ) + L  −  * sin  ÑU , 1 1 2 3 { =   * [( , L +  ·L ) cos  ÑU + ( , L −  ·L ) sin  ÑU )] + L  −  * (  cos  ÑU − , sin  ÑU ), 2 2 1 1 2 3 1 1 1J 2 1J 2 1J 2 1J 2 where , = { 3 [ # 1 + (−3 # 2 ) ]} ,   = { 3 [− # 1 + (−3 # 2 ) ]} ; the constants L 1 , L 2 , and L 3 are related by the constraint L 2 # 1 + L 1 (− # 2 − 3 # 2 )1 J 2 = 0. 2

1

1 = { 32 [(

1

Page 327

328 54.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=


1ñ

)

–1 3

+


2 ï 2ñ

)

–5 3

.

Solution in parametric form:

{

 = 55.

ñóò ò ó

=


1ï

) ñ

)

–8 3 –1 3

+


2ï

) ñ

)

–10 3 –5 3

Solution in parametric form:

{

]

{ ñóò ò ó

1

=

™

2

=

™

=


1ñ

L 1 Ÿ* + L 2  −Ÿ* + L 3 U L 1 sin  ÑU + L 2 cos  ÑU + L 3 U   ( L 1 Ÿ * − L 2  −Ÿ* ) + L 3   ( L 1 cos  ÑU − L 2 sin  ÑU ) + L

)

–1 3

+


2ñ

)

–5 3

if # if #

J

where the constants L

ñ óò ò ó

=


1ï

L 2 , and L

1,

3( # 3( #

) ñ

)

–8 3 –1 3

+

L

1

L

2 3 2 3

1


2ï

3

2)

) ñ

)

–4 3 –5 3

L 2 , and L

3( # 3( #

ñóò ò ó

=


1ñ

)

–1 3

+


2ï ñ

1

L

1

L

2 3 2 3

)

–5 3

{

1

=

1,

+ 16 # 21 L 2 2) + 4# 1(L

+# +#

 = 1,

1

where

  = | 43 # 1 |1 J 2 ,

> 0, < 0,

where

  = | 43 # 1 |1 J 2 .

3

J

3 2 2 ,

L

1 2 =0 2 2 1 + 2) =

L

if # 0 if #

1 1

> 0, < 0.

.

{

−1 1 ,

{

=

{

J

−1 3 2 1 2 ,

are related by the constraint

+ 16 # 21 L 2 2) + 4# 1(L

+# +#

{

are related by the constraint

Solution in parametric form: where the constants L

1

> 0, < 0,

.

 =

58.

{

−1 3 2 1 2 .

1

if # if #

3

Solution in parametric form:

57.

{

=

In the solutions of equations 56–59, the following notation is used:

{

56.

J

3 2 2 .

. −1 1 ,

 =

{

=

1,

L

1 2 =0 2 2 1 + 2) =

2)

L

if # 0 if #

1 1

> 0, < 0.

.

Solution in parametric form:

 = where the constants L 3# 59.

ñóò ò ó

=


1ï

) ñ

)

–8 3 –1 3

+

2 3 2 3

L

1 1

−

1

L 2 , and L

1,

3#

{

L


2ï

2

3

1

L 1L

2 1(

2 1

L

) ñ

U 2,

{

= O

−

2

#

2#

2 1

J

3 2

U P

,

are related by the constraint

2 1

+ 16 # + 4#

#

4#

)

2

+L

–7 3 –5 3

+

9 16

2 2)

#

+

−2 2 1 2 =0 −2 2 9 2 = 16 1

#

#

#

if #

1

> 0,

0 if #

1

< 0.

.

Solution in parametric form:

 = O

{

1

−

#

4#

2 1

U

2

P

−1

,

= O

{

1

−

#

4#

2 1

U

2

P

−1

O

{

2

−

#

2#

2 1

U

J

P

3 2

,

Page 328

Ï Ò˜Í Í Ò

2.4. EQUATIONS OF THE FORM

where the constants L

]

L 2 , and L

1,

3#

1

L

3#

1

L

2 3 2 3

+ 16 # + 4#

 1Ë E Ï ±

=

1

1

 2Ë E Ï ±

+

2

329

2

are related by the constraint

3 2 1

L 1L

2

2 1(

2 1

+L

L

9 16

+

2 2)

#

+

−2 2 1 2 =0 −2 2 9 2 = 16 1

#

#

#

if #

1

> 0,

0 if #

1

< 0.

In the solutions of equations 60–67, the following notation is used:

 =

A (4 < ü

U =X

− 1),

3

ü

M A (4 <

< 3

−L

− 1)

2.

The function < = < (U ) is defined implicitly in terms of the above elliptic integral of the first kind. For the upper sign, < coincides with the classical elliptic Weierstrass function < = < (U + L 2 , 0, 1). In the solution given below, one can take < as the parameter instead of U and use the explicit dependence U = U ( < ). 60.

6 ï –2 =
ï –2 ñ 2 – 25 ñ . Solution in parametric form:

ñóò ò ó

 = L 1U 5,

61.

6 ï –2 =
ï –3 ñ 2 – 25 ñ . Solution in parametric form:

−5

6 ï –2 =
ï –2 ñ 2 + 25 ñ . Solution in parametric form:

6 −1 . 25



# =A

6 −1 . 25

=  (U 2
where

# =A

6 −1 . 25

U

−3

<

where

1



6 ï –2 =
ï –3 ñ 2 + 25 ñ . Solution in parametric form:



ñóò ò ó

−5

= L

,

= 12 ï –2 ñ +
ï –2 ñ –5 ) 2 . Solution in parametric form:

1

U

−5

(U 2
# =A

where

6 −1 . 25



ñ óò ò ó

−1 J 7 < 2J 7(u  A 2' U < 2) , ñóò ò ó = 12ï –2 ñ +
ï 3 ) 2 ñ –5 ) 2 .

 =L

65.

# =A

where

ñóò ò ó

 = L 1U

64.

,

,

= L

,

 = L 1 U 5,

63.

<

ñóò ò ó

 = L 1U

62.

2

= 'U

J ( pA 2U'< 2 )−4 J 7 ,

= +<

1

−6 7

where

# = T 147  7 J 2.

Solution in parametric form:

 =L

66.

1

<

J ( pA 2U'< 2 )1 J 7 ,

−2 7

ï –2 ñ +
ï –2 ñ –5 ) 3 . = 63 4 Solution in parametric form:

1

<

J ( uA 2U'< 2 )−3 J 7 ,

−8 7

where

# = T 147  7 J 2 .

where

# = − 323  8 J 3 .

ñóò ò ó

 = L 1 (U + 2 < )−1 J 4 ,

67.

= L

=  (U + 2 < )−9 J 8 ( uA 2U'< 2 )

J

3 2

ï –2 ñ +
ï 2 ) 3 ñ –5 ) 3 . = 63 4 Solution in parametric form:

,

ñóò ò ó

 = L 1 (U + 2 < )1 J 4 ,

= L

1(

U + 2 < )−7 J 8 ( pA 2U'< 2 )

J

3 2

,

where

# = − 323  8 J 3 .

Page 329

330

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 68–73, the following notation is used:

 1 = ü A 4<

3 1

 2 = ü A 4<

3 2

− 2<

1

+ 2<

2

−L

2,

−L

2,

U =ù ü

1

− 2<

M

<

−L

1

2

−L

1;

2

− L 1. + 2< 2 − L 2 ü The functions < 1 = < 1 (U ) and < 2 = < 2 (U ) are the inverses of the above elliptic integrals. For the upper signs, they are the classical Weierstrass functions < 1 = < (U + L 1 , 2, L 2 ) and < 2 = < (U + L 1 , −2, L 2 ). 68.

=
1 ñ 2 +
2. Solutions in parametric form:

A 4<

3 2

ñóò ò ó

where # 69.

U =ù

A 4<

<

M

3 1

= A 6

1



−2 −1

, #

2

=

−2

=
1 ï –5 ñ 2 +
2 ï –3 . Solutions in parametric form:

 = U , = +<  ,   (−1) ; , = 1 and , = 2.

ñóò ò ó

 = U −1 , = 'U −1 <  , where # 1 = A 6  3  −1 , # 2 =    (−1)  ; , = 1 and , = 2. 70.

=
1 ï –15 ) 7 ñ 2 +
2 ï –9 ) 7 . Solutions in parametric form:

ñóò ò ó

 = U 7 , = 'U (U 2 <  T 1), 6 1 J 7 −1 where # 1 = A 49   , # 2 = 491  −5 J 7  (−1)  ; , = 1 and , = 2. 71.

=
1 ï –20 ) 7 ñ 2 +
2 ï –12 ) 7 . Solutions in parametric form:

ñóò ò ó

 = U −7 , = 'U −6 (U 2 <  T 1), 6 6 J 7 −1 where # 1 = A 49   , # 2 = 491  −2 J 7  (−1)  ; , = 1 and , = 2. 72.

=
1 +
2 ñ –2 ) 3 . Solutions in parametric form:

ñóò ò ó

 =  [   − (−1)  U ], = +< 3 , 1 −2 5 J 3 where # 1 = A 12  −2  , # 2 = 12   (−1)  ; , = 1 and , = 2. 73.

=
1 ï –3 +
2 ï –7 ) 3 ñ –2 ) 3 . Solutions in parametric form:

ñóò ò ó

 =  [   − (−1)  U ]−1 , = +< 3 [   − (−1)  U ]−1 , where # 1 = A 12    , # 2 = 121  1 J 3  5 J 3 (−1)  ; , = 1 and , = 2. ]

In the solutions of equations 74 and 75, the following notation is used:

J

−1 2

« = X (1 ACU 4 ) The function « 74.

U + L 2,

,

2

= A 1.

can be expressed in terms of elliptic integrals or lemniscate functions.

=
1 ï –18 ) 5 ñ 3 +
2 ï –14 ) Solutions in parametric form:

ñóò ò ó

 = L

M

5 1

«

−5

,

= L

4 1

«

−4

ñ

5 2

.

(U « − , ),

where

#

1

=A

 J 

2 8 5 −2 , 25

#

2

=A

 J 

6 4 5 −1 25

, .

Page 330

2.4. EQUATIONS OF THE FORM

75.

ñóò ò ó

=


1ï

) ñ

–12 5 3


2ï

+

) ñ

–11 5 2

Ï Ò˜Í Í Ò

=

 1Ë E Ï ± 1

1

 2Ë E Ï ±

+

2

331

2

.

Solutions in parametric form: 5 1

 = L ]

= L

,

1

« (U « − , ),

#

where

 J 

2 2 5 −2 , 25

=A

1

#

2

 J 

6 1 5 −1 25

=A

, .

In the solutions of equations 76–81, the following notation is used:

5 76.

5

«

ñóò ò ó

=L

=

1

 =

(U ) ™ ®ø 1 J 3(U ) 1J 3

=

™

for the upper sign (Bessel function), for the lower sign (modified Bessel function), n 1 J 3 (U ) for the upper sign (Bessel function), ; 1 J 3 (U ) for the lower sign (modified Bessel function),

 + L 2 + ý/ eO˜ X 


1ï ñ




+

U − X M

U

M P

  =

,

1

for the upper sign, for the lower sign.

2 ™ −1

2.

Solutions in parametric form:

 = U 2 J 3 , 77.

ñóò ò ó

=


1ï

ñ

–5

+


2ï

–3

= U 1J 3 5 ,

#

9 −3 , 4

1

=T

#

1

=T

where #

1

= − 21 

where



#

2

= 94 

−2

ý .

.

Solutions in parametric form:

 = U 78.

ñóò ò ó

=


1ï

)

–3 2

+

J ,

−2 3


2ï

J 5 ,

−1 3

=U

) ñ

)

–1 2 –1 2

where

9 3 , 4



#

2

= 94   ý .

.

Solutions in parametric form:

 = U 2 J 3 5 79.

ñóò ò ó

=


1ï

2

= 'U

,

)

–3 2

+


2ï

J ( U 5
+ 1 5 )2 , 3

−2 3

*

ñ )

–2 –1 2

J 'ý , #

−1 2

2

J  J

1 −3 2 3 2 . 3

=T



.

Solutions in parametric form:

J 5

−2 3

 = U 80.

ñóò ò ó

=


1ñ

−2

)

–3 2

J 5

−4 3

= 'U

, +


2ï ñ

–2

−2

(U 5

*


+ 1 5 )2 , 3

where #

1

= − 21 

J 'ý , #

−1 2

2

=T

 J

1 3 2 . 3

.

Solutions in parametric form:

 = U where # 81.

ñóò ò ó

=

1


1ï

J V T"U 2 5

−2 3

J ý…# 2 , #

= − 

−1 2

) ñ

)

–3 2 –3 2

+


2ï

2

2

= 92 

ñ

–2 –2

2 − (U 5
+ 13 5 ) W ,

+ 2ý±U 5

= 'U 2 J 3 5

*

2

,

 .

−3 3

.

Solutions in parametric form:

T U 25  = U 2 J 3 V where #

1

2

= − 29 

2 +2ý±U 5 −(U 5
+ 13 5 ) W

J  J ý , #

−1 2 5 2

*

2

−1

,

= 'U 4 J 3 5

2

V T"U 2 5

2

2 +2ý±U 5 −(U 5
+ 13 5 ) W

*

−1

,

= 92  3 .

Page 331

332

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 82–88, the following notation is used:

˜

™

Á =

:

Á =ƒ Ä

=

™ ™

Ä

1

=

Ä

2

=Ä

for the upper sign (Bessel function), 1 ø Á (U ) L 1 ®Á (U ) for the lower sign (modified Bessel function), L 2 n Á (U ) for the upper sign (Bessel function), L 2 ; Á (U ) for the lower sign (modified Bessel function), :
: ˜ ˜
Á + @ Á , 4IÁ = Um Á + @!m|Á , 1 ’ÑÁ + ƒ 2 Á , m|Á = ý 1 ’ÑÁ + ý 2 Á , sÁ = U : Ám|Á if ¢ = −( ƒ 1 ý 2 − ƒ 2 ý 1 )2 , ˜ ˜ 2 2 ƒt’ Á + ýs’ÑÁ Á + = Á if ¢ = 4 ƒ±= − ý 2 , : Á4IÁ + m|ÁsÁ if ¢ = −( ƒ 1 ý 2 − ƒ 2 ý 1 )2 , U Ä
+ 2 @Ä if ¢ = 4 ƒ±= − ý 2 , 2

2 2 2 +   2¢ ,   =™  for the upper sign, 1 A 4U Ä

L

™

’ÑÁ =

−1

for the lower sign.

The prime denotes differentiation with respect to U . 82.

ñ óò ò ó


1ï ñ

=


2ñ

+

–3

.

Solutions in parametric form:

 = U 2 J 3 , where @ = 13 , # 83.

ñóò ò ó


1 ƒï ÷ ñ

=

1

9 −3 , 4

=T




2ñ

+

–3

#

2

J ,

1 2

9 −2 4 16



=

ù

,

= 'U 1 J 3 Ä

   2¢ .

≠ –2.

Solutions in parametric form:

 = U 2 Á , where @ = 84.

ñóò ò ó


1ï

=

1

B +2 –5

ñ

, #

+

1


2ñ

1 −  4@ 2

=T –3

−2

, #

=

2

Á Ä

= 'U 1  16 @ 2

J ,

1 2

−2 4

   2¢ .

.

Solutions in parametric form:

J ,

−2 3

 = U where @ = 13 , # 85.

ñóò ò ó


1ï

=

–3

+

1

=T


2ï

9 3 , 4



#

) ñ

=

2

)

–1 2 –1 2

J Ä

1 2

J Ä

−1

= 'U

−1 3

= 'U

−2 3

J ,

9 −2 4 16

   2¢ .



.

Solutions in parametric form:

 = U 2 J 3 Ä , where @ = 13 , # 86.

ñóò ò ó

=




1

+

1


2ï

= −2  

ñ )

–2 –1 2

2

¢ , #

2

=T

2 1,

Ä

J  J

8 −3 2 3 2 . 3



.

Solutions in parametric form:

 = U where @ = 13 , #

1

= −2 

−2

J Ä

−2 3

  2 ¢ , #

2

=T

−1

= 'U

,

J Ä

−4 3

−2

Ä

2 1,

 J

8 3 2 . 3

Page 332

2.4. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

 1Ë E Ï ±

=

1

=
1 ï ñ –2 +
2 ñ –3 . Solutions in parametric form:

1

+

 2Ë E Ï ± 2

ñóò ò ó

87.

 = U −2 J 3 Ä 9 −3 3   , # 2 = 649  where @ = 13 , # 1 = − 128

−1

=
1 ï –2 ñ –2 +
2 ñ –3 . Solutions in parametric form:

Ä

= 'U 2 J 3 Ä ,

2,

   2¢ .

−2 4

ñóò ò ó

88.

 = U 2 J 3 Ä­Ä 2−1 , = 'U 4 J 3 Ä 9 3 where @ = 13 , # 1 = − 128  , # 2 = 649  −2  4   2 ¢ . ]

333

2

2

−1 2 ,

Ä

In the solutions of equations 89 and 90, the following notation is used: 2 2

¢ =L

− 2L

² = (36 ¢ + 54$IU − 2U 3 )1 J 2 ,

1,






£ −¢ L 2 tan  A £ − ¢ÈH  + L 1 L 1 £ ¢ L tanh  T £ È ¢ H  + 2 L 1 L 1 £ 2 1 T − £ L 1H |L 1| £ 2 1 T + £ L 1H |L 1|




ú

(H ) =



















−1

H = 3 XÈU

−1

²

if ¢

< 0,

if ¢

> 0,

if ¢

= 0, L

2

< 0,

if ¢

= 0, L

2

> 0.

=
1 ñ –5 ) 3 +
2 ï –2 ) 3 ñ –5 ) 3 . Solutions in parametric form:

U , M

ñ óò ò ó

89.

J (L 1 ú where # 1 = 24  −2  8 J 3 L 1 , # 2 = −36  −4 J 3  8 J 3 $ .  = U

J (L 1 ú

−3 2

2

− 2L

2

ú

+ 2)3 J 2 ,

=
1 ï –2 ) 3 ñ –5 ) 3 +
2 ï –4 ) Solutions in parametric form:

ñóò ò ó

90.

= 'U

ñ )

3 –5 3

−9 4

2

− 2L

2

ú

ú

+ 2)3 J 4 (6 L

1

+ 2)−3 J 4 (6 L

1

− 6L

2

T*² )3 J 2 ,

− 6L

2

Tl² )3 J 2 ,

.

ú ú ú  = U 3 J 2 ( L 1 2 − 2 L 2 + 2)−3 J 2 , = 'U −3 J 4 ( L 1 where # 1 = −36  −4 J 3  8 J 3 $ , # 2 = 24  −2 J 3  8 J 3 L 1 .

2

− 2L

2

ú

ú

]

In the solutions of equations 91–102, the following notation is used: . . The functions 1 and 2 are the general solutions of the four modifications of the first Painlev e´ equation: .

= A 6. 2 + U , .

= A 6. 2 − U 1 1 2 2 . (in the case of the upper sign, the equation for 1 is the canonical form of the first Painlev´e equation, see Paragraph 2.8.2-2). In addition,

Æ

1

=A 6

Æ

2

=A 6

.

.

2 1 2 2

+U , −U ,

{

{

1

=U

2

2

=U

2

.

.

1

T 1,

’

1

2

T 1,

’

2

. . = ( 1
)2 − 2 . . = ( 2
)2 − 2

1

Æ

1

A 8

2

Æ

2

A 8

. .

3 1, 3 2,

˜

˜


Æ
+.
−Æ .
Æ 1
− . 1
− Æ = 2 2 2 2 1

=

.

1

2 1, 2 2.

The prime denotes differentiation with respect to U . 91.

=
1ñ 2 +
2ï . Solutions in parametric form:

ñ óò ò ó

.

 = U , =  , where # 1 = A 6  −2  −1 , # 2 =  −3  (−1)  +1; , = 1 and , = 2.

Page 333

334 92.

SECOND-ORDER DIFFERENTIAL EQUATIONS =
1 ï –5 ñ 2 +
2 ï –4 . Solutions in parametric form:

ñóò ò ó

.

 = U −1 , = 'U −1  , where # 1 = A 6  3  −1 , # 2 =  2  (−1)  +1 ; , = 1 and , = 2. 93.

=
1 ï –15 ) 7 ñ 2 +
2 ï –8 ) 7 . Solutions in parametric form:

ñóò ò ó

 = U 7 , 6 1 J 7 −1 where # 1 = A 49   , # 2 = 491  −6 J 7  (−1)  94.

+1

=
1 ï –20 ) 7 ñ 2 +
2 ï –13 ) 7 . Solutions in parametric form:

{

,

; , = 1 and , = 2.

ñ óò ò ó

 = U −7 , 6 6 J 7 −1   , # 2 = 491  −1 J 7  (−1)  where # 1 = A 49 95.

= 'U

=
1 ï +
2 ñ –1 ) 2 . Solutions in parametric form:

−6

= 'U +1

{

,

; , = 1 and , = 2.

ñóò ò ó

. .  =  , =  ( 
)2 , where # 1 = A 24  −3  , # 2 = 2  −2  3 J 2 (−1)  +1 ; , = 1 and , = 2. 96.

=
1 ï –4 +
2 ï –5 ) 2 ñ –1 ) 2 . Solutions in parametric form:

ñóò ò ó

.

.

.


 =   −1 , =   −1 (  )2 , where # 1 = A 24  2  , # 2 = 2  1 J 2  3 J 2 (−1)  +1 ; , = 1 and , = 2. 97.

=
1 ñ 1 ) 3 +
2 ï ñ –5 ) 3 . Solutions in parametric form:

ñ óò ò ó

.

3J 2  = ’  , =  , 3 −3 8 J 3 where # 1 = T 8  −2 # 2 , # 2 = − 16   ; , = 1 and , = 2.

98.

=
1 ï –10 ) 3 ñ 1 ) 3 +
2 ï –7 ) Solutions in parametric form:

ñóò ò ó

ñ )

3 –5 3

.

. 3J 2  =  ’  −1 , =   ’  −1 , 3 1J 3 8J 3 where # 1 = T 8  −2 # 2 , # 2 = − 16   ; , = 1 and , = 2. 99.

=
1 ï –3 ) 2 +
2 ñ –1 ) 2 . Solutions in parametric form:

ñóò ò ó

.

 =  ( 
)2 , = Æ 2 , where # 1 = 12  −1 J 2  (−1)  , # 2 = A 6  −2  3 J 2 ; , = 1 and , = 2. 100.

=
1 ï –3 ) 2 +
2 ï –5 ) 2 ñ –1 ) 2 . Solutions in parametric form:

ñ óò ò ó

. .  =  ( 
)−2 , =  ( 
)−2 Æ 2 , where # 1 = 12  −1 J 2  (−1)  , # 2 = A 6  1 J 2  3 J 2 ; , = 1 and , = 2.

Page 334

2.4. EQUATIONS OF THE FORM

101.

Ï Ò˜Í Í Ò

=
1 ñ –4 ) 3 +
2 ï ñ –5 ) 3 . Solutions in parametric form:

=

 1Ë E Ï ± 1

+

1

 2Ë E Ï ± 2

335

2

ñóò ò ó

˜ .
 =  , =  (  )3 , where # 1 =    −1 J 3 # 2 (−1)  , # 2 = 361  −3  8 J 3 ; , = 1 and , = 2.

102.

=
1 ï –5 ) 3 ñ –4 ) 3 +
2 ï –7 ) Solutions in parametric form:

ñóò ò ó

ñ )

3 –5 3

.

˜ .
˜  =   −1 , =  (  )3  −1 , 1 −1 J 3 7 J 3 where # 1 = 36   (−1)  , # 2 = 361  1 J 3  8 J 3 ; , = 1 and , = 2. ]

In the solutions of equations 103–108, the following notation is used: . . The functions 1 and 2 are the general solutions of the four modifications of the second Painlev e´ equation (with parameter  = 0):

.

.



= −U . A 2 . 3 , 2 2 where the primes denote differentiation with respect to U . In the case of the upper sign, the equation . for 1 is the canonical form of the second Painlev´e equation (with parameter  = 0, see Paragraph

= U

1

1

A 2

.

.

3 1,

2

2.8.2-3). 103.

=
1ñ 3 +
2ï ñ . Solutions in parametric form:

ñóò ò ó

.

 = U , =  , where # 1 = A 2  −2  −2 , # 2 =  3 (−1)  +1 ; , = 1 and , = 2. 104.

=
1 ï –6 ñ 3 +
2 ï –5 ñ . Solutions in parametric form:

ñóò ò ó

.

 = U −1 , = 'U −1  , where # 1 = A 2  4  −2 , # 2 =  3 (−1)  +1 ; , = 1 and , = 2. 105.

=
1 +
2 ï –1 ) 2 ñ –1 ) 2 . Solutions in parametric form:

ñ óò ò ó

. .
= (. )
, .  =   2, =  ( 
)2 ,  *  1 −3 J 2 3 J 2 −2  +1 where # 1 = A 2   , # 2 = 2   (−1) ; , = 1 and , = 2. 106.

=
1 ï –3 +
2 ï –2 ñ –1 ) 2 . Solutions in parametric form:

ñóò ò ó

.

 =   −2 , where # 1 = A 2  , # 2 = 12  3 J 2 (−1)  107.

=
1 +
2 ï ñ –2 . Solutions in parametric form:

.
= (. )
, −2 .
2  *  ( ),  +1 ; , = 1 and , = 2. =

.

ñóò ò ó

. . . . .
= (. )
,  =  [U  2 A  4 − ( 
)2 ], =   2,  *  −2  −3 3  +1 where # 1 = T 2   (−1) , # 2 = 2   (−1) ; , = 1 and , = 2. 108.

=
1 ï –3 +
2 ï –2 ñ –2 . Solutions in parametric form:

ñóò ò ó

. −1 . . .  =  V U  2 A  4 − ( 
)2 W , =  2 V U where # 1 = A 2  , # 2 = 2  3 ; , = 1 and , = 2. .

2  A

.

.
2 4  −(  ) W

−1

,

Page 335

336

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.5. Generalized Emden–Fowler Equation í­ïwî²î ï = † ì ù í þ ( í­ï î )@ 2.5.1. Classification Table The case 0 = 0 corresponding to the classical Emden–Fowler equation is outlined in Section 2.3. In this section, the case 0 ≠ 0 is considered. Table 23 presents all solvable equations of the form "

 = #*  + (
 )- whose solutions are outlined in Subsection 2.5.2. Two-parameter families (in the space of the parameters B , , and 0 ), ¡ one-parameter families, and isolated points are presented in a consecutive fashion. Equations are arranged in accordance with the growth of 0 , the growth of (for identical 0 ), and the growth of B ¡ (for identical and 0 ). The number of the equation sought is indicated in the last column in this ¡ table. TABLE 23 Solvable cases of the generalized Emden–Fowler equation "

 = #*  + (
 )-

0

No

B ¡

Equation

Two-parameter families 1

arbitrary

arbitrary

0

2.5.2.1

2

arbitrary 2B + + 3 ¡ B + +2

0 arbitrary ( ≠ −1)

arbitrary arbitrary (B ≠ −1)

2.5.2.2

−1

−1

2.5.2.6

− 21

− 21

2.5.2.97

− 21

2.5.2.13

1

2.5.2.10

3

¡

4 5 6 7 8 9 10

¡

One-parameter families

arbitrary ( 0 ≠ 1, 2) arbitrary ( 0 ≠ 32 ) 3 ¡ 2 ¡ 3 ¡ 2 ¡ 3B 2B 3B 2B 3B 2B

2.5.2.3

arbitrary ( ≠ − 32 ) ¡ arbitrary ( ≠ − 32 )

+5 +3 +5 +3 +4 +3 +4 +3 +4 +3

¡

− 21 1 −B − 3

arbitrary ≠ −1, 0)

11

1

12

2

13

3

arbitrary ( ≠ −2)

14

3

−B − 3

(

¡

−1

¡

arbitrary (B ≠ − 23 ) arbitrary (B ≠ − 23 ) arbitrary (B ≠ − 23 )

2.5.2.11 2.5.2.12 2.5.2.107

−1

2.5.2.5

arbitrary (B ≠ −1, 0)

2.5.2.4

1

2.5.2.96

arbitrary

2.5.2.9

Page 336

2.5. GENERALIZED EMDEN–FOWLER EQUATION

Ï Ò˜Í Í Ò

=

 ËEϱ

(Ï

ÒÍ

)

Ë

337

TABLE 23 (Continued) Solvable cases of the generalized Emden–Fowler equation "

 = #*  + (
 )No

0 ¡

B

Equation

Isolated points − 12

− 25

2.5.2.33

1

2.5.2.83

1

− 15 8 − 20 13 − 45

1

0

2.5.2.78

− 12 − 52

− 67 − 21

2.5.2.53

21

1 2 1 2 1 2 1 2 1 2 2 3 4 5

22

1

−2

1

2.5.2.14

23

1

−1

−1

2.5.2.8

24

8 7 8 7 6 5 5 4 5 4 9 7 9 7 13 10 27 20 18 13 7 5 7 5 7 5 7 5 7 5 7 5 7 5 10 7 22 15 3 2 3 2 3 2 3 2

1

− 43

2.5.2.54

1

− 21

2.5.2.52

− 12

2.5.2.68

1

− 32 − 21

1

0

2.5.2.56

− 13 8

1

2.5.2.39

− 12

1

2.5.2.38

− 12 − 12 − 12 − 74 − 10 7 − 23 − 12

− 25 − 32 − 27

2.5.2.47

1

2.5.2.18

1

2.5.2.46

1

2.5.2.32

1

2.5.2.17

1

0

2.5.2.89

1

1

2.5.2.91

5

1

2.5.2.75

− 12 − 12

2.5.2.19

−2

− 25 − 32 − 21

−2

1

2.5.2.99

− 12 − 12

−2

2.5.2.100

− 21

2.5.2.29

15 16 17 18 19 20

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

1

2.5.2.86 2.5.2.80

2.5.2.76

2.5.2.58

2.5.2.72 2.5.2.40

2.5.2.70 2.5.2.106

Page 337

338

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 23 (Continued) Solvable cases of the generalized Emden–Fowler equation "

 = #*  + (
 )-

B

No

0

47

− 12

1

2.5.2.98

1

−2

2.5.2.105

1

2.5.2.103

− 52

− 21 − 21 − 21

0

1

2.5.2.73

1

2.5.2.26

1

− 47 − 10 7 − 32 − 21

1

1

2.5.2.74

1

5

2.5.2.94

− 72 − 23 − 52

2.5.2.45

1

− 21 − 21 − 21 − 13 8 − 21

− 12

1

2.5.2.51

0

1

2.5.2.50

− 23 − 34 − 12

− 21

2.5.2.81

1

2.5.2.65

68

3 2 3 2 3 2 23 15 11 7 8 5 8 5 8 5 8 5 8 5 8 5 8 5 21 13 33 20 17 10 12 7 12 7 7 4 7 4 9 5 13 7 13 7

1

2.5.2.61

69

2

−1

−1

2.5.2.7

70

2

1

−2

2.5.2.16

71

− 12 − 76 − 52 − 15 8 20 − 13 − 54

− 25 − 21 − 21

2.5.2.95

1

2.5.2.69

1

2.5.2.71

1

2.5.2.67

77

11 5 7 3 5 2 5 2 5 2 5 2 5 2

0

1

2.5.2.66

78

3

−5

2

2.5.2.79

79

3 3

− 21 − 35

2.5.2.31

80

− 72 − 10 3

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

72 73 74 75 76

¡

− 23

1 1

1

Equation

2.5.2.84 2.5.2.27

2.5.2.48 2.5.2.35 2.5.2.24

2.5.2.87 2.5.2.49 2.5.2.44 2.5.2.42

2.5.2.64 2.5.2.36

2.5.2.37

Page 338

2.5. GENERALIZED EMDEN–FOWLER EQUATION

Ï Ò˜Í Í Ò

=

 ËEϱ

(Ï

ÒÍ

)

Ë

339

TABLE 23 (Continued) Solvable cases of the generalized Emden–Fowler equation "

 = #*  + (
 )-

0

No

B ¡

Equation

82

3

83

3

84

3

85

3

− 20 7 − 52 − 13 5 − 73 − 15 7

86

3

−2

−2

2.5.2.104

87

3

−2

−1

2.5.2.15 2.5.2.101

3

81

2

2.5.2.85

− 12 − 75 − 53

2.5.2.22

2

2.5.2.82

2.5.2.43 2.5.2.25

88

3

−2

− 12

89

3

−2

1

2.5.2.28

90

3 3

92

3

93

3

94

3

− 12 − 12 − 53 − 52 − 53

2.5.2.59

91

− 43 − 76 − 56 − 12 − 12

95

3

0

−4

2.5.2.55 2.5.2.88

2.5.2.60 2.5.2.93 2.5.2.90 2.5.2.92

96

3

0

97

3

0

− 52 − 12

98

3

0

2

2.5.2.77

99

3

1

−7

2.5.2.62

100

3

1

−4

2.5.2.57

101

3

1

−2

2.5.2.102 2.5.2.23

2.5.2.20

102

3

1

103

3

1

104

3

1

− 53 − 75 − 12

105

3

1

0

2.5.2.21 2.5.2.34 2.5.2.63

106

3

2

− 53

107

3

3

−7

2.5.2.41 2.5.2.30

2.5.2. Exact Solutions 1.

=
ñ ú (ñó ò )A . 1 b . Solution in parametric form with

ñóò ò ó

 = L

1− 1

+ −- X (1 AIU +

+1

)-

1 −2

M

U + L 2,

2 b . Solution in parametric form with

 = L

1

≠ −1, 0 ≠ 2:

¡

- exp(T U 2 ) U + L , X U - −2 2 M

¡

= L

2− 1

-U ,

where # = A3¡

= −1, 0 ≠ 2: = L

1

exp(T U 2 ),

where

# =T

+ 1 −2  -  2−0



4

2

2 (2 −

0)

1−

O˜T

+ −- . 

2  P

-.

Page 339

340

SECOND-ORDER DIFFERENTIAL EQUATIONS 3 b . Solution in parametric form with 1− 1+

+

≠ −1, 0 = 2:

¡

2 +1 ,

+ exp(T"U 2 ) U + L 2 , = 'U + M 4 b . Solution for = −1, 0 = 2: ¡ a 1 1− = (L 1  + L 2) O L 2 exp( L 1  )  =L

2.

1

X U

# = A ( + 1)  ¡

where

−1−

+ .

if # ≠ 1, if # = 1.

=
ï ÷ ( ñ ó ò )A . 1 b . Solution in parametric form with B ≠ −1, 0 ≠ 1:

ñ óò ò ó

 = L

1− 1

-U ,

1 +1 1− )

 - X (1 ACU 

2+ − 1

= L

-

U + L 2, M

where

B + 1 −  1−0

# =A

−2 1−



-.

2 b . Solution in parametric form with B = −1, 0 ≠ 1: 3− 1−

- exp(T"U 2 ) U + L , 2 M 3 b . Solution in parametric form with B ≠ −1, 0 = 1: 2 1−   = U  +1 , = L 1 XZU 1+  exp(T U 2 ) U + L 2 , M 4 b . Solution for B = −1, 0 = 1:  = L

1

exp(T"U 2 ),

= L

XÈU

1

™

= 3.

2÷ +ú ñ ó ò ò ó =
ï ÷ ñ ú (ñ ó ò ) ÷ +ú

+L L 1 ln | | + L

where



4

2

2 (1 −

0)

O˜T

# = T (B + 1)

1−



2 ·P

3−

-.

 .

if # ≠ 1, if # = 1.

2 2

+3 +2

. Solution in parametric form with B ≠ −1,

U M + L 2S ,  (U )

 = exp X Q

+1

L 1 | | O

# =T

where

≠ −1:

¡

= U exp −

Q

B +1

+1

¡

B +1 U M L 2S , −  (U ) +1 ¡

X

where the function  =  (U ) is defined implicitly by the formula [  + ( . − 1)U ](  + . U ) 1−‡ = L

‡

1

+

For the case B = −1, see equation 2.5.2.5. For

4.

=
ïƒ÷ ñ –1 (ñó ò )2 . Solution in parametric form with B ≠ −1, B ≠ 0:

ñóò ò ó

1

 = U  ,

= A exp XZU

Q

1−



 O

B

B +1

#*U +

B +

¡

+2

+2

,

. =−

B +1

+1

¡

.

= −1, see equation 2.5.2.4.

¡

 +1 1 U  + B/U  + L

−1 1

P

M

U + L 2S ,

where # = −  −  . For the case B = −1, see equation 2.5.2.7. For B = 0, see equation 2.5.2.1. 5.

=
ï –1 ñ ú ñó ò . Solution in parametric form with

ñóò ò ó

1−

 = A exp X U + Q For the case

¡

+ O

¡

¡

¡

≠ −1,

¡

≠ 0:

+ +1 1 U + + U + +L 1 ¡ +1 P

= −1, see equation 2.5.2.8. For

¡

−1

M

U + L 2S ,

1

= ( #*U ) + .

= 0, see equation 2.5.2.2.

Page 340

Ï Ò˜Í Í Ò

2.5. GENERALIZED EMDEN–FOWLER EQUATION

6.

 ËEϱ

=

=
ï –1 ñ –1 (ñ ó ò )A . Solution in parametric form with 0 ≠ 1, 0 ≠ 2:

ÒÍ

(Ï

)

Ë

341

ñ óò ò ó

U

 = A*O −

−1

Œ*P

1

exp

Q Œ

X

U

M

+L

 (U )

2

S ,

= A exp X

U

M

Q

+ Œ"L

 (U )

2

S ,

Π=

0 −1 0 −2

,

where the function  =  (U ) is defined implicitly by the formula ln O

U



1

−

U

−

Œ2P

#

=A

Œ" − U

U  − ln U + L 1 , Œ

Π=

0 −1 . 0 −2

For the case 0 = 2, see equation 2.5.2.7. For 0 = 1, see equation 2.5.2.8.

7.

=
ï –1 ñ –1 (ñ ó ò )2 . Solution in parametric form:

ñ óò ò ó

 =A  * , 8.

9.

=L

2(

=
ï –1 ñ –1 ñó ò . Solution in parametric form:

ñóò ò ó

T #*U +  * + L 1 ) exp A #X (T/#*U +  * + L 1 )−1 U!S . Q M

 = L 2 (A"#*U +  * + L 1 ) exp / T #X (A #*U +  * + L 1 )−1 U!S , Q M

=A  * .

=
ïƒ÷ ñ –÷ –3 (ñó ò )3 . Solution in parametric form with B ≠ −1:

ñóò ò ó

U M  £ 1 AC U

 = L 1 +1 U·OX

+1

−1

+L

2

P

U M  £ 1 AC U

= L 1 −1 OX

,

+1

+L

−1 2

P

,

B + 1 1−   +1   . where # = T 2 For the case B = −1, see equation 2.5.2.15. 10.

ú

=
ï ñ ú (ñó ò ) 2ú +3 . Solution in parametric form with 3

ñóò ò ó

+5

 = L = L

−2 1



≠ −3 2:

¡

¦ (1 AIU 



( +1)( −2) 1

J

+1 1 2

)

Q

X (1 AIU 

X (1 AIU  Q

J

+1 −1 2

)

)

1

¡

11.

M



U + L 2S M

+3 “…  +2 (“ + 1)  where “ = − ¡ , # =− A S +1 (“ + 2) EQ 2(“ + 2)  2

J

+1 −1 2

.

÷

= L

B

,

3 +4

1

 = L 1( 

where “ = −

+2

1



=
ïƒ÷ ñ – 2 (ñó ò ) 2÷ +3 . Solution in parametric form with B ≠ −3 2:

ñóò ò ó

U + L 2 S − Uh§ ,

B +1

4 1

, # =

+1)2

U 

+1

¦ (1 ACU 

Q

X (1 AIU  J

+1 1 2

)

“ + 3   +1    (“ + 1)

Q

) J

+1 −1 2

X (1 AIU  1 2

O˜A



hP



M

U + L 2S J

+1 −1 2

)

1 +3



− −1

,

M

2

U + L 2S − U § ,

.

Page 341

342 12.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=

÷


ƒï ÷ ñ (ñ óò

3 +4 +3

) 2÷

.

Solution in parametric form with B ≠ −3 2:

 +2 , X (1 AIU  +1 )−1 J 2 U + L 2 S Q M 1J 2 −1 J 2 X (1 AIU  +1 ) U + L 2 S − Uh§ , =  L 1−2 ¦ (1 AIU  +1 ) Q M

 = L 1( 



−1)( +2)

1

2B + 3 “±  +2 (“ + 1)  , # = A S where “ = − B +1 (“ + 2) uQ 2(“ + 2)  13.

ú

=
ï – 2 ñ ú (ñ ó ò ) 2ú +3 . Solution in parametric form with 3

1

ñ óò ò ó

4 1

¡

14.

ñóò ò ó =
ï ñ –2 ñ óò

.



, # =−

2

U 

≠ −3 2:

¡

¦ (1 ACU 

( +1) 1

= L

+1

1

+5

 = L

where “ = − ¡



+1

J

+1 1 2

Q

)

Q

X (1 ACU 

X (1 ACU 

“ +3   (“ + 1)

1 2



 

+1

J

+1 −1 2

)

   P ·

OA

J

+1 −1 2

)

2

U + L 2 S − Uh§ , M

U + L 2S M

1 +3



− −1

,

.

.

Solution in parametric form:

 = L

1

¦ 2U X exp(T"U 2 ) U + L 2 S A exp(T U 2 ) § , Q M

1

Q

X exp(T"U 2 ) U + L 2 S , M

1 −2 2 . 2

where # = T 15.

= L





=
ï –1 ñ –2 (ñ ó ò )3 . Solution in parametric form:

ñ óò ò ó

 =  exp(T"U 2 ) X exp(T U 2 ) U + L 2 S Q M

−1

,

=L

1

X exp(T"U 2 ) U + L 2 S Q M

−1

,

where # = A 2  2 . 16.

=
ï –2 ñ (ñ ó ò )2 . Solution in parametric form:

ñ óò ò ó

 = L

1

Q

X exp(T U 2 ) U + L 2 S , M

where # = A  2  17.

ñóò ò ó

=


ï ñ

−2

= L

1

¦ 2U X exp(T"U 2 ) U + L 2 SrA exp(T U 2 ) § , Q M

.

) (ñ óò )7 )

–1 2

5

.

Solution in parametric form:

 = A L 1 (U where # = A 15

2

− 1)(U

  J O

−2 1 2

16 "P

3

− 3U + L

J

2 5

2)

J

−1 2

,

= L

16 4 1 (

U

− 6U

2

+ 4L

2

2

U − 3) ,

.

Page 342

Ï Ò˜Í Í Ò

2.5. GENERALIZED EMDEN–FOWLER EQUATION

18.

=
ï ñ –7 ) 4 (ñó ò )7 ) 5 . Solution in parametric form: 27 3 1 (

U

32 4 1 (

= L

U

2

− 6U

−1 3 1 (

U

27 3 1 (

U

J

2 5

9  P

− 3U + L

Ë

343

4

− 15U

J

4 3

+ 20 L

2

U

3

2

− 45U

+ 12 L

2

2 2 ),

U + 27 − 8 L

,

.

2) 2)

−1

4

− 6U

(U

6

− 15U

= L

3 3 1(

−1

J



(U

3 7

27 "P

2

+ 4L 4

2

J

2 3

U − 3)

+ 20 L

2

U

3

,

− 45U

2

+ 12 L

2

2 2 2) ,

U + 27 − 8 L

.

=
ï –1 ) 2 (ñó ò )3 . Solution in parametric form:

ñóò ò ó

2

2

− 1) ,

U

− 3U + L

2 ),

# =A

where

 J 

4 3 2 −2 . 9

ñ óò ò ó

=
ñ (ñ ó ò )3 . Solution in parametric form: 3 3 1(

 = L

U

− 3U + L

= L

2 ),

=
ï –1 ) 2 ñ –5 ) 2 (ñó ò )3 . Solution in parametric form:

1

U ,

# = −6

where

−3

.

ñóò ò ó

2

 = L 1 (U

2

− 1) (U

3

− 3U + L

2)

−1

,

= L

−3 3 1 (

8 4 1(

− 6U

=
ï –5 ) 3 ñ (ñó ò )3 . Solution in parametric form:

U

− 3U + L

2)

−1

,

 J  J

4 3 2 1 2 . 9

# =T

where

ñóò ò ó

 = L

9 3 1(

U

J

3 2 , 2)

− 3U + L

= A/L

=
ï –1 ) 2 ñ (ñó ò )8 ) 5 . Solution in parametric form:

U

2

+ 4L

2

U − 3),

where

 J 

9 8 3 −3 . 64

# =T

ñóò ò ó

 = L

16 4 1 (

U

where # = T 15  1 J 2  25.

U

U − 3)

2

− 3U + L

 = L 14 (U

24.

+ 4L

−2 7 4

where # = 28 (  )1 J 2 O

23.

)

=
ï –5 ) 2 ñ –1 ) 2 (ñ ó ò )10 ) 7 . Solution in parametric form:

= L

22.

ÒÍ

(Ï

ñ óò ò ó

 = L

21.

J

−1 2 6 ( 2)

− 3U + L

  J O

5  12

where # = A

20.

 ËEϱ

ñóò ò ó

 = A L

19.

=

2

− 6U

−2

+ 4L



O

J

2 5

16 rP

2

2

U − 3) ,

= A/L

1(

U

2

− 1)(U

3

− 3U + L

2)

J

−1 2

,

.

=
ï –5 ) 3 ñ –7 ) 3 (ñ ó ò )3 . Solution in parametric form:

ñ óò ò ó

 = A L 1 (U where # = T

3

− 3U + L

2)

J (U

3 2

4

− 6U

2

+ 4L

2

U − 3)−1 ,

= A/L

−8 4 1 (

U

− 6U

2

+ 4L

2

U − 3)−1 ,

 J  J

9 8 3 1 3 . 64

Page 343

344 26.

SECOND-ORDER DIFFERENTIAL EQUATIONS =
ï –7 ) 4 ñ (ñó ò )8 ) 5 . Solution in parametric form:

ñóò ò ó

4J 3 U − 3) , −1 J 2 6 = A/L 127 (U 3 − 3U + L 2 ) (U − 15U 32 4 1 (

 = L

U

5 7J 4   12

where # = T 27.

2

− 6U

−2

O

2

J



2 5

9 ·P

4

+ 20 L

2

3

U

− 45U

2

+ 12 L

2

U + 27 − 8 L

2 2 ),

.

=
ï –1 ) 2 ñ –5 ) 2 (ñ ó ò )11 ) 7 . Solution in parametric form:

ñ óò ò ó

−1

 = L

27 3 1 (

− 3U + L

2)

= L

−1 3 1 (

− 3U + L

−1 4 2) (

U

U

6

(U

U

J



where # = −28  (  )1 J 2 O 28.

+ 4L

3 7

27 rP

4

− 15U 2

− 6U

+ 20 L

+ 4L

2

2

U

3

2

− 45U

J

2 3

U − 3)

+ 12 L

2

2

U + 27 − 8 L 22 ) ,

,

.

=
ï ñ –2 (ñó ò )3 . 1 b . Solution in parametric form with # < 14 :

ñóò ò ó

 = U ( L 1 U Á + L 2 U − Á ),

= U 2,

@ = £ 1 − 4# .

where

2 b . Solution in parametric form with # = 14 :

 = U (L

1

ln |U | + L

= U 2.

2 ),

3 b . Solution in parametric form with # > 14 :

 = UL

29.

1

sin( @ ln U + L

=
ï –1 ) 2 ñ –1 ) 2 (ñ ó ò )3 ) 2 . Solution in parametric form:

where # = T/, 2 , @ = ,

−2

4

(,

=
ï –1 ) 2 ñ (ñó ò )3 . Solution in parametric form:

−2

= 14 U

[(1 + @ ) L

1

U Á + (1 − @ ) L 2 U − Á ]2 ,

+ 4)1 J 2 .

ñóò ò ó

 = L

2 1

= L

1

where # = −16  3 J 2  31.

@ = £ 4 # − 1.

where

ñ óò ò ó

 = A"U 2 ( L 1 U Á + L 2 U − Á )2 ,

30.

= U 2,

2 ),

exp(−2U ) V 2 exp(3U ) − L

exp(−U ) V exp(3U ) + L −3

2

2

sin  £ 3 U  + £ 3 L

sin  £ 3 U  W ,

2

2 cos  £ 3 U  W ,

.

=
ï –1 ) 2 ñ –7 ) 2 (ñó ò )3 . Solution in parametric form:

ñóò ò ó

 =

L 1  − * V 2 exp(3U ) − L

2

sin  £ 3 U  + £ 3 L

exp(3U ) + L 2 sin  £ 3 U 

2

cos  £ 3 U  W

2

,

=

L exp(3U ) + L

−1 1 2

 *

, sin  £ 3 U 

where # = −16(  )3 J 2.

Page 344

2.5. GENERALIZED EMDEN–FOWLER EQUATION

32.

Ï Ò˜Í Í Ò

=

=
ï ñ –2 ) 3 (ñó ò )7 ) 5 . 1 b . Solution in parametric form with # < 0:

 ËEϱ

(Ï

ÒÍ

)

Ë

345

ñóò ò ó

 = L 1 [cosh(U + L 2 ) cos U ]1 J 2 [tanh(U + L 2 ) − tan U ], = L 16 cosh3 (U + L 2 ) cos3 U [tanh(U + L 2 ) + tan U ]3 , where # = −5

J

  J O

2 5

−2 2 3

. 12 "P 2 b . Solution in parametric form with # > 0:

 = L 1 [cosh U − sin(U + L 2 )]−1 J 2 [sinh U − cos(U + L 2 )],  2J 5 where # = 5  −2  2 J 3 O . 6 P 33.

= L

6 1 [sinh

U + cos(U + L 2 )]3 ,

=
ï –5 ) 2 ñ –1 ) 2 (ñ ó ò )1 ) 2 . Solution in parametric form:

ñ óò ò ó

 = L

−1 1 [cosh(

U + L 2 ) cos U ]−1 ,

= L

1

cosh(U + L

2 ) cos

U [tanh(U + L 2 ) − tan U ]2 ,

where # = −4 . 34.

=
ï –5 ) 3 ñ 2 (ñó ò )3 . 1 b . Solution in parametric form with # > 0:

ñóò ò ó

 = L 13 [cosh(U + L 2 ) cos U ]3 J 2 , = L 12 cosh(U + L 2 ) cos U [tanh(U + L 2 ) + tan U ], 3 8 J 3 −4 where # = 16   . 2 b . Solution in parametric form with # < 0:  = L 3 [cosh U − sin(U + L 2 )]3 J 2 , = L 2 [sinh U + cos(U + L 2 )], 1

where # = 35.

1

 J 

− 43 8 3 −4 .

=
ï –2 ) 3 ñ (ñó ò )8 ) 5 . 1 b . Solution in parametric form with # > 0:

ñóò ò ó

 = L 16 cosh3 (U + L 2 ) cos3 U [tanh(U + L 2 ) + tan U ]3 , = L 1 [cosh(U + L 2 ) cos U ]1 J 2 [tanh(U + L 2 ) − tan U ], where # = 5  2 J 3 

−2

O



J

2 5

. 12 rP 2 b . Solution in parametric form with # < 0:

U + cos(U + L 2 )]3 ,  2J 5 where # = −5 2 J 3  −2 O . 6 ·P  = L

36.

6 1 [sinh

= L

1 [cosh

U − sin(U + L 2 )]−1 J 2 [sinh U − cos(U + L 2 )],

=
ï –1 ) 2 ñ –5 ) 2 (ñó ò )5 ) 2 . Solution in parametric form:

ñóò ò ó

 = L

1

cosh(U + L

2 ) cos

U [tanh(U + L 2 ) − tan U ]2 ,

= L

−1 1 [cosh(

U + L 2 ) cos U ]−1 ,

where # = 4  .

Page 345

346

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

37.

=


ï

) ñ

) (ñ óò

–5 3 –10 3

)3 .

1 b . Solution in parametric form with # > 0:

 = L 1 [cosh(U + L 2 ) cos U ]1 J 2 [tanh(U + L 2 ) + tan U ]−1 , = L 1−2 [cosh(U + L 2 ) cos U ]−1 [tanh(U + L 2 ) + tan U ]−1 ,  J  J

3 8 3 4 3 . 16

where # =

2 b . Solution in parametric form with # < 0:

 = L 1 [cosh U − sin(U + L 2 )]3 J 2 [sinh U + cos(U + L 2 )]−1 , = L 1−2 [sinh U + cos(U + L 2 )]−1 , where # = − 43  8 J 3  4 J 3 .

]

In the solutions of equations 38–45, the following notation is used:

8

« = exp(3U ),

8

38.

3

=2

ñ óò ò ó

8

8

1( 2)

=


ï ñ

1

=« +L


− ( 8 )
8 1

*

*

2

2

−

7

.

) ( ñ ó ò )9 )

–1 2

sin  £ 3 U  ,

8 8

8

1 2,

4

=2

8

8

2

8

= 2« − L

1( 3)

2

sin  £ 3 U  + £ 3 L


− 5( 8 )
8 1

*

*

3

+

2

8 8

cos  £ 3 U  ,

1 3.

Solution in parametric form:

J 8

−1 6

 = L 1 « 39.

ñóò ò ó

=


ï ñ

J 8

−1 2 2, 1

) (ñ óò )9 )

–13 8

7

8 1

= L

J 8 2, 3

−4 3

«

where

# = 7

J

  J O

−2 1 2

2 7

64 "P

.

.

Solution in parametric form:

 = L 40.

ñóò ò ó

=

25 1


ï

J 8

J 8

−1 2 4, 1

−5 6

«

) ñ

) (ñ óò

)18 )

–7 2 –1 2

= L 13

32 1

−16 15

8 8J 5 ,

J 8

8

J

«

3

where

# = 7

J

J O 25 

−2 13 8



2 7

256 P

.

.

Solution in parametric form:

 = L 41.

ñóò ò ó

=

−1 1


ï

J

1 15

«

8

8 J

−1 2 5 1 3 ,

) ñ (ñ óò

–7 5

25 1

= L

«

−5 3

= L

4 1

−1 2 1 4,

where

J

 # = −208 5 J 2  1 J 2 O

5 13

25 "P

.

)3 .

Solution in parametric form:

 = L 42.

ñóò ò ó

=


ï

5 1

«

) ñ (ñ óò

–1 2

J 8 5J 2, 1

−5 6

«

J 8 , 3

−2 3

where

5 # = − 1024 

J 

12 5 −3

.

)12 ) 7 .

Solution in parametric form:

 = L

8 1

«

J 8 2, 3

−4 3

= L

1

«

J 8

−1 6

J 8

−1 2 2, 1

where

# = −7 1 J 2 

−2

O



64 rP

J

2 7

.

Page 346

2.5. GENERALIZED EMDEN–FOWLER EQUATION

43.

=
ï –7 ) 5 ñ –13 ) 5 (ñó ò )3 . Solution in parametric form:

J 8 5J 2 8

−1 6

{

−1 3 ,

1

J 8

−4 1

= L

=
ï –13 ) 8 ñ (ñ ó ò )12 ) 7 . Solution in parametric form:

ÒÍ

(Ï

)

Ë

347

2 3

«

−1 3 ,

5 # = − 1024 

where

J  J .

12 5 3 5

32 1

8 8J 5 ,

J

−16 15

«

= L

3

25 1

«

J 8

25  2 J 7 . 256  P

J 8

where

# = −7

8 J

where

# = 208 1 J 2  5 J 2 O

−1 2 4, 1

−5 6

J 

13 8 −2

O

=
ï –1 ) 2 ñ –7 ) 2 (ñ ó ò )21 ) 13 . Solution in parametric form:

ñ óò ò ó

 = L ]

 ËEϱ

ñ óò ò ó

 = L 45.

=

ñóò ò ó

 = L 1 «

44.

Ï Ò˜Í Í Ò

25 1

«

J 8

8

−5 3

−1 2 1 4,

−1 1

= L

«

8

J

1 15

−1 2 5 1 3 ,

In the solutions of equations 46–49, the following notation is used:

{

{

J



5 13

25 rP

{

.

{ {

U + L 2) cos U , 2 = tanh( U + L 2) + tan U , 3 = tanh( U + L 2) − tan U , 4 = 3 2 3 − 4, 9 9 9 9 9 9 92 1 = cosh U − sin(U + L 2), 2 = sinh U + cos(U + L 2), 3 = sinh U − cos( U + L 2), 4 = 3 2 3 − 2 1.

46.

1 = cosh(

=
ï ñ –10 ) 7 (ñó ò )7 ) 5 . 1 b . Solution in parametric form with # < 0:

ñóò ò ó

 = L

{

{

J

9 3 2 4, 1 1

{

{

J

J

14 7 3 7 3 1 1 2 ,

= L

where

5 9

J

J O 9

−2 10 7

# =− 



2 5

28  P

.

2 b . Solution in parametric form with # > 0:

9 J 9

9 −1 2 4, 1 1

 = L 47.

9 J

14 7 3 1 2 ,

# =

where

=
ï –5 ) 2 ñ –1 ) 2 (ñó ò )13 ) 10 . Solution in parametric form:

5  9

J

J O 9

−2 10 7



2 5

14 "P

.

ñóò ò ó

 = L 48.

= L

{

{

J

J

−1 −1 3 2 3 1 1 2 ,

= L

{ {

9 3 2 1 1 4,

 # = −20 ( )1 J 2 O

where

J

3 10

9  P

=
ï –10 ) 7 ñ (ñó ò )8 ) 5 . 1 b . Solution in parametric form with # > 0:

.

ñóò ò ó

 = L

{

J

{

J

14 7 3 7 3 1 1 2 ,

= L

{

J

{

9 3 2 4, 1 1

where

# =

5  9

J 

9 2J 5 . 28 rP

10 7 −2

O

J 

9 2J 5 . 14 rP

2 b . Solution in parametric form with # < 0:

 = L 49.

9 J

14 7 3 1 2 ,

= L

9 J 9

9 −1 2 4, 1 1

where

5 9

# =− 

=
ï –1 ) 2 ñ –5 ) 2 (ñó ò )17 ) 10 . Solution in parametric form:

10 7 −2

O

ñóò ò ó

 = L

{ {

9 3 2 1 1 4,

= L

{

J

{

J

−1 −1 3 2 3 1 1 2 ,

where

# = 20  ( )1 J 2 O



9 ·P

J

3 10

.

Page 347

348

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 50–65, the following notation is used:

² =

A (4U 3 − 1), ü

U

where ® (U ) = X 50.

²

U M



1

=U

−1

( ²u

1



− 1),

−3 1

² ,

= L

5 −1 1, 1

U



where

2 −2 ( 3

# =T



=
ï ñ –1 ) 2 (ñ ó ò )7 ) 4 . Solution in parametric form:

3

= 4U 

2 1

Tç

2 2,

T 6   )3 J 4 .

−1 2, 1



= L

5 2 −2 1 1 ,

where

# =T

J 

where

# =T

U 

=
ï –1 ) 2 ñ (ñ ó ò )8 ) 7 . Solution in parametric form:

 J A 3   )3 J 4 .

2 −2 1 2 ( 3



ñ óò ò ó

−16 2 1 3,



5 −3 2 2, 1 1

= L



=
ï –7 ) 6 ñ –1 ) 2 (ñ ó ò )2 ) 3 . Solution in parametric form:

7  16

J 

−1 2 −1

16  1 J 7 . 3 þP

O

ñ óò ò ó

5 −3 6 1 1 3,





= L

=
ï –3 ) 4 ñ (ñ ó ò )8 ) 7 . Solution in parametric form:

−3 1 1 ( 2 3



 

2 2 1) ,

− 8

where

J  J (   )2 J 3 .

−5 6 3 2

# = T 4

ñ óò ò ó

 = L 55.

2

ñ óò ò ó

 = L

54.



=
ï ( ñ ó ò )7 ) 4 . Solution in parametric form:

 = L 53.

U2Tç² ,

ñ óò ò ó

 = L 52.

2

is the incomplete elliptic integral of the second kind in the form of Weierstrass.

 = L 51.

= 2UÈ® (U ) + L

ñóò ò ó

=


ï

−32 −4 1 3 ,



–4

J  

3 −3 2 ( 2 3 1 1

= L



7  32

2 1 ),

where

# =T

5 −1 1, 1

where

# = A 6 5 

− 8

J 

−1 4 −1

O

32  1 J 7 . 3 þP

(ñó ò )3 .

Solution in parametric form:

 = L 56.

U

= L

U



−2

.

=
ñ (ñ ó ò )5 ) 4 . Solution in parametric form:

ñ óò ò ó

5 −1 1, 1

 = L 57.

2 −1 , 1

ñóò ò ó

=


ï

–4

ñ (ñ óò

U



−3 1

= L

² ,

where

# =A

2 −2 ( 3



T 6   )3 J 4 .

)3 .

Solution in parametric form:

 = L

58.

3 −1 1 1 ,



= L

=
ï –1 ) 2 ñ (ñ ó ò )5 ) 4 . Solution in parametric form:

5 1

U 

−1 1 ,

where

# = A 6 5 

−3

.

ñ óò ò ó

 = L

5 2 −2 1 1 ,

U 

= L

−1 2, 1



where

# =A

 J 

2 1 2 −2 ( 3

A 3   )3 J 4 .

Page 348

2.5. GENERALIZED EMDEN–FOWLER EQUATION

59.

=
ï –1 ) 2 ñ –4 ) 3 (ñó ò )3 . Solution in parametric form: 4 2 1 2,



5 −3 2 1 1 2,





Ë

349

 J 

J

4 3 2 −2 3 . 3

# =T

where

9 −3 1 1 ,

= L



 J 

J

4 3 2 −5 6 . 3

# =T

where

J 

5 −3 2 2, 1 1



= L

−16 2 1 3,



# =A

where

=
ï –7 ñ (ñó ò )3 . Solution in parametric form:

7  16

J O 16  1 J 7 . 3 P

−1 −1 2



J

3 1 2 1 1 ,



8 1 3,

= L



where

# =T

=
ï –7 ñ 3 (ñ ó ò )3 . Solution in parametric form:

3 8 −3 . 64

 

ñ óò ò ó

J 

5 1 2 −1 1 1 3 ,



8 −1 1 3 ,

= L



3 8 −5 . 64

# =T

where

 

=
ï –1 ) 2 ñ –7 ) 6 (ñó ò )7 ) 3 . Solution in parametric form:

ñóò ò ó

−3 1 ( 2 3

 

− 8

2 2 1) ,

− 8

2 1 ),

5 −3 6 1 1 3,

= L

=
ï ñ –3 ) 4 (ñ ó ò )13 ) 7 . Solution in parametric form:





# = A 4 3 J 2 

where

J (   )2 J 3 .

−5 6

ñ óò ò ó

 = L ]

)

ñóò ò ó

 = L 1 

65.



=
ï ñ –1 ) 2 (ñó ò )13 ) 7 . Solution in parametric form:

 = L 64.

ÒÍ

(Ï

ñóò ò ó

 = L 63.

9 3 1 1,

= L

=
ï –1 ) 2 ñ –7 ) 6 (ñ ó ò )3 . Solution in parametric form:

 = L 62.

 ËEϱ

ñ óò ò ó

 = L 61.

=

ñóò ò ó

 = L 60.

Ï Ò˜Í Í Ò

J

3 −3 2 ( 2 3 1 1



 

−32 −4 1 3 ,

= L



where

7  32

# =A

J O 32  1 J 7 . 3 P

−1 −1 4



In the solutions of equations 66–95, the following notation is used:

U =X

M

ü

A (4 <

< 3

− 1)

−L

2,

 = ü

A (4 <

3

− 1).

The function < = < (U ) is defined implicitly. The upper sign in the formulas corresponds to the classical elliptic Weierstrass function < = < (U + L 2 , 0, 1). The solutions given below are written in parametric form. One can assume as the parameter either U , hence < = < (U ), or < , hence U = U ( < ). 66.

=
ï (ñó ò )5 ) 2 . Solution in parametric form:

ñóò ò ó

 = L

−3 1

 ,

= L

1

U ,

where

# =T

2



O˜A

6

 P

J

1 2

.

Page 349

350 67.

SECOND-ORDER DIFFERENTIAL EQUATIONS =
ï ñ –5 ) 4 (ñó ò )5 ) 2 . Solution in parametric form:

ñóò ò ó

−1 1 (

 = L 68.

<

9 −3 3 , 1

U

4 1(

= L

U  + 3U 2 I < T 1),

3 −6 3 ( 1

U

3

U <…T 1) ,

−1 3 2 ( 1

U

2 P

U

1 5

4  P

.

 J (T 3   ) 1 J 2 .

−1 7 8

# = 18 

where

J

J  J O 

−1 3 1 2

.

2

U  +3U 2 … < T 1) ,

4 −12 3 ( 1

= L

U

where # = −5 

= L

13 13 , 1

U

where

2  13

# =T

=
ï –2 ) 3 ñ –1 ) 2 (ñó ò )27 ) 20 . Solution in parametric form:

 J

−1 7 13

OA

ñóò ò ó

3

U <…T 1) ,

−9 −18 2 ( 1

U

= L

1

2

U 2(U 3  −4U 2 h < A 6) ,

where

=
ï (ñó ò )8 ) 5 . Solution in parametric form:

 ,

<

7 −2 ( 1

= L

=
ï ñ ( ñ ó ò )8 ) 5 . Solution in parametric form:

pA 2U'< 2 ),

where

6 1J 2 . 13  P

−1 3 1 2

5 −2 (3 6

# =T



.

4 "P

J

7 20

4 "P

ñóò ò ó

−3 1

7 15

J  J OA 

20  3

# =

J

J  J OA 

−1 3 1 2

=
ï ñ –20 ) 13 (ñó ò )5 ) 2 . Solution in parametric form:

.

  )3 J 5 .

ñ óò ò ó

−8 1 (

U + 2 < ),

7 1

= L

=
ï ñ 5 (ñó ò )7 ) 5 . Solution in parametric form:

< (p  A 2' U <

J

2 −1 2

)

,

where

# =

10 −2 −1 (3 3





  )3 J 5 .

ñóò ò ó

 = L

76.

5  3

1 2

ñóò ò ó

 = L

75.

4 −8 , 1

= L

=
ï –2 ) 3 ñ –1 ) 2 (ñ ó ò )22 ) 15 . Solution in parametric form:

 = L

74.

# =A

where

J

3  J OA

−1 1 4

ñ óò ò ó

 = L 73.

U − < )2 ,

=
ï ñ –15 ) 8 (ñó ò )5 ) 2 . Solution in parametric form:

 = L 1 U (U 3  − 4U 2 C < A 6), 72.

1 2

# =− 

where

ñóò ò ó

 = L 71.

U

=
ï –2 ) 3 ñ –1 ) 2 (ñó ò )6 ) 5 . Solution in parametric form:

 = L

70.

2 4 , 1

= L

ñóò ò ó

 = L 69.

U − < ),

J

−1 2

U <ÑT 1)( sA 2U'< 2)

−27 2 1 (

=
ï –1 ) 2 ñ –5 ) 2 (ñó ò )4 ) 5 . Solution in parametric form:

,

= L

8 1(

U + 2 < )−1 J 3,

where

# = −10

 (   )2 J 5.

−2 −5

ñóò ò ó

2

−1

U
27 2 1 (

= L

7 1(

pA 2U'< 2 ) (U + 2 < )4 J 3 , −1

Page 350

Ï Ò˜Í Í Ò

2.5. GENERALIZED EMDEN–FOWLER EQUATION

77.

=
ï 2 (ñó ò )3 . Solution in parametric form: 2 1

<

−3 1

= L

3 −1 1

U

=
ï –5 ) 4 ñ (ñó ò )1 ) 2 . Solution in parametric form: 2 4 , 1

U

 ,

# =A

where

<

= L

,

1

−1

U

,

2

.



J

6

OA

1 2

 P

# = T 6

where

= L

−1 1 (

U − < ),

# =

where

=
ï –1 ) 2 ñ –2 ) 3 (ñó ò )9 ) 5 . Solution in parametric form: 4 1(

U − < )2 ,

.

−1 3

 .

1 1J 4   2

−1

3 1J 2 . 2 ·P

O˜A

= L

<

9 −3 3 , 1

U

where

=
ï 2 ñ –15 ) 7 (ñ ó ò )3 . Solution in parametric form:

J

J O 

5 1J 2   3

# =T

−1 3

1 5

4 ·P

.

ñ óò ò ó

 J

7 7 , 1

where

# =T

U  + 3U 2 I < T 1),

where

# = − 18  7 J 8 

= L

U

=
ï –15 ) 8 ñ (ñó ò )1 ) 2 . Solution in parametric form:

6 −1 1 7 . 49



ñóò ò ó

4 −8 , 1

U

= L

3 −6 3 ( 1

U

=
ï –1 ) 2 ñ –2 ) 3 (ñ ó ò )23 ) 15 . Solution in parametric form:

−1

(T 3   ) 1 J 2 .

ñ óò ò ó

−12

2

(U 3  + 3U 2 <ÑT 1) ,

= L

3

U <ÑT 1) ,

−1 3 2 ( 1

U

where

# = 5 1 J 2

J O˜A

−1 3

=
ï 2 ñ –20 ) 7 (ñó ò )3 . Solution in parametric form:



J

7 15

4 3P

.

ñóò ò ó

 = L 86.



ñóò ò ó

 = L 14U 85.

−1 −2

# = T 6

where

ñóò ò ó

 = L 84.

U ,

=
ï 2 ñ –5 (ñ ó ò )3 . Solution in parametric form:

 = L 1 U (U 2 p < T 1),

83.

351

ñ óò ò ó

 = L 82.

Ë

=
ñ (ñó ò )1 ) 2 . Solution in parametric form:

 = L 81.

)

ñóò ò ó

 = L

80.

−1 1

= L

,

 = L 1 U , 79.

ÒÍ

(Ï

ñóò ò ó

 = L 78.

 ËEϱ

=

U
6 −6 2 ( 1

U

= L

7 −7 , 1

U

where

# =T

=
ï –20 ) 13 ñ (ñó ò )1 ) 2 . Solution in parametric form:

 J

6 −1 6 7 . 49



ñóò ò ó

 = L

13 13 , 1

U

= L

1

U (U 3  − 4U 2 I < A 6),

where

# =A

2 7J  13

13 −1



OA

6 1J 2 . 13 rP

Page 351

352 87.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=


ï

) ñ

) (ñ óò

–1 2 –2 3

)33 )

20

.

Solution in parametric form: 2

 = L 1 U 2 (U 3  − 4U 2 I < A 6) , 88.

ñóò ò ó

=


ï

) (ñ óò

–5 2

= L

U

J O˜A 

20 1 J 2   3

3

U
−9 −18 2 ( 1

−1 3

J

7 20

4 ·P

.

)3 .

Solution in parametric form:

<

4 −2 , 1

 = L 89.

ñóò ò ó

=


ñ (ñ óò )7 )

5

<

7 −2 ( 1

= L

pA 2U'< 2 ),

where

# = A 3 7 J 2 

−2

.

.

Solution in parametric form:

90.

ñ óò ò ó

=


ï

<

 = L

7 −2 ( 1

) ñ

) (ñ ó ò

–5 2 –1 2

pA 2U'< 2 ),

= L

−3 1

 ,

# =A

where

5 −2 (3 6



  )3 J 5 .

)3 .

Solution in parametric form: −1

 = L 13 ( pA 2U'< 2 ) , 91.

ñ óò ò ó

=


ï ñ ( ñ ó ò )7 )

5

<

= L

7 2 ( 1

= L

−8 1 (

−1

uA 2U'< 2 ) ,

# = A 3 7 J 2 

where

J .

−3 2

.

Solution in parametric form:

 = L 92.

ñóò ò ó

=


ï

7 1

) ñ

J

2 −1 2

< (p  A 2' U <

)

) (ñ óò

–5 3 –1 2

,

U + 2 < ),

where

# = − 103 

−1 −2



(3   )3 J 5 .

)3 .

Solution in parametric form:

 = L 93.

ñ óò ò ó

=


ï

9 1(

) ñ

uA 2U'< 2 )

) (ñ ó ò

–5 3 –5 6

J

3 2

= L

,

16 1 (

U + 2 < )2 ,

where

# = 16  8 J 3 

J .

−3 2

)3 .

Solution in parametric form:

J

3 2

 = L 17 ( pA 2U'< 2 ) 94.

ñóò ò ó

=


ï 5 ñ (ñ óò )8 )

5

(U + 2 < )−2 ,

= L

16 1 (

U + 2 < )−2 ,

where

# = 16  8 J 3 

J .

−7 6

.

Solution in parametric form:

 = L 95.

ñóò ò ó

=

8 1(


ï

U + 2 < )−1 J 3,

) ñ

) (ñ óò

–5 2 –1 2

J

−1 2

U
−27 2 1 (

= L

,

where # = 10 

−5 −2



(   )2 J 5.

)11 ) 5 .

Solution in parametric form:

 = L where # = 5  7 J 2 

7 1(

pA 2U'< 2 ) (U + 2 < )4 J 3 , −1

J O 

−3 2

2 rP

J

4 5

= L

2

−1

U
27 2 1 (

.

Page 352

2.5. GENERALIZED EMDEN–FOWLER EQUATION

]

Ï Ò˜Í Í Ò

=

 ËEϱ

ÒÍ

(Ï

)

Ë

353

In the solutions of equations 96 and 97, the following notation is used:

:

ø

Á (U ) + L 2 n Á (U ) for the upper sign, L 1 ®Á (U ) + L 2 ; Á (U ) for the lower sign, L

™

=

1

where ø Á (U ) and n Á (U ) are the Bessel functions, and ® Á (U ) and functions. =
ï ñ ú (ñó ò )3 . Solution in parametric form with

ñóò ò ó

96.

:  =U Á , For the case

= 'U

2

¡

) are the modified Bessel

≠ −2:

Á ,

@ =

where

¡

= −2, see equation 2.5.2.28.

¡

; Á (U

1 +2 2 , # = A O±¡ . +2 2 P

ñ óò ò ó =
ï –1 ) 2 ñ –1 ) 2 (ñ óò )A .

97.

Solution in parametric form with 0 ≠ 3 2:

:  = U 2 Á 2 ,

−2

= 'U

Á ( U :
+ @ : )2 ,

*

3 2−

1−0 1  -. , # = O˜T 3 − 20 3 − 2 0 ·  P For the case 0 = 3 2, see equation 2.5.2.29.

where @ =

]

In the solutions of equations 98–106, the following notation is used:

:

ø

L

J U ) + L 2 n 1 J 3 (U ) for the upper sign, L 1 ® 1 J 3 (U ) + L 2 ; 1 J 3 (U ) for the lower sign, :
1: : : ’ 1=U + 3 , ’ 2 = ’ 12 ACU 2 2 , ’ 3 = A 23 U 2 3 − 2 ’ 1 ’ 2 , =

™

1 1 3(

*

where ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; functions. 98.

=
ï ñ –1 ) 2 (ñó ò )3 ) 2 . Solution in parametric form:

ñóò ò ó

 = U 99.

J :

−2 3

’ 1,

2

3

J ’ 2, 2

where

# = − ˜O T  ·P

J ’ 2,

where

# = −

=U

−4 3

= 'U

−2 3

=
ï ñ –2 (ñó ò )3 ) 2 . Solution in parametric form:

J :

−4 3

J

1 2

.

−1

’ 3,

=
ï –2 ñ –1 ) 2 (ñó ò )3 ) 2 . Solution in parametric form:

−2

 (A 3  )1 J 2 .

ñóò ò ó

 = U 101.

−1

ñóò ò ó

 = U

100.

J U ) are the modified Bessel

1 3(

J :

−4 3

−2

’ 2,

=U

J :

−8 3

=
ï –1 ) 2 ñ –2 (ñó ò )3 . Solution in parametric form:

−2

’

2 3,

where

# =A

 J

2 3 2 . 3

ñóò ò ó

 = U

J :

−4 3

−2

’

2 1,

= 'U

J :

−2 3

−2

,

where

# =A

 J

1 3 2 . 3

Page 353

354 102.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=


ï

ñ (ñ óò

–2

)3 .

Solution in parametric form:

:  = U 2 J 3 2 , 103.

ñóò ò ó

=


ï

) ñ (ñ óò )3 )

–1 2

2

J ’ 2,

where

# = 92 (   )3.

−1

where

# =

−2 3

= 'U

.

Solution in parametric form:

 =U 104.

ñóò ò ó

=


ï

ñ

–2 –2

J ’

−4 3

2 2,

= 'U

J :

−2 3

’ 1,

2



J

3

O'T

1 2

±P

.

(ñó ò )3 .

Solution in parametric form:

:  = U 4 J 3 105.

ñóò ò ó

=


ï

–2

2

’

= U 2J

−1 2 ,

3

:

−1

’

−1 2 ,

# = 92  3 .

where

ñ (ñ óò )3 ) 2 .

Solution in parametric form:

J ’ 2,

−2 3

 = U 106.

ñóò ò ó

=


ï

) ñ

–1 2 –2

= 'U

J :

−4 3

−1

= 'U

−4 3

’ 3,

where

# = 

−2

1J 2  A 3   .

(ñó ò )3 ) 2 .

Solution in parametric form:

 =U 107.

ñóò ò ó

=


ƒï ÷ ñ –÷

–3

J :

−8 3

÷

−2

3 +4 +3

(ñó ò ) 2÷

’ 32 ,

J :

−2

’ 2,

# =T

where

 J

2 3 2 . 3

.

In the books by Zaitsev & Polyanin (1993, 1994) it was shown that this equation is reducible to a Riccati equation whose solution is expressed in terms of associated Legendre functions.

2.5.3. Some Formulas and Transformations For the sake of visualization, we use the symbolic notation {B ,

, 0}

¡

to denote the generalized Emden–Fowler equation



 = #*  + ( 
)- . Hereinafter we omit the insignificant parameter # (which can be reduced to A 1 by scaling the variables in accordance with the rule ED  , D  , selecting appropriate constants  and  ). 2.5.3-1. A particular solution. If

¡

+ 0 ≠ 1, the generalized Emden–Fowler equation has a particular solution: = $I



+2− 1− −

+ -- ,

where

$ =O

B +2−0

1−

¡

− 0žP

1−

+ +- - −1

B +

+1 + S Q # (1 − − 0 )

¡

1 + −1

-

.

¡

Page 354

2.5. GENERALIZED EMDEN–FOWLER EQUATION

Ï Ò˜Í Í Ò

 ËEϱ

=

(Ï

ÒÍ

)

Ë

355

2.5.3-2. Discrete transformations of the generalized Emden–Fowler equation. 1 b . Taking as the independent variable and  as the dependent one, we obtain a generalized Emden–Fowler equation for  =  ( ) with changed parameters:



= −# 

Denote this transformation by ¸ {B ,

¡

and represent it as follows:

ˆŠ‰T‰ÊD

, 0}

The twofold transformation ¸

+   (
)3−- . 

{ , B , 3−0 }

transformation ¸ .

¡

yields the original equation.

2 b . For ≠ 0, B ≠ −1, 0 ≠ 1, the transformation a = ("
 )1−- , ¨ ¡ Emden–Fowler equation for  =  ( ) with changed parameters:

¨ g

g = $Ia

1 1−

1

¨ -





− +1

+

2 (¨ g
) +

+1

= 

+1

leads to a generalized

,

# (1 − 0 ) + S . Denote this transformation by ‹ and represent it as follows: B +1 Q B +1 1 B 2 +1 {B , , 0 } Œ‰-‰-‰-‰-‰-‰D ¦ , − § transformation ‹ . , ¡ ¡ 1− 0 B +1 ¡ The threefold transformation ‹ yields the original equation. Whenever the solution of the transformed equation is obtained in the form ¨ = ¨ (a ), the solution where $

=− ¡

of the original equation can be written in parametric form as:

 =¨ 

1 +1

1

1

− = , (¨ g
) + ,

,

where

, =

B +1 + S . Q # (1 − 0 )

Different compositions of the transformations ¸ and ‹ generate six different generalized Emden–Fowler equations, whose parameters are shown in Figure 1 (see Subsection 0.6.5).

3 b . In the special case 0 = 0, the transformation = ¨u a ,  = 1 a leads to an Emden–Fowler equation with the independent variable raised to a different power:

¨ g

g = #*a −  − +

−3

¨ + .

{ and represent it as follows: ˆI‰l‰-‰-‰-‰D {−B − − 3, , 0}

Denote this transformation by {B ,

¡

, 0}

¡

¡

transformation { .

If 0 = 0, different compositions of the transformations ¸ , ‹ , and { generate 12 different generalized Emden–Fowler equations, whose parameters are shown in Figure 2 (see Subsection 0.6.5). If 0 = 0 and B = 1, different compositions of the transformations ¸ , ‹ , and { generate 24 different generalized Emden–Fowler equations, whose parameters are presented in Figure 3 (see Subsection 0.6.5). 2.5.3-3. Reduction of the generalized Emden–Fowler equation to an Abel equation. The transformation

H =




,

 = #*  −- +2 + +- −1

reduces the generalized Emden–Fowler equation to the equation (H -  − H

2

+ H ) ‘
= [(

Furthermore, using the substitution ˆ =  − H kind:

¡

2−

+ 0 − 1) H + B − 0 + 2] .

- +H

1−

- , we obtain an Abel equation of the second

ˆˆ ‘
= [( + 2 0 − 3)H + B − 2 0 + 3] H −- ˆ + [( + 0 − 1)H 2 + (B − − 2 0 + 3)H − B + 0 − 2] H ¡ ¡ ¡

1−2

-.

Page 355

356

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.6. Equations of the Form í­ïwî²î ï = † 1 ì ù 1 í þ 1 ( í­ï î )@ 1 + †

ù 2 í þ 2 ( í­ï î )@ 2

2ì

2.6.1. Modified Emden–Fowler Equation

â0äãÑã ä = M

1

æ

–1

âäã + M

2

æ è â|

2.6.1-1. Preliminary remarks. Classification table. For the sake of clarity, below in this subsection we use the conventional notation

 

 − , 
= #* 

+1

+

for the modified Emden–Fowler equation. For , = 0, see Section 2.3. For , ≠ −1, the substitution H =   +1 leads to the Emden–Fowler equation:

‘'

‘ =

#

(, +

1)2

 −2  H  +1 + ,

which is discussed in Section 2.3. The classification Table 24 represents all solvable equations whose solutions are outlined in Subsection 2.6.1. Equations are arranged in accordance with the growth of parameter . The ¡ number of the equation sought is indicated in the last column in this table. TABLE 24 Solvable cases of the modified Emden–Fowler equation  "

 − , 
 = #*  No 1 2 3 4 5

¡

arbitrary ( ≠ −1) ¡ arbitrary ( ≠ −1) ¡ arbitrary ( ≠ −1) ¡ arbitrary ( ≠ −1)

¡

−7

6

−7

7

−4

8

−4

9

−4

10

−2

11

−2

B arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2)

, B +

−2 arbitrary (B ≠ −2) −2

B

2.6.1.1

+3 +1 ¡ 2B + +3 ¡ 1−

¡

¡

2.6.1.2 2.6.1.3

−1

2.6.1.6

1 3(

B − 1)

2.6.1.45

1 5(

B − 3)

2.6.1.46

1 2

2.6.1.40

−2 arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2)

+

Equation

1 2

−

+1

1 3(

B

B − 1)

2.6.1.42

−1

2.6.1.41

B − 1)

2.6.1.28

arbitrary ( , ≠ −1)

2.6.1.29

1 3(

Page 356

2.6. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

=

 1Ë E Ï ± 1

1

(Ï

ÒÍ

Ë  2Ë E Ï ±

)1+

2

2

(Ï

ÒÍ

)

Ë

357

2

TABLE 24 (Continued) Solvable cases of the modified Emden–Fowler equation  "

 − , 
 = #* 

B

No

¡

12

− 25

13

− 25

14

− 25

15

− 35

16

− 35

17

− 35

18

− 35

19

− 35

20

− 35

21

− 35

22

− 35

23

− 35

24

− 57

25

− 57

26

−1

27

−1

28

−1

−2

29

−1

30

− 21

31

− 21

32

− 21

33

− 21

34

− 21

35

− 21

−2 arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2)

arbitrary (B ≠ −2) arbitrary (B ≠ −2) −2 arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) −2 arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2)

,

+

Equation

1 2 1 3 (2

+1

B

2.6.1.35

B + 1)

2.6.1.37

−1

2.6.1.36

−3B − 7

2.6.1.14

1 2 1 2 (3 1 3(

2.6.1.8

B + 4)

2.6.1.9

B − 1)

2.6.1.13

1 3 (2 1 4(

B

B + 1)

2.6.1.38

B − 2)

2.6.1.18

− 14 (3B + 10) 1 7 (6

1 3(

2.6.1.19

B + 5)

2.6.1.39

−1

2.6.1.22

B − 1)

2.6.1.24

− 13 (5B + 13)

2.6.1.25

B +1

2.6.1.5

1 2

B

2.6.1.4

arbitrary ( , ≠ −1)

2.6.1.7

−1

2.6.1.20

−2B − 5

2.6.1.12

1 2 1 2 (3 1 3(

B

2.6.1.11

B + 4)

2.6.1.43

B − 1)

2.6.1.16

1 3 (2

B + 1)

2.6.1.26

− 13 (2B + 7)

2.6.1.17

Page 357

358

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 24 (Continued) Solvable cases of the modified Emden–Fowler equation  "

 − , 
 = #* 

B

No

¡

36

− 21

arbitrary (B ≠ −2)

37

− 21

−2

38

− 21

39

1 2

40

1 2

−2 arbitrary (B ≠ −2) arbitrary (B ≠ −2)

41

1 2

42

2

43

2

44

2

45

2

46

2

−2 arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) arbitrary (B ≠ −2) −2

+1

+

,

Equation

B + 7)

2.6.1.44

arbitrary ( , ≠ −1)

2.6.1.27

−1

2.6.1.21

1 2

B

2.6.1.15

− 13 (2B + 7)

2.6.1.10

−1

2.6.1.23

−7B − 15

2.6.1.33

1 5 (6

1 2

B

2.6.1.30

− 13 (B + 5)

2.6.1.32

− 16 (7B + 20)

2.6.1.34

−1

2.6.1.31

2.6.1-2. Solvable equations and their solutions. 1.

– 12 ùñó ò =
ïƒ÷ +1 ñ ú , Solution in parametric form:

ï ñóò ò ó

þ

ù

≠ –1,

≠ –2. 2 +2

X (1 AIU + +1 )−1 J 2 U + L 2 S  Q M where # = A 18 ( + 1)(B + 2)2  −  −2  1− + . ¡  = L

2.

ï ñóò ò ó

ù

+

+þ

+3 ñóò =
ïƒ÷ þ +1 Solution in parametric form: +

 = L

1− 1

where # = A 3.

1− 1

ï ñóò ò ó

+ Q

X (1 AsU +

(B + 2)2 −   2( + 1)

2ù + þ

¡

J

+1 −1 2

)

−2 1−



M

ñ ú

,

þ

+ U + L 2S 

+1 +2

+1

ù

≠ –2.

= L 1 +2U

,

Q

+ .

+3 ñóò =
ïƒ÷ +1 ñ ú , –1 Solution in parametric form: 1−  = exp ¡ L XÈOL Q B +2 2 +

≠ –1,

þ

þ

1

+

≠ –1, 1 U 4

2

+

= L 1 +2 U ,

,

X (1 AsU +

ù

≠ –2.

2$

U +

+1

1 2$ ¡ + = U exp L 2 XÈOL 1 + U 2 + U Q 4 +1 ¡ 4(B + 2)2 where # = $ . ( − 1)2 ¡

+1

+1

J

J

M

−1 2

P

)

−1 2

P

M

J

+1 −1 2

M

U + L 2S

−1

,

U!S ,

U!S ,

Page 358

2.6. EQUATIONS OF THE FORM

4.

ï ñ óò ò ó

–

1 2

ùñ óò

=


ï ÷

ñ

+1 –1

Ï Ò˜Í Í Ò

ù

,

=

 1Ë E Ï ± 1

Ë  2Ë E Ï ±

ÒÍ

(Ï

1

)1+

2

2

(Ï

ÒÍ

Ë

)

359

2

≠ –2.

Solution in parametric form:

1 2(

where # = T 5.

ï ñóò ò ó

B + 2)2  − 

= L 1 +2 exp(T"U 2 ),

,

 .

−2 2


ïƒ÷

– (ù + 1)ñó ò =

2 +2

X exp(T"U 2 ) U + L 2 S  Q M

2 1

 = L

ñ

+1 –1

ù

≠ –2.

,

Solution in parametric form: −1 J 2 2# Uh§ , ln | U | S B +2 Q (B + 2)2 M −1 J 2 1 2# = U exp ¦ L 2 X L 1 + U 2 + ln |U |S Uh§ . 2 Q 4 (B + 2) M

2L

 = exp ¦

6.

ï ñóò ò ó


ï

+ ñó ò =

–1

ñ ú

2

þ

,

X

L

1

1 U 4

+

2

+

≠ –1.

Solution in parametric form:

 =L 7.

ï ñ óò ò ó

–

2

üIñ óò

exp X ( L


ï

=

AIU +

1

Q

ñ

–1 –1

) J

+1 −1 2

ü

,

U!S , M

= 'U ,

1 1− 2

# =A

where



+ ( + 1). ¡

≠ –1.

Solution in parametric form: 2# ln U + L Q ( , + 1)2

 = ¦/X 8.

ï ñóò ò ó

–

1 2

ùñóò

=


ïƒ÷

1

S

J

−1 2

M

ñ )

+1 –5 3

U +L

2

1 − +1



§

ù

,

2# ln U + L Q ( , + 1)2

= Uh¦/X

,

1

S

J

−1 2

M

U +L

2

§

−1

.

≠ –2.

Solution in parametric form:

 = L 9.

ï ñ óò ò ó

8 3 1(

U

2 +2 ,

A 3U + L 2 ) 

óò

– 12 (3ù + 4)ñ


ï ÷

=



3 +6 2 ( 1

= L

ñ )

+1 –5 3

U

ù

,

J

3 2

A 1)

,

 

 J B + 2)2 .

1 − −2 8 3 ( 12

# =A

where

≠ –2.

Solution in parametric form:

ï ñóò ò ó

U

 

2 +6

= L

,



J

3 2

3 +6 2 ( 1

U

A 1)

(U

−1

3

A 3U + L 2 ) ,

 J B + 2)2 .

3 − −2 8 3 ( 4

where # = A 10.

− A 3U + L 2 ) 3 

8 3 1(

 = L


ïƒ÷

+ 13 (2ù + 7)ñó ò =

ñ )

,

+L

2 2S

+1 1 2

ù

≠ –2.

Solution in parametric form:

 = L where # = A

1

Q

U X ü

 

A (4U

M

U 3

− 1)

 J B + 2)2 .

3 +4

,

= L



2 +4 2 1

U

Q

U X ü

A (4U

M

U 3

− 1)

+L

2

S

−1

,

16 − −2 1 2 ( 3

Page 359

360 11.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñóò ò ó

– 12 ùñó ò =
ïƒ÷ +1 ñ –1 ) 2 , Solution in parametric form:

 = L 13 (U 12.

3

− 3U + L

3 3 1(

16 9 (

ï ñóò ò ó

8 1

1 64 (

V A (U

4

2

 = L

8 1

81 64 (

ï ñóò ò ó

B + 2)2  − 

)

–5 3

ï ñóò ò ó

1

Q

U X ü

A (4U

M

U 3

− 1)

ï ñóò ò ó

16 3 ( 9

°

ù

2

)

–5 3

U

2

3

− 3U + L

−1 2) ,

3 +6 3 ( 1

− 3U + L

2)

− 1) (U

≠ –2. 3 +2

U − 3)W 

ù

,

1 +6 ,

= L

,



U

J

3 2

,

≠ –2.



3 +6 3 ( 1

= A L

+L

ù

U

− 3U + L

2)

J

3 2

(U

4

− 6U

2

+ 4L

2

−1

U − 3) ,

≠ –2.

2 +2

 2S

)

–1 2

,

= L

,

ù

À

À  = V L 1 2 * + L 2 where # =



2 +4 2 ( 1

 J .

– ù – 1)ñó ò =
ïƒ÷ +1 ñ Solution in parametric form: À

1

 J .

−2 3 2

−2 8 3

1 ( 3

= , 2L

= L

,

,

+ 4L

– 12 ùñó ò =
ïƒ÷ +1 ñ 1 ) 2 , Solution in parametric form:

 = L

17.



B + 2)2  − 

≠ –2.

,

1 2 +4

1 9(

# =A

where

 J .

V A (U 4 −6U 2 +4 L 2 U −3)W 3 

where # = A

2)

ù

2

− 1) ,

−2 8 3

+ (3ù + 7)ñ ó ò =
ï ÷ +1 ñ Solution in parametric form:

ï ñ óò ò ó

)

–1 2

U

 J .

− 6U

B + 2)2  − 



2 +4 2 ( 1

−2 3 2

– 13 (ù – 1)ñó ò =
ïƒ÷ +1 ñ Solution in parametric form:

where # = A

16.

− 3U + L

B + 2)2  − 

 = L

15.

U

≠ –2.

= L

,

+ (2ù + 5)ñ„ó ò =
ïƒ÷ +1 ñ Solution in parametric form:

where # = A

14.

2 +2



ï ñóò ò ó

 = L

13.

2)

ù

U

°  2 * + L 2° 

−

where

# = A 3 −

 J (B + 2)2 .

−2 1 2

≠ –2.

À sin  £ 3 °˜U  W 

*

−



2 +4 2 , 1

* V£

3 +2 ,

2 3 cos  £ 3 °˜U  − sin( £ 3 °˜U )W. ,

B + 2)2 .

+ 13 (2ù + 7)ñó ò =
ïƒ÷ Solution in parametric form: À

ñ )

+1 –1 2

À  = V L 1 2 * + L 2

ù

,

À

−

*

≠ –2.

À sin  £ 3 °˜U  W 2 

,

À , 2 L 1 °  * + L 2 °  * V £ 3 cos  £ 3 °˜U  − sin  £ 3 °˜U  W . 2 = , L 1  2 * + L 2  − * sin  £ 3 ˜° U  2

where # =

64 3 ( 9

°

À

3 +4

−

B + 2)2 .

Page 360

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

18.

=

ï ñóò ò ó

 1 Ë E Ï ± ( Ï ÒÍ ) Ë +  2 Ë E Ï ± ù ≠ –2. 1

1

1

2

2

(Ï

ÒÍ

)

Ë

361

2

– 14 (ù – 2)ñó ò =
ïƒ÷ +1 ñ –5 ) 3 , 1 b . Solution in parametric form with # < 0:

4

V cosh(U + L 2 ) cos U!W  +2 V tanh(U + L 2 ) + tan U!W  3J 2 = L 13  +6 V cosh(U + L 2 ) cos U!W , 8 1

 = L 3 −  where # = − 256

4 +2

,

 J (B + 2)2 .

−2 8 3

2 b . Solution in parametric form with # > 0:

 = L where # = 19.

ï ñóò ò ó

8 1

V sinh U + cos(U + L 2 )W 

 

4 +2



3 +6 1

= L

,

V cosh U − sin(U + L 2 )W 3 J 2 ,

 J B + 2)2 .

3 − −2 8 3 ( 64

+ 14 (3ù + 10)ñ„ó ò =
ïƒ÷ +1 ñ –5 ) 3 , ù ≠ –2. 1 b . Solution in parametric form with # < 0: 4

4

V cosh(U + L 2 ) cos U!W 3  +6 V tanh(U + L 2 ) + tan U!W 3  +6 , 1J 2 V −1 = L 13  +6 V cosh(U + L 2 ) cos U!W tanh(U + L 2 ) + tan U!W , 8 1

 = L 27 −   where # = − 256

 J (B + 2)2 .

−2 8 3

2 b . Solution in parametric form with # > 0: 4

V sinh U + cos(U + L 2 )W 3  +6 , 3J 2 V = L 13  +6 V cosh U − sin(U + L 2 )W sinh U + cos(U + L 2 )W 8 1

 = L where # = 20.

 

 J B + 2)2 .

+ ñ ó ò =
ï –1 ñ –1 . Solution in parametric form:

ï ñ óò ò ó

23.

exp X (2 # ln |U | + L

2

+ ñ ó ò =
ï –1 ñ –1 ) 2 . Solution in parametric form: 3

− 3L

1

1)

Q

ï ñ óò ò ó

 = exp(AU 22.

,

27 − −2 8 3 ( 64

 =L 21.

−1

U + L 2 ),

=  (AU

2

J

−1 2

−L

1)

U!S , M

2

,

=U .

where

 J

4 3 2 . 9

# =A

+ ñó ò =
ï –1 ñ –5 ) 3 . Solution in parametric form:

ï ñóò ò ó

 = exp( L 1 U

ï ñóò ò ó

+ ñó ò =
ï –1 ñ 1 ) 2 . Solution in parametric form:

 =L

1

ü

= (A 3 # L

A 3U + L 2 ),

U

exp X

Q

3

M

U

A (4U − L 1 ) 3

S ,

= 'U 2 ,

1)

J (L 1 U

3 8

where

2

J

3 2

A 1)

.

# = A 12  1 J 2 .

Page 361

362

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

À À À In the solutions ofÀ equations 24 and 25, the following notation is used:

8 8 24.

1

=L

1

 2 * + L 2

3

=

2 2

− 2 1(

8

8

ï ñóò ò ó

8

−

2)


.

8

sin  £ 3 °˜U  ,

*

*


ïƒ÷

– 13 (ù – 1)ñó ò =

= 2L

2

ñ )

+1 –7 5

°  2 * + L 2° 

1

ù

,

* V£

−

3 cos  £ 3 °˜U  − sin  £ 3 °˜U  W ,

≠ –2.

Solution in parametric form:

8  3+2 , 3

 = 25.

ï ñóò ò ó

8 5J 2 ,

=


ïƒ÷

+ 13 (5ù + 13)ñ„ó ò =

5 # = − 9216  −

where

1

ñ )

+1 –7 5

ù

,

J °

−2 12 5 −6



(B + 2)2 .

≠ –2.

Solution in parametric form:

 = ]

8 5 3

3 +10

=

,

8 5J 2 8 1

−1 3 ,

125 −  # = − 9216 

where

(B + 2)2 .

In the solutions of equations 26–29, the following notation is used:

:

ø J U ) + L 2n ®1 1 J 3 (U ) + L 2 ; L

™

=

U ) for the upper sign, ( 1 J 3 U ) for the lower sign,

1 1 3(

L

1 3(

J

where ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; functions. 26.

 J °

−2 12 5 −6

ï ñóò ò ó


ïƒ÷

– 13 (2ù + 1)ñó ò =

ñ )

+1 –1 2

ù

,

J U ) are the modified Bessel

1 3(

≠ –2.

Solution in parametric form:

 = L 27.

ï ñóò ò ó

3 1

–

U 

1 +2

üIñóò

:  =

3 +2


ï



ñ )

–1 –1 2

J

2 +4 −2 3 ( 1

= L

,

U

ü

,

U

:
*

+

1 3

:

)2 ,

where

 

 J B + 2)2 .

4 − −2 3 2 ( 27

# =T

≠ –1.

Solution in parametric form:

: −  = L 1 (U 1 J 3 )  28.

ï ñóò ò ó

2 +1


ïƒ÷

– 13 (ù – 1)ñó ò =

J :

= 'U

,

ñ

+1 –2

−4 3

−2

,

ù

(U

:
*

+

1 3

:

)2 ,

where

# =T

 J

1 3 2 ( 3

, + 1)2 .

≠ –2.

Solution in parametric form:

 = L where # = − 21  −  29.

ï ñóò ò ó

–

üIñóò

=


ï

2 3 − +2 1

U 

V (U :
+ 1 : )2 ACU 2 : 2 W  3

*

3 +2

= L 1 +2 U 2 J

,

3

:

2

,

 (B + 2)2 .

−2 3

ñ

–1 –2

,

ü

≠ –1.

Solution in parametric form:

 = L 1 U 3

2 +3

V (U :
+ 1 : )2 ACU 2 : 2 W −  3

*

1 +1

,

= 'U 4 J

3

: 2V :
(U *

+

1 3

:

)2 AIU

2

:

2 −1

W

,

where # = − 29  3 ( , + 1)2 .

Page 362

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

]

 1Ë E Ï ±

=

1

1

(Ï

ÒÍ

Ë  2Ë E Ï ±

)1+

2

2

(Ï

ÒÍ

)

Ë

363

2

In the solutions of equations 30–39, the following notation is used:

U =X

<

M A (4 < ü

−L

− 1)

3

 =

2,

A (4 < ü

− 1).

3

The function < is defined implicitly by a first integral; the upper sign in the formulas corresponds to the classical Weierstrass elliptic function < = < (U + L 2 , 0, 1). 30.

ï ñ óò ò ó

–

1 2

ùñ óò


ï ÷

=

ñ

+1 2

ù

,

≠ –2.

Solution in parametric form: −1 1

 = L 31.

ï ñóò ò ó

+ ñó ò =


ï

U 

ñ

–1 2

2 +2 ,

= L 1 +2 < ,

 

3 − −2 −1 ( 2

# =A

where



B + 2)2 .

.

Solution in parametric form:

 = L 2 * , where #

= A 6

−1

, and the elliptic Weierstrass function

by the integral U = X>Ž (4 H 32.

ï ñ óò ò ó

= +< (U , 0, L

÷

óò

+ 13 (ù + 5)ñ

3

−L


ï ÷

=

1)

ñ

+1 2

J

−1 2

H . M

ù

,

<

1 ),

= < (U , 0, L

1)

is defined implicitly

≠ –2.

Solution in parametric form: −1 1

 = L 33.

ï ñ óò ò ó

+ (7ù + 15)ñ

U 

3 +2

óò

=

= L 1 +2 U

,


ï ÷

ñ

+1 2

−1

ù

,

<

,

 

2 − −2 −1 ( 3

# =A

where



B + 2)2 .

≠ –2.

Solution in parametric form:

 = L 34.

ï ñóò ò ó

1 −1 − +2 , 1

= L 1 +2 U (U

U 

+ 16 (7ù + 20)ñ„ó ò =


ïƒ÷

ñ

+1 2

2

,


1),

ù

≠ –2.

# = A 6 −

where

−2 −1



(B + 2)2 .

Solution in parametric form:

 = L 35.

ï ñ óò ò ó

–

1 2

−1 1

ùñ óò

U

6 7( +2)



=


ï ÷

= L 1 +2 U

,

ñ )

+1 –5 2

−6

ù

,

(U 2
where

# =A

 

1 − −2 −1 ( 6



B + 2)2 .

≠ –2.

Solution in parametric form:

 = L 36.

ï ñóò ò ó

4 7 − +2 ( 1

< 

+ ñó ò =


ï

pA 2U'< 2 ) 

ñ )

–1 –5 2

2 +2

= L

,



<

2 +4 −2 , 1

where

# =T

 

 J B + 2)2 .

3 − −2 7 2 ( 4

.

Solution in parametric form:

 =L

2

exp[ <

−2

( pA 2U'< 2 )],

where # = T 3  7 J 2 , and the elliptic Weierstrass function −1 J 2 X Ž (4 H 3 − L 1 ) H . by the integral U = >

÷

<

= +<

−2

,

= < (U , 0, L

1)

is defined implicitly

M

Page 363

364 37.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñóò ò ó

– 13 (2ù + 1)ñó ò =
ïƒ÷ Solution in parametric form:

ñ )

+1 –5 2

ù

≠ –2.

,

3 +4

3

−  = L 17 <  +2 ( pA 2U'< 2 ) 2  where # = T 43  −  −2  7 J 2 (B + 2)2 .

38.

ï ñóò ò ó

– 13 (2ù + 1)ñó ò =
ïƒ÷ Solution in parametric form:

 = L

39.

]

ï ñóò ò ó

8 1(

3 +2

U + 2 < ) 

,

– 17 (6ù + 5)ñó ò =
ïƒ÷ Solution in parametric form:

ñ )

+1 –5 3

ù



ñ )

+1 –5 3

7 +6

 = L 18 (U + 2 < )− 3  6 −  −2 8 J 3 where # = − 49   (B + 2)2 .



2 +4 ( 1

−1

uA 2U'< 2 ) ,

≠ –2.

,

uA 2U'< 2 )

3 +6 ( 1

= L

= L

,

ù

J

3 2

,

# = − 272  − 

where

 J (B + 2)2 .

−2 8 3

≠ –2.

,



3 +6 ( 1

= L

,

J

3 2

pA 2U'< 2 )

(U + 2 < )−2 ,

In the solutions of equations 40–46, the following notation is used:

² = A (4U 3 − 1),  1 = 2UÈ® (U ) + L 2 U2Te² ,  2 = U −1 ( ²u 1 − 1), ü U U where ® (U ) = X M is the incomplete elliptic integral of the second kind in the Weierstrass form. ² 40.

41.

– 12 ùñ ó ò =
ï ÷ +1 ñ –4 , Solution in parametric form:

ï ñ óò ò ó ï ñóò ò ó

43.

44.

45.

46.

U





2

U

exp 2 X

Q

ï ñóò ò ó

A (4U

ü

M

U 3

−L

– 13 (ù – 1)ñó ò =
ïƒ÷ +1 ñ Solution in parametric form: 5 1(

 = L

ï ñóò ò ó ï ñóò ò ó

2 3 +2 1 1

 

,

– 15 (6ù + 7)ñó ò =
ïƒ÷ Solution in parametric form:

 = L

ï ñóò ò ó

5 3 − 2 +4 1 1





2 1

Tç

 = L

8 1 (4

U 

2 1

Tç

5 2 − +2 2)



–7

3 2 +2 , 2) +1 –7

,

ñ



ù



where

S ,

= T ( #3L

2 1

6)1 J 5 U

−1

.

ù

 

2 − −2 5 ( 3

# =T



B + 2)2 .

# = A 3 −

 J (B + 2)2 .

−2 3 2

≠ –2.



where

# =A

 

 J B + 2)2 .

48 − −2 3 2 ( 25

≠ –2.

,

= L 1 +2 

= L 1 +2 

1)

B + 2)2 .

≠ –2.

ù

,

,

−L

where

2 +4 −3 2 1 1 2,



– 15 (ù – 3)ñó ò =
ïƒ÷ Solution in parametric form:

ï ñóò ò ó

−1 1 ,

,



3



≠ –2.



ñ )

A (4U

 

3 − −2 5 ( 2

# =T

where

U ü

2 +4 2 1 2, +1 –1 2

= L

,

1

T

ù

ñ )

– 13 (ù – 1)ñó ò =
ïƒ÷ +1 ñ Solution in parametric form:

 = L 18 (4U 

2

,

+1 –1 2

= L

,

1)

+L

−1

= L 1 +2 



– 12 (3ù + 4)ñó ò =
ïƒ÷ Solution in parametric form:

 = L

–4

3 −1 +2 , 1 )

U 

≠ –2.

= L 1 +2 U

,

+ ñó ò =
ï ñ . Solution in parametric form:

 =L 42.

2 5 −1 +2 1) 1( –1 –4

 = L

ù

ù

J

1 2 1 ,

where

# =A

 

1 − −2 8 ( 192



B + 2)2 .

J

≠ –2.

1 2 1 (4

U 

2 1

Te

2 −1 2) ,

where # = A

 

− −2 8 3 ( 1600



B +2)2.

Page 364

Ë  2 Ë E Ï ± ( Ï ÒÍ ) Ë 2.6.2. Equations of the Form â$äãÑã ä = ( M 1 æ è 1 | ⠍ 1 + M 2 æ è 2 â| 2 )(âäã ) 2.6. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

=

 1Ë E Ï ± 1

1

(Ï

ÒÍ

) 1+

See Section 2.4 for the case 0 = 0; see Section 2.5 for the cases #

2

1

2

= 0 or #

2

2

365

= 0.

2.6.2-1. Classification table. Table 25 presents all solvable equations whose solutions are outlined in Subsection 2.6.2. Equations are arranged in accordance with the growth of 0 , the growth of 1 (for identical 0 ), the growth of 2 ¡ ¡ (for identical 0 and 1 , 1 ≥ 2 ), the growth of B 1 (for identical 0 , 1 , and 2 ), and the growth ¡ ¡ ¡ ¡ ¡ of B 2 (for identical 0 , 1 , 2 , and B 1 ). The number of the equation sought is indicated in the last ¡ ¡ column in this table. TABLE 25 Solvable cases of the equation 

 = ( #

0 Any ( 0 ≠ 2) +2B 1 +3 ¡ 1 + B +2 ¡ 1 1 Any ( 0 ≠ 1) Any ( 0 ≠ 2) Any ( 0 ≠ 2) 3 1 +5 ¡ 2 1 +3 ¡ +5 ¡ 1 +3 ¡ 1 3B 1 +4 B 1 +1 2(B 1 +2) B 1 +3 Any ( 0 ≠ 1, 2) 1 2

1

¡

Any 1 ≠ −1)

(

¡

(

Any 2 ≠ −1)

¡

Any

¡

0

Any 1 ≠ −1)

0

Any

−

Any

¡

¡

1 −2 1 −1

2

B

1

(B

#

2

0

1

#

2

Equation

Any

2.6.2.1

Any

Any

2.6.2.98

Any

Any

2.6.2.5

0

Any

Any

2.6.2.2

−1

Any

Any

2.6.2.6

1

0

Any

Any

2.6.2.21

1

0

Any

Any

2.6.2.94

1 −2

Any

Any

2.6.2.22

Any 1 ≠ −1)

¡

B

2 ( 1 +1)−

¡ (B

¡

1 +1

Any 2 ≠ −1)

0 (B

  2 + 2 )(
 )-

2

Any

Any

−1

0

  1 + 1 + #

0

Any

0 (

B

2

¡

1

Any ≠ −1)

1

B

1+ 1

1

0

Any

−B

1

0

Any

B 1 −1

Any

Any

2.6.2.95

1

0

0

1

Any

Any

2.6.2.20

− 41

− 74

0

1

Any

Any

2.6.2.71

− 74

Any

Any

2.6.2.81

2

1 2

1

0

− 15 8

1 2

1

0

− 15 8

− 13 8

Any

Any

2.6.2.66

1 2

1

0

20 − 13

15 − 13

Any

Any

2.6.2.68

14 − 13

Any

Any

2.6.2.84

1 2

1

0

20 − 13

1 2

1

0

− 54

− 34

Any

Any

2.6.2.64

1 2

1

0

− 54

− 12

Any

Any

2.6.2.78

Page 365

366

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 25 (Continued) Solvable cases of the equation 

 = ( # 1   1 +

0

1

¡

1 2

B

2

¡

B

1

2

1

+#

2

#

  2 + 2 )(
 )1

#

2

Equation

1

0

0

1

Any

Any

2.6.2.62

1 2

1

0

0

2

Any

Any

2.6.2.75

1

0

0

Any

Any

2.6.2.7

1

0

0

−1

Any

Any

2.6.2.8

1

0

−2

0

1

Any

Any

2.6.2.25

1

1

0

0

1

Any

Any

2.6.2.23

3 2

Any

Any

Any

Any

2.6.2.97

3 2

0

−2

1

Any

Any

2.6.2.107

3 2

0

− 21

0

1

Any

Any

2.6.2.105

3 2

1

0

−2

0

Any

Any

2.6.2.108

3 2

1

0

− 21

0

Any

Any

2.6.2.106

0

0

Any

Any

2.6.2.3

−1

0

0

Any

Any

2.6.2.4

1

0

−2

0

Any

Any

2.6.2.26

2

1

0

0

1

Any

Any

2.6.2.24

5 2

− 47

− 15 8

0

1

Any

Any

2.6.2.80

5 2

− 13 8

− 15 8

0

1

Any

Any

2.6.2.65

5 2

− 15 13

− 20 13

0

1

Any

Any

2.6.2.67

5 2

− 14 13

− 20 13

0

1

Any

Any

2.6.2.83

5 2

− 43

− 45

0

1

Any

Any

2.6.2.63

5 2

− 21

− 45

0

1

Any

Any

2.6.2.77

5 2

1

0

− 47

− 41

Any

Any

2.6.2.72

5 2

1

0

0

1

Any

Any

2.6.2.61

5 2

2

0

0

1

Any

Any

2.6.2.74

3

Any

Any

−

2 −3

Any

Any

2.6.2.9

3

Any

Any

−2

Any

Any

2.6.2.93

Any 1 ≠ −2)

Any

2 2

Any ≠ −1) ¡ 1 Any ( 1 ≠ −1) (

¡

2

3 3

(

¡

Any

(

¡

Any 1 ≠ −1) Any (B 1 ≠ −1) (B

−3

1

¡

Any 2 ≠ −1)

−

¡

¡

1 −3

2

¡

0

1 −3

¡

Any 2 ≠ −1)

(B

−

¡

−2

¡

2 −3

1

0

Any

Any

2.6.2.49

1 −3

0

Any

Any

2.6.2.19

Page 366

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

=

 1Ë E Ï ± 1

1

(Ï

ÒÍ

Ë  2Ë E Ï ±

) 1+

TABLE 25 (Continued) Solvable cases of the equation 

 = ( # 1   1 +

0 3

1

¡

2

¡

Any 1 ≠ −2)

B

1

B

2

2

1

#

+#

2

1

2

(Ï

ÒÍ

Ë

)

367

2

  2 + 2 )(
 )#

2

Equation

0

1

−3

Any

Any

2.6.2.51

−5

−6

1

3

Any

Any

2.6.2.100

3

−4

−5

0

2

Any

Any

2.6.2.76

3

−3

−5

0

1

Any

Any

2.6.2.44

3

−3

−5

0

2

Any

Any

2.6.2.58

3

−3

− 27

0

− 21

Any

Any

2.6.2.28

3

− 14 5

− 18 5

2

3

Any

Any

2.6.2.112

3

− 83

− 10 3

− 31

− 35

Any

Any

2.6.2.38

3

− 52

−4

− 21

0

Any

Any

2.6.2.86

3

− 52

− 27

− 21

− 21

Any

Any

2.6.2.13

3

− 52

−3

− 21

0

Any

Any

2.6.2.32

3

− 12 5

− 13 5

− 53

− 57

Any

Any

2.6.2.30

3

− 73

− 10 3

− 35

− 35

Any

Any

2.6.2.34

3

− 73

− 10 3

− 35

1 3

Any

Any

2.6.2.88

3

− 73

−3

− 32

0

Any

Any

2.6.2.70

3

− 73

− 38

− 35

− 31

Any

Any

2.6.2.42

3

− 11 5

− 12 5

2

3

Any

Any

2.6.2.113

3

−2

1

Any

2(B 2 +1) (B 2 +3)2

Any

2.6.2.132

3

−2

−3

−2

0

Any

Any

2.6.2.104

3

−2

−3

−1

0

Any

Any

2.6.2.12

Any

2.6.2.145

3

(

¡

−B

2 −1

3

−2

−3

1

2

6 − 25

3

−2

−3

1

2

6 25

Any

2.6.2.144

3

−2

−3

− 21

0

Any

Any

2.6.2.102

3

−2

−2

1

Any

2(B 2 +1) (B 2 +3)2

Any

2.6.2.116

3

−2

−2

1

−7

− 15 4

Any

2.6.2.117

3

−2

−2

1

−4

−6

Any

2.6.2.118

3

−2

−2

1

− 25

−12

Any

2.6.2.119

3

−2

−2

1

−2

−2

Any

2.6.2.120

3

−2

−2

1

− 35

− 63 4

Any

2.6.2.124

Page 367

368

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 25 (Continued) Solvable cases of the equation 

 = ( # 1   1 +

0 3

1

¡

−2

2

¡

−2

B

1

B

2

#

1

1

+#

2

  2 + 2 )(
 )#

2

Equation

1

− 35

− 34

Any

2.6.2.123

9 100

Any

2.6.2.122

3

−2

−2

1

− 35

3

−2

−2

1

− 35

3 16

Any

2.6.2.121

3

−2

−2

1

− 57

5 36

Any

2.6.2.125

−20

Any

2.6.2.128

3

−2

−2

1

− 21

3

−2

−2

1

− 21

4 25

Any

2.6.2.127

3

−2

−2

1

− 21

2 9

Any

2.6.2.126

12 49

Any

2.6.2.129

3

−2

−2

1

1 2

3

−2

−2

2

1

Any

6 − 25

2.6.2.131

3

−2

−2

2

1

Any

6 25

2.6.2.130

3

− 13 7

− 20 7

0

2

Any

Any

2.6.2.82

3

− 12 7

− 20 7

0

2

Any

Any

2.6.2.60

3

− 35

− 73

− 34

− 35

Any

Any

2.6.2.92

3

− 58

− 13 5

− 57

− 57

Any

Any

2.6.2.109

3

− 23

− 52

0

− 21

Any

Any

2.6.2.90

3

− 23

−2

− 23

−2

Any

Any

2.6.2.48

3

− 23

−2

0

− 21

Any

Any

2.6.2.46

3

− 23

−2

1 2

1

Any

12 49

2.6.2.143

3

− 34

− 10 3

− 35

− 35

Any

Any

2.6.2.36

3

− 34

− 83

− 35

− 31

Any

Any

2.6.2.40

3

− 34

− 73

− 35

− 35

Any

Any

2.6.2.14

3

− 34

− 43

0

− 21

Any

Any

2.6.2.15

3

− 79

− 15 7

0

2

Any

Any

2.6.2.59

3

− 67

− 53

− 21

0

Any

Any

2.6.2.16

3

− 78

− 15 7

0

2

Any

Any

2.6.2.79

3

−1

−2

−2

−2

Any

Any

2.6.2.110

3

− 32

− 43

− 35

− 35

Any

Any

2.6.2.115

3

− 21

−3

− 21

0

Any

Any

2.6.2.53

3

− 21

−2

− 21

1

Any

−20

2.6.2.141

3

− 21

−2

− 21

1

Any

4 25

2.6.2.134

Page 368

2.6. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

=

 1Ë E Ï ± 1

1

(Ï

ÒÍ

Ë  2Ë E Ï ±

) 1+

TABLE 25 (Continued) Solvable cases of the equation 

 = ( # 1   1 +

B

B

#

2

1

+#

2

2

(Ï

ÒÍ

)

Ë

369

2

  2 + 2 )(
 )#

0 ¡

3

− 12

−2

− 21

1

Any

2 9

2.6.2.137

3

− 12

− 23

− 21

0

Any

Any

2.6.2.45

3

0

−5

−3

1

Any

Any

2.6.2.52

3

0

−2

−3

−2

Any

Any

2.6.2.56

3

0

−2

0

− 21

Any

Any

2.6.2.54

3

0

− 23

− 21

0

Any

Any

2.6.2.89

3

0

− 32

− 35

− 35

Any

Any

2.6.2.114

3

0

− 21

0

− 21

Any

Any

2.6.2.101

− 35

Any

Any

2.6.2.39

1

2

¡

1

2

1

2

Equation

3

0

0

− 31

3

0

0

0

−1

Any

Any

2.6.2.11

3

0

0

0

− 32

Any

Any

2.6.2.69

Any

Any

2.6.2.31

3

0

0

0

− 21

3

0

0

2

0

Any

Any

2.6.2.57

3

2 5

−2

− 57

1

Any

5 36

2.6.2.139

3

2 3

−2

− 35

1

Any

− 63 4

2.6.2.147

3

2 3

−2

− 35

1

Any

− 34

2.6.2.136

3

2 3

−2

− 35

1

Any

9 100

2.6.2.135

3

2 3

−2

− 35

1

Any

3 16

2.6.2.138

3

1

−2

−2

1

Any

−2

2.6.2.133

3

1

0

−7

−3

Any

Any

2.6.2.17

3

1

0

−4

−3

Any

Any

2.6.2.10

3

1

0

−2

−3

Any

Any

2.6.2.55

Any

Any

2.6.2.47

3

1

0

−2

− 23

3

1

0

−2

0

Any

Any

2.6.2.103

− 34

Any

Any

2.6.2.91

3

1

0

− 35

3

1

0

− 35

− 31

Any

Any

2.6.2.41

3

1

0

− 35

1 3

Any

Any

2.6.2.87

− 53

Any

Any

2.6.2.29

3

1

0

− 57

3

1

0

− 21

0

Any

Any

2.6.2.27

3

1

0

0

− 21

Any

Any

2.6.2.85

Page 369

370

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 25 (Continued) Solvable cases of the equation 

 = ( # 1   1 +

0

1

¡

B

2

¡

B

1

#

2

1

+#

2

  2 + 2 )(
 )#

1

2

Equation

3

1

0

0

2

Any

Any

2.6.2.73

3

1

0

1

−3

Any

Any

2.6.2.50

3

1

0

1

0

Any

Any

2.6.2.43

3

1

0

1

3

Any

Any

2.6.2.99

3

3 2

−2

− 25

1

Any

−12

2.6.2.146

3

2

0

−5

−5

Any

Any

2.6.2.96

− 35

Any

Any

2.6.2.35

3

2

0

− 35

3

2

0

− 35

− 31

Any

Any

2.6.2.37

3

2

1

− 35

− 35

Any

Any

2.6.2.33

3

3

−2

−4

1

Any

−6

2.6.2.140

3

3

0

−7

−3

Any

Any

2.6.2.18

3

4

3

−7

−7

Any

Any

2.6.2.111

Any

− 15 4

2.6.2.142

3

6

−7

−2

1

2.6.2-2. Solvable equations and their solutions. 1.

ñ óò ò ó

= (


1

ñ ú

1


2ñ ú

+

ó ò )A ,

)(ñ

2

@

≠ 2,

1 b . Solution in parametric form:

 =  X (L

1

+U +

1 + 1 −2 1− + 1 − -, #  -  2−0 2 b . Solution in parametric form:

where #

1

= ¡

 =  X (L where # 2.

ñóò ò ó

= (


1

= ¡ 1

ñ ú

1

+1


2ñ

+

Solution:  = X 3.

ñóò ò ó

= (


ñ ú

Q

1

+

Solution:  = L

1

1

 - −2 

0 −2

L

1

–1

+


2ñ ú

1−

+

1 1−

1 +1

)(ñó ò )A ,

1 +1

@

2

+1

+1

)(ñó ò )2 ,

X exp O − ¡

#

1

1

+1

1

þ

+

1−

2−

1

2−

= 'U ,

2,

M

+

= 'U ,

-.

) - −2 U + L 1−

≠ –1.

2,

M

-.

≠ –1. 1

+ (2 − 0 ) #

þ +

2 +1

2

) - −2 U + L

2 + 1 −2  -  2−0

≠ 2,

# 1 (2 − 0 ) + ¡

ACU +

= A2¡

2

2 +1

þ

≠ –1,

1

2 + 1 −2  -  2−0

= A2¡

2

−U +

-, #

ACU +

þ

1 +1

1

≠ –1,

−

¡

#

2

2

S - −2

2 ln

þ

+1

M

2

≠ –1.

+

2 +1

PuM

+L

2.

+L

2.

Page 370

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

4.

ñóò ò ó

= (


1

ñ ú

Solution:  = L 5.

ñóò ò ó

= (


1


2ñ

+

ïƒ÷

−

X

1

)(ñó ò )2 ,

O

2


2 ïƒ÷

+

1

–1

2

þ #

exp O −

)(ñó ò )A ,

1

1

Ë  2Ë E Ï ±

ÒÍ

(Ï

)1+

2

2

ÒÍ

(Ï

)

Ë

2

371

≠ –1.

+

1

+1

¡

 1Ë E Ï ±

=

@

+1

+L

P M

ù

≠ 1,

2.

ù

≠ –1,

1

≠ –1.

2

1 b . Solution in parametric form:

 = U , B

= /X  L

+ 1 −  1 −2 1−   -, # 1−0 2 b . Solution in parametric form: where #

1

1

=

 = U , where # 6.

ñóò ò ó

1

= (


= 1

ñóò ò ó ñóò ò ó

= (


1

ïƒ÷

= (


1

ñóò ò ó

ïƒ÷

=L

= (


1

ï –ú

1−



-, #

L

+

# 1 (1 − 0 )   B +1

1


2 ïƒ÷ )ñ óò


2ï 1

1 –3

–1

@

ù

,

2

X exp O

#

B

 

1

1+1

)ñó ò ,

1

+


2 ï –ú

ù

1 +1

1

≠ –1,

B

1−



AIU  2 −2

- M U + L 2,

-.

2 +1

1−



1 1−



2 +1

1 1−



-

U + L 2, M

-.

≠ –1.

+ (1 − 0 ) #

+

AIU  2 −2

2 + 1 −  1−0

+1

1 +1

ù

−U 

1

≠ 1,

ù

#

2

-  + L 2. M

≠ –1.

 

2

2+1

1 1−

 S

2 ln

2 +1

 + L 2.

P3M

≠ –1.

# 1  XZ O 2 exp O  B +1

ñ ú

+ 1 −  1−0

B

=A

2

)(ñó ò )A ,

1

+

1 −2

–1

+

1

 - −

1 +1

2

= /XÊÝL


2ï Q

=L

Solution: 9.

+ =X

Solution: 8.

+1

1

0 −1

ïƒ÷

Solution: 7.

B

B

=A

2

+U 

1

2 –3

ñ ú

+1

P3M

 + L 2.

)(ñó ò )3 ,

2

þ

þ

≠ –2,

1

≠ –2.

2

1 b . Solution in parametric form:

 = U Ê X L Q =  XÊ L

1

Q

where #

1

= 12  +

1 +4

 −+

1 −2

(

1

¡

1

+ U −+

+ U −+

1 −2

+ 2), #

1 −2

2 −2

 +

2 +4

1 2

J

−1 2



2 −2

ACU − +

=A

2

ACU − +

M

J

−1 2



U + L 2S M

 −+

2 −2

(

2 b . Solution in parametric form:

 = U X  L Q = X L

1

Q

where #

1

= − 21  +

1 +4

 −+

1 −2

(

¡

1

1

−U

− U −+

+ 2), #

−

+

1 −2

1 −2

2

ACU − +

ACU − +

=A

1 2

 +

2 −2

2 −2



2 +4



U + L 2S

J

2

¡

−1 2

J

−1 2

 −+

M

(

¡

,

,

+ 2).

U + L 2S

2 −2

−1

U + L 2S M

−1

2

−1

−1

,

,

+ 2).

Page 371

372

SECOND-ORDER DIFFERENTIAL EQUATIONS = (
1 ï –4 ñ +
2 ï –3 )(ñó ò )3 . 1 b . Solution in parametric form:

ñóò ò ó

10.

 = U ” ù where #

1

L ü

= 32  5 

−3

U M

+U

1

, #

AIU

−3

−2

= A  4 

2

+L

−2

−1

= ” ù

,

2

•

L ü

1

U M

+U

−3

AIU

−1 2

+L

2

−2

+L

,

•

.

2 b . Solution in parametric form:

 = U*” ù where #

ñóò ò ó

11.

1

= (


ü

= − 23  5 


2ï

+

1

−U

−3

, #

2

–1

AIU

−3

£ L

− 2#

1

1

 − 2#

M

1

U

= (
1 ï –1 ) 2 ñ –5 ) 2 +
2 ï Solution in parametric form:

,

 = where 

=L

1

2

where #

1

•

ln  )−1 J

2

2

U

)

+L

2

2

•

−U

−3

£ L

1

M

− 2#

1

2  −  * V £ 3 cos( ÑU ) − sin( ÑU )W. ,

#

) ñ

)

–5 3 –7 3

1

, #

2

AIU

−2

−1

,

•

U

ln U − 2 #

2

U

+L

−1 2

•

.

=



1

,

= −16 , 3 ,   = , £ 3.

2

)(ñó ò )3 .

1 36

#

2

U 4 + L 1U 3 + L 2U 2 + L 3U 

= 

1 36

#

2

U 4 + L 1U 3 + L 2U 2 + L 3U 

3

− 3L

L

= ” ù

,

 =

1

1

U

)(ñó ò )3 .

#  2  * + L 2  −  * sin( ÑU ) −

= 9L

ü

L

M

 + L 2. M

−1

+L

) ñ

= (
1 ï –5 ) 3 ñ –4 ) 3 +
2 ï Solution in parametric form:

= /” ù

,

)3 .

, 2L 1  2 *



2

−1 2

.

–1 2 –7 2

ñóò ò ó

14.

óò

ln U − 2 #

ñóò ò ó

13.

−2

)(ñó ò )3 .

− 2#

1

+L

−2

= A  4 

= (
1 ï –1 ñ –2 +
2 ñ –3 )(ñ Solution in parametric form:

 = U*” ù

15.

1

ñ óò ò ó

12.

]

L

= X (L

Solution:

U M

−1 1 9 −1

 #

2

3J 2 U 3 + 3L 1 U 2 + 2L 2 U + L 3  ,

,

2 2.

In the solutions of equations 15–18, the following notation is used:

²

1=

²

2=

L

1+

U

−3

ü

L

1−

U

−3

ü

AlU

−2,

«

1=

X

AlU

−2,

«

2=

X

= (
1 ñ –4 ) 3 +
2 ï –1 ) 2 ñ Solution in parametric form:

ñ óò ò ó

U M

²

)

²

M

U

–4 3

1 2

)(ñ

+L

2,



1=

U − ² 1 « 1, 5

1=3

U 3

2 1 + 3(1

+L

2,



2=

U − ² 2 « 2, 5

2=3

U 3

2 2 + 3(−1

óò

)3 .

 = !  2 ,   −2 J 3 , # 2 = 13  3 J 2  −2 J 3 (−1)  where # 1 = A 29 

= 'U +1

−3

ArU ) « 12, ArU )« 22.

« 3 ,

; , = 1 and , = 2.

Page 372

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

ñóò ò ó

16.

= (


1

ï –1 ) 2 ñ –7 ) 6 +
2 ñ –5 ) 3 )(ñóò

=

 1Ë E Ï ± 1

1

Ë  2Ë E Ï ±

ÒÍ

(Ï

)1+

2

2

(Ï

ÒÍ

)

Ë

373

2

)3 .

Solution in parametric form:

 = U 3 « −3   2 , = 'U 3 « −3 , where # 1 = 13  3 J 2  −5 J 6 (−1)  , # 2 = T 29  −1 J 3 ; , = 1 and , = 2.

ñóò ò ó

17.

= (


ï –7 ñ

1

+


2ï

–3

)(ñó ò )3 .

Solution in parametric form:

J « 1J 2, 

−1 2

 = U where #

ñóò ò ó

18.

1

= (


=

1 8 −3 , 36

 

ï –7 ñ

1

3

+

#

2


2ï

1 4 −2 ; 36

=A

–3

 

= 'U

−3

5 ,

, = 1 and , = 2.

)(ñó ò )3 .

Solution in parametric form:

 = U 5 J 2 « 1 J 2 5  −1 , where #

]

19.

1

=

1 8 −5 , 36

 

#

2

1 4 −2 ; 36

=T

 

= 'U 3 5  −1 ,

, = 1 and , = 2.

In the solutions of equations 19–22, the following notation is used:

²

1

=

²

2

=

ñóò ò ó

L

1

ACU ;

+1

L

1

ACU ;

+1

ï –ú

–3

ü ü

= (


1

«

1

= X†M

−U ,

«

2

=X

U

+U ,

ñ ú

+

²


2ñ

–3

²

U M

1 2



+«

2 1,



−«

2 2.

+L

2,



1

= 2²

1

−«

1,

5

1

= 4(U − ²

1 1)

+L

2,



2

= 2²

2

+«

2,

5

2

= 4(U − ²

2 2)

)(ñó ò )3 ,

þ

≠ –2.

Solution in parametric form:

 = U « −1 ,

= ˜« −1 ,

= = − − 3, ¡ 1 + +4 − + −2 1   ( + 2), # 2 = 2    (−1) ; , = 1 and , = 2. where # 1 = A 2  ¡ 20.

ñóò ò ó

= (


1

ñ

+


2ï

)(ñó ò )A ,

@

@

≠ 1,

≠ 2.

Solution in parametric form:

 = !  , where # 21.

ñóò ò ó

1

= (


=  1

−1

ï ñ ú

# 2 (−1)  +

+1


2 ñ –ú

, #

–2

2

=− 3

ú

)(ñó ò ) 2ú

= ˜«  ,

=

2 2Q +5 +3

A

= =

(= + 1) 



S

1

1 , 0 −2

J; ; , = 1 and , = 2.

.

Solution in parametric form:

 = !5  , where #

1

=

=

4(= + 2)



−1

 ;

1 +2

= ˜«  ;

+2

2

1

2(= + 1)  ; S , # Q (= + 2) 

T

+3

, = =− ¡ +1 ¡

,

2

2 − +2

=  ;

# 1 (−1)  ; , = 1 and , = 2.

Page 373

374

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

22.

= (


ïƒ÷ ñ

1


2 ï –÷

+

)(ñó ò )

–2

÷

3 +4 +1

÷

.

Solution in parametric form:

 = !«  ; where #

]

=−

1

=

 ; 4(= + 2)

1 +2 −1



+2

= ˜5  ,

,

= =−

1

2(= + 1)  ; S , # Q (= + 2) 

T

=

2

2B + 3 , B +1

2 − +2

;

˜# 1 (−1)  ; , = 1 and , = 2.

In the solutions of equations 23–26, the following notation is used:

23.

²

1

=

²

2

=

ñóò ò ó

L

1

+ U3A ln U ,

«

1

=X

L

1

− U3A ln U ,

«

2

=X

ü ü

= (


ñ

+

1


2 ï )ñ óò

U M

²

² M

U

1 2



+«

2 1,



−«

2 2.

+L

2,



1

= 2²

1

−«

1,

5

1

= 4(U − ²

1 1)

+L

2,



2

= 2²

2

+«

2,

5

2

= 4(U − ²

2 2)

.

Solution in parametric form:

 = !  , = ˜«  , where # 1 =    −1 # 2 (−1)  , # 2 = A 12  −2 ; , = 1 and , = 2.

24.

ñ óò ò ó

= (


1

ñ

+


2ï

óò

)(ñ

)2 .

Solution in parametric form:

 = !«  , = ˜  , where # 1 = T 12  −2 , # 2 =  −1 ˜# 1 (−1)  ; , = 1 and , = 2. 25.

ñóò ò ó

= (


+

1


2ï ñ

–2

)ñó ò .

Solution in parametric form: where # 26.

ñóò ò ó

= (


1

=  1

ï –2 ñ

−2

# 2 (−1)  , #

+




2 )(

ñóò

 = !5  , = ˜«  ,   , = 1 and , = 2. 2 = A 1 −2 −2 ; 8

)2 .

Solution in parametric form:

]

 = !«  , = ˜5  , where # 1 = T 18  2  −2 , # 2 =  −2 ˜# 1 (−1)  ; , = 1 and , = 2.

In the solutions of equations 27–30, the following notation is used:

² ² ² Æ Æ Æ

U  1 + L 2U  2 + L 3U  3 , À  2 = ( L 1 + L 2 U )  * + L 3  Ÿ* ,  3 = L 1  * +  * ( L 2 sin  ÑU + L 3 cos  ÑU ),  1 + L 2 , 2U  2 + L 3 , 3U  3 , 1 = L 1, 1U À  2 = ( , L 1 + L 2 + , L 2 U )  * +  ·L 3  Ÿ* ,  3 = , L 1  * +  * [( °L 2 −  ·L 3 ) sin  ÑU + ( °L 8 8


1 = U (Æ 1) , 2 = (Æ 2) , 3 = (Æ 3) , 1

8

=L

1

*

*

3

+  ·L

2 ) cos

 ÑU ],

*

Page 374

2.6. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

=

 1Ë E Ï ± 1

1

Ë  2Ë E Ï ±

ÒÍ

(Ï

)1+

2

2

(Ï

ÒÍ

)

Ë

375

2

where , 1 , , 2 , and , 3 (real numbers) or , and °…AC™  (one real and two complex numbers) are roots 8 of the cubic equation Œ 3 − 12 $ 2 Œ − 12 $ 1 = 0. The subscripts of the functions ² + , Æ + , and + ( = 1, 2, 3) are selected depending on the sign of the expression ¢ = 2 $ 23 − 27 $ 12 :

¡

¢ > 0 subscript

if if if

If 2 $

3 2

= 27 $

2 1

= 1, = 2, = 3.

¡

¢ = 0 subscript ¡

¢ < 0 subscript ¡

(subscript 2), then

, = ( 16 $ 2 )1 J 2 ,   = −2( 61 $ 2 )1 J 2 if $ , = −( 61 $ 2 )1 J 2 ,   = 2( 61 $ 2 )1 J 2 if $

x3y'z|{}~/€

1

< 0,

1

> 0.

The expressions for ² + , and Æ + contain three constants L 1 , L 2 , and L 3 . One of them can be arbitrarily fixed to let it be any nonzero number (for instance, we can set L 3 = A 1), while the other constants remain arbitrary. 27.

28.

29.

= (
1 ï –1 ) 2 ñ +
2 )(ñ ó ò )3 . Solution in parametric form:

ñ óò ò ó

 = Æ 2+ ,

=² + ,

= (
1 ñ –3 +
2 ï –1 ) 2 ñ –7 ) 2 )(ñó ò )3 . Solution in parametric form:

ñóò ò ó

2  = ² −1 + Æ + ,

= ² −1 + ,

1

= −$

#

1

where

= (
1 ï –7 ) 5 ñ +
2 ï –3 ) 5 )(ñó ò )3 . Solution in parametric form:

ñóò ò ó

#

where

#

1,

= −$

2,

8  = !² 5+ J 2 , =  (2 Æ 2+ − 4 ² + + + $ 5 12 J 5 −3 −2   $ 1 , # 2 = −  −4 J 5 ˜ # 1 $ 2 . where # 1 = 32 30.

= (
1 ï –3 ) 5 ñ –12 ) 5 +
2 ï Solution in parametric form:

ñóò ò ó

) ñ

)

–7 5 –13 5

2

2

= −$

#

2

1.

)(ñó ò )3 .

8

8

+ +$

2

² 2+ 

−1

,

In the solutions of equations 31 and 32, the following notation is used:




 2= where , = Å

1 2|

$ L 1   * + L 2  − * −




 1=

31.

= −$

² 2+ ),

−1  = !² 5+ J 2  2 Æ 2+ − 4 ² + + + $ 2 ² 2+  , =  2 Æ 2+ − 4 ² + 5 8 J 5 2 J 5 −2 where # 1 = − 32   $ 1 $ 2 , # 2 = 325  12 J 5  3 J 5 $ 1−2 .

]

2.



$







L

1

sin( ,!U ) + L

2

1

U

2

cos( ,!U ) −

$ , (L 1   * − L 2  − * ) − $

, [L

1 cos(

,!U ) − L

2

$

1

$

U

2

1 2

sin( ,!U )] −

$ $

1 2

if $

2

> 0,

if $

2

< 0,

if $

2

> 0,

if $

2

< 0,

$ 2 |.

= (
1 +
2 ï –1 ) 2 )(ñ ó ò )3 . Solution in parametric form:

ñ óò ò ó

 =  22 ,

=  1,

where

#

1

= −$

2,

#

2

= −$

1.

Page 375

376 32.

SECOND-ORDER DIFFERENTIAL EQUATIONS = (
1 ï –1 ) 2 ñ –5 ) 2 +
2 ñ Solution in parametric form:

ñóò ò ó

 =

–3

)(ñó ò )3 .

−1 2 1 2,



−1 1 ,

=

#

where

= −$

1

#

1,

2

= −$

2.

]

In the solutions of equations 33–36, the following notation is used: For $ 1 > 0, { = L   * + L  −  * + L sin( ,!U ), , =  4 $  1 J 4 , 1 1 2 3 3 1

{

For $

33.

1

À

< 0,

{

2

= , (L

1

  * − L 2  − * ) + , L À

À

À

* [ L 1 sin( °˜U ) + L 2 cos(°˜U )] + L 3  − * sin( °˜U ), ° =  − 13 $ 1  1 J 4 , { = °  * [( L − L ) sin(°˜U ) + (L + L ) cos(°˜U )] − °L  − * [sin(°˜U ) − cos(°˜U 2 1 2 1 2 3 1

=

= (
1 ï –5 ) 3 ñ 2 +
2 ï –5 ) Solution in parametric form:

ñóò ò ó

3

ñ )(ñ óò

where $

1

= −#

$

1,

2

= −#

2;

{

J

3 2 2 ,

= (
1 ï –5 ) 3 ñ –7 ) 3 +
2 ï Solution in parametric form:

the constants L

 = O where $

1

= −#

$

2,

2

= −#

{

1;

) ñ

)

–5 3 –10 3

1

−

#

1,

2#

{

−2 1

#

2 2

J

the constants L

= (
1 ï –5 ) 3 ñ 2 +
2 ï –5 ) 3 )(ñ Solution in parametric form:

ñ óò ò ó

óò

1

= −#

$

1,

2

= −#

2;

1,

{

= (
1 ï –5 ) 3 ñ –4 ) 3 +
2 ï Solution in parametric form:

) ñ

1

= −#

2,

$

2

= −#

1;

)

–5 3 –10 3

 = where $

, are related by the constraint

3

if #

1

> 0,

if #

1

< 0.

−2 2

{

#

1

−1

1

−

3

are related by the constraint

2#

if #

2

> 0,

if #

2

< 0.

1,

{

=

{

2

P

,

1,

L 2 , and L

−1 1

L 1L 3 = # # 2 4 L 1 L 2 + L 32 = − 21 #

ñóò ò ó

1

L 2 , and L

J

3 2 2 ,

the constants L − 14

36.

2

)3 .

 = where $

2#

= O

L 1 L 3 = 161 # 21 # −2 2 2 1 2 4L 1 L 2 + L 3 = 4 # 1 #

35.

#

−

L 2 , and L

3 2 2 ,

P

2

1

)(ñó ò )3 .

−1

1

{

=

2 L 1 L 3 = 161 # −2 1 # 2 4 L 1 L 2 + L 32 = 14 #

ñóò ò ó

)].

)3 .

 =

34.

cos( ,!U ).

3

−1 1

#

2

3

are related by the constraint

if #

1

> 0,

if #

1

< 0.

)(ñó ò )3 .

{

J

−1 3 2 1 2 ,

the constants L

1,

L 1 L 3 = − 14 # 1 # −1 2 4 L 1 L 2 + L 32 = − 21 #

=

{

−1 1 ,

L 2 , and L 1

#

−1 2

3

are related by the constraint

if #

2

> 0,

if #

2

< 0.

Page 376

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

]

 1Ë E Ï ±

=

1

Ë  2Ë E Ï ±

ÒÍ

(Ï

1

)1+

2

2

ÒÍ

(Ï

)

Ë

377

2

In the solutions of equations 37 and 38, the following notation is used:

1 b . For $

1

{

> 0, $

2

=L

1

1

≠ 0:

  * + L 2  − * + L

3

{

sin  ÑU ,

= , (L

2

1

  * − L 2  −  * ) +  ·L

3

cos  ÑU ,

where

, =,

2 3 [(

2 2

$

2 b . For − $

2 2

=L

1

{

1

+ 3$

< 3$

1)

J + $ 2] . 1J 2,   = ,

1 2

< 0, $

1

2

2 3 [(

> 0:

2 2

$

{

U  1 + L 2 U − 1 + L 3 U  2 + L 4 U − 2 ,

2

+ 3$

1)

= , 1(L

1

J − $ 2] . 1J 2, 4, 2 L 1 L

1 2

2

2

+ 

2 3

L

= 0.

U  1 − L 2 U −  1 ) + , 2 ( L 3 U  2 − L 4 U −  2 ),

where

, 3 b . For − $

{

1

=L

< 3$

2 2

sin  

1

=

1

1

J ] . 1 J 2 , , 2 = , 2 [ $ 2 − ( $ 2 + 3 $ 1 )1 J 2 ] . 1 J 2 , 2 3 2 1J 2 ( L 1 L 2 + L 3 L 4 )( $ 2 + 3 $ 1 ) + ( L 1 L 2 − L 3 L 4 ) $ 2 = 0.

,

2 3[

1

< 0, $

U +L

$

2

2

cos  

2

2 2

+ ($

+ 3$

1)

1 2

< 0: 1

U +L

3

sin  

{

U ,

2

= 

2

1(

L

1

cos  

1

U −L

2

sin  

1

U ) +   2L

cos  

3

2

U ,

where

  1 = , − 23 [ $ 4 b . For $

{

1

2 2

= (L

1

2 +(

+ 3$

+L

1

2

2 2 +3

$

J

$ 1)1 J 2] .

= 0, $

1 2

,  

> 0:

2

U )  * + ( L

+L

3

4

1J 2 , =  23 $ 2  , 5 b . For $ 22 + 3 $ 1 = 0, $ 2 < 0:

{

1

= (L

1

+L

2

  =  − 32 $

6 b . For 3 $

{ {

1

(L

1

< −$

= (, L

4

−L

= ( ·L

2

 1J 2,

2

L 1L

3

1

2 2 +3

$

+L

2

L 3 )  23 $ 1

+L

 − 23 $

2

$ 1)1 J 2] .

+, L 2

2

,   12( L

U )  * − ( , L

 1J 2 + 2L 2 L

+  ·L

2

 1J 2 + L

2 2

3

J

1 2

4

2 1+

3

2 3

−L

4

+, L

2

−  ·L

2 3 = 0.

4

U ) −  * ,

= 0.

U ) cos  ÑU + ( L +L

L 22)−   22 L

3

U ) sin  ÑU ,

= 0.

2 2:

 −  * sin  ÑU ,  − * (  cos  ÑU − , sin  ÑU ), 2 =  * [( , L 2 +  ·L 1 ) cos  ÑU + ( , L 1 −  ·L 2 ) sin  ÑU )] + L 3  1

=   * (L

2

2 −(

2

L

{

U ) sin  ÑU + L 3 U cos  ÑU ,

where

− 23 [ $

{

U ) −  * ,

where

,

2=

1

sin  ÑU + L

cos  ÑU ) + L

2

3

where

, =, 37.

1 3[

$

2

+ (−3 $

1)

J ] . 1J 2,   = ,

1 2

$

2

= (
1 ï –5 ) 3 ñ 2 +
2 ï –1 ) 3 )(ñó ò )3 . Solution in parametric form:

+ (−3 $

1)

J ] . 1J 2, L $ 2

1 2

2

+L

1 (−

$

2 2

− 3$

1)

J = 0.

1 2

ñóò ò ó

 =

38.

1 3 [−

{

J

3 2 2 ,

= (
1 ï –1 ) 3 ñ –8 ) 3 +
2 ï Solution in parametric form:

ñóò ò ó

 =

{

{

J

−1 3 2 1 2 ,

=

{

) ñ

1,

)

where

–5 3 –10 3

=

{

$

1

= −#

$

1

$

1,

2

= −#

$

2

2.

)(ñó ò )3 .

−1 1 ,

where

= −#

2,

= −#

1.

Page 377

378

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 39–42, the following notation is used:

{ {

1

=

™

2

=

™

L 1  Ÿ* + L 2  −Ÿ* + L 3 U L 1 sin  ÑU + L 2 cos  ÑU + L 3 U   ( L 1 Ÿ * − L 2  −Ÿ* ) + L 3   ( L 1 cos  ÑU − L 2 sin  ÑU ) + L

3

if $ if $

> 0, < 0,

if $ if $

> 0, < 0,

where   = | 34 $ |1 J 2 . 39.

ñ óò ò ó

= (


1

ï –1 ) 3 +
2 ï –5 ) 3 )(ñ óò

)3 .

Solution in parametric form:

{

 = where $

= −#

1;

the constants L 3( # 3( #

40.

ñóò ò ó

= (


1

1

L

1

L

2 3 2 3

L 2 , and L

1,

{

 = 2;

the constants L 3( # 3( #

41.

ñóò ò ó

= (


1

ï –5 ) 3 ñ

+

+# 1+# 1


2ï

)

–1 3

2 3) 2 3)

L

2 2L

1,

2 1

2 2)

+L

L

1

= 0 if # =0 if #

2

1 1

> 0, < 0.

)3 .

{

J

−1 3 2 1 2 ,

L 2 , and L

1,

{

are related by the constraint

3

2)

ï –5 ) 3 ñ –4 ) 3 +
2 ï –1 ) 3 ñ –8 ) 3 )(ñóò

= −#

=

− 4 # 21 ( L 2 2 ) − 16 # 1 L

+# +#

Solution in parametric form: where $

J

3 2 2 ,

{

−1 1 ,

are related by the constraint

3

− 4 # 22 ( L − 16 # 22 L

=

2 1

2 2)

+L

1L

= 0 if # =0 if #

2

2 2

> 0, < 0.

)(ñó ò )3 .

Solution in parametric form:

 =O where $

42.

ñóò ò ó

= (


= −#

1

2;

{

2

the constants L 2

L

3#

2

L

2 3 2 3

3#

#

−

2#

2

U P

L

L L

#

ï –5 ) 3 ñ –7 ) 3 +
2 ï –1 ) 3 ñ –8 ) 3 )(ñóò

=

,

{

1

−

#

1

4#

2

U 2,

are related by the constraint

3

2 2 2 2( 1 + 2 ) − 2 9 2 1 2 − 16

L

− 16 #

J

3 2

L 2 , and L

1,

− 4#

1

9 16 2 1

2 −2 1 2 = −2 2 =0

# #

#

0 if #

2

> 0,

if #

2

< 0.

)3 .

Solution in parametric form:

 = O where $

= −#

2;

{

1

#

−

4#

1 2

U

2

−1

P

the constants L 3#

2

L

3#

2

L

2 3 2 3

− 4# − 16 #

O

{

2

−

#

2#

L 2 , and L

1,

1 2 3

2 2 2 2( 1 + 2 ) − 2 9 2 1 2 − 16

L

L

L L

#

U

J

3 2

P

,

= O

{

1

#

−

4#

1 2

U

2

P

−1

,

are related by the constraint 9 16 2 1

#

#

2 −2 1 2 = −2 2 =0

#

0 if #

2

> 0,

if #

2

< 0.

Page 378

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

]

1

1

Ë  2Ë E Ï ±

ÒÍ

(Ï

)1+

2

2

(Ï

ÒÍ

)

Ë

379

2

In the solutions of equations 43–48, the following notation is used:

™ ®ø 11J J 33((U U ))

 =

™ ;n

= 5

43.

 1Ë E Ï ±

=

ñóò ò ó

=L

1

= (


1

for the upper sign (Bessel function), for the lower sign (modified Bessel function),

J U ) for the upper sign (Bessel function), 1 J 3 ( U ) for the lower sign (modified Bessel function),

1 3(

 + L 2 + ý/ eO˜ ‹Xv

ï ñ

+




2 )(

ñóò

U − 3XÈ M

U , M P

  =

1

for the upper sign, for the lower sign.

2 ™ −1

)3 .

Solution in parametric form:

 = U 1J 3 5 , 44.

ñóò ò ó

= (


1

ñ

–3


2ï ñ

+

= 'U 2 J 3 ,

–5

#

where

1

=A

9 −3 , 4

#

1

= − 49 'ý ,

#

1



#

2

= − 94 

#

2

−2

ý .

)(ñó ò )3 .

Solution in parametric form:

 =U 45.

ñ óò ò ó

= (


1

J 5 ,

−1 3

J ,

−2 3

= 'U

ï –1 ) 2 ñ –1 ) 2 +
2 ñ –3 ) 2 )(ñ óò

where

=A

9 3 4 .



)3 .

Solution in parametric form:

 = U 46.

ñóò ò ó

J ( U 5
+ 1 5 )2 , 3

−2 3

= (


1

= 'U 2 J 3 5

*

ñ –3 ) 2 +
2 ï –1 ) 2 ñ

–2

2

,

where

 J 

J

= 12   

−1 2

1 3 2 −3 2 , 3

=A

#

= 12   

2

J ý .

−1 2

)(ñó ò )3 .

Solution in parametric form:

 = U 47.

ñ óò ò ó

J 5

−4 3

= (


−2

1

ï –2 ñ

(U 5 +

*


+ 1 5 )2 , 3


2ï

)

–3 2

)(ñ

J 5

−2 3

= 'U

óò

−2

,

J ý ,

where

#

+ 2ý±U 5

− (U 5
+ 13 5 )2 W ,

1

#

2

=A

 J

1 3 2 . 3

)3 .

Solution in parametric form:

 = U 2 J 3 5 where # 48.

ñóò ò ó

1

= (


= − 29  3  1

−3

, #

2

2

= 'U

,

= −

J V T U 2 5

−2 3

2

*

J 'ý…# 1 .

−1 2

ï –3 ) 2 ñ –3 ) 2 +
2 ï –2 ñ

–2

)(ñó ò )3 .

Solution in parametric form:

 = U 4 J 3 5 2 V T U 2 5 2 + 2ý±U 5 − (U 5
+ 13 5 )2 W *
+ 1 5 )2 W −1 , = 'U 2 J 3 V T"U 2 5 2 + 2ý±U 5 − (U 5 3

−1

,

*

where #

1

= 92  5 J 2 

J ý , #

−1 2

2

= − 92  3 .

Page 379

380 49.

SECOND-ORDER DIFFERENTIAL EQUATIONS = (
1 ï ñ ú 1 +
2 ñ ú 2 )(ñó ò )3 , Solution in parametric form:

ñóò ò ó

þ

 =U Á 5 ,

≠ –2.

1

2

= 'U

Á,

@ = ¡

where

5

=L

# ]

1

4 2/ ý   O˜ ‹XÈU   U − 3XÈU  U , ( 1 + 2)2 M M P

 + L 2 +

¡

2

, = ¡

2

¡

 =

™ ®ø Á Á ((U U ))

for the upper sign (Bessel function), for the lower sign (modified Bessel function),

=

™ n ; Á Á ((U U ))

for the upper sign (Bessel function), for the lower sign (modified Bessel function),

1 4(

=A

1

1 , 1+2

¡

1

+ 2)2  − +

1 −2

#

,

2

= −  −+

ý ,

2

1 2

™

  =



−1

−

¡

1

+1

1+2

;

for the upper sign, for the lower sign.

In the solutions of equations 50–56, the following notation is used:

˜

™

Á =

:

Á =ƒ Ä

=

™ ™

Ä

1

=

Ä

2

=Ä

for the upper sign (Bessel function), 1 ø Á (U ) L 1 ®Á (U ) for the lower sign (modified Bessel function), L 2 n Á (U ) for the upper sign (Bessel function), L 2 ; Á (U ) for the lower sign (modified Bessel function), ˜ ˜ :
:
Á + @ Á , 4IÁ = Um Á + @!m|Á , 1 ’ÑÁ + ƒ 2 Á , m|Á = ý 1 ’ÑÁ + ý 2 Á , sÁ = U : Ám|Á if ¢ = −( ƒ 1 ý 2 − ƒ 2 ý 1 )2 , ˜ ˜ 2 2 ƒt’ Á + ýs’ÑÁ Á + = Á if ¢ = 4 ƒ±= − ý 2 , : Á4IÁ + m|ÁsÁ if ¢ = −( ƒ 1 ý 2 − ƒ 2 ý 1 )2 , U Ä
+ 2 @Ä if ¢ = 4 ƒ±= − ý 2 , 2

2 2 2 +   2¢ ,   =™  for the upper sign, 1 A 4U Ä

L

™

’ÑÁ =

−1

for the lower sign,

where the prime denotes differentiation with respect to U . 50.

= (
1 ï ñ +
2 ï –3 )(ñ ó ò )3 . Solution in parametric form:

ñ óò ò ó

 = U 1 J 3 Ä where @ = 13 , # 51.

1

=A

9 −3 , 4



#

2

9 4 = − 16  

= (
1 ï ñ ú +
2 ï –3 )(ñ ó ò )3 , Solution in parametric form:

ñ óò ò ó

þ

J ,

  ¢ .

−2 2

≠ –2.

 = U Á Ä

where @ = 52.

¡

1 , # +2

1

=A

 +

1 − 4

= (
1 ï –3 +
2 ï ñ –5 )(ñó ò )3 . Solution in parametric form:

ñóò ò ó

where @ = 13 , #

1

9 4 = − 16  

−2

−2

(

¡

= 'U 2 J 3 ,

1 2

J ,

1 2

+ 2)2 , #

J Ä

 = U

−1 3

  2¢ , #

2

=A

2

Á ,

1 4 = − 16  

−2

= 'U 2

J ,

1 2

= 'U

  2 ¢ ( + 2)2 . ¡

J ,

−2 3

9 3 4 .



Page 380

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

53.

ñóò ò ó

= (


ï –1 ) 2 ñ –1 ) 2 +
2 ñ

1

=

 1Ë E Ï ± 1

1

ÒÍ

(Ï

Ë  2Ë E Ï ±

)1+

2

2

(Ï

ÒÍ

)

Ë

381

2

)(ñó ò )3 .

–3

Solution in parametric form:

J Ä

−2 3

 = U where @ = 13 , # 54.

ñ óò ò ó

= (


+

1

1


2ï

 J 

J

8 3 2 −3 2 , 3

=A

) ñ

–1 2 –2

óò

)(ñ

#

−1

= 'U 2 J 3 Ä ,

2 1,

Ä

2

= 2  

J Ä

−2

=A

8 3 2 . 3

2

¢ .

)3 .

Solution in parametric form: −4 3

 = U where @ = 13 , # 55.

ñóò ò ó

= (


ï –2 ñ

1

1

= 2 

+


2ï

  ¢ , #

−2 2 –3

2

2 1,

Ä

J Ä

−2 3

= 'U

−1

,

 J

)(ñó ò )3 .

Solution in parametric form:

 = U 2 J 3 Ä , where @ = 13 , # 56.

ñóò ò ó

= (


ï

–3

1

1

+

=

9 3 −3 , 128

 


2ï

ñ

–2 –2

#

9 4 = − 64  

2

J Ä

−2 3

= 'U −2

−1

Ä

2,

  2¢ .

)(ñó ò )3 .

Solution in parametric form:

 = U 4 J 3 Ä where @ = 13 , #

]

1

9 4 = − 64  

−2

  2¢ , #

2

2

= 'U 2 J 3 ­ Ä Ä

Ä

−1 2 ,

=

9 3 . 128

−1 2 ,



In the solutions of equations 57–72, the following notation is used:

 1 = Å A 4<

3 1

 2 = Å A 4<

3 2

− 2< + 2<

1

2

−L −L

2,

2,

U =ù

Å A 4<

U =ù

Å A 4<

< M

3 2

1

− 2<

3 1

M

<

1

−L

2

2

−L

2

2

+ 2<

−L

1;

−L

1.

The functions < 1 = < 1 (U ) and < 2 = < 2 (U ) are defined implicitly by the above elliptic integrals. For the upper signs, they are the classical elliptic Weierstrass functions < 1 = < (U + L 1 , 2, L 2 ) and < 2 = < (U + L 1 , −2, L 2 ). 57.

ñóò ò ó

= (


ï

1

2

+




2 )(

ñóò

)3 .

Solution in parametric form:

 = -<  , = 'U , where # 1 = T 6  −1  −2 , # 2 =    −2 (−1)  +1 ; , = 1 and , = 2. 58.

ñóò ò ó

= (


1

ñ

–3

+


2ï 2ñ

–5

)(ñó ò )3 .

Solution in parametric form: where #

1

=  (−1) 

+1

, #

2

= T 6

 = U

−1

< 

,

= 'U

−1

,

 ; , = 1 and , = 2.

−1 3

Page 381

382 59.

SECOND-ORDER DIFFERENTIAL EQUATIONS = (
1 ñ –9 ) 7 +
2 ï 2 ñ –15 ) 7 )(ñó ò )3 . Solution in parametric form:

ñóò ò ó

 = U (U 2 <  T 1), = 'U 7 , 1 where # 1 = 49  −5 J 7 (−1)  +1 , # 2 = T 496  −1  1 J 7 ; , = 1 and , = 2. 60.

= (
1 ñ –12 ) 7 +
2 ï 2 ñ –20 ) 7 )(ñó ò )3 . Solution in parametric form:

ñóò ò ó

−6

 = U where # 61.

1

=

1 49

J (−1) 



−2 7

+1

, #

2

2

(U

<  T  J

6 −1 6 7 ; 49

=T



= (
1 ñ +
2 ï )(ñ ó ò )5 ) 2 . Solution in parametric form:

= 'U

1),

−7

,

, = 1 and , = 2.

ñ óò ò ó

 =  [   − (−1)  U ], = 'U , where # 1 =    −1 # 2 (−1)  , # 2 = −2  −1  −1 (A 6   )1 J 2 ; , = 1 and , = 2. 62.

= (
1 ñ +
2 ï )(ñó ò )1 ) 2 . Solution in parametric form:

ñóò ò ó

=  [   − (−1)  U ],

 = U , where # 63.

1

= 2

−1 −1



( A 6   )1 J 2 , #

2

=

−1

= (
1 ñ –3 ) 4 +
2 ï ñ –5 ) 4 )(ñó ò )5 ) 2 . Solution in parametric form:

˜# 1 (−1)  ; , = 1 and , = 2.

ñóò ò ó

 =  [2U  − 2 <  + (−1)  +1 U 2 ], = 'U 4 ,   −1 J 2 # 2 (−1)  , # 2 = − 41  −1  1 J 4 (A 3   )1 J 2 ; , = 1 and , = 2. where # 1 =  64.

= (
1 ï –5 ) 4 ñ +
2 ï –3 ) 4 )(ñó ò )1 ) 2 . Solution in parametric form:

ñóò ò ó

 = U 4 , where # 1 = 14  1 J 4  −1 (A 3   )1 J 2, # 65.

U ],

J ˜# 1 (−1)  ; , = 1 and , = 2.

=

2

+1 2

−1 2

= (
1 ñ –13 ) 8 +
2 ï ñ –15 ) 8 )(ñó ò )5 ) 2 . Solution in parametric form:

ñóò ò ó

 = U where # 66.

=  [2U  − 2 <  + (−1) 

1

= 

−6

[2U 3   + 6U

J # 2 (−1)  , #

−1 4

2

=

2

<  T

2 + (−1)  U 4 ],

= 'U

−8

,

1J 2  J  T 6    ; , = 1 and , = 2.

1 −1 7 8 16



= (
1 ï –15 ) 8 ñ +
2 ï –13 ) 8 )(ñó ò )1 ) 2 . Solution in parametric form:

ñóò ò ó

 = U where #

1

1 7J 8 = − 16  

−1

−8

,

= 'U

1J 2  T 6   , #

−6

2

[2U 3   + 6U =

2

<  T

2 + (−1)  U 4 ],

J ˜# 1 (−1)  ; , = 1 and , = 2.

−1 4

Page 382

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

67.

ñóò ò ó

= (


1

)

ñ

–15 13


2ï ñ

+

)

 1Ë E Ï ±

=

1

ÒÍ

(Ï

1

)(ñó ò )5 ) 2 .

–20 13

Ë  2Ë E Ï ±

)1+

2

2

(Ï

ÒÍ

)

Ë

383

2

Solution in parametric form:

 = U [5U 3   − 20U 2 <  A 30 − (−1)  U 4 ], where # 68.

ñóò ò ó

1

= (


= 

−5 13

# 2 (−1)  , #

ï

) ñ

+

1

J

–20 13


2ï

=−

2

)

–15 13

2  65

 J

−1 7 13

= 'U

13

,

30  1 J 2 ; , = 1 and , = 2. 13 …P

OA

)(ñó ò )1 ) 2 .

Solution in parametric form: 13

 = U where # 69.

ñ óò ò ó

1

= (


2 7J  65

= 1


2ï

+

13 −1



)

–2 3

30  1 J 2 , # 13 rP

OA

)(ñ

óò

= 'U [5U 3   − 20U

,

2

J

=

−5 13

2

<  A

30 − (−1)  U 4 ],

˜# 1 (−1)  ; , = 1 and , = 2.

)3 .

Solution in parametric form: =  [   − (−1)  U ],

 = -< 3 , where # 70.

ñóò ò ó

1

= (


1 2

=T 1



−2

, #

2

 J 



1 5 3 −2 (−1) +1 ; 12

=

ï –2 ) 3 ñ –7 ) 3 +
2 ñ

–3

, = 1 and , = 2.

)(ñó ò )3 .

Solution in parametric form:

 = -< 3 V   − (−1)  U W where # 71.

ñóò ò ó

1

= (


 J  J

1



1 5 3 1 3 (−1) +1 , 12

=

#

2

=T

1 2

−1

=  V   − (−1)  U W

,

−1

,

 ; , = 1 and , = 2.

ñ –1 ) 4 +
2 ï ñ –7 ) 4 )(ñóò )1 ) 2 .

Solution in parametric form:

 =    2 − U where # 72.

ñ óò ò ó

1

= (


= 

1

J # 2, #

−3 2

ï –7 ) 4 ñ

+


2ï

=

2

)

–1 4

™

)(ñ

A  A 

2

4J 3 =  V   − (−1)  U!W ,

T 4 < 3  , J (T  J (A 

−3 11 4



−3 11 4







)1 J )1 J

2 2

if , = 1, if , = 2.

ó ò )5 ) 2 .

Solution in parametric form:

 =  [   − (−1)  U ]4 J 3 , where #

1

=

™

T/ T  /

J 

11 4 −3

(T/ 11 J 4 −3  (A 

 )1 J

 )1 J

2 2

if , = 1, if , = 2,

#

=  ( 2 − U 2

=

2

T 4 < 3 ),

J ˜# 1 .

−3 2

Page 383

384

SECOND-ORDER DIFFERENTIAL EQUATIONS

]

In the solutions of equations 73–92, the following notation is used: . . The functions 1 and 2 are the general solutions of the four modifications of the first Painlev e´ equation: .

= A 6. 2 + U , .

= A 6. 2 − U 1 2 1 2 (in the case of the upper sign, the equation for see Paragraph 2.8.2-2). In addition,

Æ

1

²

8

{ ’ ú

˜ :

=A 6

. =2

.

2 1

.

is the canonical form of the first Painlev´e equation;

1

+U ,

Æ


2 1 1 −U , . .
3 1 = 3U 1 −3 1−U , . 2 1 = U 1 T 1, . . . 3
2 1 = ( 1) − 2 1 Æ 1 A 8 1 , .

.
2 1 = 1 Æ 1 + 1 − Æ 1, 3.
2. 5 1 = U 1T 1+U , 1 + 3U . . 3
2 5 1 = 6( U 1 A 6) − U , 1 − 4U

²

’

8

ú

{

:

˜

=A 6

.

2 2

−U ,
+ U 2, 2 2 . .
3 2 = 3U 2 −3 2+U , . 2 2 = U 2 T 1, . . . 3
2 2 = ( 2) − 2 2 Æ 2 A 8 2 , .

.
2 2 = 2 Æ 2 − 2 − Æ 2, 3.
2. 5 2 = U 2T 1−U , 2 + 3U . . 3
2 5 2 = 6( U 2 A 6) + U , 2 − 4U 2

. =2

where the prime denotes differentiation with respect to U . 73.

= (
1 ñ +
2 ï 2 )(ñ ó ò )3 . Solution in parametric form:

ñ óò ò ó

where # 74.

= 

1

= 

−2

= 2

−1 −1

2

= T 6

= (
1 ñ 2 +
2 ï )(ñó ò )5 ) 2 . Solution in parametric form:



,

= 'U ,

; , = 1 and , = 2.

# 2 (−1) 

+1

 = !²  , = 'U ,

−1 −1 , # 2 = −2   (A 3   )1 J 2 ; , = 1 and , = 2.

= (
1 ñ +
2 ï 2 )(ñó ò )1 ) 2 . Solution in parametric form:

ñóò ò ó

where # 76.

1

−1 −2

ñóò ò ó

where # 75.

(−1)  , #

−3

.

 =

1



( A 3   )1 J 2 , #

= (
1 ñ –4 +
2 ï 2 ñ –5 )(ñó ò )3 . Solution in parametric form:

 = U , = ˜²  , −2 ˜# 1 (−1)  +1 ; , = 1 and , = 2. 2 = 

ñóò ò ó

.

 = U −1  , = 'U −1 , where # 1 =    2 (−1)  , # 2 = T 6  −1  3 ; , = 1 and , = 2. 77.

= (
1 ñ –1 ) 2 +
2 ï ñ –5 ) 4 )(ñó ò )5 ) 2 . Solution in parametric form:  =

ñóò ò ó

8

= 'U 4 , , where # 1 =    −3 J 4 # 2 (−1)  +1 , # 2 = − 41  −1  −1 J 4 (A 2   )1 J 2 ; , = 1 and , = 2.

78.

= (
1 ï –5 ) 4 ñ +
2 ï –1 ) 2 )(ñ Solution in parametric form:

ñ óò ò ó

ó ò )1 ) 2 . 8

 = U 4 , =  ,

1 −1 J 4 −1  (A 2   )1 J 2, # 2 =  −3 J 4 ˜ # 1 (−1)  +1 ; , = 1 and , = 2. where # 1 = 4 

Page 384

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

79.

ñóò ò ó

= (


ñ –8 ) 7 +
2 ï 2 ñ

1

)

–15 7

 1Ë E Ï ±

=

1

1

ÒÍ

(Ï

Ë  2Ë E Ï ±

)1+

2

2

(Ï

ÒÍ

)

Ë

2

385

)(ñó ò )3 .

Solution in parametric form:

{

= 'U 7 , , 1 where # 1 = 49  −6 J 7 (−1)  , # 2 = T 496  −1  1 J 7 ; , = 1 and , = 2. 80.

ñ óò ò ó

= (


ñ –7 ) 4 +
2 ï ñ

1

 = U

)

–15 8

ó ò )5 ) 2 .

)(ñ

Solution in parametric form:

where # 81.

ñóò ò ó

1

= (


J # 2 (−1)  , #

= 

−1 8

ï

) ñ

–15 8

1


2ï

+

2

)

–7 4

ú

 = U

−6

−8

= 18 

 J (T 3   )1 J 2; , = 1 and , = 2.

= 'U

 ,

,

−1 7 8

)(ñó ò )1 ) 2 .

Solution in parametric form: −8

 = U where # 82.

ñóò ò ó

1

= (


= − 81  7 J 8 

)

ñ

–13 7

1

−1

(T 3   ) 1 J 2 , #


2ï 2ñ

+

)

–20 7

83.

ñóò ò ó

1

= (


=

1 49

−1 7

)

ñ

–14 13

1

+


2ï ñ

2

)

–20 13

84.

ñ óò ò ó

1

= (


= 

−6 13

# 2 (−1) 

ï

) ñ

+

1

J

–20 13


2ï

, #

)

–14 13

85.

ñóò ò ó

1

= (


=

2 7J  13

ñ

+

1


2ï

13 −1



)

–1 2

OA

2

=−

)(ñ

J



−7

= 'U

,

,

, = 1 and , = 2.

)(ñó ò )5 ) 2 .

Solution in parametric form: where #

 J



:

 = U

+1

{

6 −1 6 7 ; 49

=T

Solution in parametric form: where #

−1 8

−6

 = U

J (−1)  , #



 ,

)(ñó ò )3 .

Solution in parametric form: where #

ú

−6

J ˜# 1 (−1)  ; , = 1 and , = 2.

=

2

= 'U

,

2  13

13 rP

 J

−1 7 13

OA

,

J



1 2

13 "P

; , = 1 and , = 2.

ó ò )1 ) 2 . 13

 = U

1 2

13

= 'U

 ,

, #

2

= 'U

,

=

J

−6 13

:

,

˜# 1 (−1) 

+1

; , = 1 and , = 2.

)(ñó ò )3 .

Solution in parametric form:

.

.

 =  ( 
)2 , =  , where # 1 = T 24  −3, # 2 = 2  3 J 2  −2 (−1)  ; , = 1 and , = 2. 86.

ñóò ò ó

= (


1

ï –1 ) 2 ñ –5 ) 2 +
2 ñ

–4

)(ñó ò )3 .

Solution in parametric form:

 = where #

1

= 2  3 J 2  1 J 2 (−1)  , #

2

.

.


2 −1  ( ),

=

.

−1  ,

= T 24  2 ; , = 1 and , = 2.

Page 385

386 87.

SECOND-ORDER DIFFERENTIAL EQUATIONS = (
1 ï –5 ) 3 ñ +
2 ï 1 ) 3 )(ñó ò )3 . Solution in parametric form:

ñóò ò ó

.  =   3J 2, = ’  , 3 8 J 3 −3 where # 1 = 16   , # 2 = T 8  −2 ˜# 1 ; , = 1 and , = 2. 88.

= (
1 ï –5 ) 3 ñ –7 ) 3 +
2 ï Solution in parametric form:

ñ óò ò ó

) ñ

)

1 3 –10 3

.

 = where # 89.

1

 J  J

3 8 3 1 3 , 16

=

#

2

= T 8

−2

óò

)(ñ



)3 .

J ’ −1 , 

3 2

= ’  −1 ,

˜# 1 ; , = 1 and , = 2.

= (
1 ï –1 ) 2 +
2 ñ –3 ) 2 )(ñó ò )3 . Solution in parametric form:

ñóò ò ó

.
 = Æ 2 , =  (  )2 , where # 1 = T 6  3 J 2  −2 , # 2 = 12  −1 J 2 (−1)  +1 ; , = 1 and , = 2. 90.

= (
1 ñ –3 ) 2 +
2 ï –1 ) 2 ñ Solution in parametric form:

ñ óò ò ó

)

–5 2

)(ñ

óò

)3 .

. .  =  ( 
)−2 Æ 2 , =  ( 
)−2 , where # 1 = 12    −1 J 2 (−1)  +1 , # 2 = T 6  3 J 2  1 J 2 ; , = 1 and , = 2. 91.

92.

= (
1 ï –5 ) 3 ñ +
2 ï –4 ) 3 )(ñó ò )3 . Solution in parametric form: ˜ .  =  ( 
)3 , =  , 1 8 J 3 −3 where # 1 = − 36   , # 2 =  −1 J 3 ˜# 1 (−1)  ; , = 1 and , = 2.

ñóò ò ó

= (
1 ï –4 ) 3 ñ –5 ) 3 +
2 ï Solution in parametric form:

ñóò ò ó

) ñ

)

–5 3 –7 3

)(ñó ò )3 .

. ˜ ˜  =  ( 
)3  −1 , =   −1 , 1 7 J 3 −1 J 3 1 8J 3 1J 3 where # 1 = 36   (−1)  +1 , # 2 = − 36   ; , = 1 and , = 2. ]

In the solutions of equations 93–96, the following notation is used:

 =L

2

exp O X

² =

93.



U







£M

² P

1 U 4 1 L 1+ U 4

L

1

+

= U + 2 £ ² + 4$

, 4

5

=X O L

2 $ 1  1 +1 2$ 2  U + U , 1+1 , 2+1 2 $ 1  +1 2 + U + 2 $ 2 ln |U | , +1 2

+

= (
1 ï –2ú 1 –3 ñ ú 1 +
2 ï Solution in parametric form:

ñóò ò ó

2,

–2

ú

2 –3

ñ ú

2

2 +1

1

−

if ,

1 2# 1

1

#

2

U

≠ −1, ,

if , = ,

1

−4

2

−U

2

J

−1 2

P M

U + L 2,

≠ −1;

≠ −1, ,

2

= −1.

)(ñó ò )3 .

 = U  1J 2, = , where , 1 = −2 1 − 3, , 2 = −2 2 − 3, # 1 = − $ 1 , # 2 = − $ 2 . ¡ ¡

Page 386

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

94.

ñóò ò ó

= O+


1

ï ñ ú


2ñ

+

ú

ú

–1 2

Solution in parametric form:

+5 +3

(ñó ò ) ú

P

95.

ñóò ò ó

1

+3 , , +1

= , =−¡

= O+


1

¡

ïƒ÷ ñ

÷
2ï

+

= 0, #

2

–1 2

(ñó ò )

P

=

÷

2 +4 +3

Ë  2Ë E Ï ±

ÒÍ

(Ï

1

)1+

2

2

Ë

ÒÍ

)

2

= −4  − 

(Ï

387

2

.

J 4 ,

1

÷

1

, 

1 +1

= ˜ +

−1 2

 = ! where ,

 1Ë E Ï ±

=

,

4 !$ 1 1 'J  S , # ( , + 1) ‚Q ( , + 1)  +1

−

1 +1

#

1

$ 2.

.

Solution in parametric form: 1 +1

 = !  where , 96.

ñ óò ò ó

1

= (


=, =−

ï –5 ñ

1

B +3 , , = 0, # B +1 2

2


2ï

+

–5

óò

)(ñ

=−

1

,

, +1



= (


ïƒú ñ ú 1

1

1


2 ïƒú ñ ú

+

−

2

1 − +1



= −4 

˜#

1

$ 2.

)3 .

U £ −#  = cos 5

ñóò ò ó

4 ˜$ 1 1 'J  S , # Q ( , + 1) 

2 +1 −1

 

Solution in parametric form:

97.

J 4 ,

−1 2

=

,

2

2

2

#

=

,

#

2

tan 5 .

1

)(ñó ò )3 ) 2 .

Solution in parametric form:



1  = L 1 U 1 J 2 exp O − X 2

where

 =

98.

ñ óò ò ó

= O+







 1

U

J

−1 2

L

Q

+

2

#

 ü

1

1 + 1)

U +

M

U 2

+4P

1 +1

=L

,

#

+

2( ¡ # ¡ 1 1 U −1 J 2 L 2 + U + 1 +1 + # Q 2( 1 + 1) 2

ï ÷ ñ ú

2(

U

¡

1

+

ú
2ï

2(

÷ ú

ú

+1)– 1 +1

1+

÷ ú ñ

2

Solution in parametric form:

 =L

1

exp O X

U

M

U H·P

,

=L

P −

1

+

2

2 + 1)

2

ln U!S

(ñ

ú óò ) ú



+1

1 +1

U +

2 +1

2

S

1 +2 1+

÷

÷

+3 +2

U exp O −

U ü



¡

1

≠ −1,

¡

1

≠ −1,

if if

where H = H (U ) is the solution of the algebraic equation

OH − ¡



1 U J exp O X

−1 1 2 1

U M

2

¡

2

≠ −1,

¡

2

= −1.

B +1 1

¡

+1

X

M

U

U H·P

,

 +1 + 1 +1 +B +2 B + 1 + 1 +  +2 # 1 − OH − = U + 1 +  +2 O L 2 + U + 1+1 P 1+1P 1+B +2 ¡ ¡
¡ # 2 ( 1 + 1) + 2 +1  ¡ U if 2 ≠ −1, ¡  =
 ( 1 + B + 2)( 2 + 1) ¡

#

1

2

+B +2

¡

ln |U |

if

,

.

1

¡

+4P

¡

2

1 +1

+

P

,

= −1.

Page 387

388

SECOND-ORDER DIFFERENTIAL EQUATIONS

]

In the solutions of equations 99–108, the following notation is used: . . The functions 1 and 2 are the general solutions of the four modifications of the second Painlev e´ equation (with parameter  = 0):

.

.



=U

1

A 2

1

.

.

3 1,

.



= −U

2

.

2

A 2

.

3 2.

In the case of the upper sign, the equation for 1 is the canonical form of the second Painlev´e equation (with parameter  = 0; see Paragraph 2.8.2-3);

Æ

1

=U

Æ

2

=U

.

2 1 2 2

.

.

A

4 1 4 2

. A

−(

.

. −(

1)


2, ²

1

2)

²

2


2,

.

=

. =


T

1

.


A

2

.

1 2

Æ

8

1,

Æ

8

2,

=2

1

.

. =2

2

1


Æ

1

2

2


Æ

−

.

. +

3 1 3 2

.

T A

.

1

Æ

2

Æ

2 1, 2 2;

where the prime denotes differentiation with respect to U . 99.

ñóò ò ó

= (


1

ï ñ


2ï

+

3

)(ñó ò )3 .

Solution in parametric form: where # 1 =  (−1)  , # 3

100.

ñóò ò ó

= (


1

ï ñ

–5

2


2 ï 3ñ

+

 =

= T 2

−2 −2

–6



where # 1 =  (−1)  , # 101.

ñóò ò ó

= (


1

+


2ï

2

) ñ

)

; , = 1 and , = 2.

 = U

−1

.

= 'U

,

−1

,

−2 4

= T 2

–1 2 –1 2

= 'U ,

,

)(ñó ò )3 .

Solution in parametric form: 3

.

 ; , = 1 and , = 2.

)(ñó ò )3 .

Solutions in parametric form:

. .
= (. )
, .  =  ( 
)2 , =   2,  *  where # 1 = T 2  −2 , # 2 = 12  3 J 2  −3 J 2 (−1)  ; , = 1 and , = 2. 102.

ñóò ò ó

= (


1

ï –1 ) 2 ñ

–2


2ñ

+

–3

)(ñó ò )3 .

Solutions in parametric form:

.

 = where # 103.

ñóò ò ó

1

= (


= 12  3 J 2 (−1)  , # 1

ï –2 ñ

+




2 )(

ñóò

2

−2 .
2  ( ),

.

where # 104.

ñóò ò ó

1

= (


= 2 3  1

ï –2 ñ

−3

–2


2ñ

=  VU

2  ,

–3

2

where #

1

= −2  3 , #

.

2V  U 2

.

 = (  )* ,

.

= A 2 

. −2

2  A

.

.
2 4  −(  ) W ,

.

(−1)  ; , = 1 and , = 2.

.

 = (  )* ,

)(ñó ò )3 .

Solutions in parametric form:

 =

.

−2  ,

)3 .

(−1)  , #

+

.

= T 2  ; , = 1 and , = 2.

Solutions in parametric form:

 =

=

2  A

.

.
2 4  −(  ) W

−1

,

=  VU

.

2  A

.

.
2 4  −(  ) W

−1

,

= T 2  ; , = 1 and , = 2.

Page 388

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

105.

ñóò ò ó

= (


1


2ï ñ

+

)

–1 2

106.

ñ óò ò ó

= (


J # 2 (−1)  , #

= T/ 1

−1 2

ï –1 ) 2 ñ

+




2 )(

 = Æ 2 ,

107.

ñóò ò ó

1

= (


1

 )1 J

 )1 J

−2  1 J 2  −2 (2  2  1 J 2  −2 (−2 

™

=


2ï ñ

+

108.

ñ óò ò ó

1

= (


= T/ 1

2

# 2 (−1)  , #

+




2 )(

109.

ñóò ò ó

1

= (


= 1

− 2  − 2 

™

−2 −2

8

 = 2

ÒÍ

(Ï

)

Ë

389

2

2

.

−1  ² ,

J ˜# 1 (−1)  .

−1 2

= T 

= Æ  ,

1J 2 if , = 1,   (2   )  −2  2 (−2   )1 J 2 if , = 2.

,

−2 2

™

=

ñ ó ò )3 ) 2 .

Solutions in parametric form: where #

2

)(ñó ò )3 ) 2 .

–2

−2

ï –2 ñ

=

if , = 1, # if , = 2,

2

Solutions in parametric form: where #

2

ñ ó ò )3 ) 2 .

Solutions in parametric form:

where #

)1+

−1 = Æ 2 ,  ² , 2  −2  1 J 2 (2   )1 J 2 if , = 1, −2  −2  1 J 2 (−2   )1 J 2 if , = 2.

™

=

2

Ë  2Ë E Ï ±

ÒÍ

(Ï

1

.

 = 1

1

)(ñó ò )3 ) 2 .

Solutions in parametric form:

where #

 1Ë E Ï ±

=

 = Æ  , if , = 1, # if , = 2,

(2   )1 J 2 (−2   )1 J 2

ï –7 ) 5 ñ –8 ) 5 +
2 ï –7 ) 5 ñ

)

–13 5

= 2

= T 

8

, −2

˜# 1 (−1)  .

)(ñó ò )3 .

Solution in parametric form:

 = L where 110.

ñ óò ò ó

8

=L

= (


1

1

5 1

8 5J 2 4 O L 1 

#

−

#

1 2

−1

P

4 1

= OL

,

 −

8 8 8  2  * + L 2  −  * sin  £ 3 ,!U  ,  = (
)2 − 2

, #

ï –2 ñ

–1

+


2ï

ñ

–2 –2

)(ñ

óò

*

*+*

2

#

1

#

2

=

−1

P

,

J 

5 12 5 −3 −6 . 1024



,

)3 .

Solution in parametric form:

:  = L 1 U 2 J 3 = ¦/L where

U

¦/L 1 U

J V (U :
+ 1 : )2 AIU 2 : 2 W − # 3

−2 3

*

J V (U :
+ 1 : )2 AIU 2 : 2 W − # 3

−2 3

:

*

L

ø

#

1 2

§

−1

#

1 2

§

−1

,

,

J U ) + L 2 n 1 J 3 (U ) for the upper sign, U ) + L 2 ; 1 J 3 (U ) for the lower sign, 1 J 3 ( U ) and n 1 J 3 ( U ) are the Bessel functions, and ® 1 J 3 ( U ) and ; 1 J 3 ( U ) are the modified Bessel #

ø

1

2

2

=

9 3   2

−3

,

=

™

L

1 1 3(

®

1 1 3(

J

functions.

Page 389

390 111.

SECOND-ORDER DIFFERENTIAL EQUATIONS = (
1 ï –7 ñ 4 +
2 ï –7 ñ 3 )(ñó ò )3 . Solution in parametric form:

ñóò ò ó

 = L where ²

# ]

2

=T

=

3 8 64

 

ü −3

A (4U

3

3 1



J OL

1 2

− 1), 

8 1

#

4 −

1

#

2

−1

P

−1

= 2U X U ²

.

M

8 1

= OL

,

4 −

U + L 2\ U T­² , 4

#

1

#

2

−1

P

= 4U 

,

T U

2

−2

( ²u

− 1)2 ,

In the solutions of equations 112 and 113, the following notation is used:

« = X (1 ACU 4 )−1 J the function « 112.

113.

= (
1 ï 2 ñ –14 ) 5 +
2 ï 3 ñ Solutions in parametric form: 4 1

«

−4

(U « − , ),

)

–18 5

= L

= (
1 ï 2 ñ –11 ) 5 +
2 ï 3 ñ Solutions in parametric form:

ñóò ò ó

 = L 1 « (U « − , ), ]

U + L 2, M

2

,

= A 1;

can be expressed in terms of elliptic integrals or lemniscate functions.

ñóò ò ó

 = L

2

)

5 1

«

–12 5

= L

5 1

«

)(ñó ò )3 . −5

,

#

where

1

=T

 J , , #

6 −1 4 5 25



2

=T

 J

2 −2 8 5 . 25



)(ñó ò )3 . 5

,

where

#

1

=T

 J , , #

6 −1 1 5 25



2

=T

 J

2 −2 2 5 . 25



In the solutions of equations 114 and 115, the following notation is used:

¢ =L

2 2

− 2L








ú

(H ) =




























114.

1,

² = (36 ¢ + 54$IU − 2U 3 )1 J 2 , H = 3 X

£ −¢ L 2 tan  A £ − ¢wH  + L 1 L 1 £ ¢ L tanh  T £ È ¢ H  + 2 L 1 L 1 £ 2 1 T − £ L 1H |L 1| £ 2 1 T + £ L 1H |L 1|

if ¢

< 0;

if ¢

> 0;

if ¢

= 0, L

2

< 0;

if ¢

= 0, L

2

> 0.

− 6L

2

Tç² )3 J 2 ,

− 6L

2

Tç² )3 J 2 ,

= (
1 ï –5 ) 3 +
2 ï –5 ) 3 ñ –2 ) 3 )(ñó ò )3 . Solutions in parametric form:

ñóò ò ó

ú ú ú  = U −9 J 4 ( L 1 2 − 2 L 2 + 2)3 J 4 (6 L 1 ú ú = 'U −3 J 2 ( L 1 2 − 2 L 2 + 2)3 J 2 , where # 1 = −24  8 J 3  −2 L 1 , # 2 = 36  8 J 3  −4 J 3 $ . 115.

U M , U ²

= (
1 ï –5 ) 3 ñ –2 ) 3 +
2 ï –5 ) Solutions in parametric form:

ñ óò ò ó

ñ )

3 –4 3

)(ñ

óò

)3 .

ú ú ú  = U −3 J 4 ( L 1 2 − 2 L 2 + 2)−3 J 4 (6 L 1 ú ú = 'U 3 J 2 ( L 1 2 − 2 L 2 + 2)−3 J 2 , where # 1 = 36  8 J 3  −4 J 3 $ , # 2 = −24  8 J 3  −2 J 3 L 1 .

Page 390

2.6. EQUATIONS OF THE FORM

116.

ñóò ò ó

=

2(ù + 1) ï

Q (ù + 3)2

+


ƒï ÷ S ñ

–2

Ï Ò˜Í Í Ò

(ñó ò )3 ,

=

 1Ë E Ï ± ù

1

1

(Ï

≠ –3,

ÒÍ

Ë  2Ë E Ï ±

)1+

ù

2

2

(Ï

ÒÍ

)

Ë

2

391

≠ –1.

Taking to be the independent variable, we obtain an equation of the form 2.4.2.4 with respect 2(B + 1) to  =  ( ): 

= −2 −  − #*  S .  Q (B + 3)2 117.

118.

119.

120.

121.

122.

123.

124.

125.

126.

127.

ï +
ï –7 )ñ –2 (ñ óò )3. = (– 15 4 Taking to be the independent variable, we obtain an equation of the form 2.4.2.35 with −7 respect to  =  ( ): 

= −2 ( 15 ). 4  − #* ñ óò ò ó

ñ óò ò ó = (–6ï

+
ï



)ñ –2 (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.31 with respect to  =  ( ): 

= −2 (6 − #* −4 ).

ñ óò ò ó = (–12ï

–4



+
ï –5 ) 2 )ñ

(ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.64 with respect to  =  ( ): 

= −2 (12 − #* −5 J 2 ).

ñ óò ò ó = (–2ï

+
ï

–2



)ñ –2 (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.6 with respect to  =  ( ): 

= −2 (2 − #* −2 ). –2



= +
ï –5 ) 3 )ñ –2 (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.26 with 3 respect to  =  ( ): 

= −2 (− 16  − #* −5 J 3 ).

ñóò ò ó

ï

3 ( 16

ï

ñ óò ò ó



+
ï –5 ) 3 )ñ

9 –2 = ( 100 ( ñ ó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.10 with 9 respect to  =  ( ): 

= −2 (− 100  − #* −5 J 3 ).

ñóò ò ó

ï



+
ï –5 ) 3 )ñ

–2 = (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.12 with respect to  =  ( ): 

= −2 ( 43  − #* −5 J 3 ).

(– 34



ï +
ï –5 ) 3 )ñ –2 (ñóò )3 . = (– 63 4 Taking to be the independent variable, we obtain an equation of the form 2.4.2.66 with −5 J 3 respect to  =  ( ): 

= −2 ( 63 ). 4  − #* ñóò ò ó ñ óò ò ó

5 ( 36

ñóò ò ó

( 29

ï



+
ï –7 ) 5 )ñ

–2 = ( ñ ó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.29 with 5 respect to  =  ( ): 

= −2 (− 36  − #* −7 J 5 ).

ï



+
ï –1 ) 2 )ñ

–2 = (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.14 with respect to  =  ( ): 

= −2 (− 92  − #* −1 J 2 ).



4 ï = ( 25 +
ï –1 ) 2 )ñ –2 (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.4.2.8 with respect 4  − #* −1 J 2 ). to  =  ( ): 

= −2 (− 25

ñóò ò ó



Page 391

392 128.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

= (–20 ï +


ï –1 ) 2 )ñ

–2

(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.33 with respect to  =  ( ): 

= −2 (20 − #* −1 J 2 ). 129.

ñóò ò ó



ï + = ( 12 49


ï 1 ) 2 )ñ

–2

(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.37 with 12  − #* 1 J 2 ). respect to  =  ( ): 

= −2 (− 49 130.

ñóò ò ó

ï

= (


2

6 25

+

ï )ñ



–2

(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.60 with 6 respect to  =  ( ): 

= −2 (− #* 2 − 25  ). 131.



6 ï 2 – 25 )ñ

ñ óò ò ó = (
ï

–2

(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.62 with 6 respect to  =  ( ): 

= −2 (− #* 2 + 25  ).



132.

ñóò ò ó

=

2(ù + 1) ï

Q (ù + 3)2

ñ

–2

+


ƒï ÷ ñ –÷ –1 S (ñ óò

)3 ,

ù

≠ –3, –1.

Taking to be the independent variable, we obtain an equation of the form 2.4.2.5 with respect 2(B + 1) −2 to  =  ( ): 

= −  − # −  −1   .  (B + 3)2 133.

ñóò ò ó

= (


ï –2 ñ

– 2ï

ñ

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.7 with respect to  =  ( ): 

= 2 −2  − #  −2 .



134.

ñóò ò ó

= (


ï –1 ) 2 ñ –1 ) 2 + 4 ï ñ 25

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.9 with respect 4 −2 to  =  ( ): 

= − 25  − # −1 J 2  −1 J 2 .



135.

ñóò ò ó

= (


ï –5 ) 3 ñ 2 ) 3 + 9 ï ñ 100

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.11 with 9 −2  − # 2 J 3  −5 J 3 . respect to  =  ( ): 

= − 100



136.

ñóò ò ó

= (


ï –5 ) 3 ñ 2 ) 3 – 3 ï ñ 4

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.13 with respect to  =  ( ): 

= 34 −2  − # 2 J 3  −5 J 3 .



137.

ñ óò ò ó

= (


ï –1 ) 2 ñ –1 ) 2 + 2 ï ñ 9

–2

)(ñ

óò

)3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.15 with respect to  =  ( ): 

= − 92 −2  − # −1 J 2  −1 J 2 .



138.

ñ óò ò ó

= (


ï –5 ) 3 ñ 2 ) 3 + 3 ï ñ 16

–2

)(ñ

óò

)3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.27 with 3 −2  − # 2 J 3  −5 J 3 . respect to  =  ( ): 

= − 16



Page 392

2.6. EQUATIONS OF THE FORM

139.

ñóò ò ó

= (


ï –7 ) 5 ñ 2 ) 5 + 5 ï ñ 36

–2

Ï Ò˜Í Í Ò

=

 1Ë E Ï ± 1

1

(Ï

ÒÍ

Ë  2Ë E Ï ±

)1+

2

2

(Ï

ÒÍ

)

Ë

393

2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.30 with 5 −2 respect to  =  ( ): 

= − 36  − # 2 J 5  −7 J 5 . 140.

ñóò ò ó

= (


ï –4 ñ

– 6ï

3



ñ

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.32 with respect to  =  ( ): 

= 6 −2  − # 3  −4 .



141.

ñóò ò ó

= (


ï –1 ) 2 ñ –1 ) 2 – 20ï ñ

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.34 with respect to  =  ( ): 

= 20 −2  − # −1 J 2  −1 J 2 . 142.

ñóò ò ó

= (


ï –7 ñ



6

15 4

–

ï ñ

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.36 with −2 respect to  =  ( ): 

= 15  − # 6  −7 . 4



143.

ñóò ò ó

= (


ï 1 ) 2 ñ –3 ) 2 + 12 ï ñ 49

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.38 with 12 −2 respect to  =  ( ): 

= − 49  − # −3 J 2  1 J 2 . 144.

ñóò ò ó

6 ï = ( 25



ñ

–2

+


ï 2ñ

–3

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.61 with 6 −2  . respect to  =  ( ): 

= − # −3  2 − 25 145.

ñ óò ò ó

6 ï = (– 25

ñ



–2

+


ï 2ñ

–3

)(ñ

óò

)3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.63 with 6 −2 respect to  =  ( ): 

= − # −3  2 + 25  .



146.

ñóò ò ó

= (


ï –5 ) 2 ñ 3 ) 2 – 12ï ñ

–2

)(ñó ò )3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.65 with respect to  =  ( ): 

= 12 −2  − # 3 J 2  −5 J 2 .



147.

ñ óò ò ó

= (


ï –5 ) 3 ñ 2 ) 3 – 63 ï ñ 4

–2

)(ñ

óò

)3 .

Taking to be the independent variable, we obtain an equation of the form 2.4.2.67 with −2 respect to  =  ( ): 

= 63  − # 2 J 3  −5 J 3 . 4



2.6.3. Equations of the Form

â$äãÑã ä = VMxæ è â| (âäã ) + Mxæ è

â

–1 |

+1 (

âäã ) –1

2.6.3-1. Classification table. Table 26 presents all solvable equations whose solutions are outlined in Subsection 2.6.3. Twoparameter families (in the space of the parameters B , , and 0 ), one-parameter families, and isolated ¡ points are presented in a consecutive fashion. Equations are arranged in accordance with the growth of 0 . The number of the equation sought is indicated in the last column in this table.

Page 393

394

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 26 Solvable cases of the equation 

 = ./#* 

0 arbitrary ( 0 ≠ 2) arbitrary ( 0 ≠ 2) arbitrary ( 0 ≠ 3) arbitrary ( 0 ≠ 1) +3 ¡ +2 3¡ B + 2 B +1

B ¡

arbitrary ( ≠ −1)

¡

¡

0

−1

−1

2.6.3.80

0

2

arbitrary ( ≠ −1) ¡ arbitrary ( ≠ −1) ¡ arbitrary ( ≠ −1) ¡ arbitrary ( ≠ −1) ¡ arbitrary ( ≠ −1)

2

−1

3 3 0

arbitrary ≠ −1, −2) ¡ arbitrary ( ≠ −2) ¡ arbitrary ( ≠ −2) ¡ arbitrary ( ≠ −1, −3)

1

+1

¡

arbitrary (B ≠ 0, −1) −

1

B

−2

¡

−1

arbitrary (B ≠ −1) arbitrary (B ≠ 0, −2) arbitrary (B ≠ 0, −1) arbitrary (B ≠ 0)

B

1 2

B B

1

2.6.3.74 2.6.3.73 2.6.3.1 2.6.3.23 2.6.3.37 2.6.3.41

−1

2.6.3.85

−1

2.6.3.82

−2

−1

2.6.3.83

+1 2

−1

2.6.3.84

0

arbitrary

2.6.3.87

arbitrary (B ≠ 0)

arbitrary

2.6.3.86

+1

¡ −2 −¡

¡

1

(

¡

2.6.3.75

2.6.3.79

3 2

¡

−1

−1

1

2

Equation

1

1

2

.

−2

0

2

( 
)- −1

2.6.3.76

1

2

+1

−1

0

¡

+

0 −2

arbitrary ( ≠ −1)

3

−1

¡

−1

1−0

1

5 2

−

arbitrary ≠ −1, −2)

(

+ ( 
)- + #* 

−

+1

−2

¡

¡

1

−1

2

+1

¡

¡

+1 2 2

2.6.3.42 2.6.3.3 2.6.3.24 2.6.3.38

−3

−1

0

−3

1 2

−4

2.6.3.61

0

−3

2

−1

2.6.3.35

0

0

−2

−1

2.6.3.48

2.6.3.65

Page 394

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

=

 1Ë E Ï ± 1

1

(Ï

ÒÍ

Ë  2Ë E Ï ±

)1+

2

TABLE 26 (Continued) Solvable cases of the equation 

 = ./#*  + (
 )- + #* 

(Ï

ÒÍ

−1

+

2

)

Ë

395

2

+1

(
 )- −1

B

.

Equation

0

−1

−2

2.6.3.50

0

−1

−1

2.6.3.33

0

0

− 25

2.6.3.59

0

0

2

− 25 1 2

1

−3

1

−1

2.6.3.15 2.6.3.51

0 0 0

¡

2.6.3.63

1

−2

−2

1 2

1

−2

−1

arbitrary

2.6.3.71

1

−2

−1

−1

2.6.3.7

1

−2

−1

1

2.6.3.5

1

−2

2

2.6.3.55

1

−2

− 12 1 2

2.6.3.13

1

−2

1

−1 arbitrary

1

−2

1

−2

2.6.3.11

1

−2

1

−1

2.6.3.29

1

−1

−1

−1

2.6.3.2

1

−2

− 41

2.6.3.53

−1

−1

2.6.3.45

− 12

−1

2.6.3.31

1

− 12 − 12 − 12 − 12

1

1 2

2.6.3.57

1

0

−1

2.6.3.77

1

0

1

−1 arbitrary

1

0

1

−1

2.6.3.9 2.6.3.21

1 1

2.6.3.69

2.6.3.67

1

1

−4

− 21

1

1

−1

−2

2.6.3.39

1

1

−2

−1

2.6.3.25

1

1

1

1

2.6.3.17

3 2 3 2 3 2

0

−1

−1

2.6.3.27

0

− 12

−2

2.6.3.43

0

1

2.6.3.19

2

−1

0

1 arbitrary

5 2 5 2 5 2

−2

1

−1

2.6.3.28

− 32

1

− 21

2.6.3.44

0

1

1

2.6.3.20

3

−5

2

−2

2.6.3.22

3

−3

−1

2

2.6.3.52

2.6.3.81

Page 395

396

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 26 (Continued) Solvable cases of the equation 

 = ./#*  + (
 )- + #* 

0

B ¡

−1

+

+1

(
 )- −1

.

Equation

−4

2.6.3.54 2.6.3.26

3

−3

1 2

3

−3

2

3

−2

−1

−1 arbitrary

3

−2

−1

−1

2.6.3.8

3

−2

−1

1

2.6.3.6

3

−2

0

−1

2.6.3.4

3

−2

1 2

−1

2.6.3.46

3

−2

1

−1

2.6.3.78

− 21 1 2

2.6.3.40

−1

2.6.3.32

2.6.3.72

3

−2

2

3

−1

3

− 32 − 32 − 12

−1

−1

2.6.3.14

3

0

−2

2.6.3.16

3

0

−1

−1 arbitrary

3

0

−1

−1

2.6.3.30

0

−1

2.6.3.12

3

0

1 2

− 21

2.6.3.58

3

0

1

2 arbitrary

3

0

1

−1

2.6.3.10

3

0

2

1

2.6.3.18

4

−3

1

−1

2.6.3.47

4

−2

−2

1 2

2.6.3.66

4

−2

1

−1

2.6.3.34 2.6.3.49

3

3

1 2

2.6.3.56

2.6.3.70

2.6.3.68

4

−2

1

4

−2

4

− 12 − 25

1

− 21 − 41 − 52

4

1

−2

−1

2.6.3.36

4

1

1

2

2.6.3.64

2.6.3.62 2.6.3.60

2.6.3-2. Solvable equations and their solutions. 1.

=
ï –ú –2 ñ ú ñ ó ò –
ï –ú Solution in parametric form:

ñ óò ò ó

–3

ñ ú

+1

U M + +1 + L 2 P 1 ACU where # = T ( + 1)  + +1  − + . ¡  = L 1+ O X

þ

, −1

,

≠ –1. = L

+

1

+1

U O X

U M + 1 ACU

+1

+L

−1 2

P

,

Page 396

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

2.

=
ï –1 ñ –1 ñó ò –
ï –2 . Solution in parametric form: 1

Q

XZU

−1

exp(T"U 2 ) U + L

M

=
ï –ú –2 ñ ú (ñó ò )3 –
Solution in parametric form:

2

ï –ú

ñóò ò ó

1

(Ï

ÒÍ

Ë  2Ë E Ï ±

)1+

2

2

(Ï

ÒÍ

)

Ë

397

2

U·OX

−2

=−

,

ñ ú

+1

+L

2

#

2

exp(T"U 2 ) XZU

Q

(ñó ò )2 , −1

P

þ

exp(T"U 2 ) U + L

M

S

2

S

−1

2

.

≠ –2.

= L 1+

,

−1

+3

U M −+ 1 AIU

OX

−2

−1

+L

2

P

,

=
ñ –2 (ñó ò )3 –
ï –1 ñ –2 (ñó ò )2 . Solution in parametric form:

ñóò ò ó

 =− ]

+2

−1

S

–3

U M −+ 1 ACU + +3 − + −2 where # = A ( + 2)   . ¡  = L 1+

4.

1

ñóò ò ó

 =L 3.

 1Ë E Ï ±

=

#

2

exp(T"U 2 ) XÈU

Q

−1

exp(T"U 2 ) U + L

M

2

S

−1

,

=L

1

Q

XZU

−1

exp(T"U 2 ) U + L

M

−1

.

In the solutions of equations 5–12, the following notation is used:

 = X exp(T U 2 ) U + L 2 . M

5.

=
ï –1 ñ –2 ñó ò +
ï –2 ñ –1 . Solution in parametric form:

ñóò ò ó

 =L

6.

1

exp(T"U 2 ) 

−1

=
ï –1 ñ –2 (ñó ò )3 +
ï –2 ñ Solution in parametric form:

=  [2U2A exp(T U 2 ) 

,

ñóò ò ó

–1

 =  [2U3A exp(T"U 2 ) 

7.

1

V 2UpA exp(T"U 2 )W

=
ï –1 ñ –2 (ñó ò )3 –
ï –2 ñ Solution in parametric form:

# = A 2 2.

=L

],

1

exp(T"U 2 ) 

−1

,

where

# = T 2 2 .

–1

−1

=
ï ñó ò –
ñ . Solution in parametric form:

=  V 2UpA exp(T U 2 )W

,

−1

,

where

# =A

1 2 2 .

,

where

# =A

1 2 . 2



(ñó ò )2 .

,

=L

1

V 2UpA exp(T U 2 )W

−1



ñóò ò ó

 = U ,

10.

−1

ñóò ò ó

 =  V 2UpA exp(T U 2 )W

9.

where

ñóò ò ó

 =L 8.

],

(ñó ò )2 .

−1

=
ï –1 ñ –2 ñó ò –
ï –2 ñ –1 . Solution in parametric form:

−1

=L

=
ï (ñ ó ò )3 –
ñ (ñ ó ò )2 . Solution in parametric form:

1 [2

UpA exp(T U 2 )],

where

# = T 2

−2

where

# = T 2

−2

.

ñ óò ò ó

 = L 1 [2UpA exp(T"U 2 )],

= 'U ,

.

Page 397

398 11.

12.

SECOND-ORDER DIFFERENTIAL EQUATIONS = 2
ï ñ –2 ñó ò –
ñ –1 . Solution in parametric form:  = L 1 [2U 2 pAIU exp(T U 2 ) Aç ],

ñóò ò ó

ñ óò ò ó =
ï

–1 ( ñ óò )3 – 2
ï –2 ñ (ñ óò

where

# =T

1 −2 2 . 2

U 2p  ACU exp(T"U 2 ) Ae ],

where

# =T

1 2 −2 . 2

1 [2

= L

1 [2

14.

15.

16.

17.

18.

19.

=
ï 1 ) 2 ñ –2 ñó ò –
ï –1 ) 2 ñ Solution in parametric form:  = L 14  −2 ,

ñóò ò ó

 

3 −1 1 2

= L

U

«I

=
ï ñ –3 ñó ò –
ñ –2 . Solution in parametric form:

 = L

 = L

2 1



−1



−1

U +1 , U

=
ï –2 (ñó ò )3 –
ï –3 ñ (ñó ò )2 . Solution in parametric form:

ñóò ò ó

3 1

= L

,

2 1

= L

−1



=
ï ññó ò +
ñ 2 . Solution in parametric form:  = L 1−1 U −1 « −1 (U  2 + U 2  − «

ñóò ò ó

ñ óò ò ó =
ï 2 (ñ óò

3

+
ï ñ ( ñ óò

,

where

# = −

−2

,

where

# = − 2 



−1

ñóò ò ó =
ï (ñ óò )3 ) 2 +
ñ (ñóò )1 ) 2 . Solution in parametric form:

U +1 − «pU U

” 

−1

•

−1

«

−1

,

=
ï (ñó ò )5 ) 2 +
ñ (ñó ò )3 ) 2 . Solution in parametric form:

ñóò ò ó

3 1



−1

ñ óò ò ó =
ï –4 ñ ñóò

U +1 , U

= L

– 2
ï –5 ñ

J .

−3 2

−2 3

,

where

# = −2

U +1 , U

where

# = −2 3 

= L

),

−3 2 2

2 1

−1



 .

−2

.

where

# =

),

where

# = −

U +1 , U

where

# = 2

where

# = −2 

,

−2 −1



.

2

) ). Solution in parametric form:  = L 12  −1 , = L 1−1 U

−1 1

2

J  .

−2

ñóò ò ó

3 1

U +1 −U . U

 =«

.

=
ï –1 ñ –1 ) 2 (ñó ò )3 –
ï –2 ñ 1 ) 2 (ñó ò ) . Solution in parametric form:  = L 13 U −1 «I −2 , = L 14 

 = L 21.

–1

ñóò ò ó

 = L 20.



In the solutions of equations 13–22, the following notation is used:

« = £ U (U + 1) − ln  £ U + £ U + 1  + L 2 ,

13.



)2 .

Solution in parametric form:  = L 1 [2UpA exp(T"U 2 )],

]

UuA exp(T U 2 )],

= L

2

−1 1

(U 

2

= L

3 1

” 



+ U 2 − «

−1

U +1 − «pU U

−1

•

2

,

−1 −2



−2

O−

−2



O−

.

J

1 2

…P

.

 1J 2 .  P ·

.

Solution in parametric form:

 = L 1 U «­U  where # = −2 3  −1 .

2

+ U 2 − «

2 −1



,

= L

3 1

U «I

−1

U 

2

+ U 2 − «

2 −1



,

Page 398

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

22.

= 2
ï 2 ñ –5 (ñó ò )3 –
ï ñ Solution in parametric form:

ñóò ò ó

3 1

 = L where # = −2 23.

=

1

(Ï

Ë  2Ë E Ï ±

ÒÍ

)1+

2

2

(Ï

ÒÍ

)

Ë

399

2



= L

,

1

U «­U 

2

2 −1



+ U 2 − «

,

 .

1 +1

=L

,


ï ñ ú (ñ óò

)3 +

1

ù




þ

≠ –1.

  : * O X U −  +1  −: *

ñ ú

+1 Solution in parametric form:

 : *

+1 +2 −

M

+1

M

U +L

(ñó ò )2 ,

U +L

2

P

,

2

P

,

þ

≠ –2.

= 'U +

1 +2 ,

where

# = (B + 1) − 

where

# = −¡

+ 2 −+  +1

¡

−1

ý .

−2

ý .

In the solutions of equations 25–36, the following notation is used:

a U Á

² =

Á

+ L 2U − for the upper sign, sin( @ ln U ) + L 2 cos( @ ln U ) for the lower sign, ln U + L 2 for @ = 0, Á −Á (1 + @ )U + (1 − @ ) L 2 U for the upper sign, (1 − @ L 2 ) sin( @ ln U ) + ( L 2 + @ ) cos( @ ln U ) for the lower sign, ln U + 1 + L 2 for @ = 0.

a Æ =

25.

2 −1

+ U 2 − «

=
ïƒ÷ ñó ò + ù
ïƒ÷ –1 ñ , Solution in parametric form:

ñóò ò ó

1

−1 3

+  = L 1 : * O X U − + ]

2

U 

 1Ë E Ï ±

(ñó ò )2 .

ñóò ò ó

 = U  24.

−1

U «I

–4

=

=
ï –2 ññó ò –
ï –3 ñ 2 . Solution in parametric form:

ñóò ò ó

 = U

−2

= 'U

,

−2

²

−1

Æ ,

where

@ = L 1 , # = 

−1

.

The solution is valid for all three cases of the functions ² and Æ given above. 26.

=
ï 2 ñ –3 (ñó ò )3 –
ï ñ –2 (ñó ò )2 . Solution in parametric form:

ñóò ò ó

 = U

−2

−1

²

Æ ,

= 'U

−2

,

where

@ = L 1, # = 

−1

.

The solution is valid for all three cases of the functions ² and Æ given above. 27.

=
ï –1 (ñó ò )3 ) 2 –
ï –2 ñ (ñó ò )1 ) 2 . Solution in parametric form:

ñóò ò ó

 = U 28.

−2

,

= 12 ' U

−2

 2 Æp²

=
ï ñ –2 (ñ ó ò )5 ) 2 –
ñ –1 (ñ Solution in parametric form:

ñ óò ò ó

 = 12  U

−2

(2 Æp²

−1

−1

− 1 Aç@ 2 ),

where

@ = L 1 , # = (2   )1 J 2.

where

@ = L 1 , # = (2   )1 J 2.

ó ò )3 ) 2 .

− 1 Ae@ 2 ),

= 'U

−2

,

Page 399

400 29.

30.

31.

SECOND-ORDER DIFFERENTIAL EQUATIONS =
ï ñ –2 ñó ò –
ñ –1 . Solution in parametric form:

ñóò ò ó

1

UÆ ,

where

# =

 = L 1 UÆ ,

= L

1

U ² ,

where

# =  2

2

,

=
ï –1 (ñó ò )3 –
ï –2 ñ (ñó ò )2 . Solution in parametric form:

ñóò ò ó

=
ï –1 ) 2 ñ –1 ) 2 ñó ò –
ï Solution in parametric form:

ñóò ò ó

2

,

)2 .

@ = L 1, # =  1J 2 

where

2

2

+ (1 Te@ 2 ) ²

2

=
ï 2 ñ –3 –
ï ñ –2 (ñ ó ò )–1 . Solution in parametric form: = L

+ (1 Te@ 2 ) ²

2

−1

],

where

@ = L 1 , # = 2

2

where

@ = L 1 , # = 2 

.

= 'U 2 ²

],

,

−1

.

1

U VÆ

2

J

2 1 2

+ (1 Te@ 2 ) ²

,

where

# = 4(1 Te@ 2 ) 

U ² ,

where

# = 4(1 Te@ 2 )  4 

W

=
ï –2 ñ (ñó ò )4 –
ï –3 ñ 2 (ñó ò )3 . Solution in parametric form:

−4 4

 .

ñóò ò ó

2

+ (1 Te@ 2 ) ²

J

2 1 2

W

= L

,

1

−4

.

In the solutions of equations 37–50, the following notation is used:

:

=

™ ®ø Á Á ((U U )) ++ L L 2 n Á Á ((U U )) 2;

for the upper sign, for the lower sign,

where ø Á (U ) and n Á (U ) are the Bessel functions, and ® Á (U ) and functions. 37.

J .

−1 2

ñ óò ò ó

 = L 1 U V Æ ]

= 'U 2 [ Æ

=
ï ñ –2 (ñó ò )4 –
ñ –1 (ñó ò )3 . Solution in parametric form:

 = L 1 U ² ,

36.

2

J  J .

−1 2 1 2

@ = L 1, # = 

where

ñóò ò ó

 = U 2 [ Æ

35.

= 'U 2 ²

=
ï –1 –
ï –2 ñ (ñ ó ò )–1 . Solution in parametric form:

(1 Te@ 2 ).

ñ óò ò ó

 = U 2 ² 2 ,

34.

ï –1 ) 2 ñ –1 ) 2 (ñ óò

,

−2

.

= 'U 2 Æ

,

=
ï 1 ) 2 ñ –3 ) 2 (ñó ò )3 –
Solution in parametric form: 2

) ñ )

–3 2 1 2

ñóò ò ó

 = U 2 Æ

33.

 (1 Te@ 2 ).

= L

 = U 2 ²

32.

−2 2

 = L 1 U ² ,

= 2
ï ÷ ññ ó ò + ù
ï ÷ –1 ñ 2 , Solution in parametric form:

ñ óò ò ó

 = L where @ =

−1 2−2 1

U

B +1 B + 2 − , # =−  B +2 2

ù

≠ 0,

Á ,

ù



) are the modified Bessel

≠ –2.

= L 1 +1 U

−1 −1

; Á (U

−2

Á :

−1

(U

:
*

+@

:

),

.

Page 400

 1Ë E Ï ± ñ óò ò ó = (þ + 1)
ï 2 ñ ú (ñ óò )3 + 2
ï ñ ú +1 (ñóò )2 , Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

38.

Solution in parametric form: − 1

 = L where @ = ¡ 39.

+

¡

U

 = L 1 U 2,

40.

=
ï 2 ñ –2 (ñ ó ò )3 – 2
ï ñ Solution in parametric form:

ñ óò ò ó

:

 = U

41.

–1

:
*,

=
ïƒ÷ (ñó ò )3 ) 2 + ù
ïƒ÷ Solution in parametric form:

where @ =

−1 4−2 1

U

−3 −2

U

−1

:

Ë  2Ë E Ï ± þ ≠ –1, þ

ÒÍ

)1+

2

= L

),

2

(Ï

ÒÍ

)

Ë

401

2

≠ –3.

1

U

2−2

Á ,

.

where

@ = 0, # = − 12 

−1

U 2,

where

@ = 0, # = − 12 

−1



2 +1 −2 1

.

U

≠ 0,

ù

Á : Q

−1

:
*

(U

≠ –1.

+@

:

1 U 2S , 2(1 − @ )

)A

−

+1

:
*

(U

ù

.

1J 2 2 S . Q (B + 1) 

−1

+ 2)  − +

−2

(ñó ò )3 ) 2 ,

:

+@

)A

þ

≠ –1,

1 U 2S , 2(1 − @ )

þ

≠ –2.

= L

1

U

4−2

Á ,

1J 2 2 S . Q ( + 2) 

−

¡

ñóò ò ó = 2
ï –1 ) 2(ñ óò )3 ) 2 –
ï –3 ) 2 ñ (ñ óò )1 ) 2 . Solution in parametric form:

 = L 1U 4,

=  (U

:

=
ï ñ –3 ) 2 (ñ ó ò )5 ) 2 – 2
ñ Solution in parametric form:

ñ óò ò ó

:

−1

:
* A

=L

Solution in parametric form: 1

:

−2

= 'U

,

=
ï 1 ) 2 ñ –2 (ñó ò )3 –
ï Solution in parametric form:

) ñ

ñóò ò ó

 = U

2

:

−2

2

–1 2 –1

: (
)2 , *

U 2 ),

) ( ñ ó ò )3 )

−1

 =L

1 2

–1 2

:
1 2 * A 2 U ), ñóò ò ó =
ï –1 ñ –1 ) 2 ñóò –
ï –2 ñ 1 ) 2 .  =  (U

46.

(Ï

:
*,

1


ñ ú

+

Á : Q

+3 , # =( ¡ +2

+@

1

)2 .

= L

2

:
*

(U

ñ (ñ óò )1 ) 2 ,

Á ,

= (þ + 1)
ï ñ ú (ñó ò )5 ) Solution in parametric form:

¡

45.

−1

=L

–1

ñóò ò ó

2 where @ = ¡

44.

óò

(ñ

−2

:

2B + 1 , # = −(B + 1)  −  B +1

 = L 12 +

43.

−1

ñóò ò ó

 = L

42.

 +

= 'U

1

−1

−1 −

= 2
ï –1 ññó ò –
ï –2 ñ 2 . Solution in parametric form:

ñóò ò ó

Á :

−2 −2

+2 +3 , # = ¡  +3 2

=

:

−2

2

: (
)2 ,

(ñó ò )2 . =L

1

:

@ = 0, # = − 21 (−  )−1 J 2.

where

@ = 0, # = − 21 (−  )−1 J 2 .

.

U 4,

1

where

*

−2

,

where

@ = 0, # = −  1 J 2 .

where

@ = 0, # = − 1 J 2 .

Page 401

402 47.

SECOND-ORDER DIFFERENTIAL EQUATIONS =
ï ñ –3 (ñó ò )4 –
ñ –2 (ñó ò )3 . Solution in parametric form:

ñóò ò ó

:

 =

48.

−1

=
ï –2 –
ï –3 ñ (ñó ò )–1 . Solution in parametric form:

ñóò ò ó

 =L

49.

:
2: A U * C

(2U

1

:

−1

:

=

,

=L

),

−1

=
ï ñ –2 (ñó ò )4 – 2
ñ –1 (ñó ò )3 . Solution in parametric form:

1

ñóò ò ó

]

1

:

2

= L

,

1[

where

@ = 0, # = 4 .

),

where

@ = 0, # = 4  .

:

2

,

where

@ = 0, # = 12  

: :q:
2 : 2 U 2 (
)2 + 2 U AIU ],

where

@ = 0, # = 12 

= 2
ï –1 –
ï –2 ñ (ñ ó ò )–1 . Solution in parametric form:

 = L

,

= L

*

*

ñ óò ò ó

−1

:
2: A U * I

(2U

:q:
2 : 2 :  = L 1 [U 2 (
)2 + 2U AIU ],

50.

:

*

1

*

−1

−1

.

.

In the solutions of equations 51–66, the following notation is used:

:

(U ) + L n ™ ®ø 1 J 3(U ) + L 2 2; 1J 3

J U ) for the upper sign, 1 J 3 ( U ) for the lower sign, : : :
1: 2 ’ 1=U + 3 , ’ 2 = ’ 1 ACU 2 2 , ’ 3 = A 23 U 2 3 − 2 ’ 1 ’ 2 , =

1 3(

*

where ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; functions. 51.

=
ï –2 ñ –2 ñó ò + 2
ï –3 ñ Solution in parametric form:

ñóò ò ó

: U J

−2 4 3 1

 = L

52.

= 2
ï –1 ñ –3 (ñó ò )3 +
ï Solution in parametric form: −1

ñ

−1 2

’

=
ï –2 ñ –1 ) 2 ñ ó ò – 4
ï –3 ñ Solution in parametric form:

J :

−1 −4 3 1

U

−2

= 4
ï 1 ) 2 ñ –3 (ñó ò )3 –
ï Solution in parametric form:

J :

−2 −4 3 1

U

−2

1 2

’

= 2
ï –1 ) 2 ñ –2 ñ ó ò +
ï –3 ) Solution in parametric form:

J :

4 −4 3 1

U

−2

’

2 1,

’

−1 2

: U J

2

−2

’

−2 2

J :

−2

−2

’ 2,

2 4 3 1

J :

2 −4 3 1

U

) ñ

(ñó ò )2 .

2 3,

= L

ñ

−1

where

# = 2 2 .

where

# = −2 2  .

2 3,

where

# =T

2 3

’ 2,

where

# =T

2 1 2 3

’ 3,

−1 2 ,

’

.

–1 2 –2

−2 2

’

J :

−2 3

U

= L

= L

ñ óò ò ó

 = L

)

1

(ñó ò )2 .

’ 3,

’ 2,

ñóò ò ó

 = L 55.

U

= L

–2 –2

ñ óò ò ó

 = L 54.

J :

−1 −2 3 1

.

−1 2 ,

’

ñóò ò ó

 = L

53.

2

–1

J U ) are the modified Bessel

1 3(

2 –1

1

U

−4 3

’

 1 J 2 .

 J .

.

= L

1

U

J :

−4 3

where

# =A

J 

1 −1 2 2 . 6



Page 402

 1Ë E Ï ± ñ óò ò ó =
ï –1 ñ –3 ) 2 (ñ óò )3 + 2
ï –2 ñ –1 ) 2 (ñóò )2 . Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

56.

=

Solution in parametric form:

J :

−4 3

 = L 1 U

57.

= L

=
ï ñ –1 ) 2 ñó ò + 2
ñ 1 ) 2 . Solution in parametric form:

J :

−2 3

−1

’ 1,

= 2
ï 1 ) 2 (ñó ò )3 +
ï –1 ) Solution in parametric form:

ñóò ò ó

J :

4 −8 3 1

 = L

59.

’ 2,

U

−4

2

ñ (ñ óò

U

J

J ’

5 2 1 ,

−5 2

J :

8 −8 3 1

U

−4

2 2

(’

J :

8 −4 3 1

U

4 2 3

A

U

:

3

−2

2 1,

’

=
ï –2 ñ –1 ) 2 (ñó ò )4 – 4
Solution in parametric form:

J :

5 −4 3 1

U

−2

2 2

’

4 2 3

A

U

=
ï 2 + 2
ï ñ (ñó ò )–1 . Solution in parametric form:

−1

−4

where

2

(Ï

ÒÍ

)

Ë

403

2

# =T

J

1 2 −1 2 . 6

 

2 2,

’

 J .

−2 1 2

where

# = 2

’ 1,

where

# = −2 1 J 2 

3

’ 1 ),

where

# =

32 −8 5 125

J

where

# =

32 125

−2

.

2 2

(’

4 2 3

A

U

:

J .



)3 . = L

J :

J ’

5 −5 3 1

U

5 2 1 ,

−5 2

J .

−8 5



(ñó ò )–1 .

J :

U

3

J :

−2 3

U

J :

U

−2

ï –3 ñ 1 ) 2 (ñ óò

:

−4

8 −8 3 1

5 −4 3 1

= L

J :

1

’ 1 ),

–2

ñóò ò ó

2 2

(’

4 3

A

2

U

:

3

’ 1 )1 J 2 ,

J 

where

# =A

1 −5 2 4 . 3

where

# =A

1 4 −5 2 . 3



)3 .

1J 2 ’ 1 ,

J :

8 −4 3 1

= L

U

−2

’

2 1,

J

 

ñóò ò ó

J

−1 2 , 2

J :

4 −4 3 1

= L

U

= 2
ï ñ (ñ ó ò )4 +
ñ 2 (ñ ó ò )3 . Solution in parametric form:

−2

’

−2 2 2 ( 3

’

3 2 ),

− 4’

32 −4 9

where

# =



.

where

# = − 329 

−4

where

# = − 98 

−1 4

where

# = 89  4 

ñ óò ò ó

 = L

J :

4 −4 3 1

U

−2

−2 2 2 ( 3

’

’

3 2 ),

− 4’

= 2
ï –1 ñ –3 +
ï –2 ñ –2 (ñó ò )–1 . Solution in parametric form:

= L

1

: U 2J 3 ’

J

−1 2 , 2

.

ñóò ò ó

 = L

66.

2 1,

2

) .

) (ñ óò

= 4
ï 1 ) 2 ñ –3 –
ï –1 ) 2 ñ Solution in parametric form:

:  = L 1 U 2 J 3 ’

65.

’

)1+

–1

–7 5

ñóò ò ó

 = L

64.

−2

ÒÍ

4 −8 3 1

U

= L

= L

= 2
ï ñ –2 ) 5 (ñó ò )4 – 5
ñ Solution in parametric form:

 = L

63.

J :

5 −5 3 1

ñóò ò ó

 = L

62.

U

Ë  2Ë E Ï ±

(Ï

)2 .

2 2,

’

ñóò ò ó = 5
ï –2 ) 5 – 2
ï –7 ) 5 ñ (ñ óò  = L

61.

J :

4 −4 3 1

= L

Solution in parametric form:

60.

1

ñóò ò ó

 = L 1 U

58.

−2

1

: U J

4 4 3 1

2

’

−1 2 ,

= L

=
ï –2 ñ –2 (ñó ò )4 + 2
ï Solution in parametric form:

ñóò ò ó

 = L 1 U

J :

−2 3

−1

’

−1 2

֒

2 3

1

J :

−2 3

U

ñ

(ñó ò )3 .

− 4’

3 1 2 , 2

–3 –1

 J

−1

’

−1 2

’

= L

2 3

− 4’

: U J

4 4 3 1

 J

3 1 2 , 2

2

’

−1 2 ,

 .

−1

.

Page 403

404

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 67–72, the following notation is used:

8

=L

1

` ( Π, 12 ; A"U ) + L 2 i ( Π, 12 ; A"U ),

where ` and i are linearly independent solutions of the degenerate hypergeometric equation:

U8

+ ( 21 ACU ) 8
− Œ"8

*+*

= 0.

*

The function ` = ` ( Π, 12 , AU ) can be expressed in terms of a degenerate hypergeometric series (see equation 2.1.2.70). 67.

68.

=
1 ï ñó ò +
2 ñ . Solution in parametric form:

ñóò ò ó

ñ óò ò ó =
1 ï (ñ óò

 = U 1 J 2 ,

=8

ñóò ò ó =
1 ï ñ –2 ñ óò

= 'U 1 J 2 ,

where

+
2ñ . Solution in parametric form: = A/'U 1 J 2 8

=
1 ï –1 ñ –2 ñó ò +
2 ï –2 ñ Solution in parametric form:

ñóò ò ó

−1

–1

,

where

= A/'U 1 J 2 8

,

–2 (ñ ó ò )3 +
2 ï –2 ñ

 = A U 1 J 2 8

−1

÷

*

3 +2



−2 −1 1

O X

–1

M

ý±U

= A 2

−2

, #

1

= T 4

−2

Π, #

U

+1

=
(þ + 1) ï ñ ú (ñó ò ) ú Solution in parametric form:

+

 = L 1( + where , = − ¡

¡

+1)2

OU − X

+3 +2

ý±U

M

U

+L

 

)2 .

−1

,

−1 2

−2


ñ ú

+1

−L

+2 +2 , # =− ¡  +1 ( + 1)2 ý

¡

1

= T± 2 Œ , #

1

= T/

2

#

#

#

where

*

÷ –1 ñ (ñ óò ) 2÷ ÷ +1+1 ,

1−

ú

óò

(ñ

8
,

=8

B +1 B +1 where , = − , # = 2  B B ý

ñóò ò ó

−1

8
,

=
ï ÷ (ñ ó ò ) ÷ +1 + ù
ï Solution in parametric form:

ñ óò ò ó

 = L

74.

1

2

−2

= A 4

Π.

= T 2

2

−2

.

2

= A/ 2 Ý Œ + 12  .

݌ + 12  , #

2

= A" 2 Π.

1

= A/ 2 Π, #

2

=A

1

=T

2

= T/ 2 Π.

.

Solution in parametric form:

73.

where

=8

*

ñ óò ò ó =
1 ï –1 ñ

#


,

=
1 ï –1 (ñó ò )3 +
2 ï –2 ñ (ñó ò )2 . Solution in parametric form:

 =8

72.

*

ñóò ò ó

 = A U 1 J 2 8
,

71.

#

–1

 =8 ,

70.

where

)3 +
2 ñ (ñó ò )2 . Solution in parametric form:

 =8 ,

69.

,

J

P

ù

1 +1

(ñó ò ) ú

1 +2

2

,

= L

 +



≠ –1.

þ

,

U −L 2 , M ý±U  + 1 P

−2 1

+

−3

þ

≠ –1,

OX

 ( + 1)ý + ¡ S  Q −

#



.

+1

−2 −

ù

≠ 0,

1 2 , 2

2 = L 1 O U − X

,

B…'ý  O−  P

P

#

where

1 2 2 .

ý±U 1 +2

M

U

+1

≠ –2.

+L

− 2

P

+

1 +1

,

.

Page 404

 1 Ë E Ï ± ( Ï ÒÍ ) Ë +  2 Ë E Ï ± ñ óò ò ó =
ï –ú –1 ñ ú (ñ óò )A –
ï –ú –2 ñ ú +1 (ñ óò )A –1 , þ ≠ –1, @ Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

75.

=

1

1

1

2

2

(Ï

ÒÍ

)

Ë

405

2

≠ 2,

þ

+ @ – 1 ≠ 0.

Solution in parametric form:

 = L

exp O

1

where

0 −2 X +0 −1 ¡

U

M

= L

,

 P

1

−2 U + - +- −1 exp O

¡

0 −2 X +0 −1

U

M

 P

,

1 ( + 1)( 0 − 2) +0 −1 ,  ý + L 2 U   2−- − U , , = ¡ 0 −2 +0 −1 ( 0 −¡ 2)  ( + 1)( + 0 − 1) + −1 1− + ý S -. # =− ¡ ¡ 3   Q ( + 0 − 1)  ( 0 − 2)

 = ¡

76.

ñóò ò ó =
ï A –2 ñ 1–A (ñ óò )A –
ï A –3 ñ 2–A (ñ óò )A –1 , @

¡

≠ 2.

Solution in parametric form:

 =L where  77.

ñóò ò ó

=

ñ óò ò ó

=

1 2−

= (2 − 0 ) V ý +  (- −2) * W


ï

–1

ñóò


ï ñ

–2


ï

–

(ñ

–2

ñ

óò

)3 –
ñ

–1

=L

2

exp OU + X

M

U ,  P

- − 1, # = (2 − 0 )2−- ý .

(ñ

óò

if # ≠ 1, if # = 1.

)2 .

™

L

1

ÝL

+ L 2 | |O + L 2 ln | | 

if # ≠ 1, if # = 1.

1

In the solutions of equations 79 and 80, the following notation is used:

 =

79.

U ,  P

.

Solution in implicit form:  =

]

M

L 1  + L 2 | | O   L 1 + L 2 ln | | 

™

=

Solution: 78.

exp O X

1

ñóò ò ó

=


ï ñ

–2

(ñó ò )A –


ñ



1

ý +1

U :

+1

+

U + ln |U | + L –1

(ñó ò )A –1 ,

ý

1 :

U

+L

2

if ý = 0.

2

@

if ý ≠ 0,

≠ 3.

Solution in parametric form:

 = L 1 [U : where ý = 80.

ñ óò ò ó

=


ï

+1

− (ý + 1)  ] exp O − XÈU

2−0 , # = − - −3  0 −3 –1

(ñ

3−

:

−1 −1



U , M P

:

−1 −1

U , M P

− (ý + 1)  ] exp O − X U :

−1 −1

U , M P

= L

1

exp O − XwU



-.

óò )A –
ï –2 ñ (ñ óò )A –1 , @

≠ 1.

Solution in parametric form:

 = L where ý =

1

exp O − X U :

−1 −1



0 −2 , # = −  - −1  1−0

U , M P

1−

= L

1[

U :

+1



-.

Page 405

406 81.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó


1ñ

=

–1

=

Solution:

]


2ï

(ñó ò )2 +







–1

ñóò

.

A ( L 1 | | O + L 2 ) O 1 −1 A ( L 1 ln | | + L 2 ) O 1 −1 L 2 exp( L 1 | | O 2 +1 ) L 2 | | k 1 2 +1

#

if if if if

1

#

1

#

#

≠ 1, ≠ 1, = 1, = 1,

1

#

1

≠ −1; = −1; ≠ −1; = −1.

2

#

2

#

2

#

2

In the solutions of equations 82–84, the following notation is used:

ú

82.

=
ïƒú +1 ñ ú (ñó ò )2 –
ïƒú Solution in parametric form:

ñ ú

ñóò ò ó

 = L 1 U 1J 2 ’

83.

84.

=
ï –2ú –2 ñ ú (ñ ó ò )2 –
Solution in parametric form:

+1

 =L 85.

=
ïƒ÷ ñ ú (ñó ò )2 –
ïƒ÷ Solution in parametric form:

ñóò ò ó

 =L where 

86.

÷ ñ

=
1ï Solution: 1

–1

(ñ

óò

Q

ú (ñ óò

=
1ñ Solution: 1

ñóò

,

þ

, =L

ñóò

+1

1

’ ,

, =

þ

,

J ’

Q

#

1 2

þ

=L

2

ù

,

.

¡

.

¡

+3 . 2

, = −¡

ù

≠ –1,

≠ 0.

U , U Ê  P

, =−

M

  (  + L 2 )−1  S , B P M

#


2ï

–1

+

+1

1

¡

ñ ú

+1

M

ñóò

+1 P  +L

2.6.4. Other Equations (‘

,

2

S ,

U

−1

#

exp O

¡

≠ 0.

2

)2 +

exp X exp O −


2 ï ÷ –1 ñ óò

)2 +

−1

where

þ

S .

≠ –1.

J ,

−1 2

U

2 1

, =

≠ –1.

= L 1 U exp O,lX

M

( + , )  (  + , − 1) 

exp X exp O

ñóò ò ó

 =L

ñ ú

=L +1

ñ óò

+1

≠ –1.

U J ’ 1J 2,

J ,

+3 2

ñ ú

+1

# U  Q 2( , + 1)

−1 1 2 1

1 2

U , U ¥  P

exp OX

1

J ’

−1 2

’ , –1

þ =L

ñ ú

J exp

−1 2

U

=  (U ) is the solution of the transcendental equation

ñ óò ò ó

=L 87.

2 1

–3

2

,

J ,

ï –ú

=
ï – 2 ñ ú (ñó ò )2 –
Solution in parametric form:

ñóò

=L

−1 2

 = L 1U

ú

+1

ï –2ú

ñ óò ò ó

ñóò ò ó

ú

U , ú M2 U £ +4P

’ = exp OX

+1

U +

+1

P

¡

B

+1

,

.

#   ) exp O

 = X (1 − #

1

+

+1

 

2

B

PuM

 .

≠ –1. where

 = X (1+ #

2

) exp O −

# ¡

1

+

+1

+ 1 PuM

.

1 ≠ ‘ 2)

2.6.4-1. Classification table. Table 27 presents all solvable equations whose solutions are outlined in Subsection 2.6.4. Equations are arranged in accordance with the growth of 0 1 ( 0 1 > 0 2 ). The number of the equation sought is indicated in the last column in this table.

Page 406

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

 1Ë E Ï ±

=

1

1

Ë  2Ë E Ï ±

ÒÍ

(Ï

)1+

2

TABLE 27

+
Solvable cases of the equation   = # 1   1 1 (  )- 1 + #

01

02

arbitrary

arbitrary

arbitrary

arbitrary

arbitrary

0

arbitrary

arbitrary

arbitrary

1

¡

¡

B

2

¡

2

 

2

2

ÒÍ

(Ï

)

407

2

+ 2 ( 
)- 2 , 0 ≠ 0 1 2 B

1

Equation

2

0

1

Ë

0

2.6.4.3

1

2.6.4.4

B

0

arbitrary

1−01

1− 02

01−2

02−2

2.6.4.9

3−0 1

1−01

01−2

01−2

1−01

2.6.4.8

1

0

− 21

− 12

0

0

2.6.4.1

1

0

− 21

− 12

0

1

2.6.4.14

1

0

− 21

0

0

0

2.6.4.16

2

0

−1

1

arbitrary

arbitrary

2.6.4.18

2

1

arbitrary

arbitrary

0

−1

2.6.4.13

2

1

0

0

arbitrary (B 2 ≠ −1)

2.6.4.5

2

1

0

0

−1

2.6.4.7

2

1

−1

0

arbitrary

arbitrary

2.6.4.12

2

1

−1

0

0

arbitrary (B 2 ≠ −1)

2.6.4.6

5 2

1 2

arbitrary

3

0

arbitrary* ( 1 ≠ −2)

3

1

arbitrary

arbitrary ( 1 ≠ −1) ¡ arbitrary ( 1 ≠ −1)

¡

¡ ¡

¡

1

+2

1

+3

¡

1

+2

¡

1

+3

¡

1

2.6.4.11

¡

1

2.6.4.10

arbitrary

1

−1

2.6.4.19

− 21

2.6.4.2

3

2

0

0

− 21

3

2

0

0

0

− 21

2.6.4.17

0

− 21

− 21

2.6.4.15

3 * For

2

¡

1

1

= −2, see Equation 2.6.4.8 with 0 = 3.

2.6.4-2. Solvable equations and their solutions. 1.

=
1 ñ –1 ) 2 ñó ò +
2 ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó

 =L 2.

exp( #

1

1

U )−

#

4#

2

=
1 ï –1 ) 2 (ñ ó ò )3 +
2 ï –1 ) 2 (ñ Solution in parametric form:

ñ óò ò ó

 = # Q

2

L

1

exp(− #

2

U )+

1

2

U

óò

+L

2,

= #

1

Q

L

1

exp( #

1

U )−

#

2#

2 1

2

U!S .

)2 .

#

2#

1 2

2

U!S ,

=L

1

exp(− #

2

U )−

#

4#

1 2

U

2

+L

2.

Page 407

408 3.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

=


1 ñ ú ( ñ ó ò )A


2 ñ ú ( ñ ó ò )A

+

1

2

.

1 b . Solution in parametric form with

 =L where  = L

1

+(

+ XÊ #

2

1

+ 1) X U  #

¡

1

U -1 +#

where  = L 4.

ñóò ò ó

=

1

X U #


1 ïƒ÷ (ñ óò )A

1

1

U -1 +#

2

1

−1

U - 2


2 ïƒ÷ (ñ óò )A

+

−1

U -1 +#

2

+1

M

U ,

= +

U ,

= ,

1 +1

,

U . −1

U - 2

– 

M

–

U . M

2

M

+ +



= −1:

¡

+ XÊ #

2

−1 −

U - 2

2

U - 1 + # 2U - 2 

2 b . Solution in parametric form with

 =L

≠ −1:

¡

.

1 b . Solution in parametric form with B ≠ −1:

 =  where  = L

1

1 +1

=L

,

+ (B + 1) XÊ #

2

U -1 +#

1

2

+X U  # −1

U - 2

M

1



U - 2

−1 − +1

U - 2

−1

U -1 +#

2

U -1 +#

2





M

U ,

U .

2 b . Solution in parametric form with B = −1:

–  = , where  = L 5.

ñóò ò ó

=

1

X¥ # 1 U - 1 + #


1 ñ ú (ñ óò

)2 +

Solution: X exp O − 6.

ñ óò ò ó

=


1ñ

–1

(ñ

óò

1 b . Solution for #

# ¡

)2 + 1

2

=L


2 ƒï ÷ ñ óò 1

+1

+

+1


2 ï ÷ ñ óò

7.

ñóò ò ó

=


1 ñ ú (ñ óò

1 b . Solution for #

2 b . Solution for #

1

2

1



– M

U ,

ù

≠ –1,

PuM

=L

ù

,

1

þ

≠ –1.

#

X exp O

2

B +1

 

+1

 + L 2.

P3M

≠ –1.

≠ 1: = L

1

X exp O

=L

2

exp L

–1

ñóò

,

þ

 = L Q

1

#

2

B +1

 

+1

P3M

1 −1

 + L 2S O

.

= 1:

)2 + 2

+X U  #

U . M

,

Q

2 b . Solution for #

−1

U - 2

2


2ï

Q

#

X exp O

1

2

B +1

 

+1

P3M

 S .

≠ –1.

≠ −1:

X exp O −

= −1:

 =L

2

exp L

Q

1

¡

#

1

+1

+

X exp O −

+1

¡

#

PuM

+L

1

+1

+

+1

−

2

S O

PuM

2 −1

.

S .

Page 408

Ï Ò˜Í Í Ò

2.6. EQUATIONS OF THE FORM

8.

=
1 ï A –2 ñ 1–A (ñ ó ò )A +
2 ï Solution in parametric form:

ñ óò ò ó

 =L 9.

#

U - +#

1

2

M

=
1 ï A 1 –2 ñ 1–A 1 (ñ ó ò )A 1 +
Solution in parametric form:

U U

exp OX

2

#

1

U - + # 2U - − U

ñ ú

J exp O

−1 2

=U

2

3#

+4

¡

=
ï ú +2 ñ ú (ñ ó ò )5 ) 2 +
Solution in parametric form:

ñ óò ò ó

Here, the function 1 b . For

≠ −1,

¡

˜

=

M2

+2

þ

,

U

PpQ

ï ú ñ ú

2

˜

+2

M2

U

Ë  2Ë E Ï ±

)1+

=L

1

exp OX

=L

1

exp OX

ÒÍ

(Ï

)

2 +3

˜

,

U

1

U - +#

2

U M 3− U

#

U

1

- −U

2

+ U*P

2

M

U -2 −U

2

+ U*P

U

3#

+2

U +

¡

+2

˜

:

−1

: 


(U

+@

:

U £ =

+4P

J

˜

2

U

M2

−1 2

U!S

P3M

1 U J exp O X

−1

˜

U £

2

−1 1 2 1

=L

,

˜

1 U J exp O X

=L

 (2 + 3) ¡ ‡ 2( + 1) ¡

=

409

2

U - +#

≠ −3 2:

¡

Ë

1

−1 1 2 1

ó ò )1 ) 2 .

+4P

#

2

.

.

≠ –2.

− # X U + exp O

(ñ

2

.

,

+4P

L

ÒÍ

(Ï

(U ) is defined in parametric form, U = U (‡ ),

U = ‡ 2 +

:

U +

U £

2

˜

˜

˜

1  = L 1 U J exp O − X 1 2

˜

2

,

+ U*P

+3

U £

2

,

2

2

1

)3–A .

+ U*P

2

2

=
ïƒú +3 ñ ú (ñó ò )3 –
ïƒú Solution in parametric form:

˜

- −U 2

ñóò ò ó

where

A ñ A –2 (ñ óò

U

1  = L 1 U J exp O − X

1

ï A –2 ñ 1–A (ñ óò )A

2

M

1

1 2

11.

3−

ñ óò ò ó

 =L 10.

exp OX

2

1–

 1Ë E Ï ±

=

˜

˜

M2

U

,

.

+4P

.

(‡ ), as follows:

),

for the upper sign, 2 ø Á ( ‡ ) + n Á ( ‡ ) Á (‡ ) and n Á (‡ ) are the Bessel functions, L 2 ®Á (‡ ) + ; Á (‡ ) for the lower sign, ø and ®Á (‡ ) and ; Á (‡ ) are the modified Bessel functions,

where

L

=™

+1 , 2 +3

@ = ¡ 2 b . For

¡

+2

#3

P

−

+

1 +1

= −1:

¡

U = 3 b . For

2

 =O− ¡

¡

U =#

where , =

‡

2#

2 2

˜

,

= −3 2:

2 −4

ü

‡

,

|1 − 8 #

˜




=















#

= £ 2

:

−1

: 


,

,

A

(2 ¡ 8(

¡

where

+ 3)2 2+ +  + 1)2

:

=L

+3 +1

ø

2 = O − ¡

2 0(

#

+2 +

P

1 +1

.

‡ ) + n 0 (‡ ).

1 (1 + , ) L 2 ‡  + (1 − , )‡ −  2# L 2 ‡  + ‡ − 1 L 2 ln ‡ + L 2 + 1 2# L 2 ln ‡ + 1 1 ( L 2 − , ) sin( , ln ‡ ) + (1 + , L 2 ) cos( , ln ‡ ) 2# L 2 sin( , ln ‡ ) + cos( , ln ‡ )

if #

2

< 18 ,

if #

2

= 18 ,

if #

2

> 18 ,

2 |.

Page 409

410 12.

SECOND-ORDER DIFFERENTIAL EQUATIONS =
1 ïƒ÷ 1 ñ –1 (ñó ò )2 +
1 b . Solution for B 2 ≠ −1:

ñóò ò ó

2

ïƒ÷ ñóò

.

2

−1 # 2  2+1 #   2+1 L 2 + X (1− # 1  1) exp O 2  S . M P B 2 +1 PpQ B 2 +1 P M 2 b . Solution for B 2 = −1, # 2 ≠ −1, # 2 ≠ −B 1 − 1: −1 1 # 1 = L 1 exp X  O 2 OL 2 +  O 2 +1 −   1 + O 2 +1  S . Q # 2+1 B 1+# 2+1 P M 3 b . Solution for B 2 = −1, # 2 = −1:

=L

1 exp

O Xv



where  = exp O

,

=L 4 b . Solution for B

2

= −1, # =L

13.

2

Q

= −B

1 −1

Q

=
1 ñ ú 1 ( ñ ó ò )2 +
2 ï 1 b . Solution for 1 ≠ −1:

–1

ñ ú ñ óò

M P

2 b . Solution for

 =L

1

¡

1

3 b . Solution for

4 b . Solution for

¡

= −1, # −

exp X

Q

1

¡

1

≠ 1, #

1

O OL 1

1

= 1:

 =L

1

exp X

= −1, #

1

=

1

exp X

Q

2

−

B

1+1

1 +1

¡

≠

2

¡

−1

#

1

1

B

1

 

1

 − 1 − #

−1

1

+

1

PpQ

L 2 + X (1+ #

P

 S . M

ln 

−1

P

 S . M

2 −1

2

+ −1 + 2) exp O − # 1 1+1 S . +1 PuM ¡ 1

+ 1: 1−

1

O

+

1

#

2

¡

OL

2

+ ln +

OL

2

−

#

¡

1 2

−

+

−# 2

2

+

2

+#

+

+1

1

2

2

¡

+ 1:

2 −

OL

1 1−#

Q

¡

+ ln  −

# 1+

1

+

2

= −1, #

 =L ]

 =exp O −

,

2

.

2

¡

1 exp O X 

OL

− 1:

1

exp XZ − 

1

ñ óò ò ó

 =L

−1

exp XÈ

1

2

−1

O

1 +1

P

−1

M

S .

M

S .

P ln

2−

−1

P

M

S .

In the solutions of equations 14 and 15, the following notations are used:

² =



Æ =



L 1U  1 + L 2U  2 + L 3U  3 L 1U   * + L 2  ‡ * L 1   * + L 2  ~ * cos  ÑU L 1 , 1U  1 + L 2 , 2U  2 + L 3 , L 1 (1 + ,!U )   * + L 2 .  ‡ * L 1 ,   * + L 2  ~ * ( cos  ÑU

if $ 2 (8 $ 13 + 27 $ 2 ) < 0, if 8 $ 13 + 27 $ 2 = 0, if $ 2 (8 $ 13 + 27 $ 2 ) > 0,

U 

if $ 2 (8 $ 13 + 27 $ 2 ) < 0, if 8 $ 13 + 27 $ 2 = 0, −   sin  ÑU ) if $ 2 (8 $ 13 + 27 $ 2 ) > 0,

3

3

where , 1 , , 2 , and , 3 (real numbers) or , and *AC™  (one real and two complex numbers) are roots of the cubic equation Œ 3 − $ 1 Œ 2 − 12 $ 2 = 0. 3 In the special case 8 $ 1 = −27 $ 2, we have , = 23 $ 1 (multiple root) and . = − 13 $ 1 (simple root). x3y'z|{}~/€ In the expressions for ² and Æ , the constant L 3 can be set to any nonzero number (for example, one can set L 3 = A 1).

Page 410

2.7. EQUATIONS OF THE FORM

14.

ñóò ò ó

) ñóò


1ñ

=

–1 2


2ï ñ

+

)

–1 2

Ï Ò˜Í Í Ò

= â (Ë )ã (Ï ) (Ï

›

ÒÍ

411

)

.

Solution in parametric form:

 =² , 15.

ñ óò ò ó


1ï

=

) ñ (ñ ó ò

–1 2


2ï

)3 +

2

=Æ

,

) (ñ ó ò

–1 2

$

where

=#

1,

$

2

=#

= −#

2,

$

2

= −#

1

2.

)2 .

Solution in parametric form: 2

 =Æ ]

=² ,

,

$

1

1.

In the solutions of equations 16 and 17, the following notations are used:


 U P J 2 ( L 1 U1  =
 L 1 U exp( 2 $

U −  ) + L 3 if $ 12 + 2 $ 2 > 0, if $ 12 + 2 $ 2 = 0, 1U ) + L 2 1 if $ 12 + 2 $ 2 < 0, 1 exp( 2 $ 1 U ) cos( ÑU ) + L 2 P 1 J 2 [( L 1 ($ 1 + 2 , )U  + L 2 ($ 1 − 2 , )U −  ] if $ 1 if $ 1 ( $ 1 U + 2) exp( 2 $ 1 U ) 1 if $ 1 exp( 2 $ 1 U )[ $ 1 cos( ÑU ) − 2  sin( ÑU )] +L

1

²

L




Æ =

U




L L

where , = 16.

where

ñóò ò ó

1 2

Å $

2 1


1ñ

=

+ 2$

2

) ñóò

–1 2

and   = +




1 2

2

Å −( $

2 1

= 14 Æ

2

+ 2$

2 1 2 1 2 1

+ 2$ + 2$ + 2$

2

> 0, = 0,

2

< 0,

2

2 ).

2.

Solution in parametric form:

 =² , 17.

ñóò ò ó

=




1(

ñóò

)3 +


2ï

) (ñ óò

–1 2

,

$

where

=#

1,

$

2

=#

= −#

2,

$

2

= −#

1

2.

)2 .

Solution in parametric form:

 = 14 Æ 18.

19.

ñóò ò ó

=


1 ïƒ÷ ñ

–1

ñóò ò ó

=


1ï ñ ú

1

1

(ñó ò )2 +

2

=² ,

,


2 ïƒ÷ ñ 2

where

$

1

1.

.

¨2
 Solution: = L 1 exp − X S , where ¨ = ¨ ( ) is the general solution of the M Q ( # 1   1 − 1)¨ second-order linear equation ( # 1   1 − 1)¨ 

 − # 1 B 1   1 −1 ¨ 
+ # 1   2 ( # 1   1 − 1)2 ¨ = 0. (ñó ò )3 +


2ï

–1

ñ ú ñóò 2

.

¨
 M Solution:  = L 1 exp X S , where ¨ = ¨ ( ) is the general solution of the Q ( # 2 + 2 + 1)¨ ¨
− # + 1 ( # + 2 + 1)2 ¨ = 0. second-order linear equation ( # 2 + 2 + 1)¨2

− # 2 2 + 2 −1 2 1 2   ¡

2.7. Equations of the Form í ïwî²î ï = ð ( ì )ë ( í ) ’ ( í­ï î ) See Section 2.3 for the case  ( ) = const   , ( ) = const + , % (¨ ) = const.

See Section 2.5 for the case  ( ) = const   , ( ) = const + , % (¨ ) = const ¨ - .

Page 411

412

SECOND-ORDER DIFFERENTIAL EQUATIONS

â$äãÑã ä = å ( æ )ç (â )

2.7.1. Equations of the Form 1.

ï

2(þ


ñ ú S

+ 1) ñ + + 3)2 See equation 2.4.2.4.

ñ óò ò ó

=

–2

–

Q (þ

2.

= ï –2 ( 15 ñ +
ñ –7 ). 4 See equation 2.4.2.35.

3.

= ï –2 (6ñ +
ñ –4 ). See equation 2.4.2.31.

4.

= ï –2 (12ñ +
ñ –5 ) 2 ). See equation 2.4.2.64.

5.

= ï –2 (2ñ +
ñ –2 ). See equation 2.4.2.6.

6.

3 = ï –2 (– 16 ñ +
ñ See equation 2.4.2.26.

7.

9 = ï –2 (– 100 ñ +
ñ See equation 2.4.2.10.

8.

= ï –2 ( 34 ñ +
ñ –5 ) 3 ). See equation 2.4.2.12.

9.

= ï –2 ( 63 ñ +
ñ –5 ) 3 ). 4 See equation 2.4.2.66.

10.

5 = ï –2 (– 36 ñ +
ñ See equation 2.4.2.29.

11.

= ï –2 (– 29 ñ +
ñ –1 ) 2 ). See equation 2.4.2.14.

12.

4 = ï –2 (– 25 ñ +
ñ See equation 2.4.2.8.

13.

= ï –2 (20ñ +
ñ –1 ) 2 ). See equation 2.4.2.33.

14.

= ï –2 (– 12 ñ +
ñ 49 See equation 2.4.2.37.

15.

6 = ï –2 (
ñ 2 – 25 ñ ). See equation 2.4.2.60.

16.

6 = ï –2 (
ñ 2 + 25 ñ ). See equation 2.4.2.62.

,

þ

≠ –3,

þ

≠ –1.

ñóò ò ó ñóò ò ó ñ óò ò ó ñóò ò ó ñ óò ò ó ñóò ò ó

)

–5 3

).

)

–5 3

).

ñóò ò ó ñóò ò ó ñóò ò ó

)

–7 5

).

ñ óò ò ó ñóò ò ó

)

–1 2

).

ñóò ò ó ñóò ò ó

)

1 2

).

ñóò ò ó ñ óò ò ó

Page 412

2.7. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

= â (Ë )ã (Ï ) (Ï

17.

= ï –4 ) 3 (
+ (­ñ –1 ) 2 ). See equation 2.4.2.40.

18.

= (
ï 4 + ( ï 3 )ñ –7 . See equation 2.4.2.39.

19.

= (
ï 2 + ( )ñ –5 . See equation 2.4.2.16.

20.

= (
ï –1 + ( ï –2 )ñ See equation 2.4.2.28.

21.

= (
ï –7 ) 3 + ( ï –10 ) 3 )ñ See equation 2.4.2.48.

22.

= (
ï –4 ) 3 + ( ï –10 ) 3 )ñ See equation 2.4.2.49.

23.

= (
ï –4 ) 3 + ( ï –7 ) 3 )ñ See equation 2.4.2.24.

24.

= (
ï –2 ) 3 + ( ï –4 ) 3 )ñ See equation 2.4.2.90.

25.

= (
+ ( ï –2 ) 3 )ñ See equation 2.4.2.89.

26.

= (
ï 2 + ( )ñ –5 ) 3 . See equation 2.4.2.47.

27.

= (
ï 2 + ( ï )ñ –5 ) 3 . See equation 2.4.2.46.

28.

=
( ô ï –2 ) 3 + õ ï –5 ) 3 )2 ñ –5 ) 3 . This is a special case of equation 2.7.1.37 with ( = 1 and

)

413

ñóò ò ó ñóò ò ó ñ óò ò ó

–2

.

ñ óò ò ó ñ óò ò ó ñ óò ò ó ñ óò ò ó ñóò ò ó

)

–5 3

)

.

)

.

–5 3

–5 3

)

.

)

.

–5 3

–5 3

.

ñóò ò ó ñóò ò ó ñóò ò ó

= (
ï –8 ) 5 + ( ï –13 ) 5 )ñ See equation 2.4.2.25.

30.

= (
ï –5 ) 2 + ( ï –7 ) 2 )ñ See equation 2.4.2.23.

31.

=
( ô ï 5 + õ ï 4 )–1 ) 2 ñ –1 ) 2 . This is a special case of equation 2.7.1.38 with ( = 1 and

33.

ÒÍ

ñóò ò ó

29.

32.

›

ñóò ò ó ñóò ò ó

)

–7 5

)

–1 2

= 0.

M

.

.

ñóò ò ó

= 0.

=
( ô ï 15 ) 8 + õ ï 7 ) 8 )–4 ) 3 ñ –1 ) 2 . This is a special case of equation 2.7.1.39 with ( = 1 and

M

ñóò ò ó

=
( ô ï 7 ) 3 + õ ï 4 ) 3 )–15 ) 7 ñ 2 . This is a special case of equation 2.7.1.40 with ( = 1 and

= 0.

ñóò ò ó

M M

= 0.

Page 413

414 34.

35.

36.

37.

38.

39.

40.

41.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

= (ô

ï

2

+

õï

+ ö )ñ

)

–5 3

.

J

The transformation  =  (a ), = ( g
) leads to a third-order equation: 2 g
 g

g
g − ( 4 2 + ' + ( ). Differentiating the latter equation with respect to a and dividing it by  3 (  obtain a fourth-order constant coefficient linear equation: 3 g

g
g
g = 4  + 2  . 3 2

= ( ô ï –10 ) 3 + õ ï –7 ) 3 + ö ï The transformation  = 1 a , ( a 2 + 'a + ( )¨ −5 J 3 .

) )ñ

ñóò ò ó

)

–4 3

–5 3

g

g ) = g
, we 2

.

= ¨u a leads to an equation of the form 2.7.1.34: ¨ g

g =

= ü ( ô ï 2 + õ ï + ö )÷ ñ –2÷ –3 . This is a special case of equation 2.9.1.21 with  (‡ ) = ,!‡ −2  . Setting ‡ ( ) = (  2 + ' + ( )−1 J 2 and integrating the equation, we obtain a first-order separable equation: (  2 + ' + ( )2(‡
 )2 = ( 14  2 − !( )‡ 2 −   +1 ‡ −2  −2 + L 1 .

ñ óò ò ó


(ô ï

ï

+ ý )–10 ) 3 ñ –5 ) 3 .  +  ¨ The transformation ˆ = , = leads to an Emden–Fowler equation of the form (˜ + (˜ + 3 ¨

¨ −2 2 −5 J 3 2.3.1.9: ŠªŠ = #3¢ ˆ ,M where ¢ = M  − ˜( .

ñóò ò ó

=

ñ óò ò ó =
(ô ï

+ õ )2 ( ö

M

)

+ý ) ñ .   +  ¨ The transformation ˆ = , = leads to an Emden–Fowler equation of the form (˜ + ˜(  + 3 ¨

−2 −1 J 2 ¨ M −1 J 2 M ª Š Š 2.3.1.25: = #3¢ ˆ , where ¢ =  − ˜( .


(ô ï

+õ )

) (ö ï

–1 2

M

ï

+ ý )–7 ) 6 ñ –1 ) 2 .   +  ¨ , = leads to an Emden–Fowler equation of the form The transformation ˆ = (˜ +¨ ˜(  + 3 ¨

−2 −4 J 3 −1 J 2 M M , where ¢ =  − ˜( . 2.3.1.17: ŠªŠ = #3¢ ˆ

ñóò ò ó

=

ñ óò ò ó =
(ô ï

+ õ )–4 ) 3 ( ö

–2 –1 2

+õ )

) (ö ï

M

) ñ

+ý ) .  +  ¨ The transformation ˆ = , = leads to an Emden–Fowler equation of the form (˜ + ˜(  +

¨ −2 −15 J 7 ¨ M 2 M ª Š Š 2.3.1.20: = #3¢ ˆ , where ¢ =  − ˜( . –15 7

–20 7 2

M

=
exp( ô ï 2 + õ ï ) exp( üIñ ). The substitution , ¨ = , +  2 + '  leads to an autonomous equation of the form 2.9.1.1: ? ¨2

 = #   + 2 , −1 .

ñóò ò ó

2.7.2. Equations Containing Power Functions ( “1 const) 1.

2(ù + 1) ï +
Q (ù + 3)2 See equation 2.6.2.116.

2.

ï +
ï –7 )ñ = (– 15 4 See equation 2.6.2.117.

3.

= (–6 ï +
ï –4 )ñ –2 (ñ See equation 2.6.2.118.

ñ óò ò ó ñóò ò ó ñ óò ò ó

=

ï ÷ Sñ –2

–2

(ñ

óò

)3 ,

ù

≠ –3,

ù

≠ –1.

(ñó ò )3 .

óò

)3 .

Page 414

2.7. EQUATIONS OF THE FORM

4.

= (–12 ï +
ï –5 ) 2 )ñ See equation 2.6.2.119.

5.

= (–2 ï +
ï –2 )ñ –2 (ñó ò )3 . See equation 2.6.2.120.

6.

3 ï = ( 16 +
ï –5 ) 3 )ñ See equation 2.6.2.121.

7.

9 ï = ( 100 +
ï –5 ) 3 )ñ See equation 2.6.2.122.

8.

= (– 34 ï +
ï –5 ) 3 )ñ See equation 2.6.2.123.

9.

ï +
ï –5 ) 3 )ñ = (– 63 4 See equation 2.6.2.124.

10.

5 ï = ( 36 +
ï –7 ) 5 )ñ See equation 2.6.2.125.

11.

= ( 29 ï +
ï –1 ) 2 )ñ –2 (ñ See equation 2.6.2.126.

12.

4 ï = ( 25 +
ï –1 ) 2 )ñ See equation 2.6.2.127.

13.

= (–20 ï +
ï –1 ) 2 )ñ See equation 2.6.2.128.

14.

ï +
ï 1 ) 2 )ñ –2 (ñóò )3 . = ( 12 49 See equation 2.6.2.129.

15.

6 ï = (
ï 2 + 25 )ñ –2 (ñó ò )3 . See equation 2.6.2.130.

16.

6 ï = (
ï 2 – 25 )ñ –2 (ñó ò )3 . See equation 2.6.2.131.

17.

= (
+ ( ï –1 ) 2 )ñ See equation 2.6.2.15.

18.

= ï –7 (
ñ 4 + (­ñ 3 )(ñó ò )3 . See equation 2.6.2.111.

19.

= ï –5 (
ñ 2 + ( )(ñó ò )3 . See equation 2.6.2.96.

20.

= ï –2 (
ñ –1 + (­ñ –2 )(ñó ò )3 . See equation 2.6.2.110.

ñóò ò ó

–2

Ï Ò˜Í Í Ò

= â (Ë )ã (Ï ) (Ï

›

ÒÍ

)

415

(ñó ò )3 .

ñóò ò ó ñóò ò ó ñ óò ò ó ñ óò ò ó ñ óò ò ó ñ óò ò ó

–2

(ñó ò )3 .

–2

ñóò ò ó ñóò ò ó

–2

)3 .

óò

(ñ

óò

(ñ

ñ óò ò ó

)3 .

óò

(ñ

–2

–2

óò

(ñ

–2

óò

)3 . )3 .

)3 .

(ñó ò )3 .

–2

(ñó ò )3 .

ñóò ò ó ñóò ò ó ñóò ò ó ñóò ò ó

) (ñ óò

–4 3

)3 .

ñóò ò ó ñóò ò ó ñóò ò ó

Page 415

416

SECOND-ORDER DIFFERENTIAL EQUATIONS

21.

ñóò ò ó

= ï –5 ) 3 (
ñ –7 ) 3 + See equation 2.6.2.34.

(­ñ

–10 3

)

)(ñó ò )3 .

22.

ñóò ò ó

= ï –5 ) 3 (
ñ –4 ) 3 + See equation 2.6.2.36.

(­ñ

–10 3

)

)(ñó ò )3 .

23.

ñóò ò ó

= ï –5 ) 3 (
ñ –4 ) 3 + See equation 2.6.2.14.

(­ñ

–7 3

24.

= ï –5 ) 3 (
ñ –2 ) 3 + (­ñ See equation 2.6.2.115.

25.

= ï –5 ) 3 (
+ (­ñ –2 ) 3 )(ñ See equation 2.6.2.114.

26.

= ï –5 ) 3 (
ñ 2 + ( )(ñó ò )3 . See equation 2.6.2.35.

27.

= ï –5 ) 3 (
ñ 2 + (­ñ )(ñó ò )3 . See equation 2.6.2.33.

28.

ñ óò ò ó ñ óò ò ó

)

)(ñó ò )3 .

)

)(ñ

–4 3

óò

óò

)3 .

)3 .

ñóò ò ó ñóò ò ó

=
ï –5 ) 3 ( ôñ –2 ) 3 + õÿñ –5 ) 3 )2 (ñó ò )3 . This is a special case of equation 2.7.2.37 with ( = 1 and

ñóò ò ó

29.

= ï –7 ) 5 (
ñ –8 ) 5 + (­ñ See equation 2.6.2.109.

30.

= ï –1 ) 2 (
ñ –5 ) 2 + See equation 2.6.2.13.

31.

=
ï –1 ) 2 ( ôñ 5 + õÿñ 4 )–1 ) 2 (ñó ò )3 . This is a special case of equation 2.7.2.38 with ( = 1 and

32. 33. 34.

35.

36.

ñ óò ò ó ñ óò ò ó

(­ñ

)

–13 5

)

–7 2

óò

)(ñ

)(ñ

óò

= 0.

)3 .

M

)3 .

ñóò ò ó

= 0.

=
ï –1 ) 2 ( ôñ 15 ) 8 + õÿñ 7 ) 8 )–4 ) 3 (ñó ò )3 . This is a special case of equation 2.7.2.39 with ( = 1 and

M

ñóò ò ó

=
ï 2 ( ôñ 7 ) 3 + õÿñ 4 ) 3 )–15 ) 7 (ñó ò )3 . This is a special case of equation 2.7.2.40 with ( = 1 and

= 0.

ñóò ò ó

M M

= 0.

= ï –5 ) 3 ( ôñ 2 + õÿñ + ö )(ñ ó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.7.1.34 with respect to  =  ( ): 

= −(  2 +  + ( ) −5 J 3 .

ñ óò ò ó

ñ óò ò ó = ï –5 ) 3 (ô ñ

)



+ õÿñ –7 ) 3 + ö^ñ –4 ) 3 )(ñ ó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.7.1.35 with respect to  =  ( ): 

= −(  −10 J 3 +  −4 J 3 + ( −4 J 3 ) −5 J 3 . –10 3



= ï –2÷ –3 ( ôñ 2 + õÿñ + ö )÷ (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.9.1.21 (for  (ˆ ) = −ˆ −2  ) with respect to  =  ( ): 

= −(  2 +  + ( )   −2  −3 .

ñóò ò ó



Page 416

2.7. EQUATIONS OF THE FORM

37.

38.

39.

40.

41.



ñ óò ò ó =
ï –1 ) 2 (ô ñ

M



ñ óò ò ó =
ï –1 ) 2 (ô ñ

M

+ õ )–4 ) 3 ( ö^ñ + ý )–7 ) 6 (ñ ó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.7.1.39 with respect to  =  ( ): 

= − # (  +  )−4 J 3 ( ( + )−7 J 6  −1 J 2 .



ñ óò ò ó =
ï 2 (ô ñ

) (^ö ñ

) (ñ ó ò

M

+õ ) +ý ) ). Taking to be the independent variable, we obtain an equation of the form 2.7.1.40 with respect to  =  ( ): 

= − # (  +  )−15 J 7 ( ( + )−20 J 7  2 . –15 7

–20 7



ñ óò ò ó =
ï –1 ) 2 ñ

–2

2

2 1 2

U

2

ü

− 2U

−1

+L

2

3

M

)

[(ñó ò ) + ( ] Solution in parametric form:

.

− 1)−1 (U‡IAC² )2 ,

‡ = T tanh OL

1,

2

= 'U −1

²

+X

=
ï –1 ) 2 ñ –2 [(ñó ò )2 – ( 2 ]1 ) 2 . Solution in parametric form:

ñóò ò ó

where ² =

L ü

1

−U

2

− 2U

2

+ 1)−1 (U‡IAC² )2 ,

‡ = A tan OL

−1 ,

2

+ Xv²

−1

=
ï –1 ) 2 ñ –2 [ ( 2 – (ñó ò )2 ]1 ) 2 . Solution in parametric form:

ñóò ò ó

where ² =

U

− 2U

−1

+L

‡ = A tanh OL

1,

=
ï –2 ñ –1 ) 2 (ñ ó ò )2 [(ñ ó ò )2 + Solution in parametric form:

ñ óò ò ó

where ² =

U

2

ü

− 2U

 = U

−1

+L

1,

−1

U ü

2

− 2U

 = U

−1

+L

1,

−1

(

2

(‡

)

2 1 2

]

(

2

− 1)−1 J 2 ,

)

2 1 2

]

−1

=  (‡

 = U where ² =

ü

−U

2

− 2U

−1

+L

−1

2

J  , $ = 1 2

−1

(‡

2

−1

−1 2 2

−1

.

+ 1)−1 J 2 ,

J  , $ = 1 2

−1 2 2

(1 − ‡ 2 )−1 J 2 ,

U , # = − 12  M P

−1

.

J  , $ = 1 2

−1

J , $ = 2

−1

.

J , $ = 2

−1

.

J , $ = 2

−1

.

−1 2 2

.

+ Xv²

− 1)−1 (U‡pAC² )2 ,

−1

U , # = − 14  3  M P

−3 2

.

(1 − ‡ 2 )−1 J 2 ,

‡ = A tanh OL

=  (1 − ‡ 2 )−1 (U‡uTç² )2 , 2

(1 + ‡ 2 )−1 J 2 ,

1,

− 1)−1 J 2 ,

2

+ XZ²

−1

=
ï –2 ñ –1 ) 2 (ñó ò )2 [ ( 2 – (ñó ò )2 ]1 ) 2 . Solution in parametric form:

ñóò ò ó

2

U , # = − 21  M P

= 'U

+X ²

(‡

.

‡ = T tanh O L

=
ï –2 ñ –1 ) 2 (ñó ò )2 [(ñó ò )2 – Solution in parametric form:

ñóò ò ó

where ² = 46.

2

ü

−1

U , # = − 12  M P

= 'U

 =  (1 − ‡ 2 )−1 (U‡pTç² )2 ,

45.

417

)

+ õ )–1 ) 2 ( ö^ñ + ý )–2 (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.7.1.38 with respect to  =  ( ): 

= − # (  +  )−1 J 2 ( ( + )−2  −1 J 2 .

 =  (‡

44.

ÒÍ

=
ï –5 ) 3 ( ôñ + õ )2 ( ö^ñ + ý )–10 ) 3 (ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.7.1.37 with respect to  =  ( ): 

= − # (  +  )2 ( ( + )−10 J 3  −5 J 3 .

where ² =

43.

›

ñóò ò ó

 =  (‡

42.

= â (Ë )ã (Ï ) (Ï

Ï Ò˜Í Í Ò

‡ = A tan OL

U , # = − 14  3  M P

=  (1 + ‡ 2 )−1 (U‡pAC² )2 , 2

+X ²

−1

U , # = − 14  3  M P

−3 2

−3 2

Page 417

418

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.7.3. Equations Containing Exponential Functions ( “Ã1 const) 2.7.3-1. Preliminary remarks. 1 b . If 0 ≠ 1 −

¡

, the equation

 +
(  )-



 = # 

(1)

has a particular solution:



=$   , 2 b . If

¡

1

Π=

where

1−

¡

$ = ( #3Œ - −2 )  .

,

−0

≠ 0 and 0 ≠ 1, equation (1) can be reduced with the aid of the transformation

¨ = 

a = ( 
)1−- ,

to a generalized Emden–Fowler equation with respect to ¨ = ¨ (a ):

¨ g

g = $Ia

1 1−

-

¨

−1

+

2 (¨ g
) +

+1

,

(2)

1

where $ = − V # (1 − 0 )W + . Equations of the form (2) are outlined in Section 2.5. ¡ Whenever the general solution ¨ = ¨ (a ) of the Emden–Fowler equation (2) is obtained, the solution of the original equation (1) can be written out in parametric form as:

 = ln ¨ ,

1

− = , (¨ g
) + ,

3 b . If 0 ≠ B + 2, the equation

1

− , = V # (1 − 0 )W + .

where



 = #*    ( 
)-

(3)

has a particular solution: = Πln( $I ),

where

$ = O−

Œ = 0 − B − 2,

Π#

1−

P

1

 .

4 b . Taking to be the independent variable and  to be the dependent one, we obtain from equation (3) an equation of the form (1) for  =  ( ):



= − #     (
)3−- .   5 b . If B ≠ −1 and 0 ≠ 1, equation (3) can be reduced with the aid of the transformation

a = ( 
)1−- ,

‡ = 

+1

to a generalized Emden–Fowler equation for ‡ = ‡ (a ): 1 ‡ g

g = − a B +1

1 1−

- ‡





− +1 (

‡ g
)2 .

(4)

Equations of this form are outlined in Section 2.5. Whenever the general solution ‡ = ‡ (a ) of the Emden–Fowler equation (4) is obtained, the solution of the original equation (3) can be written out in parametric form as:

 =‡ 

1 +1

,

B +1 . = − ln(‡ g
) + ln # (1 − 0 )

Page 418

2.7. EQUATIONS OF THE FORM

= â (Ë )ã (Ï ) (Ï

›

Ï Ò˜Í Í Ò

ÒÍ

419

)

2.7.3-2. Solvable equations and their solutions. 1.

ñóò ò ó

ô]

=

ñóò ò ó








−






−






−

ó (ñ óò )A .


B

=






=

Solution:

2.

.

1

Œ

1

Œ

1

Œ

Œ

ln ¬

2L

Π2L

ln ¬

1

sinh2 ( L

2 1

Π2L

ln ¬ −

sin2 ( L

2 1

 + L 2 )® 1

cosh2 ( L

2 1

if Π> 0,

 + L 2 )® 1

if Π> 0,

 + L 2 )®

if Π< 0.

1 b . Solution in parametric form with 0 ≠ 1: 1

1− 1

 = ln A L Q # (1 − 0 )

- U!S ,

=L

1

1

X U

1 1−

 1 ACU 

- M U + L 2.

2 b . Solution in parametric form with 0 = 1:

U , #­P

 = ln OA 3.

ñóò ò ó

ó
B ñ ú (ñ óò

=

=L

1

1

X

exp(AU ) U + L

U

M

2.

)2 .

1 b . Solution in parametric form with

 =X M



U

+L

≠ −1:

¡

= U exp −

2,

Q

¡

1 OX +1

M 

U

+L

2

P

S ,

where the function  =  (U ) is defined implicitly by the relation



ln O 2 b . Solution for

4.

ñ óò ò ó

ó


B ñ

=

¡

−

U

= −1:

1 − +1P (

¡

=L

®

=L where ® 0 ( H ) and ;

0(

1 0

=L

5.

ñóò ò ó

=

0(

2

exp OX

# ¡

+1

 M   +#  +L

U +

1

P

+1

− ln U + L

1.

.

  2£ #  J 2  + L 2 ;

0

  2£ #  J 2  ,

H ) are the modified Bessel functions.

2 b . Solution for # < 0:

ø

¡

+ 1)  − U

=

.

1 b . Solution for # > 0:

where

U

ø

1 0

   2 £ −#  J 2  + L 2 n 0  2 £ −#  J 2  ,

H ) and n 0 ( H ) are the Bessel functions.

ó


B ñ

) (ñ óò )3 )

–1 2

2

.

Solution in parametric form:

 =U

2

− ln( #3 ),

=L

1

V 2U − exp(U 2 )W 2 ,

where

 = X exp(U 2 ) U + L 2 . M

Page 419

420 6.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ó

=
B ñ (ñó ò )3 ) 2 . Solution in parametric form:

ñóò ò ó

 = − ln V #3L 7.

3 1

£ U

2

+ U −  W ,

2 1

= 2L

U +1 , U P

O 1− 

=
B] (ñó ò )A . 1 b . Solution in parametric form with 0 ≠ 2:

 = ln  £ U + £ U + 1  + L 2 .

where

ñóò ò ó

 =L

1

X

1

1

(1 AIU ) - −2 U + L U M

1 L - −2 U!S . # (2 − 0 ) 1

= ln A

2,

Q

2 b . Solution in parametric form with 0 = 2:

 =L 8.

1

1

X

exp(T"U ) U + L

U

ñ óò ò ó =
ïƒ÷  ]Èñóò

M

U . # P 

= ln O˜A

2,

. 1 b . Solution in parametric form with B ≠ −1: 1

 = U exp − OX Q B +1 M



U

+L

2

S , P

=X

M 

U

+L

2,

where the function  =  (U ) is defined implicitly by the relation ln O

U



−

1

B +1P

U

−

=−

(B + 1)  − U

# U  B +1

+1

− ln U + L

1.

2 b . Solution for B = −1:

 =L 9.

M

−#   +L

=
ï –1 ) 2  ] (ñ ó ò )3 ) 2 . Solution in parametric form:

1

P

.

1

V 2U − exp(U 2 )W 2 ,

=U

2

− ln(− #3 ),

= − ln V #3L

3 1

Ý − £ U

where

=
ï ] (ñó ò )3 ) 2 . Solution in parametric form:

 = X exp(U 2 ) U + L 2 . M

ñóò ò ó

 = 2L 11.

exp OX

ñ óò ò ó

 =L 10.

2

2 1

” 1− 

U +1 , U •

2

+U  W ,

where  = ln  £ U + £ U + 1  + L

2.

=
ï ] (ñó ò )3 . 1 b . Solution in parametric form with # > 0:

ñóò ò ó

 =L

ø

ø

1 0 (2

U ) + L 2 n 0 (2U ),

= ln  U £ #  ,

H ) and n 0 ( H ) are the Bessel functions. 2 b . Solution in parametric form with # < 0:

where

0(

 =L where ® 0 ( H ) and ;

0(

®

1 0 (2

U ) + L 2;

0 (2

U ),

= ln  U £ − #  ,

H ) are the modified Bessel functions.

Page 420

2.7. EQUATIONS OF THE FORM

12.

›

ó

ÒÍ

421

)

=
B  ] (ñ ó ò )A . Solution in parametric form:

ñ óò ò ó

where  =



U M + L 2, "U -

 =X









1 2− 1 U - + U 2−0 1−0 U + ln |U | + L 1 1 ln |U | − + L 1

1−

- +L

= ln O

 −X # P

U M − L 2, "U -

if 0 ≠ 1, 2;

1

if 0 = 1; if 0 = 2.

U

13.

= â (Ë )ã (Ï ) (Ï

Ï Ò˜Í Í Ò

=
exp( ü ï ) exp( ôñ 2 + õÿñ )(ñó ò )3 . Taking to be the independent variable, we obtain an equation of the form 2.7.1.41 with respect to  =  ( ): 

= − # exp(  2 +  ) exp( ,! ).

ñóò ò ó



14.

ó ñóò ò ó = B
 ñ –1 ) 2 (ñ óò )3 ) 2 ñ óò ü

– 2( .

Solution in parametric form:

= $ cosh2 ‡ (U tanh ‡pAC² )2 ,

 = ln[ U (cosh ‡ )−1 ], where  = − # 15.

J , ² =

$

−1

2 ln U + U

−1 2

ó

ü

=
B ñ –1 ) 2 (ñó ò )3 ) 2 2 ( ü Solution in parametric form:

ñóò ò ó

–ñ

óò

2

+L

1,

16.

−1

$

J , ² =

−1 2

−1

TeXZ²

U . M

2 ln U − U

= $ cos2 ‡ (U tan ‡pAC² )2 , 2

+L

1,

‡ =L

2

−1

AçXZ²

=
ï –1 ) 2 ]¦ñó ò ñ óò – ( . ü Solution in parametric form:

M

U .

ñóò ò ó

 = where  = # 17.

ü

2

.

 = ln[ U (cos ‡ )−1 ], where  = − #

‡ =L

−1

1 cos2 ‡ (U tan ‡pAC² )2 , 2$

£ 2, ² =

2 ln U − U

ü

2

+L

1,

‡ =L

= ln[ 'U (cos ‡ )−1 ], 2

−1

AeXZ²

=
ï –1 ) 2 ]¦ñó ò ( – ñ óò . ü Solution in parametric form:

M

U .

ñóò ò ó

1 cosh2 ‡ (U tanh ‡pAç² )2 , 2$

 = where  = #

−1

£ 2, ² = ü

2 ln U + U

2

+L

1,

‡ =L

= ln[ 'U (cosh ‡ )−1 ], 2

−1

T X ²

M

U .

2.7.4. Equations Containing Hyperbolic Functions ( “1 const) 1.

=
ï [cosh( ƒñ )]–2 ñó ò . Solution in parametric form:

ñóò ò ó

=‡ Œ ,

 =  cosh ‡ (U tanh ‡pAç² ), where # = 

−2

, ²

=

ü

2 ln U + U

2

+L

1,

‡ =L

2

T X ²

−1

M

U .

Page 421

422 2.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó


ï

=

[sinh( ƒñ )]–2 ñó ò .

Solution in parametric form: =‡ Œ ,

 =  sinh ‡ (U coth ‡pAç² ), where # =  3.

ñóò ò ó


ï

=

−2

, ²

=

2 ln U + U

ü

+L

2

‡ =L

1,

2

−1

TeXZ²

cosh( ƒñ )(ñ„ó ò )3 ) 2 .

U . M

Solution in parametric form:

 =  (‡ where # = 2  4.

ñóò ò ó


ï

=

−2

J

−1 2

2

+ 1)

£ Œ , ² =

L ü

(U‡pAç² ),

−U

1

− 2U

2

−1 ,

=Œ

−1

ln  ‡ + £ ‡

‡ = A tan OL

2

+ Xv²

2

+1 ,

−1

U . M P

2

−1 ,

sinh( ƒñ )(ñ„ó ò )3 ) 2 .

Solution in parametric form:

 =  (‡ where # = A 2  5.

ñóò ò ó




=

−2

− 1)

£ Œ , ² =

ï )ñ („ñ óò )3 ) 2 .

cosh(

J

−1 2

2

L ü

1

(U‡IAC² ), +U

2

− 2U

= A/Œ −1 ,

−1

ln ‡ + £ ‡

‡ = T tanh OL

2

+ XZ²

−1

U . M P

Solution in parametric form:

 =Œ where # = −2  6.

ñóò ò ó




=

−2

−1

ln(‡ + £ ‡

£ Œ , ² =

L

2

=  (‡

+ 1),

−U

2

− 2U

ln(‡ + £ ‡

2

− 1),

2

− 2U

ü

1

−1 ,

2

J

−1 2

+ 1)

‡ = A tan OL

2

(U‡pAç² ),

²

+X

ï )ñ („ñ óò )3 ) 2 .

sinh(

−1

U . M P

Solution in parametric form:

 = A/Πwhere # = T 2  7.

ñóò ò ó




=

−2

£ Œ , ² =

ï

[cosh(

−1

ü

L

1

+U

−1 ,

=  (‡

2

J

−1 2

− 1)

‡ = T tanh OL

2

(U‡IAC² ),

+X ²

−1

U . M P

)]–2 ñ (ñó ò )2 .

Solution in parametric form:

 =‡ Œ , where # = −  8.

ñóò ò ó

=




−2

[sinh(

, ²

ï

2 ln U + U

=

ü

2

=  cosh ‡ (U tanh ‡pAç² ), +L

1,

‡ =L

2

T X ²

−1

U . M

)]–2 ñ (ñó ò )2 .

Solution in parametric form:

 =‡ Œ , where # = − 

−2

, ²

=

ü

2 ln U + U

2

=  sinh ‡ (U coth ‡pAç² ), +L

1,

‡ =L

2

T X ²

−1

M

U .

Page 422

2.7. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò

= â (Ë )ã (Ï ) (Ï

›

ÒÍ

423

)

2.7.5. Equations Containing Trigonometric Functions ( “1 const) ]

In the solutions of equations 1–4, the following notation is used:

² = 1.

2.

3.

4.

]

2

AeXZ²

−1

U . M

=‡ Œ ,

where

# =

−2

 =  sin ‡ (U cot ‡uTe² ),

=‡ Œ ,

where

# =

−2

=  cos ‡ (U tan ‡pAC² ),

where

# = − 2.

=  sin ‡ (U cot ‡uTç² ),

where

# = − 2.

=
ï [sin( ƒñ )]–2 ñó ò . Solution in parametric form:

.

ñóò ò ó

=
[cos( ï )]–2 ñ (ñ ó ò )2 . Solution in parametric form:

.

ñ óò ò ó

 =Œ

−1

 =Œ

−1

‡ ,

=
[sin( ï )]–2 ñ (ñó ò )2 . Solution in parametric form:

ñóò ò ó

‡ ,

In the solutions of equations 5–8, the following notation is used:

ü

U

2

− 2U

−1

+L

‡ = A tanh OL

1,

2

+X

²

−1

=
ï cos( ƒñ ) (ñ„ó ò )3 ) 2 . Solution in parametric form:

U . M P

ñóò ò ó

J

−1 2

(U‡uTç² ),

−1

=Œ

arccos ‡ ,

−2

(− Œ )1 J 2 .

where

# = 2

where

# = 2

−2

( Π)1 J 2 .

where

# = −2 

−2

(− Œ )1 J 2 .

where

# = −2 

=
ï sin( ƒñ ) (ñ ó ò )3 ) 2 . Solution in parametric form:

ñ óò ò ó

J

−1 2

(U‡uTç² ),

−1

=Œ

arccos ‡ ,

=
cos( ï ) ñ (ñ ó ò )3 ) 2 . Solution in parametric form:

ñ óò ò ó

 =Π8.

‡ =L

1,

 =  cos ‡ (U tan ‡pAç² ),

 = s 1 − ‡ 2  7.

+L

=
ï [cos( ƒñ )]–2 ñó ò . Solution in parametric form:

 =   1 − ‡ 2 6.

2

ñóò ò ó

² = 5.

2 ln U − U

ü

−1

arccos ‡ ,

=   1 − ‡ 2

J

−1 2

(U‡uTç² ),

=
sin( ï ) ñ (ñ„ó ò )3 ) 2 . Solution in parametric form:

ñóò ò ó

 =Œ

−1

arccos ‡ ,

=   1 − ‡ 2

J

−1 2

(U‡uTç² ),

−2

( Œ )1 J 2 .

Page 423

424

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.7.6. Some Transformations For the sake of visualization, we also use the symbolic notation {  , , % } to denote the equation



 =  1 ( ) 1 ( ) % 1 ( 
).

(1)

1 b . Taking to be the independent variable and  to be the dependent one, we obtain an equation of similar form for  =  ( ):



= 1 ( )  1 ( ) % 1ø (
), where % 1ø (¨ ) = −¨ 3 % 1 (1 ¨ ).   Denote this transformation by ¸ . 2 b . The B¨acklund transformation ¨ ¯ = X M ¨ , ¯ = Xv 1 ( )  , where ¨ = 
, % 1( ) M

(2)

leads to an equation of similar form for the function ¯ = ¯ ( ¯ ): ¯ 
¯
 ¯ =  2 ( ¯ ) 2 ( ¯) % 2 ( ¯ 
¯ ), where the functions  2 , 2 , and % parametrically by the relations

2

are defined in terms of the original functions  1 , 1 , and %

 2 ( ¯ ) = ¨ , 2 ( ¯) =

¯ = X 1

¨ M ¨ ; % 1( )

¯ = Xv 1 ( )  ;

,

 1 ( )

1

1 % 2 (¨ ¯ ) = − M 3 [ 1( )]

1

M

,

¨¯ = 1 . 1( )

M

Denote transformation (2) by ‹ . For equations of the form (1) in which  1 , 1 , and % 1 are power functions of their arguments, the transformation ‹ (up to a constant factor) is considered in Subsection 2.5.3. For equations (1) with exponential functions  1 and 1 , the transformation ‹ is discussed in Subsection 2.7.3. Whenever the solution ¯ = ¯ ( ¯ ) of the transformed equation is found, the formulas ¯=X

 1 ( )  , M

¯
¯ =

1 , 1( )

can be used to obtain the solution of the original equation (1) in parametric form,  =  ( ¯ ),

= ( ¯ ).

3 b . The twofold application of the transformation ‹ to the original equation yields an equation of similar form: ¯
 ¯
 ¯ =  3 ( ¯ ) 3 ( ¯) % 3 ( ¯
 ¯ ), where the functions  3 , 3 , and % parametrically by

3

are defined in terms of the original functions  1 , 1 , and %

 3 ( ¯) =

1

1( ) 1

3 ( ¯) = ¨ , % 3 (¨ ¯) = M M

 

1

,

,

1

 ¯ = X 1( ) ; M ¨ ¨ ¯=X M ; % 1 (¨ ) ¨ ¯ =  ( ). 1

The threefold transformation ‹ yields the original equation. Different compositions of the transformations ¸ and ‹ generate six different equations of the analogous form, which are shown in Figure 4 (see Subsection 0.6.5).

Page 424

425

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

1 4 b . In the special case ( ) = + , % = 1, the transformation  = , U of similar form: 1 + ‡ . ‡

= U − + −3 *O

*+*

=

‡ U

leads to an equation

U*P

Denote this transformation by { . For ( ) = + and % = 1, different compositions of the transformations ¸ , twelve different equations of the form (1).

‹

, and

{

generate

2.8. Some Nonlinear Equations with Arbitrary Parameters 2.8.1. Equations Containing Power Functions 2.8.1-1. Equations of the form  ( , )

 + ( , ) = 0. 1.

2.

3. 4.

ñóò ò ó

=

ôñ

2

+

õï

.

The transformation = , 3 ¨ ,  = ,H , where , = 61 J 5 (  )−1 J 5 , leads to the first Painlev´e transcendent: ¨3‘'

‘ = 6¨ 2 + H (see Paragraph 2.8.2-2).

= ôñ 3 + õ ï ñ + ö . The transformation = (2  )1 J 2  1 J 3 ¨ ,  =  −1 J 3 H leads to the second Painlev´e transcendent: ¨2‘'

‘ = 2¨ 3 + H ¨ + (  2)1 J 2  −1 ( (see Paragraph 2.8.2-3).

ñ óò ò ó ñóò ò ó

=

õ ƒï ÷ ñ

+ ôñ

–3

. This is a special case of equation 2.9.1.2 with  ( ) = − '  .

= ô ï ÷ ñ + õ ü ï  –1 ñ –1 – õ 2 ï 2  ñ –3 . This is a special case of equation 2.9.1.3 with  ( ) = −   and ( ) = '  .

ñ óò ò ó

5.

= ( ô ï 2 + õ ï + ö )ñ –5 . This is a special case of equation 2.7.1.36 with B = 1.

6.

ñóò ò ó

7.

ñ óò ò ó

=

ôƒõ ï –1 ñ –1 ) 2 + ô ï

–2

+ 2õ 2.

The solution is determined by the first-order equation  ("
 + 2  1 J 2 ) = ¨ ( , , L ), where the function ¨ is defined implicitly by ¨ − ln |¨ +  1 J 2 | =  −1 J 2 ( 1 J 2 + '  )2 + L . = ( ôñ 2 + õ ï ñ + ö ï 2 + ñ +  The substitution 2  ¨ = 2  + ' + ƒ

ï

ñ óò ò ó

¨ 

 = ¨ where  (ˆ ) = ˆ 3 ( ˆ 8.

ñóò ò ó

=

ƒñ –1 ) 3 + (ô ï

2

J

−3 2

+ 1) 2

+

õï

, # =

+ ö )ñ

)

+ )–3 ) 2 , ô ≠ 0. leads to an equation of the form 2.9.1.21: −3

u” £ #*

4 !( −  2 , $ 4

–5 3

¨ 2

+ $I + L •

=

,

2 ý − ˜ƒ , L 2

=

4 ´= − ƒ 4

2

.

.

3J = (  g
)

leads to a third-order equation: 2 g
 g

g
g − ( g

g ) = The transformation  =  (a ), 2
4 4 2 + ' + ( ). Differentiating the latter equation with respect to a and dividing it 3 Π(  g ) + 3 (  by  g
, we arrive at a fourth-order constant coefficient linear equation: 3 g

g
g
g = 2 Π g

g +4  +2  . 2

2

Page 425

426 9.

10.

11.

12.

13.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï –8 ) 3 ñ –1 ) 3 + (ô ï

ñóò ò ó

=

ñóò ò ó

= ( ôñ +

õï

ñóò ò ó

= ( ôñ +

õ ï 2 )÷

)

–10 3

õ ï –7 ) 3 + ö ï –4 ) 3 )ñ –5 ) 3 .

+

The transformation  = 1 a , Œ ¨ −1 J 3 + ( a 2 + 'a + ( )¨ −5 J 3 .

¨ g

g = = ¨u a leads to an equation of the form 2.8.1.8: I

+ ö )÷ .

This is a special case of equation 2.9.1.4 with  (ˆ ) = ˆ  . +ö .

The substitution  ¨ =  ¨2

 =   ¨  + ( + 2  −1  .

ñ óò ò ó

=

ï –2÷

ñóò ò ó

=


(ô ï

ñóò ò ó

=

–3

(ï

+ '

2

leads to an autonomous equation of the form 2.9.1.1:

+ ô )÷ .

ñ

This is a special case of equation 2.9.1.15 with  (ˆ ) = Œ ˆ  and  = ( = 0. + õ )÷ ( ö

+ ý )– ÷

ï

–

ú

–3

ñ ú

.

 +  ¨ , = leads to the Emden–Fowler equation The transformation ˆ = (˜ + (˜ + ¨ + , whose ¨2Šª

Š = # (  − ˜( )−2 ˆ  2 M solvable cases are M outlined in Section 2.3. M 14.

15.

ö ïƒú ñ –'÷  –ú

–3

÷

( ôñ

+

õ ïƒ÷ )

.

  −+ This is a special case of equation 2.9.1.8 with  (ˆ ) = (˜ˆ − ´

ñóò ò ó

=

ö ïƒú ñ –2'÷  –2ú

–3

( ôñ

2

÷

+

õ ïƒ÷ )

.

This is a special case of equation 2.9.1.9 with  (ˆ ) = (˜ˆ 16.

ô ƒï ÷ ñ ú

ï 2 ñ óò ò ó

=

ï 2 ñ óò ò ó

=

+ 1)ñ + ô

ï 3÷

+2

ï 2 ñ óò ò ó

= ü ( ü + 1)ñ + ô

ï  ú

+3 +2

+1

+

õÿñ

−2

´

−2

−3

+

( ˆ  +  )  .

−3

 ( ˆ 2  +  ) .

.

This is a special case of equation 2.9.1.11 with  ( H ) = !H +  . 17.

18.

19.

ù (ù

+

õ ïƒ÷¿ú

÷ ñ ú

+3 +2

.

This is a special case of equation 2.9.1.12 with  (ˆ ) =  + 'ˆ + .



(õ

ï 2 +1 + ö )÷ ñ ú

.

The transformation ˆ = '   + ( , ¨ =   leads to the Emden–Fowler equation ¨ Šª

Š =  −2 (2 , + 1)−2 ˆ  ¨ + , whose solvable cases are outlined in Section 2.3. ( ôñ +

õ ï 2 )ñ óò ò ó

2 +1

= 1.

This is a special case of equation 2.8.1.11 with B = −1 and ( = 0.

20.

( ï + ô )2 ñ

2

ñ óò ò ó

=

õï

.

 +

The transformation ˆ = ln û û  2.2.1.7: ¨3Šª

Š − ¨2Š
=  −2 ¨ −2û . 21.

(ñ

2

+ô

ï

2

+ 2õ

ï

+ö )

2

ñóò ò ó

û

û,

¨ =

û



leads to an autonomous equation of the form

+ sñ = 0.

Dividing by the coefficient of 

 and multiplying by  ( 
− ) +  (2 
 − ) + ( 
 , we arrive at an exact differential equation. Integrating the latter, we obtain a first-order equation: ( 

2

+ 2 ' + ( )( 
)2 − 2(  +  )! 
+ 

2

+

2

+ 

°

2

2

+ 2 ' + (

=L .

Page 426

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

22.

ï

(ô

+ õ )2 ( ö

ï

óò ò ó

+ ý )2 ñ

= sñ +


(ô ï

+ õ ) (ö

ï

+ ý )1–ú

–



ñ ú

427

.

 +   +  + −1 , ¨ = O leads to an autonomous P ˜ (  + (˜ + P ¨ + (˜ +


¨ ¨ 2M −2 ¨ −2 M M ª Š Š Š equation: − (2B + 1) + (B + B − °¢ ) = #3¢ , where B = +  −1 , ¢ =  − ˜( . M The transformation ˆ = ln O

23. 24.

(ô

ï

(ô

ï

+

+ ö )÷

ñóò ò ó

= ü (

ï

+ ñ + )÷

–1

õÿñ

+ ö )÷

ñóò ò ó

= ü (

ï

+ ñ + )÷

–3

+

õ ïƒ÷ )ñ óò ò ó

+

ö ïƒ÷

–3

= 0.

+

õ ïƒ÷ )ñ óò ò ó

+

ö^ñ ÷

–3

.

This is a special case of equation 2.9.1.16 with  (¨ ) = , ¨ +

( ôñ

÷

26.

( ôñ

÷

29.

3−

 .

= 0.

This is a special case of equation 2.9.1.8 with  (ˆ ) = − (˜ˆ 

( ôñ

÷

2

+

õ ïƒ÷ )ñ óò ò ó

+

ö^ñ 2÷

–3

−3

( ôñ

÷

+

õ ïƒ÷ )ñ óò ò ó

+

( ˆ  +  )−1 .

= 0.

ö ïƒú ñ ÷ –ú

–3

This is a special case of equation 2.9.1.8 with  (ˆ ) = − (˜ˆ  − +

( ôñ

2

÷

+

õ ïƒ÷ )ñ óò ò ó

+

ö ïƒú ñ 2÷ –2ú

−1 ( ˆ 2  +  ) .

−3

= 0.

–3

= 0.

This is a special case of equation 2.9.1.9 with  (ˆ ) = − (˜ˆ 2 

]

 .

This is a special case of equation 2.9.1.8 with  (ˆ ) = − ( ( ˆ  +  )−1 .

This is a special case of equation 2.9.1.9 with  (ˆ ) = − (˜ˆ 2 

28.

1−

.

This is a special case of equation 2.9.1.17 with  (¨ ) = , ¨

25.

27.

õÿñ

−2

−3

+

( ˆ  +  )−1 .

−3

−1 ( ˆ 2  +  ) .

See also equations 2.7.1.1–2.7.1.40. 2.8.1-2. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0.

30. 31. 32. 33. 34. 35.

ñóò ò ó

+ 3ññó ò + ñ

ñóò ò ó

+ ( ôñ +

ñóò ò ó

+ (2 ôñ +

3

ïƒ÷ ñ

+ô

= 0.

This is a special case of equation 2.9.2.1 with  ( ) =   .

õ ïƒ÷ )ñ óò

+

õÿù ïƒ÷ –1 ñ

= 0.

This is a special case of equation 2.9.2.4 with  ( ) = '  .

õ ïƒ÷ )ñ óò

+

ôƒõ ƒï ÷ ñ

+

ô ï 2÷ +1 ñ

2

=

ö ïƒú

.

This is a special case of equation 2.9.2.5 with  ( ) = '  and ( ) = (˜ + .

ï ñóò ò ó

=

ùñóò

+

õ ïƒú ñ

ï ñ óò ò ó

=

ùñ óò

+ô

ï 2÷

ï ñóò ò ó

= –(ù + 1)ñ„ó ò + ô

–3

. This is a special case of equation 2.9.2.9 with  ( ) = − ' + . +1

+

õ ï 2÷ +1 ñ ú

.

This is a special case of equation 2.9.2.20 with  ( ) =  +  "+ .

ïƒ÷

–1

+

õ ïƒ÷¿ú +÷ –1 ñ ú

. This is a special case of equation 2.9.2.30 with  (ˆ ) =  + 'ˆ + .

Page 427

428 36.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñ ÷

ï ñ óò ò ó

= (ô Solution:

+

38. 39.

– 1)ñ

óò

M  ( )+ L

ù 37.

ü

ï 2 ñ óò ò ó

. =L

1

2

1   ,

+

,

where

+ ï ñó ò = ôñ ÷ + õ . This is a special case of equation 2.9.2.23 with  ( ) = 

 ( )=

1



B +1

+1

.

 + .

= –(ù + þ + 1) ï ñ ó ò – ùþxñ + ô ï ÷' +÷ –2ú ñ  . This is a special case of equation 2.9.2.31 with  (ˆ ) = ˆ  .

ï 2 ñ óò ò ó ï 2 ñ óò ò ó

+

ô ï ñóò

+

õÿñ

=

ö ƒï ÷ ñ ú

.

1 1 1− = ˆ : ¨ , where ƒ = A £ , ý =A £ − , c = (1 −  )2 − 4  , 2 c 2 c leads to the Emden–Fowler equation ¨ Šª

Š = (ƒ 2 ˆ  9 + +r: −: −2 ¨ + , whose solvable cases are outlined in Section 2.3. The transformation  = ˆ 9 ,

40.

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + ö^ñ ÷ = 0. This is a special case of equation 2.9.2.24 with  ( ) = (  .

2.8.1-3. Equations of the form  ( , )

 + ( , )(
 )2 + % ( , )
 + & ( , ) = 0. 41.

42. 43. 44. 45. 46. 47.

48.

49.

= (ñó ò )2 – 2 ô ï ñó ò + 2 ôñ + õ . = ¨ + 1  2 leads to an autonomous equation of the form 2.9.3.25: The substitution ¨2

 = (¨2
)2 + 2  ¨ −  +  . 2

ñóò ò ó

= ô (ñó ò + õÿñ + ö ï )2 + õ 2 ñ + ü ï + s. ¨ 
=  ¨ The substitution ¨ = 
+  + (˜ leads to a Riccati equation: u

ñóò ò ó

2

+  ¨ + ( , − ˜( ) + ( + ° .

= ô ïƒ÷ ( ï ñó ò – ñ )2 + õ ïƒú . This is a special case of equation 2.9.3.2 with  ( ) = ' + , ( ) = 0, and % ( ) =   .

ñóò ò ó

= ô ï ÷ (ñ ó ò + õÿñ )2 + õ 2 ñ + ö ï ú . ¨ 
=   ¨ The substitution ¨ = 
+  leads to a Riccati equation: 3

ñ óò ò ó

= ( ô 2 ï 2 + ô )ñ + õ ïƒ÷ (ñó ò – ô ï ñ )2 + ö ïƒú . ¨ 
= '  ¨ The substitution ¨ = 
−  leads to a Riccati equation: 3

2

ñóò ò ó

+  ¨ + (˜ + . 2

= ( ô ï + õÿñ + ö )÷ [  (ñó ò )2 +  ]  . This is a special case of equation 2.9.4.37 with  (‡ ) = ‡  and ( ) = (ƒ…

ñóò ò ó

ï ñ óò ò ó

+

ô ï (ñ ó ò

)2 +

1 2

ñ óò

+

õÿñ

2

+

ö^ñ

+

ü

−  ¨ + (˜ + . + ý ) .

2

= 0.

¨
+ 2 ¨ + 2  The substitution ¨ ( ) =  (
 )2 leads to a first-order linear equation: u  2 ( + 2 , = 0.

ï ñóò ò ó

+ô Solution:

ï ñóò ò ó

ï (ñ óò ù

+ ô ï ( ñ óò

)2 – õÿñ

 ñóò

 

M  ( )+ L

Solution:

)

ù

2

1

2

+

= 0. =L

= ( õ ï ñ ÷

 

M  ( )+ L

1

2

+ ln | |,

+

ü

=L

– 1)ñó ò . 2

+

1   ,

,

where

 ( )= ù     M

where

 ( )= ù

  

+

M

1   .



.

Page 428

429

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

50. 51. 52.

ï 2 ñ óò ò ó

= 2ñ + ô ïƒ÷ ( ï The substitution ¨ = 

ñóò

ï 2 ñ óò ò ó

+ ñ )2 + õ ïƒú . ¨ 
=   ¨ 
+ leads to a Riccati equation:  3

= ô ( ô + 1)ñ + õ ïw÷ ( ï The substitution ¨ =  
+ 

ñóò

+ ôñ )2 + ö ïƒú . ¨ 
= '  ¨ leads to a Riccati equation:  3

= (ñó ò )2 – ô . 1 b . Solution (  is any): 1

exp( L

2 b . Solution for  < 0: =L

1

sin ”

2

 £ 

 )−

4L +L

 1

L

2 2

2

55. 56. 57. 58.

ññóò ò ó

– 14 (ñó ò )2 =

ô ï

2

+

õï

L

2 1

exp(− L

2

+ (  + 1)¨ + (˜ + .

 ).

 £ 

cos ”

+ L 22 • ü There are also singular solutions: = A" £  + L .

54.

2

ññóò ò ó

=L

53.

+ 2¨ + ' + .

2

ü

L

2 1

+L

2 2

•

.

+ö .

The substitution = ¨ 4 J 3 leads to a special case of the equation 2.8.1.8 with Œ = 0: ¨42

 = 3(  2 + ' + ( )¨ −5 J 3 .

3ññó ò ò ó – 2(ñó ò )2 = ô ï The substitution = ¨

+õ ï +ö . 3 leads to an equation of the form 2.8.1.5: 9¨u

 = ( 

2

2ññó ò ò ó = (ñó ò )2 + õ ïƒ÷ ñ 2 – ô . This is a special case of equation 2.9.3.5 with  ( ) = − '  .

+ ' + ( )¨

−5

.

= ù (ñ ó ò )2 – ôñ 4÷ –2 + õ ï ú ñ 2 . This is a special case of equation 2.9.3.8 with  ( ) = − ' + .

ññ óò ò ó

= ù (ñ ó ò )2 + ô ï  ñ 2 + õ ï ú ñ ÷ +1 . This is a special case of equation 2.9.3.9 with  ( ) = −   and ( ) = − ' + .

ññ óò ò ó

(ù + 2)ññ„ó ò ò ó – (ù + 1)(ñ„ó ò )2 = ( ô ï 2 + õ ï + ö )÷ . The substitution = ¨  +2 leads to an equation of the form 2.7.1.36: 1 ¨ 

 = (  2 + ' + ( )  ¨ −2  −3 . (B + 2)2

59.

= (ñó ò )2 + ô ïƒ÷ ññó ò + õ ïƒú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = − ' + .

60.

ôññóò ò ó

ññóò ò ó

+ õ (ñó ò )2 + ( ïƒ÷ + )ú

Solution: 61.

2

 +  =L

ññóò ò ó

– (ñó ò )2 = 1 b . Solution:

1

ôñóò ò ó

ññóò 1

= 0.

X exp − X (  + Œ ) +  S  + L 2 . M M Q  +

õÿñ

+ö .

= L 1 sinh( L 3  ) + L 2 cosh( L 3  ) +  + L 3−2 , where the constants L 1, L 2 , and L 3 are related by the constraint ( L 12 − L 22 ) L 2 b . Solution:

= L 1 sin( L 3  ) + L 2 cos( L 3  ) +  − L where the constants L 1, L 2 , and L 3 are related by the constraint ( L There is also a singular solution: = − (  .

−2 3 , 2 1+

L 22 ) L

2 3+

 + ( +  2 L

−2 3 = 0.

2 3+

 + ( −  2 L

−2 3 = 0.

Page 429

430 62.

63.

64.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ññóò ò ó

– (ñó ò )2 = ô 2 ñó ò ò ó + ô 1 ñó ò + ô 0 ñ + õ .  Particular solutions: = L ´ − ˜ −1 0 , where L is an arbitrary constant and Œ = Œ of the quadratic equation (  2  0 +  ) Œ 2 +  1  0 Œ +  20 = 0.

66.

67.

(2ñ + ô ï + õ )ñ„ó ò ò ó – (ñó ò )2 – ôñó ò + Solution: 1,

L 2 , and L

ö

= 0. =L

1



2

+L

2

 + L 3,

3 are related by the constraint

4L

1

L 3−L

2 2 +2

L 1 − L 2 + ( = 0.

= ï (ñó ò )2 – ññó ò + ô ï ñ s . This is a special case of equation 2.9.4.64 with  (ˆ ) = ˆ , (ˆ ) = 1, , = B − 1, and ° =

ï ññóò ò ó

ñ 2 ñ óò ò ó

+ ñ (ñ

óò

)( õ

–ñ

)2 =

ô ï

+õ .

¡

+ 2.

2 5 Having set 1  = ‡
 ( ), we obtain a third-order equation: −‡
 ‡


 +3(‡

 ) = (  +  )(‡
 ) . Taking ‡ to be the independent variable, we obtain a constant coefficient linear equation for  =  (‡ ): 


  =  +  .

(ô

2

–ï

2

2

2

)ñó ò ò ó + ( ô

Solution: arcsin 68.

are roots

(ñ + ô ï )ñ„ó ò ò ó = õ ïƒ÷ ( ï ñó ò – ñ )2 . The substitution = −  + "H leads to the equation "  HH 

 + 2 HH 
− '  +3 ( H 
)2 = 0. Having ¨


u ¨
set = H  H , we obtain a Bernoulli equation:   + 2¨ +  (1 − '  +2 )¨ 2 = 0.

where the constants L 65.

1,2



=L

1

2

+L

–ï 2

2

arcsin

( ï ñó ò – ñ + ô )ñó ò ò ó = õ 2 (ñó ò )2 + The contact transformation

õ 1 ñ óò



2

–ñ

2

)ñó ò .

.

 +

õ

0.

n ol

o = 1  

 , where n = n (m ), leads to a linear equation: (  2 m 2 +  1 m +  0 ) n ol

o − n = 0. m

= 
,

)ñ (ñó ò )2 = ï ( õ

n =  
− +  ,

n o
= ,

Inverse transformation:

 = n o
,




= mn o − n +  ,


= m ,



 = 1 n ol

o .

2.8.1-4. Other equations. 69. 70. 71. 72. 73.

= ô ïƒ÷ ( ï ñó ò – ñ )ú . This is a special case of equation 2.9.4.58 with  ( ) =   and (ˆ ) = ˆ + .

ñóò ò ó

= ô 2 ñ + õ ïƒ÷ (ñó ò + ôñ )ú . ¨ 
=  ¨ + '  ¨2+ . The substitution ¨ = 
+  leads to a Bernoulli equation: u

ñóò ò ó

= ( ô 2 ï 2 + ô )ñ + õ ï ÷ (ñ The substitution ¨ = 
− 

ñ óò ò ó

óò

– ô ï ñ )ú . leads to a Bernoulli equation: ¨ 
= −  ¨ + '  ¨ + .

= ô ï –÷ –3 ñ ÷ ( ï ñó ò – ñ )ú . This is a special case of equation 2.9.4.59 with  (ˆ ) = ˆ + .

ñóò ò ó

= ô ï –1 ñ ÷ ñó ò ( ï ñó ò – ñ )ú . This is a special case of equation 2.9.4.60 with  ( ) = 

ñóò ò ó

 and (ˆ ) = ˆ + .

Page 430

431

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

74.

75.

76.

ñ óò ò ó

=

ô ï ÷' –1 ñ úI –1 (ï ñ óò

ñóò ò ó

=

ü ï ^ (ñ óò )_ (ï ñ óò

–ñ )

÷ ú ÷

2 +

.

This is a special case of equation 2.9.4.24 with  (ˆ ) = ˆ  . – ñ )` .

The Legendre transformation  = ¨ug
, 1 Emden–Fowler equation: ¨ g

g = a −: ¨ , in Section 2.3 and Section 2.5.

ñóò ò ó

=

ô ïƒ÷ –1 ñ ú

–1

(ñó ò )

¨ g
− ¨ , where ¨ = ¨ (a ), leads to the generalized =a2 −

÷ ú –÷' ï ÷ +ú ( ñ óò

2 +

; (¨ g
)− 9 . Solvable equations of this type are outlined

– ñ ) .

This is a special case of equation 2.9.4.25 with  (ˆ ) = ˆ . 77.

78.

79.

80.

81.

ñ óò ò ó

=

ô ï ÷ (ï ñ óò

–ñ )+

õ ï ú (ï ñ óò

– ñ ) .

This is a special case of equation 2.9.4.4 with  ( ) =   and ( ) = ' + .

ô ï ÷ (ï ñ óò

+ ñ )ú .

ï 2 ñ óò ò ó

= 2ñ +

The substitution ¨ = 


+

ï 2 ñ óò ò ó

=

ïw÷ (ï ñ óò

ù (ù

– 1)ñ + ô

leads to a Bernoulli equation:  ¨ 
= 2¨ +   ¨ + . – ùñ )  .

This is a special case of equation 2.9.4.3 with  ( ) =  

ï 2 ñ óò ò ó

= ô ( ô + 1)ñ +

The substitution ¨ =  (ñ

óò ò ó

)2 =

 (ï ñ óò

õ ïw÷ (ï ñ óò 
+ 

óò

– ñ ) + ñ

+

ôñ )ú

−2

.

.

leads to a Bernoulli equation:  ¨u
= (  + 1)¨ + '  ¨2+ .

+ .

Differentiating the equation with respect to  yields:



 (2 


 − ƒ… − ý ) = 0.

(1)

Equating the second factor to zero and integrating, one obtains: 1 48

=

ƒ…

4

+

1 12

3

ý±

+L

2

2



+L

1

 + L 0.

(2)

The integration constants L‹© and the parameters ƒ , ý , and = are related by the constraint 4 L 22 = ýhL 1 − ƒsL 0 + = , which is obtained by substituting the above solution (2) into the original equation. In addition, there is a singular solution, which corresponds to setting the first factor in (1) equal to zero: =L 82.

(ñó ò ò ó )2 + 2 ôññó ò ò ó +

õ ï ñóò ò ó

¶

1

¶  + L 0,

+

ö^ñóò ò ó

ý L

where

¶

1

– ô (ñó ò )2 – õÿñó ò +

−ƒ L

ü

¶

0

+ = = 0.

= 0.

Solution: =L where the constants L 2 (L 1 + , = 0.

1,

L 2 , and L

3

1



2

+L

2

 + L 3,

are related by the constraint 4 L

2 1+

 (4 L 1 L

3−

L 22 ) − L

2+

Page 431

432

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.8.2. Painleve´ Transcendents 2.8.2-1. Preliminary remarks. Singular points of solutions. 1 b . Singular points of solutions to ordinary differential equations can be fixed or movable. The coordinates of fixed singular points remain the same for different solutions of an equation.* The coordinates of movable singular points vary depending on the particular solution selected (i.e., they depend on the initial conditions). Listed below are simple examples of first-order ordinary differential equations and their solutions having movable singularities: Equation

‘
= − 2 ‘
= 1  ‘
=  − ‘
= − ln2

Solution

Solution’s singularity type

= 1 (H − H 0) = 2£ H − H 0 = ln( H − H 0 ) = exp[1 (H − H 0 )]

movable pole algebraic branch point logarithmic branch point essential singularity

Algebraic branch points, logarithmic branch points, and essential singularities are called movable critical points. 2 b . The Painlev´e equations arise from the classification of the following second-order differential equations over the complex plane: ‘'

‘ = ² ( H , , ‘
), ¨ where ² = ² ( H , , ) is a function rational in and ¨ and analytic in H . It was shown by P. Painlev e´ (1897–1902) and B. Gambier (1910) that all equations of this type whose solutions do not have movable critical points (but are allowed to have fixed singular points and movable poles) can be reduced to 50 classes of equations. Moreover, 44 classes out of them are integrable by quadrature or admit reduction of order. The remaining 6 equations are irreducible; these are known as the Painlev e´ equations or Painlev´e transcendents, and their solutions are known as the Painlev´e transcendental functions. The canonical forms of the Painlev´e transcendents are given below in Paragraphs 2.8.2-2 through 2.8.2-7. Solutions of the first, second, and fourth Painlev e´ transcendents have movable poles (no fixed singular points). Solutions of the third and fifth Painlev e´ transcendents have two fixed logarithmic branch points, H = 0 and H = F . Solutions of the sixth Painlev e´ transcendent have three fixed logarithmic branch points, H = 0, H = 1, and H = F . It is significant that the Painlev´e equations often arise in mathematical physics. 2.8.2-2. First Painlev´e transcendent. 1 b . The first Painlev´e transcendent has the form

‘'

‘ = 6

2

+H .

(1)

The solutions of the first Painlev´e transcendent are single-valued functions of H . The solutions of equation (1) can be presented, in the vicinity of movable pole H p , in terms of the series: ½ 1 = + ÷   (H − H p) , ( H − H p )2

 where H p and L of H p and L .

2

1 = − 10 H p, 

3

= − 61 , 



4

=2

=L , 

5

are arbitrary constants; the coefficients 

=¾ 0, 

6

=

1 2 300 p ,

H

(Â ≥ 7) are uniquely defined in terms

* Solutions of linear ordinary differential equations can only have fixed singular points, and their positions are determined by the singularities of the equation coefficients.

Page 432

433

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2 b . In a neighborhood of a fixed point H = H 0 , the solution of the Cauchy problem for the first Painlev´e transcendent (1) can be represented by the Taylor series (see Paragraph 0.3.3-1): = # + $ ( H − H 0 ) + 12 (6 #

2

+ H 0 )( H − H 0 )2 + 16 (12 #2$ + 1)( H − H 0 )3 + 12 (6 #

‘ ‘

3

+$

2

+ #2H 0 )( H − H 0 )4 + ))) ,

‘ ‘ ‘

and $ are initial data of the Cauchy problem, so that | = 0 = # and
| = 0 = $ . x3y'z|{}~/€ The solutions of the Cauchy problems for the second and fourth Painlev´e transcendents can be expressed likewise (fixed singular points should be excluded from consideration for the remaining Painlev´e transcendents).

where #

3 b . For large values of | H |, the following asymptotic formula holds:

H 1J  2 ‹ <  45 H 5 J 4 −  ; 12,   ,

š where the elliptic Weierstrass function

<

( ; 12,  ) is defined implicitly by the integral

=X

4<

ü

 and  are some constants.

3

M

<

− 12 < − 

;

The first Painlev´e transcendent (1) is invariant under scaling of variables, H = Œ H ¯, where Œ 5 = 1.

=Œ

3

¯,

2.8.2-3. Second Painlev´e transcendent. 1 b . The second Painlev´e transcendent has the form

‘'

‘ = 2

3

+H

+ƒ .

(2)

The solutions of the second Painlev´e transcendent are single-valued functions of H . The solutions of equation (2) can be represented, in the vicinity of a movable pole H p , in terms of the series: =

1= 5=

− 16 1 3024

H ,  = ¡ V p 2 (27 + 81 ƒ

where = A 1; H p and L ¡ in terms of H p and L .

¡

H −H

− 41 ( 2

¡

− 2H

½

+ ÷



p

=1

+ ƒ ),  3 p)

¡

3

  (H − H p ) , =L , 

4

=

1 72 p (

H

+ 108 ƒ − 216 LuH p W ,

¡

+ 3 ƒ ),

are arbitrary constants; the coefficients   (B ≥ 6) are uniquely defined

2 b . For fixed ƒ , denote the solution by ( H , ƒ ). Then the following relation holds: ( H , − ƒ ) = − ( H , ƒ ),

(3)

while the solutions ( H , ƒ ) and ( H , ƒ − 1) are related by the B¨acklund transformations: 2ƒ − 1 , ‘
2 (H , ƒ ) − 2 2 (H , ƒ ) − H 2ƒ − 1 ( H , ƒ ) = − ( H , ƒ − 1) − ‘
. 2 ( H , ƒ − 1) + 2 2 ( H , ƒ − 1) + H ( H , ƒ − 1) = − ( H , ƒ ) +

(4)

Therefore, in order to study the general solution of equation (2) with arbitrary ƒ , it is sufficient to construct the solution for all ƒ out of the band 0 ≤ Re ƒ < 12 . Three solutions corresponding to ƒ and ƒ\A 1 are related by the rational formulas:

9 where

+1

=−

9 stands for (H , ƒ ).

(

9

−1

+

3 9 )(4 9 + 2 H 9 + 2 ƒ + 1) + (2ƒ − 1) 9 , 2( −1 + )(2 2 + H ) + 2 ƒ − 1 9 9 9

Page 433

434

SECOND-ORDER DIFFERENTIAL EQUATIONS The solutions ( H , ƒ ) and ( H , − ƒ − 1) are related by the B¨acklund transformations: 2ƒ + 1 , ‘
2 (H , ƒ ) + 2 2(H , ƒ ) + H 2ƒ + 1 ( H , ƒ ) = ( H , − ƒ − 1) − ‘
. 2 ( H , − ƒ − 1) + 2 2 ( H , − ƒ − 1) + H ( H , − ƒ − 1) = ( H , ƒ ) +

3 b . For ƒ = 0, equation (2) has the trivial solution = 0. Taking into account this fact and relations (3) and (4), we find that the second Painlev´e transcendent with ƒ = A 1, A 2,  has the rational particular solutions: ( H , A 1) = T

1

H

1

( H , A 2) = A2O

,

3H

−

H

2



,

H 3+4P

For ƒ = 12 , equation (2) admits the one-parameter family of solutions:

¨2‘


( H , 12 ) = − ¨

,

¨ =£ H

where

Q

ø

L

2

J 

1 1 3

2

H J  + L 2 n 1J 3 

2 3 2 3

H J S .

2 3 2 3

(5)

‘'‘

(Here, the function ¨ is a solution of the second-order linear equation ¨

+ 12 H ¨ = 0; see 2.1.2.2 and 2.1.2.7 with B = 1.) It follows from (3)–(5) that the second Painlev e´ transcendent for all ƒ = B + 12 with B = 0, A 1, A 2,  has a one-parameter family of solutions that can be expressed in terms of Bessel functions. 2.8.2-4. Third Painlev´e transcendent. 1 b . The third Painlev´e transcendent has the form


‘ 2 
‘ 1 ‘'

‘ = ( ) − + (ƒ  H

2

3

+ý )+=

>

+

.

(6)

û



In terms of the new independent variable defined by H =  , the solutions of the transformed equation will be single-valued functions of . Any solution of the Riccati equation

‘
= ,

2

+

ƒ −, ,H

+( ,

(7)

where ,

2 = = , ( 2 = −> , ! , ý + ( ( ƒ − 2 , ) = 0, is a solution of equation (6). Substituting H = Œ U , ‘
‡ 1 = − , where Œ 2 = , into (7), we obtain a linear equation: ,!‡ , ( , −ƒ
‡

+ ‡ + ‡ = 0, *+* * ,!U

whose general solution is expressed in terms of Bessel functions:

‡ = U 29  L Q

ø

1

9  (U ) + L 2 n 29  (U )S .

2

2 b . In some special cases, equation (6) can be integrated by quadrature. Rewrite equation (6) in the form of integro-differential relations in two ways:

O


û P

2

+O

>

2


û

−= =X

2

Q

P



û 2

O

>

2

+2O +=

ý 2

P

−ƒ



2

û

P +O



û ý

=2X +ƒ

P

Q

> O

û

 S

2

M



−= ,

2

P



û 2

+O

û

H = .

ý

−ƒ

P

û

 S M



;

(8) (9)

Page 434

435

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

It is obvious from (8) that for ƒ = ý = = = > = 0, the general solution has the form: Adding (9) multiplied by 2 to (8), we obtain


û

O

2

P

+2


û

>

+O

2

−=

2

û 2 

P

ý

+2O

−ƒ

û 

P

> 

=4X

2

û

+ý 

O

2

P


û

−2

>

+O

2

−=

2

2

 P

û

ý

+2O

−ƒ

û

.

M

2

 P

= −4 X (= 

2

û

2

+ƒ 

1

H k 2.



û

Subtracting (9) times 2 from (8) yields


û

=L

û

(10)

 )

.

M

(11)

Substituting > = ý = 0 into equation (10) and = = ƒ = 0 into equation (11), we arrive at


û

O

2


û O

+2

P

2


û

P

−2

û

− 2ƒ


û

>

+

2



2

û

+

û

2 2

 −=



2ý

û 

=L

1,

(12)

=L

2.

(13)

Equations (12) and (13) are integrable by elementary functions. Substituting we obtain an autonomous equation: 2 ( û
) = 2 ƒ… + = + (1 + L

1)

=

−

û

 2.

 into (12), (14)

As a result, we find:





=







H (ƒ










2ƒ

2 ln2

1

H + 2 ƒsL ln H + L

H  £ = ln H + L  H + −1 L 2 H 2+ + ; 1 H + + ;

2

;

≠ 0,

H

1

=−

,

;

2

=

ƒ

−= )

2

if L

1

= −1, ý = > = 0;

if L

1

= −1, ƒ = ý = > = 0;

if L

1

≠ −1, ý = > = 0,

− = (1 + L 1 ) , 2 = 1 + L 1. L 1+1 4 L 2 (1 + L 1 )2 ¡ û Accordingly, equation (13) is reduced to equation? (14) with the substitution =   . © If ý = − ƒ and > = −= , the substitution =  − brings equation (6) to the following form: ¨ '‘

‘ + 1 ¨ ‘
= 2 ƒ sin ¨ + 2= sin 2¨ .

where L

ƒ

2

2

H

2.8.2-5. Fourth Painlev´e transcendent. 1 b . The fourth Painlev´e transcendent has the form


‘ 2 ‘'

‘ = ( ) + 3 2

2

3

+ 4H

2

+ 2( H

2

−ƒ ) +

ý

.

(15)

The solutions of the fourth Painlev´e transcendent are single-valued functions of H . The Laurent-series expansion of the solution of equation (15) in the vicinity of a movable pole H p is given by: ¾ =

¡ H −H

p

−H

= A 1; H p and L where ¡ of ƒ , ý , H p , and L .

p

− ¡

3

(H

2 p

¾

½¾

+ 2 ƒ − 4 )(H − H p ) + L ( H − H p )2 + ÷

¡

are arbitrary constants; and the 

¾

 (H − H p) , =3

(Â ≥ 3) are uniquely defined in terms

Page 435

436

SECOND-ORDER DIFFERENTIAL EQUATIONS

= A 1, is satisfied, then every solution of the 2 b . If the condition ý + 2(1 + ƒ )2 = 0, where ¡ ¡ Riccati equation ‘
= 2 + 2 H − 2(1 + ƒ )

¡

¡

¡

is simultaneously a solution of the fourth Painlev´e equation (12). Equation (15) is invariant under the transformation = Œ ¯ , H = Œ H ¯ , ƒ = ƒt ¯ Œ 2 , ý = ý ¯ , where Œ 4 = 1. 3 b . Two solutions of equation (15) corresponding to different values of the parameters ƒ are related to each other by the B¨acklund transformations:

¶

1 ( ‘
−  − 2 °H − ° 2 ), 2° ¶ ¶ 1 ¶ ‘
= − ¶ ( −  + 2 °H + ° 2 ), 2°

¶

2ý = −( ƒ…° − 1 − where

= ( H , ƒ , ý ),

¶

 2 = −2ý

=

¶

¶ ¶

1 2





and ý

,

¶

2

= −2ý ,

¶

4 ƒ = −2 ° − 2ƒ − 3 °µ ,

2

),

= ( H , ƒ , ý ), and ° is an arbitrary parameter.

2.8.2-6. Fifth Painlev´e transcendent. 1 b . The fifth Painlev´e transcendent has the form


‘ ( − 1)2 ‘'

‘ = 3 − 1 ( ‘
)2 − O ƒ + 2 ( − 1) H H 2

+

ý

û

+=

P

+

H

> ( + 1) −1

.

(16)



If we pass on to the new independent variable H =  , the solutions are single-valued functions of . Solutions of the fifth Painlev´e transcendent (16) corresponding to different values of parameters are related by: ( H , ƒ , ý , = , > ) = (− H , ƒ , ý , −= , > ), 1 (H , ƒ , ý , = , > ) = . ( H , − ý , − ƒ , −= , > ) 2 b . On setting H = 

g in (16), we obtain

3 −1 2

( g
) + ( − 1)2 Oƒ gg =

+

2 ( − 1)

ý

+=

P

g > ( + 1) 2g .  +  −1

(17)

If = = > = 0, equation (17) is reduced, by integration, to a first-order autonomous equation:


g = ( − 1) ü 2 ƒ which is readily integrable by quadrature. If the condition = = £ −2 >3 1 + is satisfied, any solution of the Riccati equation

H ‘
= £ 2ƒ

2

ü

2

+L

− 2ý ,

−2ý − £ 2 ƒ 

+  £ −2 >hH − £ 2 ƒ −

ü

−2ý 

+

ü

−2ý

(18)

is simultaneously a solution of the fifth Painlev´e transcendent (16). Equation (18) can be reduced to the degenerate hypergeometric equation 2.1.2.70.

Page 436

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

437

2.8.2-7. Sixth Painlev´e transcendent. 1 b . The sixth Painlev´e transcendent has the form ‘'

‘ = 1 O 1 + 1 + 1 ( ‘
)2 − O 1 + 1 + 1 ‘
2 −1 − HrP H H −1 − HrP H H −1 H ( H − 1) ( − 1)( − H ) + +> S . (19) ƒ +ý 2 += H 2 ( H − 1)2 Q ( − 1)2 ( − H )2 In equation (19), the points H = 0, H = 1, and H = F are fixed logarithmic branch points. Painlev´e found two integrable cases of the equation. First, if ƒ = ý = = = > = 0, the general solution of equation (19) has the form: = « (L

 

1 1

+L

 

2 2,

H ),

where « (‡ , H ) is the elliptic function, defined by the integral

‡ = X•” £ 0

M

( − 1)( − H )

,

(20)

with periods 2  1 and 2  2 , which are functions of H . Secondly, if ƒ = ý = = = 0, > = 12 , the general solution of equation (19) has the form: = « (¨ + L

1   1 + L 2   2 , H ), ¨ where 1 0 is any particular solution of the linear equation 1 ¨ ‘'

‘ − 2 H − 1 ¨ ‘
+ ¨ =0 H ( H − 1) 4 H ( H − 1) and « (‡ , H ) is the elliptic function defined by formula (20). 2 b . Solutions of the sixth Painlev´e transcendent (19) corresponding to different values of parameters

are related by:

( H , −ý , − ƒ , = , > ) =

O

( H , − ý , −= , ƒ , > ) = 1 −

1

H

1 , ƒ , ý , = , > 1

,

P

, 1 , ƒ , ý , = , > 1−H P 1 1 H = OH , −ý , − ƒ , −> + , −= + . 2 2P (H , ƒ , ý , = , > ) The successive application of these relations yields 24 equations of the form (19) with different values of parameters related by known transformations. 3 b . Every solution of the Riccati equation

‘
= £ 2ƒ H ( H − 1)

where

2

£ 2ƒ − (ƒ + ý + = + > ) £ 2 ƒ − £ −2ý − 1 ,

+

O

£ −2ý ŒH + “ + , H ( H − 1) H −1

£ −2ý − ( ƒ + ý − = − > ) £ 2 ƒ − £ −2ý − 1 , is simultaneously a solution of equation (19) if £ 2 ƒ − £ −2ý ≠ 1 and the condition 2 £ 2 ƒ (3ý − ƒ + = − > ) + 2 −2ý (3 ƒ − ý − = + > ) + 4 − ƒ…ý (ý − ƒ + = − > − 1) ü ü + ( ƒ + ý + = + > )2 + 2( ƒ − ý − = − 4 ƒ…ý − 2 ƒ±= − 2ý…> ) = 0 £ £ £ ^is_ satisfied (one should take the value of − ƒ…ý that coincides with ƒ −ý ). Œ =

“ =

References for Subsection 2.8.2: P. Painlev´e (1900), B. Gambier (1910), G. M. Murphy (1960), E. L. Ince (1964), A. S. Fokas and M. J. Ablowitz (1982), V. I. Gromak and N. A. Lukashevich (1990), R. Conte (1999), A. R. Chowdhury (2000).

Page 437

438

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.8.3. Equations Containing Exponential Functions 2.8.3-1. Equations of the form  ( , )

 + ( , ) = 0. 1.

2.

ñóò ò ó

ô ó +_ ]

=

+õ .

¨ The substitution ? ¨2

 =   : +  .

ô ó ñ ÷

.

ô ï ÷  ]

.

ñóò ò ó

=

ñ óò ò ó

=

ñóò ò ó

=

ñóò ò ó

=

2

ñóò ò ó

=

2

+ ( Œ ý )

=

 

The transformation H =   !H − ¨ 2 . 3.

4.

The transformation H =   !H + ¨ .

õ ó ñ

+

ôñ

–3

−1

+2

leads to an autonomous equation of the form 2.9.1.1:

‘ , ¨ = 
 leads to a first-order equation: H [(B −1)¨ + Œ ]¨
=

 , ¨ =  
 leads to a first-order equation: H ( Œ ¨ + B + 2)¨u‘
=

.



This is a special case of equation 2.9.1.2 with  ( ) = −   . 5. 6.

ñ

+ ô exp[ (ù + 3) ï ]ñ

ñ

+ ô

.

This is a special case of equation 2.9.1.29 with  (ˆ ) = ˆ  .

ó  ñ ú

,

≠ 0.

  The transformation ˆ =   , ‡ =  leads to the Emden–Fowler equation ‡ Šª

Š = 4Œ “ − 3Œ − Œ where B = ¡ , whose special cases are given in Section 2.3. 2Œ 2

7.

÷

ñóò ò ó

=

ñóò ò ó

=

ñ

+ ô

ñ

+

2

(

ú

+3)



ó ( õ  2 ó



+ ö )÷

ñ ú

+ (, ¨

,



2

ˆ  ‡ + ,

≠ 0.

The transformation ˆ =    =  leads to the Emden–Fowler equation ¨ Šª

Š =  ˆ  ¨ + , whose special cases are given in Section 2.3. 4 2 Œ 2 8.

2

2


B (ú

+3)

ó (ô  2 ó 

+ õ )÷ ( ö

2



ó

+ý )

÷ ú

– –

–3



ñ ú

.

   + ¨  , = 2  leads to the Emden–Fowler equation  2 (  + (  + ¨ Šª

Š = # (2 ¢eŒ )−2 ˆ  ¨ + , where M M ¢ =  − ˜( (see Section 2.3). M 2

The transformation ˆ =

9.

10.

ñ óò ò ó

=

ô

exp( 

ï

2

+

ï

) exp( ƒñ ) + õ .

¨ = + ( ƒ… The substitution ? ¨2

 =   ; +  + 2 ƒ±= −1 .

ï 2 ñ óò ò ó

=

ô ïƒ÷ +2  ]

2

+ ý± ) = leads to an autonomous equation of the form 2.9.1.1:

+ù .

This is a special case of equation 2.9.1.31 with  (ˆ ) = ˆ . 2.8.3-2. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0. 11.

ñóò ò ó

=

ôñóò

+

õ 2D ó ñ ÷

. This is a special case of equation 2.9.2.17 with  ( ) =   .

Page 438

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

12.

13.

ñóò ò ó

= – ôñó ò +

õD ÷ ó ñ ÷

15.

.

This is a special case of equation 2.9.2.36 with  (ˆ ) = 'ˆ 

ôñóò

+

ñóò ò ó

= –( + b )ñó ò – b “ñ +

+

õÿñ

ö ó ñ ú

ñóò ò ó

=



The substitution ˆ =  14.

–1

−1

=

ñ óò ò ó

=

ƒñóò

+

ƒñ óò

+

õï ñ

.

.

leads to an equation of the form 2.8.1.39: ˆ 2 
Šª
Š +(  +1)ˆ 
Š + 

ô (÷  –2 )ó ñ ÷

–1

ô 2 ó ñ

+

–3

= (˜ˆ  + .

.

This is a special case of equation 2.9.2.37 with  (ˆ ) = ˆ 

ñóò ò ó

439

−1

.

.

This is a special case of equation 2.9.2.14 with  ( ) = − ' . 16.

õ ó ñ

+ ô

2

ó

 ñ

–3

.



This is a special case of equation 2.9.2.14 with  ( ) = −   17.

18.

ñóò ò ó

=

ôñóò

ñóò ò ó

+ 3ññó ò + ñ

ñ óò ò ó

=

ï

+ õ exp(2 ô

÷

+ ö^ñ

.

).

This is a special case of equation 2.9.2.17 with  ( ) =  exp( (  ). 3

óñ

+ ô

= 0.



This is a special case of equation 2.9.2.1 with  ( ) =   . 19.

ô ï  ] ñ óò =L

Solution: 20.

ñóò ò ó

+ ô 1

ó ññóò

= 2 ô

]

.

  − ln O −  X   k 1  + L 2 . M P

óñ

+ ô

2

.

Solution in parametric form:

U

 = ln O

2

2L

:

1

Here, = L 1 ø 1 (U ) + L 2 n 1 (U ) or Bessel functions, and ® 1 (U ) and ; 21.

ñ óò ò ó

=

ô ï ÷  ] ñ óò =L

Solution: 22.

ñóò ò ó

=

+ ôù 1

ï ÷ –1  ]

 − ln L Q

ô ó ñ –1 ) 2 ñ óò

2

P

:

,

−2

L 1U

:

−1

(U

:
*

:

+ ).

= L 1 ® 1 (U ) + L 2 ; 1 (U ), where ø 1 (U ) and n 1 (U ) are the ( U 1 ) are the modified Bessel functions.

.

−  X   exp( L

ó ñ 1) 2.

+ 2 ô

−1

= −

1

 )  S . M

Solution in parametric form:

 = ln OA 23.

24.

L 

1



ñóò ò ó

+ (2 ôñ +

ñ óò ò ó

= 

P

TCU 2 ,

=L

õ ó )ñ óò

+

2 1

V 2U3A exp(T"U 2 )  W 2 ,

ˆõ ó ñ

= 0.

Integrating yields a Riccati equation: 
 + 

ô _ ó ñ

Solution: 2 X



+ 

M

2

ñ óò

+L

2

+  



where

 = X exp(T"U 2 ) U + L 2 S Q M

−1

.

=L .

. 1

=L

2

+

ý

1 :   .

Page 439

440 25.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

=

ô ó +] (ñ óò

Solution: = − − ln( L

26.

+ 1).

= − ln O L 1

1



−

k 2 −

−  ).





where

 (a ) =

28. 29.

30. 31. 32. 33. 34.

2



 P

. To the limiting case L

2

D

−1 there corresponds

+ ñ ó ò = ô ï ÷  ] . This equation is encountered in combustion theory and hydrodynamics. The transformation ? ˆ = ln  , ¨ = Œ +(B +1) ln  leads to an autonomous equation of the form 2.7.3.1: ¨IŠª

Š = Œ  . Solution in parametric form:

ï ñ óò ò ó

 = exp[ L

27.



1+ L

ï ñóò ò ó






1

Aç (a )],

=

Œ

a

−

B +1 [L Œ

£ L 2 + 2 Œ  g − £ ln £ L 2 £ L 2 + 2 Œ  g + £ 2 −£ 2 Œ  g £ L 2 + 2 Œ 2 £ − L 2 arctan £ −L 2

L

1

















2

L

2

 g

1

Aç (a )],

if L

2

> 0,

if L

2

= 0,

if L

2

< 0.

= ùñó ò + ô ï 2÷ +1 ] . This is a special case of equation 2.9.2.20 with  ( ) =   .

= ùñ ó ò + ô ï 2÷ +1 exp( ƒñ ú ). This is a special case of equation 2.9.2.20 with  ( ) =  exp( Œ "+ ).

ï ñ óò ò ó

ï 2 ñ óò ò ó

+ ï ñó ò = ô] . The substitution a = ln | | leads to an equation of the form 2.7.3.1: Solution: 1 Œ = − ln ¬ sin2 ( L 1 ln | | + L 2 )® if Œ Œ 2 L 12 1 Œ = − ln ¬ sinh2 ( L 1 ln | | + L 2 )® if Œ Œ 2 L 12 1 Œ = − ln ¬ − 2 cosh2 ( L 1 ln | | + L 2 )® if Œ Œ 2L 1


g
g =   . > 0, > 0, < 0.

ï 2 ñ óò ò ó

+ ï ñó ò = ô] + õ . This is a special case of equation 2.9.2.23 with  ( ) =   +  . + ï ñ ó ò = ü ï ÷  Df] + õ . This is a special case of equation 2.9.2.40 with  (ˆ ) = ,!ˆ +  .

ï 2 ñ óò ò ó

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + ö] = 0. This is a special case of equation 2.9.2.24 with  ( ) = (   .

ó

ó

ó

ó

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò + ö^ñ ÷ = 0.  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = − (  . ( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò + ö] = 0.  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = − ( ž .

Page 440

441

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.8.3-3. Equations of the form  ( , )

 + ( , )(
 )2 + % ( , )
 + & ( , ) = 0. 35.

36.

ïƒ÷

ñóò ò ó

= ô (ñó ò )2 –

õ 4fD ]

+ö

ñóò ò ó

= ô (ñó ò )2 –

õ 4fD ]

+ ö

ñóò ò ó

= ô (ñó ò )2 +

õ ƒï ÷  Df]

+

ö ïƒú

.

ñóò ò ó

= ô (ñó ò )2 +

õDf] +} ó

+

ü ïƒú

.

ñóò ò ó

= ô (ñó ò )2 +

õDf] + ó

+ ö

.

This is a special case of equation 2.9.3.18 with  ( ) = − (˜  .

ó

.



This is a special case of equation 2.9.3.18 with  ( ) = − (  . 37.

38.

This is a special case of equation 2.9.3.17 with  ( ) = − '  and ( ) = − (˜ + .

v

This is a special case of equation 2.9.3.17 with  ( ) = −   39.



ó

.

This is a special case of equation 2.9.3.17 with  ( ) = −   40.

ôñ ÷ (ñ óò

ñóò ò ó

+

õ]

+

ñóò ò ó

+

ô] (ñ óò

)2 +

õÿñ ÷

+

ñóò ò ó

+

ô] (ñ óò

)2 +

õ  ]

ñóò ò ó

=

ôñ ÷ (ñ óò

)2 +

õ ó ñóò

.

ñóò ò ó

=

ô] (ñ óò

)2 +

õ ƒï ÷ ñ óò

.

ñóò ò ó

=

ô] (ñ óò

)2 +

õ  ó ñóò

)2 +

ö

= 0.

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  41.

42.

43.

45.



and ( ) = − ( 

+

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =   and ( ) =     + ( .

 and ( ) =    .

This is a special case of equation 2.9.3.38 with  ( ) =    and ( ) = '  . .

ñóò ò ó

=

ô ó (ï ñ óò

– ñ )2 +

õ  ó

.

This is a special case of equation 2.9.3.2 with  ( ) =   47.

ññóò ò ó

– (ñó ò )2 =

ô ó

The substitutions ? ¨2

 = 2   − . 48.

.

 and ( ) =   + ( .

This is a special case of equation 2.9.3.38 with  ( ) =    and ( ) =   46.



This is a special case of equation 2.9.3.25 with  ( ) =   and ( ) =   + ( .

This is a special case of equation 2.9.3.38 with  ( ) =  44.

and ( ) = − ,! + .

2ññó ò ò ó = (ñó ò )2 +

.

= A exp 

õ ó ñ

2

1 2

  , (



.



) = 0, and % ( ) =    .

¨ + 1 Œ   leads to an autonomous equation of the form 2.7.3.1: 2

–ô .



This is a special case of equation 2.9.3.5 with  ( ) = −   .

49.

ññóò ò ó

=

ù (ñ óò

)2 – ôñ

÷

4 –2

+

õ ó ñ

2

.



This is a special case of equation 2.9.3.8 with  ( ) = −   .

Page 441

442 50.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ññ óò ò ó

– (ñ

óò

ô  ó ñ 

)2 =

.

1 b . For , ≠ 2, the substitution

= exp O ¨ +

?



the form 2.7.3.1: ¨  =   (  −2) . 2 b . Solution for , = 2: ln | | = L 1  + L 2

+

õ ó ñ ÷

+1

 ñ

2

+

õï ú ñ ÷

+1

óñ

2

õ  ó ñ ÷

+1

ïƒú ñ

+ Œ

2

Œ

2−, −2

 P

leads to an autonomous equation of

  .

51.

ññóò ò ó

=

ù (ñ óò

)2 + ô

.  This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  ž .

52.

ññ óò ò ó

=

ù (ñ ó ò

)2 + ô

ññóò ò ó

=

ù (ñ óò

)2 + ô

ó

.



and ( ) = − ' + .

This is a special case of equation 2.9.3.9 with  ( ) = −   53.

+

.

This is a special case of equation 2.9.3.9 with  ( ) = −   54.

ññóò ò ó

– (ñó ò )2 =

ô

 ï

ï



and ( ) = −  



.

. ¨ The substitutions =? A exp  + 12 ý± 2 + 12 Œ   leads to an autonomous equation of the form 3 ¨

− 2.9.1.1:  = 2   − 2ý . exp 

2

+ 1 2

55.

ññ óò ò ó

– (ñ

óò

ôñ

õ

ï

exp –

ï

. ¨ 1 2 The substitutions =? A exp  + 2 ý± + 12 Œ   leads to an autonomous equation of the form 3 ¨

− 2.9.1.1:  = 2   − 2(  + ý ). )2 +

2

=

2

+

1 2

56.

ññóò ò ó

– (ñó ò )2 =

ô

 ï

exp 

2

ï ñ 

+

.

1 ý± = exp ¨ + Q 2−, ? 2ý

. of the form 2.9.1.1: ¨   =   (  −2) − 2−, 2 b . Solution for , = 2: 1 b . For , ≠ 2, the substitution

ln | | = L 57.

ññóò ò ó

– (ñó ò )2 +

ôñ

2

=

õ

1

2

+ Π  S leads to an autonomous equation



 +L

 ï

exp 

2

2

+

+  X  ( − a ) exp  ý±a 2 + Œ a  a . M 0

ï ñ 

.

1  ý± 2 + Œ   S leads to an autonomous 2−, 2ý equation of the form 2.9.1.1: ¨ 

 =   (  −2) −  − . 2 −, ? 2 b . For , = 2, the substitutions = A  leads to a linear equation: ¨ 

 =  exp  ý± 2 + Œ   −  . Solution: = exp ¨ +

1 b . For , ≠ 2, the substitution



ln | | = −  2 58.

59.

ñóò ò ó

+ ô (ñó ò )2 –

1 2

ñóò

=



2

ó (õ ñ 2

Q?

+L

1

2

õ 1ñ

+



2

+  X  ( − a ) exp  ý±a 2 + Œ a  a . M 0

õ

0 ).

 +L +

 The substitution ¨ ( ) =  − (
 )2 leads to a first-order linear equation: ¨u
+ 2  ¨ = 2   2  1 + 2  0.

ññóò ò ó

= (ñó ò )2 +

ô ƒï ÷ ñ ñóò

+

õ ó ñ

2

2

+

2

.  This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  ž .

Page 442

443

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

60.

ññ óò ò ó

= (ñ ó ò )2 + ô  ññ ó ò + õ ï ÷ ñ 2 .  This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = − '  .

ó

61.

ññóò ò ó

ó

62.

ó

= (ñó ò )2 + ô ññó ò + õ  ñ 2 .   This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −    .

ó

+  (ñó ò )2 =  õ _ +` ] +   1 b . Solution with = ≠ − ƒ :

ñóò ò ó

 9  M  ( )+ L 2 b . Solution with = = − ƒ : ù

=L

1

ñóò 2

. +

ó

= ô _ (ñó ò + õÿñ )2 + õ The substitution ¨ = 
+ 

ñóò ò ó

2

ñ

1 : 



,

 9  M  +L ù

63.

ý

=L

1

ó

2

+

ý

  ( 9 +; )  . ƒ +=

 ( )=

where

1 : 



.

+ ö . ¨ 
=   :  ¨ leads to a Riccati equation: 3

2

 +  ¨ + (  .

2.8.3-4. Other equations. 64.

ñóò ò ó

+ õD ñ ú (ñó ò )3 + ôñó ò = 0. This is a special case of equation 2.9.3.35 with  ( ) =  "+ .

ó

65.

ñóò ò ó

+ õD + ] (ñó ò )3 + ôñó ò = 0. This is a special case of equation 2.9.3.35 with  ( ) =   .

ó

66.

ñóò ò ó

ô ó (ñ óò

=

)3 + ô

Solution:  = L 67.

1

ó ñ (ñ óò

)2 .

 k 1

− ln O X

+L

M

= ô ï  ] (ñ ó ò )3 + ô ] (ñ ó ò )2 . Solution in parametric form:

 *

(U

:
*

M

= ô ï 2 ] (ñó ò )3 + 2 ô ï ] (ñó ò )2 . Solution in parametric form:

 =

:

−1

L 1U

= 2 ô ï 1 ) 2 ] (ñó ò )3 + ô ï –1 ) Solution in parametric form:

ñóò ò ó

 =L 70.

−1

ñóò ò ó

Here, = L 1 ø 1 (U ) + L 2 n 1 (U ) or Bessel functions, and ® 1 (U ) and ; 69.

.

P

ñ óò ò ó

 = L 1  −  * O XZU 68.

2

2 1

V 2U2A exp(T"U 2 ) W 2 ,

ôùJ ó ñ ÷

–1

Solution:  = L

1

ñóò ò ó

=

(ñó ò )3 +

2

−2

:

:

−1

:

+ ),

2

P

= ln U .

,

= ln O

U

2L

2 1

P

.

= L 1 ® 1 (U ) + L 2 ; 1 (U ), where ø 1 (U ) and n 1 (U ) are the ( U 1 ) are the modified Bessel functions.

] (ñ óò

  k 1

)2 .

L

= ln O˜T

ô ó ñ ÷ (ñ óò

− ln O X

U +L



1



TCU 2 , P

where

 = X exp(T"U 2 ) U + L 2 S Q M

−1

.

M

)2 . +L

2

P

.

Page 443

444 71.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=

ô ó +] [(ñ óò

)3 + (ñó ò )2 ].

Solution:  = − ln OL

 = − − ln( L 72.

ñóò ò ó

=

1

+

1



 k 2 +

).

1−L

2

  . To the limiting case L P

D

2

1 there corresponds

ô ó (ñ óò )3 ) 2 + ô ó ñ (ñ óò )1 ) 2 .

Solution in parametric form:

 = ln U 2 , where

:

= −2 

−2 −4

U

V:

−1

(U

:
*

:

+2 )T

1 2

U 2W ,

ø

L

ø

U ) + L 2 n 2 (U ) for the upper sign, L 1 ® 2 (U ) + L 2 ; 2 (U ) for the lower sign, 2 ( U ) and n 2 ( U ) are the Bessel functions, and ® 2 ( U ) and ; 2 ( U ) are the modified Bessel 1 2(

™

=

functions. 73.

ñóò ò ó

=

ô ï ] (ñ óò )5 ) 2 + ô] (ñ óò )3 ) 2 .

Solution in parametric form:

 = −2  where

ø

:

U

ñóò ò ó

= – ôñó ò +

=

™

L

V:

−1

(U

:
*

:

+2 )T

1 2

U 2W ,

ø

1 2(

õD ú ó ñ  (ñ óò )ú

+2

.

This is a special case of equation 2.9.4.17 with  ( ) = −   and B =

75.

ñóò ò ó

=–

2–ü ñóò + õD ó ñ ú – +1 (ñóò ) . þ 1–ü This is a special case of equation 2.9.4.31 with  (ˆ ) = 'ˆ .

76.

ñóò ò ó

=

ñóò ò ó

= –(ñó ò )2 +

77.

= ln U 2 ,

U ) + L 2 n 2 (U ) for the upper sign, L 1 ® 2 (U ) + L 2 ; 2 (U ) for the lower sign, 2 ( U ) and n 2 ( U ) are the Bessel functions, and ® 2 ( U ) and ; 2 ( U ) are the modified Bessel

functions. 74.

−2 −4

ñóò

ô

+




exp[(ù + 2 – @ ) ï ]ñ



¡

+ 2.

ú (ñ óò )A .

The substitution ˆ =  leads to the generalized Emden–Fowler equation 
Šª
Š = #*ˆ  + (
Š )- , which is discussed in Section 2.5.


ïƒ÷

exp[(þ

+ @ – 1)ñ ] (ñ„ó ò )A .

The substitution ¨ =   leads to the generalized Emden–Fowler equation ¨ 

 = #*  ¨ + (¨ 
)- ,

which is discussed in Section 2.5. 78.

ñóò ò ó

=

ô

1–ü (ñó ò )2 + õ ïƒ÷ +  –2 Df] (ñó ò )  . ù 2–ü This is a special case of equation 2.9.4.30 with  (ˆ ) = 'ˆ .

79.

ñóò ò ó

=

ñóò ò ó

=

ôñ ÷ (ñ óò

)2 +

õ] (ñ óò )

.

This is a special case of equation 2.9.4.13 with  ( ) =  80.

ô] (ñ óò

)2 +

õÿñ ÷ (ñ óò )

 and ( ) =   .

. This is a special case of equation 2.9.4.13 with  ( ) =   and ( ) =   .

Page 444

445

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

81. 82.

83.

ñ óò ò ó

=

ô ] (ñ óò

ñóò ò ó

=

ô 2ñ

ñóò ò ó

=

ô ó +] [(ñ óò )

)2 +

õ ] (ñ óò )

.

This is a special case of equation 2.9.4.13 with  ( ) =   and ( ) =     . +

õ _ ó (ñ óò

+

ôñ )

The substitution ¨ = 
+ 

+ (ñó ò ) 

.

¨ 
=  ¨ +   :  ¨  . leads to a Bernoulli equation: u

–1

ü

],

≠ 2.

Solution in parametric form:

 =U −X 84. 85.

M 

U

−L

=X

2,

M

õ  ó (ï ñ óò



ñ óò ò ó

=

ô ï ÷ (ï ñ óò

ñóò ò ó

=

ô ó (ï ñ óò

–ñ )+

õ ïƒ÷ (ï ñ óò

ñóò ò ó

=

ô ó (ï ñ óò

–ñ )+

õ  ó (ï ñ óò

–ñ )+

U

+L

2,

where

 = V  (2 − , )  * + L 1 W 

 This is a special case of equation 2.9.4.4 with  ( ) =   and ( ) =   . – ñ ) .

– ñ ) .

This is a special case of equation 2.9.4.4 with  ( ) =    87.

ï ñ óò ò ó + ñóò = ô ïƒ÷  ] (ñóò )ú .  The transformation =  
, ¨ Œ

88.

+ 1.

– ñ ) .



and ( ) = '  .

This is a special case of equation 2.9.4.4 with  ( ) =   86.

1 −2

+B −

ï ñóò ò ó

+ 1.

¡

+ þxñó ò + ô

ïƒ÷¿ú

–2

ú

+1

=   −+



and ( ) =  



.



+1

] (ñ óò )÷

 leads to a first-order linear equation:  +2¨2û
=

= 0.

This is a special case of equation 2.9.4.14 with  ( ) =   . 89.

90. 91.

ï ñ óò ò ó + ñóò = (ô ïƒ÷  ] + õ ïƒú –1 )(ñóò )ú .   The transformation =  
, ¨ =   − + +1  + (  ¨ +  )¨2û
= ( Œ + B − + 1)¨ . ¡ ó ÷ 2 ññ óò ò ó = (ñ óò ) + õ D ñ (ñ óò ) .

leads to a first-order separable equation:





This is a special case of equation 2.9.4.68 with  (ˆ ) = 'ˆ , ( ) =  , and B =

ññóò ò ó

= (ñó ò )2 + ( ô

óñ ÷

+

õÿñ 2–ú

)(ñó ò )ú .

The transformation ˆ = 
 , ¨ =  ˆ + (  ¨ +  )¨2Š
= [(B + − 2)ˆ + Œ ]¨ .

  ++

−2

¡

− , + 2.

leads to a first-order separable equation:

¡

2.8.4. Equations Containing Hyperbolic Functions 2.8.4-1. Equations with hyperbolic sine. 1.

2.

ñóò ò ó

=

ñ óò ò ó

=

2

ñ

+ ô (sinh

ï )–÷ –3 ñ ÷

.

This is a special case of equation 2.9.1.34 with  (ˆ ) = ˆ  .

õ

sinh(

ï )ñ

+ ôñ

–3

.

This is a special case of equation 2.9.1.2 with  ( ) = −  sinh( Œ  ).

Page 445

446 3.

4.

5.

SECOND-ORDER DIFFERENTIAL EQUATIONS sinh÷ ( ôñ +

õï

ñ óò ò ó

=

ñóò ò ó

= ô (ñ + õ sinh ï ) ÷ – õ sinh ï + ö .



)+ .

This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ sinh  ¨ + ý and ( = 0. The substitution ¨ ¨2

 =  ¨  + ( .

+  sinh 

=

leads to an autonomous equation of the form 2.9.1.1:

+ 3ññó ò + ñ

ñóò ò ó

+

ñ óò ò ó

+

ô

sinh ÷

ñ (ñ ó ò

)2 +

ñóò ò ó

+

ô

sinh ÷

ñ (ñ óò

)2 + õ sinh ú ( ƒñ ) +

ñ óò ò ó

=

3

+ ô sinh(

ï )ñ

ñóò ò ó

= 0.

This is a special case of equation 2.9.2.1 with  ( ) =  sinh( Π ). 6.

)2 + õ sinh ú

ôñ ÷ (ñ óò

ñ

+ ö = 0.

This is a special case of equation 2.9.3.25 with  ( ) =  7.

8.

9.

õÿñ ú

+

ö

= 0.

ö

= 0.

11.

+

This is a special case of equation 2.9.3.25 with  ( ) =  sinh 

ôñ ÷ (ñ óò

)2 + õ sinhú

ñ (ñ ó ò ) 

ñóò ò ó

=

ô

sinh ÷

=

ùñóò

ñ (ñ óò

õÿñ ú (ñ óò )

)2 +

.

and ( ) =  sinh ( Œ

 and ( ) =  sinh+

.

ï 2÷

+ô

+1

sinhú ( ƒñ ).

+

This is a special case of equation 2.9.2.20 with  ( ) =  sinh ( Π12.

2ññ

óò ò ó

= (ñ

óò

)2 + õ sinhú (

ï )ñ

2

)+ ( .

.

and ( ) =  + .

This is a special case of equation 2.9.4.13 with  ( ) =  sinh 

ï ñóò ò ó

+( .

and ( ) =  + + ( .

This is a special case of equation 2.9.3.25 with  ( ) =  sinh 

This is a special case of equation 2.9.4.13 with  ( ) =  10.

 and ( ) =  sinh+

–ô .

).

+

This is a special case of equation 2.9.3.5 with  ( ) = −  sinh ( Œ  ).

13.

ù (ñ óò

)2 – ôñ

ù (ñ óò

)2 + ô

÷

ññóò ò ó

=

ññóò ò ó

=

ññóò ò ó

= (ñó ò )2 +

ô ƒï ÷ ñ ñóò

ññóò ò ó

= (ñó ò )2 +

ô

4 –2

+ õ sinh ú (

ï )ñ

+ õ sinh  (

ï )ñ ÷

+ õ sinh ú (

ï )ñ

2

õ ïƒú ñ

2

2

.

+

This is a special case of equation 2.9.3.8 with  ( ) = −  sinh ( Œ  ). 14.

15.

16.

17.

18.

ïƒú ñ

2

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  sinh  ( Œ  ). .

+ This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  sinh ( Œ  ). sinh ÷ (

ï ) ñ ñóò

+

.

This is a special case of equation 2.9.3.7 with  ( ) = −  sinh  ( Œ  ) and ( ) = − ' + .

ï 2 ñ óò ò ó

+

ï ñ óò

=

ô

sinh ÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  sinh  ( Œ (ô

ï

2

+ õ )ñó ò ò ó +

ô ï ñóò

+ sinh÷ ( ƒñ ) + ö = 0.

This is a special case of equation 2.9.2.24 with  ( ) = sinh  ( Œ

)+  . )+( .

Page 446

447

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.8.4-2. Equations with hyperbolic cosine. 19. 20. 21. 22.

23. 24. 25. 26. 27. 28. 29. 30.

= 2 ñ + ô (cosh ï )–÷ –3 ñ ÷ . This is a special case of equation 2.9.1.35 with  (ˆ ) = ˆ  .

ñóò ò ó

= õ cosh( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  cosh( Œ  ).

ñóò ò ó

=  cosh÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ cosh  ¨ + ý and ( = 0.

ñóò ò ó

= ô (ñ + õ cosh ï ) ÷ – õ cosh ï + ö . The substitution ¨ = +  cosh  leads to an autonomous equation of the form 2.9.1.1: ¨2

 =  ¨  + ( .

ñóò ò ó

+ 3ññó ò + ñ 3 + ô cosh( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  cosh( Œ  ).

ñóò ò ó

+ ôñ ÷ (ñó ò )2 + õ coshú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñóò ò ó

 and ( ) =  cosh+

+ ô cosh÷ ñ (ñó ò )2 + õÿñ ú + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  cosh 

ñóò ò ó

+ ô cosh÷ ñ (ñó ò )2 + õ coshú ( ƒñ ) + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  cosh 

ñóò ò ó

= ôñ ÷ (ñó ò )2 + õ coshú ñ (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) = 

ñóò ò ó

= ô cosh÷ ñ (ñó ò )2 + õÿñ ú (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) =  cosh 

= ùñó ò + ô ï 2÷ +1 coshú ( ƒñ ). + This is a special case of equation 2.9.2.20 with  ( ) =  cosh ( Œ

ï ñóò ò ó

and ( ) =  + + ( .

+

and ( ) =  cosh ( Œ

 and ( ) =  cosh+

ñóò ò ó

.

).

2ññó ò ò ó = (ñó ò )2 + õ coshú ( ï ) ñ 2 – ô . + This is a special case of equation 2.9.3.5 with  ( ) = −  cosh ( Œ  ). = ù (ñó ò )2 – ôñ 4÷ –2 + õ coshú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.8 with  ( ) = −  cosh ( Œ  ).

32.

ññóò ò ó

34.

)+( .

and ( ) =  + .

31.

33.

+( .

ññóò ò ó

=

ù (ñ óò

)2 + ô

ïƒú ñ

2

+ õ cosh  (

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  cosh  ( Œ  ).

= (ñó ò )2 + ô ïƒ÷ ññó ò + õ coshú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  cosh ( Œ  ).

ññóò ò ó

= (ñó ò )2 + ô cosh÷ ( ï ) ññó ò + õ ïƒú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −  cosh  ( Œ  ) and ( ) = − ' + .

ññóò ò ó

Page 447

448 35. 36.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ï ñó ò = ô cosh÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.23 with  ( ) =  cosh  ( Œ

ï 2 ñ óò ò ó

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + cosh÷ ( ƒñ ) + ö = 0. This is a special case of equation 2.9.2.24 with  ( ) = cosh  ( Œ

)+ . )+( .

2.8.4-3. Equations with hyperbolic tangent. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

= õ tanh( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  tanh( Œ  ).

ñ óò ò ó

=  tanh÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ tanh  ¨ + ý and ( = 0.

ñ óò ò ó

+ 3ññ ó ò + ñ 3 + ô tanh( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  tanh( Œ  ).

ñ óò ò ó

+ ôñ ÷ (ñó ò )2 + õ tanhú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñóò ò ó

 and ( ) =  tanh+

+ ô tanh÷ ñ (ñó ò )2 + õÿñ ú + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  tanh 

ñóò ò ó

+ ô tanh÷ ñ (ñó ò )2 + õ tanhú ( ƒñ ) + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  tanh 

ñóò ò ó

= ôñ ÷ (ñó ò )2 + õ tanhú ñ (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) = 

ñóò ò ó

= ô tanh÷ ñ (ñó ò )2 + õÿñ ú (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) =  tanh 

= ùñó ò + ô ï 2÷ +1 tanhú ( ƒñ ). + This is a special case of equation 2.9.2.20 with  ( ) =  tanh ( Œ

ï ñóò ò ó

and ( ) =  + + ( .

+

and ( ) =  tanh ( Œ

 and ( ) =  tanh+

ñóò ò ó

.

).

2ññó ò ò ó = (ñó ò )2 + õ tanhú ( ï ) ñ 2 – ô . + This is a special case of equation 2.9.3.5 with  ( ) = −  tanh ( Œ  ). = ù (ñó ò )2 – ôñ 4÷ –2 + õ tanhú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.8 with  ( ) = −  tanh ( Œ  ).

48.

ññ óò ò ó

50.

)+( .

and ( ) =  + .

47.

49.

+( .

ññóò ò ó

=

ù (ñ ó ò

)2 + ô

ï ú ñ

2

+ õ tanh  (

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  tanh  ( Œ  ).

= (ñ ó ò )2 + ô ï ÷ ññ ó ò + õ tanhú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  tanh ( Œ  ).

ññ óò ò ó

= (ñ ó ò )2 + ô tanh÷ ( ï ) ññ ó ò + õ ï ú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −  tanh  ( Œ  ) and ( ) = − ' + .

ññ óò ò ó

Page 448

449

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

51. 52.

+ ï ñó ò = ô tanh÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.23 with  ( ) =  tanh  ( Œ

ï 2 ñ óò ò ó

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + tanh÷ ( ƒñ ) + ö = 0. This is a special case of equation 2.9.2.24 with  ( ) = tanh  ( Œ

)+ . )+( .

2.8.4-4. Equations with hyperbolic cotangent. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.

= õ coth( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  coth( Œ  ).

ñ óò ò ó

=  coth÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ coth  ¨ + ý and ( = 0.

ñ óò ò ó

+ 3ññ ó ò + ñ 3 + ô coth( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  coth( Œ  ).

ñ óò ò ó

+ ôñ ÷ (ñó ò )2 + õ cothú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñóò ò ó

 and ( ) =  coth+

+ ô coth÷ ñ (ñó ò )2 + õÿñ ú + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  coth 

ñóò ò ó

+ ô coth÷ ñ (ñó ò )2 + õ cothú ( ƒñ ) + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  coth 

ñóò ò ó

= ôñ ÷ (ñó ò )2 + õ cothú ñ (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) = 

ñóò ò ó

= ô coth÷ ñ (ñó ò )2 + õÿñ ú (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) =  coth 

= ùñó ò + ô ï 2÷ +1 cothú ( ƒñ ). + This is a special case of equation 2.9.2.20 with  ( ) =  coth ( Œ

ï ñóò ò ó

and ( ) =  + + ( .

+

and ( ) =  coth ( Œ

 and ( ) =  coth+

ñóò ò ó

.

).

2ññó ò ò ó = (ñó ò )2 + õ cothú ( ï ) ñ 2 – ô . + This is a special case of equation 2.9.3.5 with  ( ) = −  coth ( Œ  ). = ù (ñó ò )2 – ôñ 4÷ –2 + õ cothú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.8 with  ( ) = −  coth ( Œ  ).

64.

ññ óò ò ó

66.

)+( .

and ( ) =  + .

63.

65.

+( .

ññóò ò ó

=

ù (ñ ó ò

)2 + ô

ï ú ñ

2

+ õ coth (

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  coth  ( Œ  ).

= (ñ ó ò )2 + ô ï ÷ ññ ó ò + õ cothú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  coth ( Œ  ).

ññ óò ò ó

= (ñ ó ò )2 + ô coth÷ ( ï ) ññ ó ò + õ ï ú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −  coth  ( Œ  ) and ( ) = − ' + .

ññ óò ò ó

Page 449

450 67. 68.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ï ñó ò = ô coth÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.23 with  ( ) =  coth  ( Œ

ï 2 ñ óò ò ó

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + coth÷ ( ƒñ ) + ö = 0. This is a special case of equation 2.9.2.24 with  ( ) = coth  ( Œ

)+ . )+( .

2.8.4-5. Equations containing combinations of hyperbolic functions. 69.

70.

ñóò ò ó

=

72.

73. 74. 75. 76.

) cosh–÷

ï

ú

–

–3

The transformation ˆ = tanh( Œ  ), ¨

¨2Šª

Š = Œ

−2

ñóò ò ó

2

=

ñ

−2

(

ï )ñ ú

.

=

cosh( Œ  ) ˆ  ¨2+ , which is discussed in Section 2.3. + ô cosh÷ (

ï

) sinh–÷

–

ú

The transformation ˆ = coth( Œ  ), ¨

¨2Šª

Š = Œ

71.

+ ô sinh÷ (

ñ

2

–3

( =

ï )ñ ú

leads to the Emden–Fowler equation

.

sinh( Œ  ) ˆ  ¨2+ , which is discussed in Section 2.3.

leads to the Emden–Fowler equation

= ô (ñó ò sinh ï – ñ cosh ï )  + ñ . The substitution ¨ = 
sinh  − cosh  leads to a first-order separable equation: ¨ 
=  sinh  ¨  .

ñóò ò ó

= ô (ñó ò cosh ï – ñ sinh ï )  + ñ . ¨ 
= The substitution ¨ = 
cosh  − sinh  leads to a first-order separable equation: p ¨  cosh   .

ñóò ò ó

sinh ï ñ„ó ò ò ó + 12 cosh ï ñó ò = ô sinh( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sinh  and  ( ) =  sinh( Œ

cosh ï ñó ò ò ó + 21 sinh ï ñó ò = ô cosh( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = cosh  and  ( ) =  cosh( Œ

)+ . )+ .

+ (ñó ò )2 + ô sinh( ï )ññ„ó ò + õ cosh( ï ) + ö = 0. This is a special case of equation 2.9.3.6 with  ( ) =  sinh(ý± ) and ( ) =  cosh( Œ  ) + ( .

ññóò ò ó

– (ñó ò )2 + ô sinh( ï )ññ„ó ò + õ cosh( ï )ñ 2 = 0. This is a special case of equation 2.9.3.7 with  ( ) =  sinh(ý± ) and ( ) =  cosh( Œ  ).

ññóò ò ó

2.8.5. Equations Containing Logarithmic Functions 2.8.5-1. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0. 1. 2. 3.

= õÿñ ( ô ï + þ ln ñ ). This is a special case of equation 2.9.1.27 with  ( H ) =  ln H .

ñ óò ò ó

= õ ï –2 ( ôñ + ù ln ï ). This is a special case of equation 2.9.1.28 with  ( H ) =  ln H .

ñóò ò ó

= ô ï –3 (ln ñ – ln ï ). This is a special case of equation 2.9.1.8 with  (ˆ ) =  ln ˆ .

ñ óò ò ó

Page 450

451

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

4.

ñóò ò ó

=

ñóò ò ó

=

ñ óò ò ó

=

– ln ï ).

ô ï –3 ) 2(2 ln ñ

This is a special case of equation 2.9.1.9 with  (ˆ ) = 2  ln ˆ . 5. 6. 7. 8.

ln÷ ( ôñ +

ü

õï

) + s.

This is a special case of equation 2.9.1.4 with  (¨ ) = , ln  ¨ + ° and ( = 0.

ñ

–3

ï

V 2 ln ñ – ln(ô

2

+ ö )W .

This is a special case of equation 2.9.1.21 with  (¨ ) = 2 ln ¨ and  = 0.

ï 2 ñ óò ò ó

=

ï 2 (ñ

+

ln ï + õ )÷ + ô .

ï 2 ñ óò ò ó

=

ù (ù

+ 1)ñ + ô

ï 2 ñ óò ò ó

+

ô

This is a special case of equation 2.9.1.36 with  (ˆ ) = ˆ  .

ï 3÷

(ln ñ + ù ln ï ).

+2

This is a special case of equation 2.9.1.12 with  (ˆ ) =  ln ˆ . 9.

10.

1 4

ñ

+

ú


ï

( ô ln ï + õ )÷

1– 2

ñ ú

= 0.

The transformation ˆ =  ln  +  , ¨ =  ¨2Šª

Š + #2 −2 ˆ  ¨2+ = 0 (see Section 2.3).

ï ñóò ò ó

=

ùñóò

ï 2÷

+ô

+1

J

−1 2

leads to the Emden–Fowler equation:

lnú ( ƒñ ).

+

This is a special case of equation 2.9.2.20 with  ( ) =  ln ( Π11.

ï ÷

ï ñ óò ò ó

= –(ù + 1)ñ

ï ñóò ò ó

= ( ôñ + ù ln ï )ñ„ó ò .

ï ñóò ò ó

+ ï (2 ôñ + ln ï + õ )ñ„ó ò + ñ = 0.

ï ñ óò ò ó

=

óò

+ô

–1

).

(ln ñ + ù ln ï ).

This is a special case of equation 2.9.2.30 with  (ˆ ) =  ln ˆ . 12.

This is a special case of equation 2.9.2.39 with  (ˆ ) = ln ˆ . 13. 14.

15. 16.

Integrating yields a Riccati equation: 
 +  ln  ( õÿñ )ñ

ô

óò

2

+ (ln  +  ) = L .

.

This is a special case of equation 2.9.2.21 with  ( ) =  ln  (  ).

ï 2 ñ óò ò ó

+

ï ñóò

=

ï

+ õ )ñ

óò ò ó

+

ô

ln ÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  ln  ( Œ (ô

2

ô ï ñ óò

+ ö ln ÷ ( ƒñ ) = 0.

This is a special case of equation 2.9.2.24 with  ( ) = ( ln  ( Œ

)+ . ).

2.8.5-2. Other equations. 17.

ñóò ò ó

+

ñ óò ò ó

=

ôñ ÷ (ñ óò

)2 + õ lnú

ôñ ÷ (ñ óò

)2 + õ lnú (

ñ

+

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  18.

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) = 

 and ( ) =  ln+

+( .

 and ( ) =  ln+ ( Π ).

Page 451

452 19. 20. 21. 22.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

+

ñ óò ò ó

=

ñóò ò ó

=

ñóò ò ó

=

ln ÷

ñ (ñ óò

)2 +

õÿñ ú

ô

ln÷

ñ (ñ ó ò

)2 +

õ ï ú ñ óò

ô

ln÷

ñ (ñ óò

)2 + õ lnú (

ô

+

ö

= 0.

and ( ) =  + + ( .

This is a special case of equation 2.9.3.25 with  ( ) =  ln  .

and ( ) = ' + .

This is a special case of equation 2.9.3.38 with  ( ) =  ln 

ï ) ñ óò

.

+

This is a special case of equation 2.9.3.38 with  ( ) =  ln 

ô ï –2 ñ

–1

ï

(ñó ò )4 – 2 ô

–3

and ( ) =  ln ( Π ).

ln ñ (ñó ò )3 .

Solution in parametric form: 1J 2  = ΠV ( wA 2U )2 A 4 ln( L 1  )W ,

= exp(T"U 2 ) X exp(T"U 2 ) U + L

where  23.

ñóò ò ó

Q

= 2 ô ln ï

ñ

–3

ï –1 ñ

–ô

M

–2

−1

S

2

, Π= A

1 2

=L

 ,

1

 J

2 1 4 . 1

L

(ñó ò )–1 .

Solution in parametric form: = ΠV ( wA 2U )2 A 4 ln( L

 = L 1 ,

= exp(T"U 2 ) X exp(T"U 2 ) U + L

where  24.

ñóò ò ó

ôñ ÷ (ñ óò

=

Q

)2 + õ lnú

ñ (ñ óò )

M

2

S

−1

, Π= A

1 2

.

This is a special case of equation 2.9.4.13 with  ( ) =  25. 26.

27.

ô

ln÷

ññóò ò ó

=

ù (ñ óò

)2 + ô

ññ óò ò ó

= (ñ

óò

)2 +

ô

óò

)2 +

ô ï ÷ ñ ñ óò

ñóò ò ó

=

ñ (ñ óò

õÿñ ú (ñ óò )

)2 +

1

2

ï )ñ ÷

+1

,

 J

 and ( ) =  ln+

.

+ õ ln  (

J

1 2

2 1 4 . 1

L

.

and ( ) =  + .

This is a special case of equation 2.9.4.13 with  ( ) =  ln 

ïƒú ñ

 )W

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  ln  ( Œ  ). ln ÷ (

ï ) ñ ñ óò

+

õï ú ñ

2

.

This is a special case of equation 2.9.3.7 with  ( ) = −  ln  ( Œ  ) and ( ) = − ' + . + õ lnú (

ï )ñ

28.

ññ óò ò ó

= (ñ

. + This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  ln ( Œ  ).

29.

ññóò ò ó

= (ô

ï

2

+ ù ln ñ )(ñ„ó ò )2 .

This is a special case of equation 2.9.3.40 with  (ˆ ) = ln ˆ .

2.8.6. Equations Containing Trigonometric Functions 2.8.6-1. Equations with sine. 1.

ñóò ò ó

=–

2

ñ

+ ô (sin

ï )÷ ñ –÷

–3

. This is a special case of equation 2.9.1.40 with  (ˆ ) = ˆ − 

−3

.

Page 452

453

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.

3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

+ ô ) sin ú ( ï + õ ) ñ –÷ –ú –3 . sin( Œ  +  ) ¨ The transformation ˆ = , = leads to the Emden–Fowler equation: sin( Œ  +  ) sin( Œ  +  ) ¨2Šª

Š = # [ Œ sin(  −  )]−2 ˆ  ¨ −  − + −3 (see Section 2.3).

ñóò ò ó

=–

2

ñ

+




sin÷ (

ï

= õ sin( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  sin( Œ  ).

ñóò ò ó

=  sin÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ sin  ¨ + ý and ( = 0.

ñóò ò ó

= ô (ñ + õ sin ï ) ÷ + õ sin ï + ö . The substitution ¨ = +  sin  leads to an autonomous equation of the form 2.9.1.1: ¨2

 =  ¨  + ( .

ñóò ò ó

+ 3ññó ò + ñ 3 + ô sin( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  sin( Œ  ).

ñóò ò ó

+ ôñ ÷ (ñ ó ò )2 + õ sin ú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñ óò ò ó

= ôñ ÷ (ñó ò )2 + õ sinú ( ï ) ñó ò . This is a special case of equation 2.9.3.38 with  ( ) = 

ñóò ò ó

= ôñ ÷ (ñó ò )2 + õ sinú ñ (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) = 

ñóò ò ó

 and ( ) =  sin+

 and ( ) =  sin + ( Œ  ).  and ( ) =  sin+­ .

+ ô sin ÷ ñ (ñó ò )2 + õÿñ ú + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  sin 

ñóò ò ó

+ ô sin ÷ ñ (ñó ò )2 + õ sin ú ( ƒñ ) + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  sin 

ñóò ò ó

= ô sin ÷ ñ (ñó ò )2 + õ ïƒú ñó ò . This is a special case of equation 2.9.3.38 with  ( ) =  sin 

ñóò ò ó

= ô sin ÷ ñ (ñó ò )2 + õ sinú ( ï ) ñó ò . This is a special case of equation 2.9.3.38 with  ( ) =  sin 

ñóò ò ó

= ô sin ÷ ñ (ñ ó ò )2 + õÿñ ú (ñ ó ò )  . This is a special case of equation 2.9.4.13 with  ( ) =  sin 

ñ óò ò ó

= ùñó ò + ô ï 2÷ +1 sinú ( ƒñ ). This is a special case of equation 2.9.2.20 with  ( ) =  sin + ( Œ

ï ñóò ò ó

+( .

and ( ) =  + + ( . and ( ) =  sin + ( Œ

)+( .

and ( ) = ' + . and ( ) =  sin + ( Π ). and ( ) =  + . ).

2ññ ó ò ò ó = (ñ ó ò )2 + õ sin( ï ) ñ 2 – ô . This is a special case of equation 2.9.3.5 with  ( ) = −  sin( Œ  ). = ù (ñó ò )2 – ôñ 4÷ –2 + õ sin( ï ) ñ 2 . This is a special case of equation 2.9.3.8 with  ( ) = −  sin( Œ  ).

ññóò ò ó

Page 453

454 18. 19. 20. 21. 22. 23. 24.

SECOND-ORDER DIFFERENTIAL EQUATIONS = ù (ñ ó ò )2 + ô ï ú ñ 2 + õ sin  ( ï ) ñ ÷ +1 . This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  sin  ( Œ  ).

ññ óò ò ó

= ù (ñó ò )2 + ô sin( ï ) ñ 2 + õ sin( ï ) ñ ÷ +1 . This is a special case of equation 2.9.3.9 with  ( ) = −  sin( Œ  ) and ( ) = −  sin(“/ ).

ññóò ò ó

= (ñó ò )2 + ô ïƒ÷ ññó ò + õ sin ú ( ï ) ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  sin + ( Œ  ).

ññóò ò ó

= (ñó ò )2 + ô sin ÷ ( ï ) ññó ò + õ ïƒú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −  sin  ( Œ  ) and ( ) = − ' + .

ññóò ò ó

ï 2 ñ óò ò ó

+ ï ñó ò = ô sin ÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.23 with  ( ) =  sin  ( Œ

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + sin÷ ( ƒñ ) + ö = 0. This is a special case of equation 2.9.2.24 with  ( ) = sin  ( Œ

sin2 ï

)+ . )+( .

= ù (ù + 1 – ù sin2 ï )ñ + ô (sin ï ) ÷¿ú +3÷ +2 ñ ú . This is a special case of equation 2.9.1.44 with  (ˆ ) = ˆ + .

ñóò ò ó

2.8.6-2. Equations with cosine. 25. 26.

27. 28. 29.

30. 31. 32.

= – 2 ñ + ô (cos ï )÷ ñ –÷ –3 . This is a special case of equation 2.9.1.41 with  (ˆ ) = ˆ − 

ñóò ò ó

.

+ ô ) cosú ( ï + õ ) ñ –÷ –ú –3 . cos( Œ  +  ) ¨ The transformation ˆ = , = leads to the Emden–Fowler equation: cos( Œ  +  ) cos( Œ  +  ) ¨ Šª

Š = # [ Œ sin(  −  )]−2 ˆ  ¨ −  − + −3 (see Section 2.3).

ñóò ò ó

=–

2

ñ

+




cos÷ (

−3

ï

= õ cos( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  cos( Œ  ).

ñ óò ò ó

=  cos÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ cos  ¨ + ý and ( = 0.

ñ óò ò ó

= ô (ñ + õ cos ï ) ÷ + õ cos ï + ö . The substitution ¨ = +  cos  leads to an autonomous equation of the form 2.9.1.1: ¨2

 =  ¨  + ( .

ñóò ò ó

+ 3ññ ó ò + ñ 3 + ô cos( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  cos( Œ  ).

ñ óò ò ó

+ ôñ ÷ (ñ ó ò )2 + õ cosú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñ óò ò ó

= ôñ ÷ (ñ ó ò )2 + õ cosú ( ï ) ñ ó ò . This is a special case of equation 2.9.3.38 with  ( ) = 

ñ óò ò ó

 and ( ) =  cos+­ + ( .  and ( ) =  cos + ( Œ  ).

Page 454

455

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

33.

ôñ ÷ (ñ óò

)2 + õ cosú

ñ (ñ ó ò ) 

ñ óò ò ó

=

ñ óò ò ó

+

ô

cos÷

ñ (ñ ó ò

)2 +

ñ óò ò ó

+

ô

cos÷

ñ (ñ ó ò

)2 + õ cosú ( ƒñ ) + ö = 0.

ñ óò ò ó

=

ô

cos÷

ñ (ñ ó ò

)2 +

ñ óò ò ó

=

ô

cos÷

ñ (ñ ó ò

)2 + õ cosú (

ñ óò ò ó

=

ô

cos÷

ñ (ñ ó ò

)2 +

=

ùñ óò

+ô

.

This is a special case of equation 2.9.4.13 with  ( ) =  34.

35.

36.

37.

38.

39.

40.

õÿñ ú

ö

+

 and ( ) =  cos+­ .

= 0.

and ( ) =  + + ( .

This is a special case of equation 2.9.3.25 with  ( ) =  cos 

and ( ) =  cos + ( Œ

This is a special case of equation 2.9.3.25 with  ( ) =  cos 

õ ï ú ñ óò

.

and ( ) = ' + .

This is a special case of equation 2.9.3.38 with  ( ) =  cos 

ï ) ñ óò

.

and ( ) =  cos + ( Π ).

This is a special case of equation 2.9.3.38 with  ( ) =  cos 

õÿñ ú (ñ óò )

.

and ( ) =  + .

This is a special case of equation 2.9.4.13 with  ( ) =  cos 

ï ñ óò ò ó

ï 2÷

+1

cosú ( ƒñ ).

This is a special case of equation 2.9.2.20 with  ( ) =  cos + ( Œ 2ññ

óò ò ó

= (ñ

óò

)+( .

ï )ñ

)2 + õ cos(

2

).

–ô .

This is a special case of equation 2.9.3.5 with  ( ) = −  cos( Œ  ).

41.

ññ óò ò ó

=

ññóò ò ó

÷

ù (ñ ó ò

)2 – ôñ

=

ù (ñ óò

)2 + ô

ññóò ò ó

=

ù (ñ óò

)2 + ô cos(

ññóò ò ó

= (ñó ò )2 +

ô ƒï ÷ ñ ñóò

ññóò ò ó

= (ñó ò )2 +

ô

4 –2

+ õ cos(

ï )ñ

2

.

This is a special case of equation 2.9.3.8 with  ( ) = −  cos( Œ  ). 42.

43.

ïƒú ñ

2

+ õ cos  (

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  cos  ( Œ  ).

ï )ñ

2

+ õ cos(

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  cos( Œ  ) and ( ) = −  cos(“/ ). 44.

45.

46.

47.

48.

+ õ cosú (

ï )ñ

2

õ ïƒú ñ

2

.

This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  cos + ( Œ  ). cos÷ (

ï ) ñ ñóò

+

.

This is a special case of equation 2.9.3.7 with  ( ) = −  cos  ( Œ  ) and ( ) = − ' + .

ï 2 ñ óò ò ó

+

ï ñóò

=

ô

cos÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  cos  ( Œ (ô

ï

2

+ õ )ñó ò ò ó +

ô ï ñóò

+ cos÷ ( ƒñ ) + ö = 0.

This is a special case of equation 2.9.2.24 with  ( ) = cos  ( Œ

cos2 ï

ñóò ò ó

=

ù (ù

+ 1 – ù cos2 ï )ñ + ô (cos ï )÷¿ú

÷ ñ ú

+3 +2

 ˆ + . This is a special case of equation 2.9.1.45 with  (ˆ ) = 

)+ . )+( .

.

Page 455

456

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.8.6-3. Equations with tangent. 49.

ñóò ò ó

õ

=

ï )ñ

tan(

+ ôñ

–3

.

This is a special case of equation 2.9.1.2 with  ( ) = −  tan( Œ  ). 50.

51.

tan÷ ( ôñ +

ñóò ò ó

=



ñ óò ò ó

+

ôñ ÷ (ñ óò

õï

)+ .

This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ tan  ¨ + ý and ( = 0. )2 + õ tanú

ñ

+

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  52.

ñóò ò ó

ôñ ÷ (ñ óò

=

)2 + õ tanú (

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  53.

ñ óò ò ó

ôñ ÷ (ñ óò

=

)2 + õ tanú

ñ (ñ ó ò ) 

.

This is a special case of equation 2.9.4.13 with  ( ) =  54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

õÿñ ú

ñóò ò ó

+

ô

tan÷

ñ (ñ óò

)2 +

ñóò ò ó

+

ô

tan÷

ñ (ñ óò

)2 + õ tanú ( ƒñ ) +

+

ö

 and ( ) =  tan+

 and ( ) =  tan+ ( Π ).  and ( ) =  tan+

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  tan 

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  tan 

ñ óò ò ó

=

ô

tan÷

ñ (ñ ó ò

)2 +

ñóò ò ó

=

ô

tan÷

ñ (ñ óò

)2 + õ tanú (

ñ óò ò ó

=

ô

tan÷

ñ (ñ ó ò

)2 +

=

ùñóò

+ô

õ ï ú ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  tan 

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  tan 

õÿñ ú (ñ óò )

.

This is a special case of equation 2.9.4.13 with  ( ) =  tan 

ï ñóò ò ó

ï 2÷

+1

=

ù (ñ óò

ññ óò ò ó

= (ñ

ï 2 ñ óò ò ó

+

)2 + ô

.

and ( ) =  + + ( . and ( ) =  tan + ( Œ

)+( .

and ( ) = ' + . and ( ) =  tan+ ( Π ). and ( ) =  + .

tanú ( ƒñ ).

This is a special case of equation 2.9.2.20 with  ( ) =  tan + ( Œ

ññóò ò ó

+( .

ïƒú ñ

2

+ õ tan  (

ï )ñ ÷

+ õ tanú (

ï )ñ

+1

).

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  tan  ( Œ  ).

óò

)2 +

ô ï ÷ ñ ñ óò

2

.

This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  tan+ ( Œ  ).

ï ñóò

=

ô

tan÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  tan  ( Œ (ô

ï

2

+ õ )ñ

óò ò ó

+

ô ï ñ óò

+ tan÷ ( ƒñ ) +

ö

)+  .

= 0.

This is a special case of equation 2.9.2.24 with  ( ) = tan  ( Œ

)+( .

Page 456

457

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.8.6-4. Equations with cotangent. 64.

ñóò ò ó

õ

=

ï )ñ

cot(

+ ôñ

–3

.

This is a special case of equation 2.9.1.2 with  ( ) = −  cot( Œ  ). 65.

66.

cot÷ ( ôñ +

ñóò ò ó

=



ñ óò ò ó

+

ôñ ÷ (ñ óò

õï

)+ .

This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ cot  ¨ + ý and ( = 0. )2 + õ cotú

ñ

+ ö = 0.

This is a special case of equation 2.9.3.25 with  ( ) =  67.

ñóò ò ó

ôñ ÷ (ñ óò

=

)2 + õ cotú (

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  68.

ñ óò ò ó

ôñ ÷ (ñ óò

=

)2 + õ cotú

ñ (ñ ó ò ) 

.

This is a special case of equation 2.9.4.13 with  ( ) =  69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

õÿñ ú

ñóò ò ó

+

ô

cot÷

ñ (ñ óò

)2 +

ñóò ò ó

+

ô

cot÷

ñ (ñ óò

)2 + õ cotú ( ƒñ ) + ö = 0.

 and ( ) =  cot+­ + ( .  and ( ) =  cot+ ( Œ  ).  and ( ) =  cot+­ .

+ ö = 0.

This is a special case of equation 2.9.3.25 with  ( ) =  cot  This is a special case of equation 2.9.3.25 with  ( ) =  cot 

ñ óò ò ó

=

ô

cot÷

ñ (ñ ó ò

)2 +

ñóò ò ó

=

ô

cot÷

ñ (ñ óò

)2 + õ cotú (

ñ óò ò ó

=

ô

cot÷

ñ (ñ ó ò

)2 +

=

ùñóò

+ô

õ ï ú ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  cot 

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  cot 

õÿñ ú (ñ óò )

.

This is a special case of equation 2.9.4.13 with  ( ) =  cot 

ï ñóò ò ó

ï 2÷

+1

=

ù (ñ óò

ññ óò ò ó

= (ñ

ï 2 ñ óò ò ó

+

)2 + ô

and ( ) =  cot + ( Œ

)+( .

and ( ) = ' + . and ( ) =  cot + ( Π ). and ( ) =  + .

cotú ( ƒñ ).

This is a special case of equation 2.9.2.20 with  ( ) =  cot + ( Œ

ññóò ò ó

and ( ) =  + + ( .

ïƒú ñ

2

+ õ cot (

ï )ñ ÷

+ õ cotú (

ï )ñ

+1

).

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  cot  ( Œ  ).

óò

)2 +

ô ï ÷ ñ ñ óò

2

.

This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  cot + ( Œ  ).

ï ñóò

=

ô

cot÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  cot  ( Œ (ô

ï

2

+ õ )ñ

óò ò ó

+

ô ï ñ óò

+ cot÷ ( ƒñ ) +

ö

)+ .

= 0.

This is a special case of equation 2.9.2.24 with  ( ) = cot  ( Œ

)+( .

Page 457

458

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.8.6-5. Equations containing combinations of trigonometric functions. 79.

80.

81.

82.

83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93.

ñ óò ò ó

ñ

=–

2

¨2Šª

Š = Œ

−2

cos÷ (

ô

+

ï

) sinú (

ï ) ñ –÷ –ú

The transformation ˆ = cot( Œ  ), ¨

ñ óò ò ó

=–

2

ñ

ˆ  ¨

+

 +

ô

− −

−3

sin ÷ (

=

(see Section 2.3).

ï

)[sin(

ï

–3

.

sin( Π )

) + õ cos(

leads to the Emden–Fowler equation:

ï )]ú ñ –÷ –ú

–3

.

The transformation ˆ = 1 +  cot( Œ  ), ¨ = leads to the Emden–Fowler equation: sin( Œ  ) ¨2Šª

Š =  ( Œ )−2 ˆ +3¨ −  − + −3 (see Section 2.3). = ô (ñ ó ò sin ï – ñ cos ï )  – ñ . The substitution ¨ = 
sin  − cos   sin  ¨  .

ñ óò ò ó

= ô (ñ ó ò cos ï + ñ sin ï )  – ñ . The substitution ¨ = 
cos  + sin   cos  ¨  .

ñ óò ò ó

leads to a first-order separable equation: ¨ 
=

leads to a first-order separable equation: ¨ 
=

= 2(cos ï )–2 ñ + ô (cot ï )÷ +3 ñ ÷ . This is a special case of equation 2.9.1.43 with  (ˆ ) = ˆ  .

ñóò ò ó

= 2(sin ï )–2 ñ + ô (tan ï ) ÷ +3 ñ ÷ . This is a special case of equation 2.9.1.42 with  (ˆ ) = ˆ  .

ñóò ò ó

= (ù + 1)(tan ï )ñ ó ò + ùñ + ô (cos ï )÷¿ú –2 ñ ú –1 . This is a special case of equation 2.9.2.44 with  (ˆ ) = ˆ +

ñ óò ò ó

+ (2 ôñ + õ sin ï )ñ„ó ò + õ (cos ï )ñ = 0. Integrating yields a Riccati equation: 
 + 

ñóò ò ó

2

.

+  (sin  ) = L .

+ ô ï 2 tan ï ñ ó ò + ù ( ô ï tan ï – ù – 1)ñ = õ ï ÷¿ú This is a special case of equation 2.9.2.47 with  (ˆ ) = 'ˆ +

ï 2 ñ óò ò ó

−1

(cos ï )2 D −3 .

+2

ñ ú

sin ï ñó ò ò ó + 12 cos ï ñó ò = ôñ ÷ + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) = 

–3

.

 + .

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô sin ÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  sin  ( Œ

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô cos÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  cos  ( Œ sin ï ñó ò ò ó + 12 cos ï ñó ò = ô tan÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  tan  ( Œ

)+  . )+ . )+ .

+ (ñ ó ò )2 + ô sin( ï )ññ ó ò + õ cos( ï ) + ö = 0. This is a special case of equation 2.9.3.6 with  ( ) =  sin(ý± ) and ( ) =  cos( Œ  ) + ( .

ññ óò ò ó

– (ñó ò )2 + ô sin( ï )ññ„ó ò + õ cos( ï )ñ 2 = 0. This is a special case of equation 2.9.3.7 with  ( ) =  sin(ý± ) and ( ) =  cos( Œ  ).

ññóò ò ó

Page 458

459

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.8.7. Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

ó

= 2 ñ + ô 3  (ln ñ + ï ). This is a special case of equation 2.9.1.29 with  (ˆ ) =  ln ˆ .

ñóò ò ó

ó

= – ôñó ò + õD (ln ñ + ô ï ). This is a special case of equation 2.9.2.36 with  (ˆ ) =  ln ˆ .

ñóò ò ó

ó

= ôñó ò + õ 2 D ln÷ ( ƒñ ). This is a special case of equation 2.9.2.17 with  ( ) =  ln  ( Œ

ñóò ò ó

).

ó

= ôñó ò + õ 2 D sin ÷ ( ƒñ ). This is a special case of equation 2.9.2.17 with  ( ) =  sin  ( Œ

ñóò ò ó

).

ó

= ôñó ò + õ 2 D tan÷ ( ƒñ ). This is a special case of equation 2.9.2.17 with  ( ) =  tan  ( Œ

ñóò ò ó

ó

+ ô tan ï ñó ò + õ ( ô tan ï – õ )ñ = ö  ú (cos ï )2 D³ñ This is a special case of equation 2.9.2.46 with  (ˆ ) = (˜ˆ +

ñóò ò ó

ú

–3 −3

). .

.

= ô (ñó ò )2 – õ 4 Df] + ö sinh( ï ). This is a special case of equation 2.9.3.18 with  ( ) = − ( sinh( Œ  ).

ñóò ò ó

= ô (ñó ò )2 + õ cosh÷ ( ï ) Df] + ö ïƒú . This is a special case of equation 2.9.3.17 with  ( ) = −  cosh  ( Œ  ) and ( ) = − (˜ + .

ñóò ò ó

= ô (ñó ò )2 + õ ln÷ ( ï ) Df] + ö ïƒú . This is a special case of equation 2.9.3.17 with  ( ) = −  ln  ( Œ  ) and ( ) = − (˜ + .

ñóò ò ó

ó

= ô (ñó ò )2 + õ ln÷ ( ï ) Df] + ö . Á This is a special case of equation 2.9.3.17 with  ( ) = −  ln  ( Œ  ) and ( ) = − (  .

ñóò ò ó

= ô (ñó ò )2 – õ 4 Df] + ö sin( ï ). This is a special case of equation 2.9.3.18 with  ( ) = − ( sin( Œ  ).

ñóò ò ó

= ô (ñó ò )2 + õ sin÷ ( ï ) Df] + ö ïƒú . This is a special case of equation 2.9.3.17 with  ( ) = −  sin  ( Œ  ) and ( ) = − (˜ + .

ñóò ò ó

ó

= ô (ñó ò )2 + õ sin÷ ( ï ) Df] + ö . Á This is a special case of equation 2.9.3.17 with  ( ) = −  sin  ( Œ  ) and ( ) = − (  .

ñóò ò ó

+ ô] (ñó ò )2 + õ ln÷ ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =   and ( ) =  ln 

ñóò ò ó

+ ô] (ñó ò )2 + õ sin÷ ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =   and ( ) =  sin 

ñóò ò ó

+( . +( .

= ô ] (ñ ó ò )2 + õ ln ÷ ( ï )ñ ó ò . This is a special case of equation 2.9.3.38 with  ( ) =   and ( ) =  ln  (“/ ).

ñ óò ò ó

Page 459

460 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

=

ô ] (ñ óò

)2 + õ sin ÷ (

ï )ñ ó ò

.

ñóò ò ó

=

ô] (ñ óò

)2 + õ tan÷ (

ï )ñ óò

.

ñóò ò ó

+

ñ óò ò ó

=

ñóò ò ó

=

ñóò ò ó

+

ñ óò ò ó

=

ñóò ò ó

=

ñóò ò ó

=

ñ óò ò ó

+

This is a special case of equation 2.9.3.38 with  ( ) =   and ( ) =  sin  (“/ ). This is a special case of equation 2.9.3.38 with  ( ) =   and ( ) =  tan  (“/ ). ln ÷

ô

ñ (ñ óò

)2 +

ñ (ñ ó ò

)2 +

õ]

+

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  ln 

õ  ó ñ óò

ô

ln÷

ô

ln÷

ñ (ñ óò

)2 + õ sinú (

sin ÷

ñ (ñ óò

)2 +

ñ (ñ ó ò

)2 +

ñ (ñ óò

)2 + õ lnú (

ñ (ñ óò

)2 +

and ( ) =   + ( .

.



This is a special case of equation 2.9.3.38 with  ( ) =  ln 

ï )ñ óò

and ( ) =   .

.

and ( ) =  sin + ( Π ).

This is a special case of equation 2.9.3.38 with  ( ) =  ln 

ô

õ]

ö

+

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  sin 

ô

sin ÷

ô

sin ÷

ô

tan÷

õ  ó ñ óò

and ( ) =   + ( .

.



This is a special case of equation 2.9.3.38 with  ( ) =  sin 

ï )ñ óò

and ( ) =   .

.

+

This is a special case of equation 2.9.3.38 with  ( ) =  sin 

õ ó ñóò

.

)3 + ôñ

óò

and ( ) =  ln ( Π ).



This is a special case of equation 2.9.3.38 with  ( ) =  tan 

õ D ó

óò

cosh( ƒñ )(ñ

and ( ) =    .

= 0.

This is a special case of equation 2.9.3.35 with  ( ) =  cosh( Π27.

ñóò ò ó

+

õD ó

sin( ƒñ )(ñ„ó ò )3 +

ôñóò

= 0.

This is a special case of equation 2.9.3.35 with  ( ) =  sin( Π28.

ï ñóò ò ó

=

ô ï

Solution: 29.

ññóò ò ó

=

ln ï

]Èñóò

ô ó (ñ óò

)3 + ô

Solution:  = − ln 

Q

30.

ññóò ò ó

+

ô]

.

2

−

k  = − ln  1 O L Q

= (ñó ò )2 +

óñ

k 1  O L

ô ó ññóò

 L

1

X 

+

L



1

X

+ õ sin÷ ( b

k 1

−1 −



ln ñ (ñó ò )2 . 2

−1 −



ï )ñ

k 1

2



 + M P L

M P

−

L

( ô

2

( ô

2

ó

+ õ )ñó ò ò ó +



ln S .

1

.

ô 2ó ñ óò

= cosh÷ ( ƒñ ) + ö .

ô 2ó ñ óò

= tanh÷ ( ƒñ ) + ö .

This is a special case of equation 2.9.2.34 with ( ) =  

32.

ó

+ õ )ñ

óò ò ó

+

This is a special case of equation 2.9.2.34 with ( ) =  

).

ln  S .

1



This is a special case of equation 2.9.3.7 with  ( ) = −   31.

).

and ( ) = −  sin  ( @! ).

2



+  and  ( ) = cosh  ( Œ

)+( .

2



+  and  ( ) = tanh  ( Œ

)+( .

Page 460

461

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

33. 34. 35. 36.

ó

ó

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò = ln÷ ( ƒñ ) + ö .  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = ln  ( Œ

ó

ó

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò = sin÷ ( ƒñ ) + ö .  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = sin  ( Œ

ó

)+( . )+( .

ó

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò = tan÷ ( ƒñ ) + ö .  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = tan  ( Œ sin ï

ñóò ò ó

+

1 2

cos ï

ñóò

=

ô]

)+( .

+õ .

This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  ! +  .

37. 38. 39. 40.

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô cosh÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  cosh  ( Œ sin ï ñó ò ò ó + 12 cos ï ñó ò = ô sinh ÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  sinh  ( Œ

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô tanh÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  tanh  ( Œ

sin ï ñ ó ò ò ó + 12 cos ï ñ ó ò = ô ln ÷ ñ + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  ln 

)+ . )+ . )+ .

+ .

2.9. Equations Containing Arbitrary Functions ]

Notation:  , , % , Y , and j are arbitrary composite functions of their arguments indicated in parentheses just after the function name (the arguments can depend on  , , 
).

2.9.1. Equations of the Form — ( æ , â ) â$äãÑã ä + ˜ ( æ , â ) = 0 2.9.1-1. Arguments of arbitrary functions are algebraic and power functions of  and . 1.

ñóò ò ó

= ! (ñ ). This autonomous equation arises in mechanics, combustion theory, and the theory of mass transfer with chemical reactions. The substitution ‡ ( ) = "
 leads to a first-order separated equation: ‡"‡
=  ( ).



J

−1 2

Solution: X L 1 + 2 Xv ( ) S = L 2 AI . Q M M Particular solutions: = #  , where #  are roots of the algebraic (transcendental) equation  ( #  ) = 0.

2.

+ ! ( ï )ñ = ôñ –3 . Yermakov’s equation. Let ¨

ñ óò ò ó

= ¨ ( ) be a nontrivial solution of the second-order linear

¨ 

 +  ( )¨ = 0. The transformation ˆ = X equation 2

equation of the form 2.9.1.1: H Šª

Š = !H −3 . Solution: L 1 2 =  ¨ 2 + ¨ 2 O L 2 + L 1 X ^_ Reference: V. P. Yermakov (1880).

¨M



2 2

P

¨M



2

, H = ¨

leads to an autonomous

.

Page 461

462 3.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ! ( ï )ñ = "Ió ò ( ï )ñ –1 – " 2 ( ï )ñ –3 . Generalized Yermakov’s equation. ( )  1 J 2 ¨ Solution: = ¨ L +2X ¨ 2M S , where

ñóò ò ó

Q

¨ 

 −  ( )¨ = 0. linear equation 3 4. 5. 6. 7.

8. 9.

= ! ( ôñ + õ ï + ö ). ¨ 

 =  (¨ ). The substitution ¨ =  + ' + ( leads to an equation of the form 2.9.1.1: I

ñóò ò ó

= ! (ñ + ô ï 2 + õ ï + ö ). ¨ 

 =  (¨ )+2  . The substitution ¨ = +  2 + ' + ( leads to an equation of the form 2.9.1.1: p

ñóò ò ó

= ! (ñ + ô ï ÷ + õ ï 2 + ö ï ) – ôù (ù – 1) ï ÷ –2 . ¨ 

 =  (¨ )+2  . The substitution ¨ = +   + ' 2 + (˜ leads to an equation of the form 2.9.1.1: u

ñ óò ò ó

= ï –1 ! (ñ ï –1 ). Homogeneous equation. The transformation a = − ln | |, H =   leads to an autonomous equation: H g

g − H g
=  ( H ). Reducing its order with the substitution ¨ ( H ) = ¨pg
, we arrive at ‘ the Abel equation ¨*¨3
− ¨ =  (¨ ), which is discussed in Subsection 1.3.1.

ñóò ò ó

= ï –3 ! (ñ ï –1 ). ¨ ªŠ

Š =  (¨ ). The transformation ˆ = 1  , ¨ =   leads to an equation of the form 2.9.1.1: I

ñóò ò ó

= ï –3 ) 2 ! (ñ ï –1 ) 2 ). Having set ¨ =  −1 J 2 , we obtain x x  ( ¨2
)2 = equation, we arrive at a separable equation.

ñóò ò ó

Solution: X 10.

= ¨ ( ) is the general solution of the

Q

L

1

+ 14 ¨

2

+2X

 (¨ ) ¨ S M

= ï –2 ! ( ï –  ñ ). Generalized homogeneous equation.

ñóò ò ó

1 b . The 2.9.6.2:

J

−1 2

M

1 2

¨*¨2
+ 2  (¨ )¨2
. Integrating the latter

¨ =L

2

A ln  .

transformation a = ln  , H =  −  leads to an autonomous equation of the form H g

g + (2 , − 1)H g
+ , ( , − 1) H =  (H ).

2 b . The transformation H =  −  , ¨ =  
  leads to a first-order equation: H (¨ − , )¨u‘
= H −1  ( H ) + ¨ − ¨ 2 .

11.

12. 13. 14.

= ñ ï –2 ! ( ïƒ÷ ñ ú ). Generalized homogeneous equation. The transformation H =   + , ¨ ‘ first-order equation: H ( ¨ + B )¨u
=  ( H ) + ¨ − ¨ 2 .

ñóò ò ó

=  
  leads to a

¡

= ù (ù + 1) ï –2 ñ + ï 3÷ ! (ñ ïƒ÷ ). This is a special case of equation 2.9.1.46 with j =  −  .

ñóò ò ó

= ï –3 ) 2 ! ( ôñ ï –1 ) 2 + õ ï 1 ) 2 ). ¨ 

 =  The substitution ¨ =  + ' leads to an equation of the form 2.9.1.9: I

ñ óò ò ó ï (ï

+ ô )2 ñó ò ò ó = ! (ñ

ï

J  (¨ 

−3 2

J ).

−1 2

).

 + û , H =  leads to an autonomous equation of the form 2.9.6.2: û  û
H Šª
Š − H Š
=  −2  ( H ). Reducing û itsû order with the substitution ¨ ( H ) = H Š
, we arrive at an Abel * ¨ equation of the second kind: ¨u‘
− ¨ =  −2  ( H ). See Subsection 1.3.1 for information about The transformation ˆ = ln û

an Abel equation of the second kind.

Page 462

463

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

15.

16.

ñ óò ò ó

=

1

ï 3 !IO ï

ñ

ï

+

ô

ï

+

õ

+ö

.

P ¨ ªŠ

Š = The transformation ˆ = 1  , ¨ =   leads to an equation of the form 2.9.1.5: I  (¨ + ˆ 2 + 'ˆ + ( ). 2

ô ï  ï

+ õÿñ + ö . + õÿñ + ö + ñ + P This is a special case of equation 2.9.1.18.

ñóò ò ó

=

1

ô ï

!IO

1 b . For ý − ˜ƒ = 0, we have an equation of the form 2.9.1.4. 2 b . For ý − ˜ƒ ≠ 0, the transformation

¨ = − , 0

H =  −  0, where 

0

and

0

are the constants determined by the linear algebraic system of equations



0

+

0

+ ( = 0,

ƒ…

0

+ý

0

+ = = 0,

leads to a homogeneous equation of the form 2.9.1.7:

¨ ¨ '‘

‘ = 1  ,  O H HCP

17.

1  + 'ˆ . *O  + 'ˆ ƒ + ý±ˆÑP

 (ˆ ) =

where

ô ï  ï

+ õÿñ + ö . + ñ + P + õÿñ + ö ) This is a special case of equation 2.9.1.20.

ñ óò ò ó

=

1

(ô ï

!IO 3

1 b . For ý − ˜ƒ = 0, we have an equation of the form 2.9.1.4. 2 b . For ý − ˜ƒ ≠ 0, the transformation

¨ = − , 0

H =  −  0, where 

0

and

0

are the constants determined by the linear algebraic system of equations



0

+

0

+ ( = 0,

ƒ…

0

+ý

0

+ = = 0,

leads to a solvable equation of the form 2.9.1.8:

¨ ¨ '‘

‘ = 1  O , H 3 H P

18.

ñóò ò ó

=

ô 1ï

1

+

õ 1ñ

+ö

1

!IO

ô 2ï

ô 3ï

+ +

õ 2ñ

+ö +ö

õ 3ñ

Let the following condition be satisfied: For  2  3 −  3 

2

2 3



P

û û û û

û ≠ 0, the transformation H =  −  0,

where 

0

and

0

1  + 'ˆ  O . 3 (  + 'ˆ ) ƒ +± ý ˆ P

 (ˆ ) =

where .



1 2 3

(

1

 

1

( 2 ûû = 0. ( 3 ûû û

2 3

¨ = − , 0

are the constants determined by the linear algebraic system of equations

 2

0

+

2 0

+(

2

= 0,

 3

0

+

3 0

+(

3

= 0,

leads to a homogeneous equation of the form 2.9.1.7:

¨ ¨ '‘

‘ = 1   O , H HCP

where

 (ˆ ) =

1

 1 +  1ˆ

*O

 2 +  2ˆ .  3 +  3Ñ ˆ P

Page 463

464 19.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ô ï  ï

+ õÿñ + ö . + ñ + P + õÿñ + ö ) For ý − ˜ƒ ≠ 0, the transformation

ñ óò ò ó

=

1

ï 2(ô ï

!IO

¨ = − , 0

H =  −  0, where 

0

and

are the constants determined by the linear algebraic system of equations

0



+

0

0

+ ( = 0,

ƒ…

+ý

0

0

+ = = 0,

leads to an equation of the form 2.9.1.14:

¨

H ( H +  0 )2 ¨ '‘

‘ = O 20.

,

HCP

ô 2ï

ö

õ 3ñ

Let the following condition be satisfied:

û û

=

1

(ô 1 ï

+

õ 1ñ

ö

+

For  2  3 −  3 

2

1)

!IO 3

+ +

õ 2ñ

+ +

ñóò ò ó

ô 3ï



ö

û û

û ≠ 0, the transformation

2 3

P

2 3

0

and

1

 + 'ˆ

*O

 + 'ˆ . ƒ + ý±ˆÑP

.



1

(

1

 

1

( 2 ûû = 0. ( 3 ûû û

2 3

¨ = − , 0

H =  −  0, where 

 (ˆ ) =

where

are the constants determined by the linear algebraic system of equations

0

 2

0

+

2 0

+(

2

 3

= 0,

0

+

3 0

+(

(

1   O +  1 ˆ )3 

3

= 0,

leads to a solvable equation of the form 2.9.1.8:

¨ ¨ '‘

‘ = 1  O , 3 H H P 21.

22.

ñ óò ò ó

ñ –3 e ! ” 7

ï

2

+

õï

ô ï

2

+

( 

2


+ ' + ( )2 (‡  )2 = ( 14 

+ ö )3 )

Setting ¨ = ƒ

2 3

+  2ˆ . +  3ˆ P

(ô

ï ÷

+ õ )ñ

óò ò ó

2

ñóò ò ó

õï

=e ! ”

ñ 7

ô ï

+

ï

2 + õ ï

2

=

−3

YIO £ 

ôù (ù

– 1) ï

¨

2

+ ' + ( P

÷ –2 ñ

+ñ

–2

2

− !( )‡

+

.

! O ï ÷ ô

+ 2 Xȇ

−3

 (‡ ) ‡ + L 1 . M

.

+ö •

+ ý± + = and denoting  ( H ) =

2.9.1.21: ¨ 

 = ¨ 23.

ñ

1

. +ö • Setting ‡ ( ) = (  2 + '  + ( )−1 J 2 and integrating the equation, we obtain a first-order separable equation:

(ô

=

 (ˆ ) =

where

1

ƒhH

ñ

3

Y ( H ), we obtain an equation of the form

+õ P

.

 , ¨ = leads to an autonomous equation of the M  (   +  )2   +  3 ¨

¨ ¨ −2  ( ). form 2.9.1.1: ŠªŠ =

The transformation ˆ = X

Page 464

465

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

24.

ñóò ò ó

ï

= (ô

) ! (ñ 1 )

+ õ )–2 ñ

–1 2

2

öï

+

) + 2ö 2.

The solution is determined by the first-order equation


(  +  )(  + 2 ( £

) = Y ( (˜ + £

, L ),

(1)

where the function Y = (‡ , L ) is the general solution of the Abel equation of the second kind:

YsY
 − 2( ‡ + ˜( ) Y = 2  (‡ ). Abel equations of this type are discussed in Subsection 1.3.3. By the change of variable 1J 2 = ‡ − (˜ , equation (1) is reduced to the form 2(  +  )(‡ − (˜ )‡
 = Y (‡ , L ). 25.

ñóò ò ó

=

ï

(ö

+ ý )÷

(ô

–1

!e” ï ( ô ï + õ )÷ +2 (ö

ï

+ õ )÷

ñ

+ ý )÷

+1

•

.

 +  (  +  )  , ¨ = leads to an autonomous equation of (˜ + P ( (˜ + )  +1 M M

The transformation ˆ = ln O the form 2.9.6.2:

¨ Šª

Š − (2B + 1)¨ Š
+ B (B + 1)¨ = ¢

−2

 (¨ ),

where

¢ = M

− ˜( .

2.9.1-2. Arguments of the arbitrary functions are other functions. 26.

27.

ó

ó

ñ óò ò ó

=

 –D ! ( D ñ

ñóò ò ó

=

ñ! ( D ó ñ ú

).

 1 b . The substitution H =  

leads to an autonomous equation: H 

 − 2 !H 
+  2 H =  ( H ).  ‘ 2 b . The transformation H =  , ¨ = 
  leads to a first-order equation: H (¨ +  )¨u
= ¨ −1 2 H  (H ) − . ).

Equation invariant under “translation–dilatation” transformation. The transformation  H =   + , ¨ = 
  leads to a first-order equation: H ( ¨ +  )¨u‘
=  ( H ) − ¨ 2 . 28.

¡

ñóò ò ó = ï –2 ! (ƒï ÷  Df]

).

Equation invariant under “dilatation–translation” transformation. The transformation H =      , ¨ =  
leads to a first-order equation: H (  ¨ + B )¨ ‘
=  ( H ) + ¨ . 29.

ñóò ò ó

=

ñóò ò ó

= ! (ñ + ô

2

ñ

+

3

ó

 ! (ñ 

ó

).



This is a special case of equation 2.9.1.46 with j =  −  . 30. 31.

33.

+ õ ) – ô

2



ó

.

 ¨ 

 =  (¨ ). The substitution ¨ = +  ž +  leads to an equation of the form 2.9.1.1: u

ï 2 ñ óò ò ó

=

ï 2 ! (ƒï ÷  ]

The substitution 32.

ó

)+ù . ¨ 

 =  (  = ¨ − B ln  leads to an equation of the form 2.9.1.1: I

ñóò ò ó

= ! (ñ + ô sinh ï + õ ) – ô sinh ï .

ñóò ò ó

= ! (ñ + ô cosh ï + õ ) – ô cosh ï .

The substitution ¨ = The substitution ¨ =

?

).

¨ 

 =  (¨ ). +  sinh  +  leads to an equation of the form 2.9.1.1: I

¨ 

 =  (¨ ). +  cosh  +  leads to an equation of the form 2.9.1.1: I

Page 465

466

SECOND-ORDER DIFFERENTIAL EQUATIONS =

. sinh ï P This is a special case of equation 2.9.1.46 with j = sinh Œ  .

35.

ñ óò ò ó

=

36.

+ (sinh

ñ

+ (cosh

ñ

ñóò ò ó

2

ñ

ï

34.

)–3 !IO

ï

ñ

)–3 ! O

. cosh ï P This is a special case of equation 2.9.1.46 with j = cosh Œ  .

ï 2 ñ óò ò ó

2

ï 2 ! (ñ

=

+ ô ln ï + õ ) + ô .

The substitution ¨ =

¨ 

 =  (¨ ). +  ln  +  leads to an equation of the form 2.9.1.1: I

37.

ñóò ò ó

= –ï

1 ñ + !IO ï . ln ï (ln ï )3 ln P This is a special case of equation 2.9.1.46 with j = ln  .

38.

ñóò ò ó

= ! (ñ + ô sin ï + õ ) + ô sin ï .

ñóò ò ó

= ! (ñ + ô cos ï + õ ) + ô cos ï .

39.

ñ

2

The substitution ¨ =

¨ 

 =  (¨ ). +  sin  +  leads to an equation of the form 2.9.1.1: I

The substitution ¨ = + (sin

ñ

+ (cos

ï

ñ

40.

ñóò ò ó

=–

. sin ï P This is a special case of equation 2.9.1.46 with j = sin Œ  .

41.

ñ óò ò ó

=–

. cos ï P This is a special case of equation 2.9.1.46 with j = cos Œ  .

42.

ñ óò ò ó

= 2(sin ï )–2 ñ + (tan ï )3 ! (ñ tan ï ).

ñóò ò ó

= 2(cos ï )–2 ñ + (cot ï )3 ! (ñ cot ï ).

2

ñ

¨ 

 =  (¨ ). +  cos  +  leads to an equation of the form 2.9.1.1: I

2

)–3 !IO

ï

ñ

)–3 ! O

This is a special case of equation 2.9.1.46 with j = cot  . 43.

This is a special case of equation 2.9.1.46 with j = tan  .

44.

sin2 ï

ñóò ò ó

=

+ 1 – ù sin2 ï )ñ + sin3÷

ù (ù

+ z2

The substitution  = ˆ cos2 ï

ñóò ò ó

=

The transformation ˆ = X cos2    , ¨ M form 2.9.1.1: ¨3Šª

Š =  (¨ ). 46.

ñóò ò ó

=

s s

óò ò ó ñ

+

–3

s

!IO

The transformation ˆ = X Solution: X

Q

L

1

sin ÷

ï

).

leads to an equation of the form 2.9.1.45:

+ 1 – ù cos2 ï )ñ + cos3÷

ù (ù

ï ! (ñ

Šª

Š = B (B + 1 − B cos2 ˆ ) + cos3 

cos2 ˆ 45.

+2

ñ s

P

+2X

j

M

,



2

, ¨ =

 (¨ ) ¨ S M

j

ï ! (ñ

cos÷

ï

ˆ´ ( cos ˆ ).

).

= cos   leads to an autonomous equation of the

( ï ).

=

s

+2

+2

s

¨ ªŠ

Š =  (¨ ). leads to an equation of the form 2.9.1.1: u J

−1 2

M

¨ =L

2

A X

 . M j 2 ( )

Page 466

467

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

47.

ñóò ò ó

=

r

–3

!IO r

ñ

+

r ò òó ó r ñ

+

s P

The transformation a = X 2.9.1.1: ¨3g

g =  (¨ ). Solution: X

M¨

M Y



2

–

, ¨

¨

£ 2 ( ) + L

r

=

= AuX

1

– 2r

óò ò ó s

+j

Y M Y



2

òó óò s

r

,

=

r (ï

),

s

=

s

( ï ).

leads to an autonomous equation of the form +L

where  (¨ ) = X¯ (¨ ) ¨ .

2,

M

2.9.2. Equations of the Form — ( æ , â ) â$äãÑã ä + ˜ ( æ , â ) âä ã + ™

(æ , â ) = 0

2.9.2-1. Argument of the arbitrary functions is  . 1.

ñóò ò ó

+ 3ññó ò + ñ

3

+ ! ( ï )ñ = 0. = ¨ OX ¨

The substitution 2.

3.

ñóò ò ó

+ 6ññó ò + 4ñ

ñ óò ò ó

+ 3 !„ññ

5.

6.

+ 4 ! ( ï )ñ +

3

óò

+

ñóò ò ó

+

! óò ñ

=¨ O X

 ¨

! 2ñ

3

+ [ ôñ + ! ( ï )]ñ„ó ò +

¨ 

 +  ( )¨ = 0. leads to a second-order linear equation: u

!„ó ò (ï

) = 0.

2

+ " ( ï )ñ = 0,



−1

M P

!„ó ò (ï )ñ

!

= ! ( ï ).

leads to a second-order linear equation: ¨ 

 + ( )¨ = 0.

= 0.

Integrating yields a Riccati equation: 
 +  ( ) + 12 

ñ óò ò ó

+ [2 ôñ + ! ( ï )]ñ

ñ óò ò ó

+ [3ñ + ! ( ï )]ñ

óò

+ ô! ( ï )ñ

2

2

=L .

= " ( ï ).

On setting ‡ = 
+  2 , we obtain ‡
 +  ( )‡ = ( ). Thus, the original equation is reduced to a first-order linear equation and a Riccati equation.

ñóò ò ó

óò

+ñ

3

+ [ñ + 3 ! ( ï )]ñ„ó ò – ñ

3

The substitution % ( )‡ = 0. 7.

M P

The substitution = ¨3
 ¨ leads to a third-order equation of the form 3.5.3.11: ¨*¨p


 + 3¨2
¨2

 + 4  ( )¨*¨2
+  
( )¨ 2 = 0.

The substitution 4.

−1



+ ! ( ï )ñ

2

+ ! ( ï )ñ

2

+ " ( ï )ñ + % ( ï ) = 0.

= ‡
 ‡ leads to a third-order linear equation: ‡


 +  ( )‡

 + ( )‡
 + + [ !„ó ò ( ï ) + 2 !

2

( ï )]ñ = 0.

The transformation =  ( )¨ ( H ),

H =ù

 ( )  , M

where

 ( ) = exp ¬ − ù

leads to an autonomous equation of the form 2.2.3.2: ¨p‘'

‘ + ¨*¨2‘
− ¨ 8.

ñóò ò ó

– (ù + 1)" (ï )ñ

÷ –1 ñ óò

= ! ( ï )ñ + "Ió ò ( ï )ñ

÷

– " 2 ( ï )ñ 1 1−

÷

2 –1

 , where ¨ = ¨ L + (1 − B ) X ( )¨  −1  S Q M of the second-order linear equation ¨u

 =  ( )¨ .

Solution:

3

 ( )  ® , M = 0.

.

= ¨ ( ) is the general solution

Page 467

468 9.

10.

11.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñóò ò ó

– ùñó ò + ! ( ï )ñ =

ô ï 2÷ +1 ñ

–3

.

 J 2 leads to Yermakov’s equation 2.9.1.2: The substitution ¨ =  − ´ ¨ 

 +  −2 [  ( ) − 1 B (B + 2)]¨ =  ¨ −3 . 4

ñóò ò ó

+ (2 !„ñ + " )ñó ò +

ñ óò ò ó

+ O 3 !„ñ +

!„óò ñ

2

+ "Ió ò ñ = 0,

!

Integrating yields a Riccati equation: 
 + 

"

ñ

P

ñ óò

+

! 2ñ

3

+

! óò ñ

2

2

= ! ( ï ),

+

=L .

+ (2 !" + % )ñ + "

= " ( ï ).

"

óò

"

+

Here,  =  ( ), = ( ), % = % ( ).

2

= 0.

ñ

‡
 , where ‡ = ‡ ( ) is the general solution of the linear equation:  ( )‡ ¨2
 
‡

 − O + ¨ ‡
 +  !‡ = 0, P  and ¨ = ¨ ( ) is the general solution of another linear equation: ¨p

 + % ( )¨ = 0. =

Solution:

12.

ñóò ò ó

+ [3 ! ( ï )ñ + 2" ( ï ) + % ( ï )ñ –1 ]ñó ò + ! 2 ( ï )ñ 3 + [ !„ó ò ( ï ) + 2 ! ( ï )" ( ï )]ñ 2 + ["Ió ò ( ï ) + " 2 ( ï ) + 2 ! ( ï )" ( ï ) –  ( ï )]ñ + %ƒòó ( ï ) + 2" ( ï ) % ( ï ) +

%

2

( ï )ñ

–1

= 0.

The solution satisfies the Riccati equation 
 +  ( ) 2 + [ ( ) − H ( , L )] + % ( ) = 0, where H = H ( , L ) is the general solution of another Riccati equation: H 
+ H 2 =  ( ). 13.

ñóò ò ó

+ O 2 !„ñ + 2" –

ñ

!

–

2"

ñ

ñóò P

2

+

%„ñ

+

!„ó ò

+

!

2

+"

ñ

óò

+

2 !"

ñ

+

2

"

ñ

2 3

= 0.

Here,  =  ( ), = ( ), and % = % ( ). The solution is determined by the Abel equation of the second kind ! 
= (ln ¨ )
 2 −  ( ) − ( ), where ¨ = ¨ ( ) is the general solution of the linear equation: ¨I

 + % ( )¨ = 0. Abel equations of the second kind are discussed in Section 1.3. 14.

ñóò ò ó

–

ƒñóò

+ ! ( ï )ñ =

ô 2 ó ñ

 The substitution ¨ =  −  J

2

–3

.

leads to Yermakov’s equation 2.9.1.2:

¨ 

 + V  ( ) − 1 Œ 2 W ¨ =  ¨ 4 15.

16.

ñóò ò ó

+ "Ió ò ñóò +

!„ñ

=

ô –2$ ñ

The substitution ¨ =  — J

ñóò ò ó

+

ƒþ

tan(

ï )„ñ óò

2

–3

,

!

= ! ( ï ),

−3

.

= " ( ï ).

"

leads to Yermakov’s equation 2.9.1.2:

¨ 

 + [  − 1 ( 
)2 − 1 

 ]¨ =  ¨ 4 2

+ ! ( ï )ñ = ô [cos(

ï

)]2ú

ñ

–3

This is a special case of equation 2.9.2.15 with = −

¡

−3

.

. ln cos( Π ).

2.9.2-2. Argument of the arbitrary functions is . 17.

ñ óò ò ó

=

ôñ óò

+

ó

 2 D ! (ñ

).

  , we obtain an equation of the form 2.9.2.34. −1 J 2 1   Solution: X L 1 + 2 X  ( ) S =L 2A  . Q M M 

Multiplying both sides by 

−2

Page 468

469

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

18.

ñ óò ò ó

= ! (ñ )ñ

óò

Solution: X 19.

20.

= V

ñóò ò ó

.

M  ( )+ L

^ ó ! (ñ

=L

1

+  , where  ( ) = X

2

 ( )

.

M

)+*  Wñóò .

 ¨
=  ( ). The substitution ¨ ( ) =  − 9 
 leads to a first-order separable equation: u  1  = L 2 +  9 , where  ( ) = Xv ( ) . Solution: X M  ( )+ L 1 ƒ M

ï ñóò ò ó

+ï

ùñóò

=

÷ ! (ñ

2 +1

).



Multiplying both sides by 

−2 −1

, we obtain an equation of the form 2.9.2.34.

1 b . Solution for B ≠ −1:

X Q

L

1

+2X  ( )

1

+2X  ( )

J

−1 2

S M

M

=A

 

+1

+L

2.

= A ln | | + L

2.

B +1

2 b . Solution for B = −1:

X 21.

22.

ï ñóò ò ó

= ! (ñ )ñó ò .

ï ñóò ò ó

= V ï

Solution: X 23. 24.

ï 2 ñ óò ò ó

+

)+

ü

M  ( )+ L

ï ñ óò

= ! (ñ ).

=L

1

S M

J

−1 2

M


 leads to a first-order linear equation: !¨
= −¨ + 1 +  ( ). 

– 1WUñó ò .

+

2

1   , where  ( ) = X¯ ( )

,

.

g leads to an equation of the form 2.9.1.1: 

=  ( ). gg

(ô

+ ! (ñ ) = 0.

ï

2

+ õ )ñó ò ò ó +

ô ï ñóò

£ M


Šª
Š +  ( ) = 0.

( ô

2



ó

+ õ )ñó ò ò ó + ô

2

ó

 ñóò



2

+

leads to an autonomous equation of the form 2.9.1.1:

+ ! (ñ ) = 0.

This is a special case of equation 2.9.2.34 with ( ) =   2 

26.

M

The substitution  = A 

The substitution ˆ = X

25.

Q

The substitution ¨ ( ) = 

! (ñ

L

sin ï

ñ óò ò ó

+

ñóò ò ó

–

1 2

cos ï

1 2

sin ï

ñ óò

= ! (ñ ).

ñóò

= ! (ñ ).



+ .

This is a special case of equation 2.9.2.34 with = sin  .

27.

cos ï

This is a special case of equation 2.9.2.34 with = cos  . 2.9.2-3. Other arguments of the arbitrary functions. 28.

ñ óò ò ó

= ! (ñ )ñ

óò

+ " ( ï ).

Integrating yields a first-order equation:


= X  ( ) M

+ X ( )  + L .

M

Page 469

470 29.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

+ [ ! ( ï ) + " (ñ )]ñ

óò

! ó ò ( ï )ñ

+

= 0.


+  ( ) + Xw ( )

Integrating yields a first-order equation: 30.

31.

32.

ï ñóò ò ó

+ (ù + 1)ñó ò =

ïƒ÷ –1 ! (ñ ïƒ÷

=L .

M

).

The transformation ˆ =   , ¨ =   leads to an autonomous equation of the form 2.9.1.1: B 2 ¨2Šª

Š =  (¨ ).

ï 2 ñ óò ò ó

+ (ù + þ

+ 1)ï

ï 2 ñ óò ò ó

= ! (ñ

ï )(ï ñ óò

– ñ ).

ï 3 ñ óò ò ó

= ! (ñ

ï )(ï ñ óò

– ñ ).

ñ óò

+ ùþxñ =

ï ÷ –2ú ! (ñ ï ÷

).

1 b . For B ≠ , the transformation ˆ =   − + , ¨ =   leads to an autonomous equation of ¡ the form 2.9.1.1: (B − )2 ¨ Šª

Š =  (¨ ). ¡ 2 b . For B = , the transformation ˆ = ln  , ¨ =   leads to an autonomous equation of ¡ the form 2.9.1.1: ¨3Šª

Š =  (¨ ).

This is a special case of equation 2.9.3.42 with ( ) = % ( ) = 0. 33.

This is a special case of equation 2.9.4.61 with ( H ) = H . 34.

"Iñ óò ò ó

+

1 2

" óò ñ óò

= ! (ñ ),

= " (ï ).

"

Integrating yields a first-order separable equation: ( )( 
)2 = 2 X  ( ) M Solution for ( ) ≥ 0:

X 35.

ñ óò ò ó

–

r óò r ñ óò

+

Q

L

r óò óò O r s s

The transformation ˆ = X 2.9.1.1: ¨3Šª

Š =  (¨ ).

36.

37.

ó ! (ñ D ó

ñóò ò ó

= – ôñó ò + D

ñóò ò ó

+ ( + b )ñó ò + b “ñ =

óò ò ó s

–

j

 ( )

+2X

1

Y s

2

P

ñ

S M

=

r

M

s  , ¨ =

!IO

ñ s

2

r

,

P

 £ M ( ) .

A X =

r (ï

),

s

=

s

( ï ).

leads to an autonomous equation of the form

j

).

 ( –2

)

ó ! (ñ   ó

ï ñ óò ò ó

– ùñ

ï ñóò ò ó

= ! ( ïƒ÷

óò

– ô (ô

ï

+ ù )ñ =

).

 ˆ =  Á , ¨

2 b . For “ = @ , the substitution ¨ ¨ ªŠ

Š =  (¨ ). 2.9.1.1: 3

39.

3

=L

M

1.

  The transformation ˆ =   , ¨ =   leads to an equation of the form 2.9.1.1: ¨uŠª

Š =  1 b . For “ ≠ @ , the transformation of the form 2.9.1.1: ( “ − @ )2 ¨2Šª

Š =  (¨ ).

38.

2

J

−1 2

+L

( − )

= 



= 



−2

 (¨ ).

leads to an autonomous equation

leads to an autonomous equation of the form

ï 2÷ +1  3D ó ! (ñ  D ó

).

 This is a special case of equation 2.9.2.35 with Y =   and j =  −  .

Df] )ñóò . The transformation H =      , ¨ H (  ¨ + B )¨2‘
= [  ( H ) + 1]¨ .

=  


leads to a first-order separable equation:

Page 470

471

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

40.

41.

42.

ï 2 ñ óò ò ó

+ ï ñó ò = ! ( ïƒ÷ The transformation H H (  ¨ + B )¨2‘
=  ( H ).

Df]

).

=      , ¨

44. 45. 46. 47. 48.

leads to a first-order separable equation:

ó

ï 2 ñ óò ò ó

– ô ï 2 ñó ò – ù ( ô ï + ù + 1)ñ = ï 3÷ +2  2 D ! (ñ ïƒ÷ ).  This is a special case of equation 2.9.2.35 with Y =   and j =  −  .

ñóò ò ó

–

r óò r ñóò

r óò – ôpO r

+ô

ñ P

=

ó ó  3D r 2 !  ñD 

 The transformation ˆ = XvY  2   , ¨ M ¨2Šª

Š =  (¨ ). 43.

=  


,

r

=

r (ï

).

 =   leads to an equation of the form 2.9.1.1:

ï 2 ñ óò ò ó

+ ï ñó ò = ! (ñ + ô ln ï + õ ln2 ï ). g The substitution  =  leads to an equation of the form 2.9.1.5: 
g
g =  ( + a + 'a 2 ). – (ù + 1) tan ï ñ„ó ò – ùñ = cos÷ –2 ï ! (ñ cos÷ ï ). This is a special case of equation 2.9.2.35 with Y = cos − 

ñóò ò ó

 and j = cos−   .

−1

+ (þ – ù ) tan ï ñ„ó ò – ù [(þ + 1) tan2 ï + 1]ñ = cos2ú +÷ï ! (ñ cos÷ ï ). This is a special case of equation 2.9.2.35 with Y = cos + −   and j = cos−   .

ñóò ò ó

ó

ó

+ ô tan ï ñó ò + õ ( ô tan ï – õ )ñ = cos2 D ï  3  ! (ñ  ). This is a special case of equation 2.9.2.35 with Y = cos   and j = 

ñóò ò ó

ï 2 ñ óò ò ó

−





.

+ ô ï 2 tan ï ñó ò + ù ( ô ï tan ï – ù – 1)ñ = ï 3÷ +2 cos2 D ï ! (ñ ïƒ÷ ). This is a special case of equation 2.9.2.35 with Y = cos   and j =  −  .

ï 2 ñ óò ò ó

– ô ï 2 cot ï ñó ò – ù ( ô ï cot ï + ù + 1)ñ = ï 3÷ +2 sin2 D ï ! (ñ ïƒ÷ ). This is a special case of equation 2.9.2.35 with Y = sin   and j =  −  .

2.9.3. Equations of the Form

› — ( æ , â )âäãÑã ä + š 

=0

˜  ( æ , â )(âäã )

= 0 (œ

= 2, 3, 4)

2.9.3-1. Argument of the arbitrary functions is  . 1. 2. 3. 4.

= ! ( ï )(ñó ò + ôñ )2 + ô The substitution ¨ = 
+ 

ñóò ò ó

2

ñ

. ¨ 
=  ¨ +  ( )¨ leads to a Bernoulli equation: u

2

.

= ! ( ï ) + " ( ï )( ï ñ„ó ò – ñ ) + % ( ï )(ï ñ„ó ò – ñ )2 . ¨ 
=   ( ) +  ( )¨ +  % ( )¨ The substitution ¨ ( ) =  
− leads to a Riccati equation: 3

ñóò ò ó

ï ñóò ò ó

+ ( ô + 1)ñó ò = ! ( ï )( ï The substitution ¨ =  
+ 

ñóò

– (ñó ò )2 = ! ( ï ).  1 b . The substitution =  ž  ¨

ññóò ò ó

2 b . The substitutions see 2.7.1.41.

+ ôñ )2 . ¨ 
=  ( )¨ leads to a first-order separable equation: u

−



.

.

   ( ). 2  . For  ( ) = ,  9 +: ,

leads to a similar equation: ¨*¨u

 − (¨2
)2 = 

 = A  J 2 leads to the equation ‡

 = 2  ( ) 

2

2

−2 −2



Page 471

472 5.

SECOND-ORDER DIFFERENTIAL EQUATIONS 2ññó ò ò ó – (ñó ò )2 + ! ( ï )ñ 2 + ô = 0. 1 b . Differentiating with respect to  , we obtain a third-order linear equation: 2 


 + 2  ( ) 
+  
( ) = 0.

2 b . The substitution

=H

2

leads to Yermakov’s equation 2.9.1.2: H 

 + 14  ( ) H = − 14 !H

−3

.

3 b . If ‡ and  are two solutions of the second-order linear equation 4

 +  ( ) = 0, which satisfy the condition ( ‡" 
− ‡
  )2 =  , then = ‡" is a solution of the original equation. 6. 7. 8.

+ (ñó ò )2 + ! ( ï )ññó ò + " ( ï ) = 0. The substitution ‡ = 2 leads to a linear equation: ‡

 +  ( )‡
 + 2 ( ) = 0.

ññóò ò ó

– (ñ ó ò )2 + ! ( ï )ññ ó ò + " ( ï )ñ 2 = 0. The substitution ‡ = 
 leads to a first-order linear equation: ‡
 +  ( )‡ + ( ) = 0.

ññ óò ò ó

– ù (ñó ò )2 + ! ( ï )ñ 2 + ôñ 4÷ –2 = 0. 1 b . For B = 1, this is an equation of the form 2.9.3.7.

ññóò ò ó

2 b . For B ≠ 1, the substitution ¨ (1 − B )  ( )¨ +  (1 − B )¨ −3 = 0. 9.

=

1−

¨ 

 +  leads to Yermakov’s equation 2.9.1.2: u

– ù (ñó ò )2 + ! ( ï )ñ 2 + " ( ï )ñ ÷ +1 = 0. ¨ 

 + (1 − B )  ( )¨ + The substitution ¨ = 1−  leads to a nonhomogeneous linear equation: u (1 − B ) ( ) = 0.

ññóò ò ó

10.

+ ô (ñó ò )2 + ! ( ï )ññó ò + " ( ï )ñ 2 = 0. ¨ 

 +  ( )2 ¨ 
+ (  + 1) ( )¨ = 0. The substitution ¨ =  +1 leads to a linear equation: 3

11.

ññ óò ò ó

– 2(ñ ó ò )2 – ( !„ñ + 2" )ñ ó ò + ! ó ò ñ Integrating yields a Riccati equation:

12.

ññóò ò ó

– (ñó ò )2 + ( !„ñ 2 + " )ñó ò + !„óò ñ 3 – "Ió ò ñ = 0, 
Integrating yields a Riccati equation:  +  2 + L

13.

ñ óò ò ó

14. 15.

ññóò ò ó

+ " óò 
+ L

ñ

= 0, 2 +

= ! ( ï ), + = 0.

!

= ! ( ï ), − = 0.

!

= " ( ï ).

" "

= " ( ï ).

+ 2( ! + " )ñ + " + 2 % ]ñ ó ò + ! 2 ñ 4 + 2 !"Iñ 3 + (2 ! ó ò + " 2 + 2 !% )ñ 2 + (" ó ò + 2"/% )ñ + % òó + % 2 –  = 0. Here,  =  ( ), = ( ), % = % ( ),  =  ( ). The solution satisfies the Riccati equation 
 +  ( ) 2 + ( ) + % ( ) − H ( , L ) = 0, where H = H ( , L ) is the general solution of another Riccati equation: H 
+ H 2 =  ( ). + (ñ

óò

2

)2 + [2 !„ñ

2

= ! ( ï )(ñ ó ò )2 . The substitution ¨ ( ) = 

ññ óò ò ó

¨ 
= ¨ + [  ( ) − 1]¨ 2. 
 leads to a Bernoulli equation 1.1.5:  u

+ ! ( ï )(ñó ò )2 + " ( ï )ññó ò + % ( ï )ñ 2 = 0. The substitution ‡ = 
 leads to a Riccati equation: ‡
 + (1 +  )‡

ññóò ò ó

ô ï )ñ ó ò ò ó

= ! ( ï )( ï

ñ óò

2

+ !‡ + % = 0.

16.

(ñ +

– ñ )2 .

17.

– ô (ñó ò )2 + ! ( ï ) Df] + " ( ï ) = 0. ¨ 

 − ´ ( )¨ =  ( ). The substitution ¨ =  −   leads to a nonhomogeneous linear equation: u

2 The substitution = −  + "H leads to the equation "HH 

 + 2 HH 
−  3  ( )( H 
) = 0. On


u ¨
¨ ¨ ¨ setting = H  H , we obtain a Bernoulli equation:   + 2 + [ −  3  ( )] 2 = 0.

ñóò ò ó

Page 472

473

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

18.

19.

20.

21.

22.

ñóò ò ó

– ô (ñó ò )2 +

+ ! ( ï ) = 0.

õ 4fD ]

¨ 

 −  ( )¨ =  ¨ The substitution ¨ =  −   leads to Yermakov’s equation 2.9.1.2: u

ñóò ò ó

= ! ( ï )(ñó ò sinh ï – ñ cosh ï )2 + ñ .

ñ óò ò ó

= ! ( ï )(ñ

óò

cosh ï – ñ sinh ï )2 + ñ .

ñ óò ò ó

= ! ( ï )(ñ

óò

sin ï – ñ cos ï )2 – ñ .

ñóò ò ó

= ! ( ï )(ñó ò cos ï + ñ sin ï )2 – ñ .

The substitution ¨ sinh 3 ( )¨ 2 .

= 
sinh  − cosh 

The substitution ¨ cosh 3 ( )¨ 2 .

= 
sin  −

The substitution ¨ cos 3 ( )¨ 2 .

cos 

= 
cos  +

sin 

.

¨ 
= leads to a first-order separable equation: p

= 
cosh  − sinh 

The substitution ¨ sin 3 ( )¨ 2 .

−3

¨ 
= leads to a first-order separable equation: p

leads to a first-order separable equation: ¨p
=

¨ 
= leads to a first-order separable equation: p

2.9.3-2. Argument of the arbitrary functions is . 23.

24.

ñóò ò ó

+ ô (ñó ò )2 –

1 2

27.

ó

 ! (ñ

 9  M  ( )+ L ù

26.

=

).

 The substitution ¨ ( ) =  − (
 )2 leads to a first-order linear equation: ¨u
+ 2  ¨ = 2  ( ).  ó ñóò ò ó +  (ñóò )2 = V  _ ! (ñ ) + ·WUñóò . Solution:

25.

ñóò

1

=L

ñóò ò ó

+ ! (ñ )(ñó ò )2 + " (ñ ) = 0.

ñóò ò ó

+ ! (ñ )(ñó ò ) –

ñóò ò ó

=

2

+

ý

1 :   ,

where

 ( )=ù

 9   ( ) M

.

The substitution ¨ ( ) = (
 )2 leads to a first-order linear equation: ¨u
+ 2  ( )¨ + 2 ( ) = 0. 2

1 2

ñ óò =  ó " (ñ



).

 The substitution ¨ ( ) =  − (
 )2 leads to a first-order linear equation: ¨u
+ 2  ( )¨ = 2 ( ). 

ï ! (ñ )(ñ óò

)3 .

Taking to be the independent variable, we obtain a linear equation with respect to  =  ( ): 

= −  ( ) .



28.

ñóò ò ó

+ [ï

! (ñ

) + " (ñ )](ñ„ó ò )3 + % (ñ )(ñó ò )2 = 0.

Taking to be the independent variable, we obtain a linear equation with respect to  =  ( ): 

− % ( )
−  ( ) − ( ) = 0. 29.





ñ óò ò ó

+ ! (ñ )(ñ

óò

)4 + " (ñ )(ñ

óò

)2 + % (ñ ) = 0.

¨
+ 2  ( )¨ The substitution ¨ ( ) = (
 )2 leads to a Riccati equation: 3  (see Section 1.2).

2

+ 2 ( ) ¨ + 2 % ( ) = 0

Page 473

474 30.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñóò ò ó

ï (ñ óò

+ô Solution:

 

M  ( )+ L ù

31.

32.

ï ñóò ò ó

+

ï ñóò ò ó

+ ô ï ( ñ óò

1 2

)2 – ! (ñ )ñó ò = 0.

ñóò

=

2

+ ln | |,

where

   ( )

 ( )=ù

+

M

1   .



)2 + " (ñ ).

The substitution ¨ ( ) =  (
 )2 leads to a first-order linear equation: ¨u
= 2  ( )¨ + 2 ( ).



) = V ï ! (ñ ) + ü – 1WUñó ò . 2

 

M  ( )+ L

ù

ï 3 ñ óò ò ó

=L

1

2

+

1   ,

where

,

   ( )

 ( )=ù

M

.

+ [ ï 4 ! (ñ ) + ô ](ñó ò )3 = 0. Taking to be the independent variable, we obtain an equation of the form 2.9.1.2 for  =  ( ): 

−  ( ) −  −3 = 0.



34.

1

ï ! (ñ )(ñ óò

Solution:

33.

=L

= ï –1 [ ! (ñ ) + " (ñ )( ï The substitution ¨ ( ) = 

ñ óò ò ó

ó



2

.

35.

+ D ! (ñ )(ñó ò )3 + ôñó ò = 0.  The substitution ˆ =  −  leads to an equation of the form 2.9.4.36 with ( H ) = !H 3 : 
Šª
Š −  ( )(
Š )3 = 0.

36.

ñ óò ò ó

37.

ñóò ò ó

– ñ ) + % (ñ )(ï ñ ó ò – ñ )2 ]ñ ó ò . 
− leads to a Riccati equation: ¨
=  ( ) + ( )¨ + % ( )¨

ñ óò

+ ! (ñ )(ñ ó ò )4 + " (ñ )(ñ ó ò )2 + % (ñ ) = 0. The substitution ¨ ( ) = (
 )2 leads to a Riccati equation: ¨3
+2  ( )¨



ï ñóò ò ó

2

+2 ( )¨ +2 % ( ) = 0.

+ ï 2ú +1 ! (ñ )(ñó ò )4 + þxñó ò = 0. This is a special case of equation 2.9.4.18 with B = 4 and Y =  − + .

2.9.3-3. Other arguments of arbitrary functions. 38.

= ! (ñ )(ñó ò )2 + " ( ï )ñó ò . Dividing by 
, we obtain an exact differential equation. Its solution follows from the equation:
ln |  | = Xv ( ) + Xw ( )  + L .

ñóò ò ó

M

M

Solving the latter for 
, we arrive at a separable equation. In addition, solution, with L 1 being an arbitrary constant. 39.

ñóò ò ó

=

! (ï ÷ ñ ú

ï ñ

)

(ï

ñóò

The transformation H

=   + , ¨

ó

2

= ! ( D ñ ÷ )(ñó ò )2 .  The transformation H =   ‘ ¨ 2 ¨
¨ H (B +  ) = [  ( H ) − 1] 2 .

ññóò ò ó

1

is a singular

– ñ )ñó ò .

H ( ¨ + B )¨2‘
= [  ( H ) − 1](¨ ¡

40.

=L

=

 


leads to a first-order separable equation:

− ¨ ).

 , ¨

= 
 

leads to a first-order separable equation:

Page 474

475

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

41.

ï 2 ñ óò ò ó

ï )(ï ñ óò

= ! (ñ

– ñ )(ñó ò )2 .

This is a special case of equation 2.9.4.20 with , = 2. 42.

ñóò ò ó

ï

–2

=

(ï

ñóò

–ñ ) I ! O

Q

ñ ï

P

The transformation H =   , ¨ [ Hž ( H ) − 1]¨ +  ( H ). 43.

ñóò ò ó

ï –5 ) 2 r (ñ ï –3 ) 2 )(ñóò

=

ñ ï P ñ óò

+ "3O

+I % O

ñ ï

P

(ñó ò )2 S .


 leads to a Riccati equation: H ¨3‘
= H 2 % ( H )¨

=

2

+

)4 .

This is a special case of equation 2.9.4.21 with B = −3 2,

¡

= 1, and  (H ) = HY ( H ).

2.9.4. Equations of the Form — ( æ , â , â$ä ã ) âäãÑã ä + ˜ ( æ , â , âä ã ) = 0 2.9.4-1. Arguments of the arbitrary functions depend on  or . 1. 2. 3.

4. 5. 6.

7.

8.

9.

ñóò ò ó

= ! ( ï )(ñó ò + ôñ )  +

ñóò ò ó

+ ! ( ï )ñó ò + " ( ï )(ñó ò )  = 0.

ñóò ò ó

=

ñóò ò ó

= ! ( ï )( ï

ô 2ñ

.

The substitution ¨ = 
+ 

¨ 
=  ¨ +  ( )¨  . leads to a Bernoulli equation: u

The substitution ‡ ( ) = 
leads to a Bernoulli equation: ‡
 +  ( )‡ + ( )‡  = 0.

ù (ù

– 1) ï

–2

The substitution ¨   +  − ´  ( )¨  .

ñ

+ ! ( ï )( ï

=   
 − B/ 

– ñ ) + " ( ï )( ï

ñóò

The substitution ¨ ( ) = 

ï ñóò ò ó

The substitution ¨ =  
+ 

ô

+ ! ( ï ) (ñ

=

ñóò ò ó

= ! (ï )

ü

óò

ñ„óò


−

+ ( ô + 1)ñó ò = ! ( ï )( ï

ñóò ò ó

– ùñ )  .

ñóò

ñóò

¨ 
= leads to a first-order separable equation: u

−1

– ñ ) .

¨ 
=   ( )¨ +  ( )¨  . leads to a Bernoulli equation: u

+

ôñ )

.

¨ 
=  ( )¨  . leads to a first-order separable equation: u

)2 – 2 ôñ .

Setting ‡ = 
, we rewrite the equation as follows: (‡
 −  )2 [  ( )]−2 = ‡ 2 −2  . Differentiating both sides with respect to  and dividing by (‡
 −  ), we obtain a second-order linear equation: "‡

 =  
‡
 +  3 ‡ −  
. There is also the solution: = 12  ( + L )2 .

ü

ñ óò (ï ñ óò

– ñ ).

The Legendre transformation  = ¨ug
, 1 ¨ g

g = . ¨
 ( g )£ a ¨

! 1 (ï )ñ óÈò ñóò ò ó

+

! 2 (ï )ñ ñóò ò ó

+

! 3 (ï

¨ g
− ¨ =a2

)(ñó ò )2 +

leads to an equation of the form 2.9.4.40:

! 4 (ï )ñ ñóò

+

! 5 ( ï )ñ

2

= 0.

The substitution ¨ ( ) = 
 leads to the Abel equation (  1 ¨ +  2 )¨2
+  1 ¨  4 ¨ +  5 = 0, where   =   ( ).

ñóò ò ó

= ! ( ï )(ñó ò sinh ï – ñ cosh ï )  + ñ .

The substitution ¨ sinh 3 ( )¨  .

= 
sinh  − cosh 

3

+ (  2 +  3 )¨

2

+

¨ 
= leads to a first-order separable equation: p

Page 475

476 10.

11.

12.

13.

14.

15.

16.

17.

18.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

= ! ( ï )(ñ

óò

cosh ï – ñ sinh ï )  + ñ .

ñ óò ò ó

= ! ( ï )(ñ

óò

sin ï – ñ cos ï )  – ñ .

The substitution ¨ cosh 3 ( )¨  .

= 
cosh  − sinh 

The substitution ¨ sin 3 ( )¨  .

ñóò ò ó

= 
sin  −

= ! ( ï )(ñó ò cos ï + ñ sin ï )  – ñ .

The substitution ¨ cos 3 ( )¨  .

ñóò ò ó

cos 

= 
cos  +

sin 

¨ 
= leads to a first-order separable equation: p

¨ 
= leads to a first-order separable equation: p

¨ 
= leads to a first-order separable equation: p

= ! (ñ )(ñó ò )2 + " (ñ )(ñó ò )  .

The substitution ¨ ( ) = 
leads to a Bernoulli equation: ¨
=  ( )¨ + ( )¨ 

ï ñ óò ò ó

+

ï ÷¿ú

ï

–1

–2

ú

+1



! (ñ )(ñ óò )÷

+ þxñ

óò

−1

.

= 0.

This is a special case of equation 2.9.4.18 with Y =  − + .

V ! (ñ )( ï ñó ò – ñ ) + " (ñ )(ï ñ„ó ò – ñ ) Wñó ò . The substitution ¨ ( ) =  
− leads to a Bernoulli equation: ¨u
=  ( )¨ + ( )¨  . 

ñóò ò ó

=

ñ óò ò ó

= ! (ñ )( ï

ñóò ò ó

+

D (÷

ñóò ò ó

+

r

ñ óò

–ñ )

) (ñ ó ò

1 2

)2 .

¨ g
− ¨ , The transformation  = a 3 1 . 2.9.4.40: ¨ g

g =  (−¨ g
) £ a ¨ –2)

ó ! (ñ )(ñ óò )÷

= −¨2g
, where ¨ = ¨ (a ), leads to an equation of the form

+ ôñó ò = 0.

 This is a special case of equation 2.9.4.18 with Y =  −  . 2–

÷ ! (ñ )(ñ óò )÷

–

r óò r ñóò

= 0,

r

=

r (ï

).

The substitution ˆ = X Y ( )  leads to an equation of the form 2.9.4.36: "
Šª
Š +  ( )(
Š )  = 0.

M

19.

!„ñóò ò ó

+

1 2

!„óò ñóò

=

!" (ñ )(ñ óò

)2 +

! ÷ % (ñ )(ñ óò )2÷

,

!

= ! ( ï ).

 £ M  ( ) leads to an autonomous equation of the form 2.9.4.13: 
Šª
Š = ( )(
Š )2 + % ( )(
Š )2  .

The substitution ˆ = X

2.9.4-2. Arguments of the arbitrary functions depend on  and . 20.

ñ óò ò ó

=

ï –2 ! (ñ  ï )(ï ñ óò

– ñ )(ñ

óò )

.

The transformation H =   , ¨ =  
 leads to a Bernoulli equation: H ¨u‘
= −¨ + H   ( H )¨  . There are particular solutions: = L3 and = L 1 (for , > 0).

Page 476

477

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

21.

ñóò ò ó

! (ï ÷ ñ ú

ï ñ

=

)

(ñó ò )

÷ ú ÷ ú

2 + +

.

The transformation H =   + , ¨ =  
  yields: H ( ¨ + B )¨ ‘
= H

¡

 +

−

+

2 ++  ( H )¨  + + + ¨ − ¨ 2 .

1 +1

 divide both sides of this equation by ¨  ++ and introduce the new dependent variable We +  = ¨  + + − ¨ −  + + . As a result, we obtain a first-order linear equation: 2 +



22.

ñóò ò ó

ù

–

=



‘
= − + H −  +1+  ( H ). (B + ) H ¡

üIù – Iü þ ï –1 ñ óò üIþ

+ ï

ñ  ! (ƒï ÷ ñ ú

–1 –

)(ñó ò ) 

+1

.

Passing on to the new variables H =   + and ¨ =  
  , we arrive at a first-order equation:

 The substitution 23.

ñ óò ò ó

þ

=

+

üIþ üIù

+

B (1 − , ) ¨ H ( ¨ + B )¨ ‘
= − ¨ 2 +  ( H )¨  +1 . ¡ ,   ¡ B , − 1 ¨ 1−  − ¨ −  leads to a linear equation: H ‘
= = ¡ +  ( H ). 1−, , ¡

üIù

ñ

–1

(ñ

óò

 ñ – –1 ! (ï ÷ ñ ú

)2 + ï

)(ñ

óò ) +2 .

Passing on to the new variables H =   + , ¨ =  
  , we arrive at a first-order equation:

 The substitution 24.

ñóò ò ó

! (ï ÷ ñ ú

ï ñ

=

)

(1 + , ) ¨ H ( ¨ + B )¨ ‘
= ¨ + ¡ ¡ ,!B = ¡ ¨ , (ï

−

 + B ¨ , +1

–ñ )

ñóò

÷ ú ÷

2 +

ñóò ò ó

! (ï ÷ ñ ú

ï ñ

=

)

(ñó ò )

+  ( H )¨ 

+2

.





, +1 leads to a linear equation: H ‘
= − −  ( H ). B



− −1

.

This is a special case of equation 2.9.4.25 with , = 25.

2

÷ ú –÷' ï ÷ +ú ( ñ óò

2B +

B

¡ .



2 +

–ñ ) .

The transformation H =   + , ¨ =  
  leads to a first-order equation:

 −1 2  + + − ´  ++ H ( ¨ + B )¨ ‘
= H  + +  ( H )¨ (¨ − 1)  + ¨ − ¨ 2 . ¡  + 2 ++  on to the new variable  Multiplying both sides by ¨ −  + + and passing = ¨  ++ − ¨  −1 we arrive at a Bernoulli equation: (B + ) H ‘
= − + H  + +  ( H )  . ¡ 26.

ñóò ò ó

=

ñóò ò ó

=

ï

–2

(ï

ñóò

–ñ ) I ! O

ñóò

–ñ ) I ! O

ñ ñóò ï P

Q

+ "3O

ï

(ï

2

ñ

+ï

P

(ñó ò )  S .

(ï

ñóò



–ñ ) 3 " O

ñ

ï P

u ¨

The transformation  = −1 a , = − a leads to an autonomous equation of the form 2.9.4.13: ¨2g

g =  (¨ )(¨2g
)2 + (¨ )(¨2g
)  . –3

ï

ñ


 leads to a Bernoulli equation: H´‡ ‘
= [ H! ( H ) − 1]‡ +

The transformation H =   , ‡ =  H  ( H )‡  . 27.

ï

  + ,

− +

P

–3

.

Page 477

478

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

28.

=

! (ï ÷ ñ ú

ï ñ

)

(ñó ò )

÷ ú –÷' ï ÷ +ú ( ñ óò



2 +

–ñ ) +

" (ï ÷ ñ ú

ï ñ

)

ñóò (ï ñ óò

– ñ ).

 The transformation H =   + , ¨ =  
   , followed by the substitution leads to a Bernoulli equation: (B +

ñóò ò ó

29.

=

! (ï ÷ ñ ú

ï ñ

)

÷ ú (ñó ò ÷ ú 2 + ) +

" (ï ÷ ñ ú

+

The transformation H =   + , ¨



ï ÷ ú ñóò (ï ñ óò – ñ ) + % ( ï ñ ) (ñóò ) ÷ ñ

 , followed  by the  substitution

)

ï ñ

=  


33.

34.

 +ú  (H

(ï

ñóò

–ñ ) . 2

+ ) = ¨  ++

−

1–ü (ñó ò )2 + ï –2 ! ( ïƒ÷ Df] )(ñó ò )  . ù 2–ü Passing on to the new variables H =      and ¨ =  
 , we have

ñóò ò ó

=

ô

¨

2−

 + B

¨

1−

 ,

 2−, ¨ H ( ¨ +  )¨ ‘
= −¨ 2 − +  ( H )¨  . ¡ 1−, ¡ ¨ Multiplying both sides by ¨ −  and introducing the new variable  = ¡ 2 − , H´ ‘
= ( , − 2) +  (H ). we obtain a first-order linear equation: ¡ ¡

2−

 + 

¨

1−

 ,

‘ 1 − ,  +  ( H ). we obtain a first-order linear equation: H´
=

32.

ú

  + ,

− +

− +

 1−, ¨ 2 ¨ H (  ¨ + B )¨ ‘
= + +  ( H )¨  . B 2− , Multiplying both sides by ¨ −  and introducing the new variable  =

31.

+

= ¨  ++ − ¨

1 1  + , leads to a Riccati equation: (B + )H ‘
= H  + + % ( H ) 2 + [ ( H ) − 1] + H −  + +  ( H ). ¡

¨

30.

‘
= [ ( H ) − 1] + H   +−1+  ( H )  .

)H

¡





2−,

1−,

B

2–ü ñóò + ñ 1– ! (D ó ñ ú )(ñóò ) . þ 1–ü  Passing on to the new variables H =   + and ¨ = 
  , we have

ñóò ò ó

ô

=–

ñóò ò ó

=–

ñóò ò ó

=–

þ

ñ óò ñ

ô ñóò

ln O

ô (ñ óò

)2 ln( ï

P

+ ! ( D

ó ñ ú )ñ óò

1−,

.

 The transformation H =   + , ¨ = 
  leads to a first-order equation:  H ( ¨ +  )¨ ‘
= − ¨ ln ¨ − ¨ 2 +  ( H )¨ . ¡ ¡ Dividing both sides by ¨ and passing on to the new variable  = ¨ +  ln ¨ , we obtain a ¡ ‘ first-order linear equation: H´
= − +  ( H ). ¡ ¡ ñóò ò ó = ï –2 ! (ïƒ÷ Df] ) exp O – ô ï ñóò P . ù The transformation H =      , ¨ =  
 leads to the first-order equation H (  ¨ + B )¨ ‘
= ¨ +  ( H ) exp O −  ¨ , which  can  be reduced, with the aid of the substitution = ¨ exp O  ¨ , B P B P to a linear equation: B±H ‘
= +  ( H ).

ù

ñóò

) + ! ( ïƒ÷

Df] )(ñ óò

)2 .

The transformation H =      , ¨ =  
 leads to the equation

 H (  ¨ + B )¨ ‘
= ¨ − ¨ B

2

ln ¨ + ¨

2

 ( H ).

Dividing both sides by ¨ 2 and passing on to the new variable  =  ln ¨ − B ¨ a first-order linear equation: B±H´ ‘
= − + B… ( H ).

−1

, we obtain

Page 478

479

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2.9.4-3. Arguments of the arbitrary functions depend on  , , and
 . 35.

= ! ( ï )" (ñó ò ). The substitution ‡ ( ) = 
leads to a first-order separable equation: ‡
 =  ( ) (‡ ).

36.

ñ óò ò ó

37.

38.

ñóò ò ó

= ! (ñ )" (ñ ó ò ). The substitution ‡ ( ) = 
leads to a first-order separable equation: ‡"‡
=  ( ) (‡ ).  In addition, there may exist solutions of the form = #* + L , where # are roots of the equation ( # ) = 0, L is an arbitrary number, or = $ , where $ are roots of the equation  ( $ ) = 0. = ! ( ô ï + õÿñ + ö )" (ñ„ó ò ). For  = 0, we have an equation of the form 2.9.4.35. For  ≠ 0, the substitution ‡ ( ) =  + (  + ( )  leads to an equation of the form 2.9.4.36: ‡

 =  ( '‡ ) rO˜‡
 − .

ñóò ò ó

hP

ñ óò ò ó = ï –÷ –1 ! (ï ÷ ñ óò

).

¨ 
=  (¨ ) + B ¨ . The substitution ¨ ( ) =   
 leads to a first-order separable equation:  u

39.

= ñ –2÷ –1 ! (ñ ÷ ñó ò ). The substitution ¨ ( ) = 

40.

ñóò ò ó

ñóò ò ó

=

! (ñ ó ò 7 ï

ñ

)


leads to a first-order separable equation: !¨*¨
=  (¨ ) + B ¨ 2 . 

.

Setting ‡ = 
and passing on to the new variables a = X

‡ and ¨ = 2 £  , we have M  (‡ )

= (¨2g
)2 . Differentiating the latter with respect to  , we obtain a second-order linear equation:

¨2g

g = (a )¨ . Here, the function (a ) is defined parametrically: = 1 ‡ , a = X 4 41.

ñóò ò ó

=

! (ñ ó ò

7

ô ï ñ

) +

õ

−2

=

+ '

1 ating both sides with respect to  and passing on to the new variables a = X 2 we obtain an equation of the form 2.9.1.2: H g

g = ‡ (a ) H − ˜H −3 .

ñ óò ò ó

! (ñ ó ò

=

ü

ôñ

+

)

õï

‡

 (‡ )

.

.

Setting ‡ = 
, we rewrite the equation as follows: V £ 2‡
  (‡ )W

42.

M

2

−1

‡

. Differenti-

M , H =£  ,  (‡ )

.

Setting ‡ = 
, we rewrite the equation as follows: V ‡
  (‡ )W

−2

=

both sides with respect to  and passing on to the new variable a = X

+ ' 2 . Differentiating

M

‡

, we obtain a

 (‡ ) second-order linear equation for  =  (a ) integrable by quadrature: 2 g

g = 2 ' + ‡ (a ). Here, ‡ the function ‡ = ‡ (a ) is defined implicitly: a = X M .  (‡ ) 43.

ñóò ò ó

=

! (ñ ó ò

)

. + õÿñ 2 ü Taking to be the independent variable, we obtain an equation of the form 2.9.4.42 for −1 J 2  =  ( ): 

= −(  +  2 )  (1 
)(
)3 .

ô ï







Page 479

480 44.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

= (ô

ñóò ò ó

=

ï

2

+

õï ñ

+ ö^ñ

! (ñ ó ò

)

2

+

 ï

+ ñ + )

) ! (ñ óò

–1 2

).

. +

The transformation  = #*a + $I‡ + L , = c\a . ‡ + Æ , where ‡ = ‡ (a ), reduces this equation by selecting appropriate constants # , $ , L , c , , and Æ , to an equation of the form 2.9.4.41, 2.9.4.42, or 2.9.4.43. 45.

ü

ô ï ñ

+ õ ï 3) 2 + ö ï

.

Setting ‡ = 
, we rewrite the equation as follows: V £ 2‡
  (‡ )W

+ £  +(. ‡ 1 Differentiating both sides with respect to  and passing to the new variables a = X M 2  (‡ ) and H = £  , we obtain a second-order linear equation: 2 H g

g = 2 ‡ (a ) H +  . Here, the function 1 ‡ ‡ = ‡ (a ) is defined implicitly: a = X M . 2  (‡ ) 46.

ñ óò ò ó

=

48.

49.

50.

51.

= 

)

. + õÿñ 3 ) 2 + ö^ñ ü Taking to be the independent variable, we obtain an equation of the form 2.9.4.45 for −1 J 2  =  ( ): 

= −(  +  3 J 2 + ( )  (1 
)(
)3 .

ô ï ñ 

47.

! (ñ ó ò

−2

ñ óò ò ó

=

! (ñ ó ò





)

. + ö ï 3) 2 + ý ï ü The substitution  ¨ =  + ' leads to an equation of the form 2.9.4.45 for ¨  (¨2
−   ) ¨ 

 = £  ¨ + (˜ 3 J 2 +  .

ñóò ò ó

=

ô ï ñ

+õ ï

2

! (ñ ó ò

= ¨ ( ):

M

)

. + õÿñ 2 + ö^ñ 3 ) 2 + ýIñ ü Taking to be the independent variable, we obtain an equation of the form 2.9.4.47 for −1 J 2  =  ( ): 

= −(  +  2 + ( 3 J 2 + )  (1 
)(
)3 .

ñ óò ò ó = ï

ô ï ñ 

–2

(ï ñ óò

óò

– ñ ) ! (ñ

M

).

The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ of the form 2.9.3.14: ¨*¨3g

g = [  (a )]−1 (¨2g
)2 . = Vï

ñ óò ò ó

! (ñ ó ò

) + ñ" (ñ

óò

) + % (ñ

óò

)W

–1





(
 = a , 

 = 1 ¨2g

g ) leads to an equation

.

The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ (
 = a , 

 = 1 ¨2g

g ) leads to a second-order linear equation: ¨3g

g = [  (a ) + aÝ (a )]¨3g
− (a )¨ + % (a ).

(ï

ñóò

+ ôñó ò – ñ + õ )ñó ò ò ó = ! (ñó ò ).

The contact transformation

n o
=  +  , n ol

o = 1  

 ,

− n = 0. where n = n (m ), leads to a linear equation:  (m ) n ol o m

= 
,

n =  
+  
− +  ,

Inverse transformation:

 = n o
− ,

= mn o
− n +  ,


= m ,





 = 1 n l o o .

Page 480

481

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

52.

53. 54.

+ " (ñ )ñ ó ò + % ( ï ) = 0. Integrating yields a first-order equation:

! (ñ ó ò )ñ ó ò ò ó

Xv (‡ ) ‡ + Xw ( ) M M


where ‡ =  .

+ Xv% ( )  = L ,

M

= ô 2 ñ + ! ( ï )" (ñó ò + ôñ ). ¨ 
=  ¨ +  ( ) (¨ ). The substitution ¨ = 
+  leads to a first-order equation: u

ñóò ò ó

+ ô ï )ñó ò ò ó + (ñó ò 2 + 2 ôñ )(ñó ò ò ó + ô ) = 0. The contact transformation

! (ñ óò


=  +  , m

n = 12 ( 
)2 +  ,

n o
= 
,



+ 2n where n = n (m ), leads to a linear equation:  (m ) n ol o Inverse transformation: 1

 = 55.

ñ óò ò ó

=

ï –1 I ! Oñ óò

–

ï

ñ P



 m −n o
 ,

(ï

– ñ )( ï

ñóò

2

ñóò ò ó

ô ï ñóò

+

= 
+ 



,

– ôñ ) =

ï 2I ! Oiñóò

n = 

+1


−   ,

+

ô ñï P

n o
= 

.

+1

,

n ol

o =



leads to a linear equation:  (m ) n l o o = (  + 1) n . Inverse transformation:  = (n o
)  57.

58. 59.


= n o
.

¨ 
= −¨ +  (¨ ). leads to a first-order separable equation:  u 

The contact transformation (  ≠ −1)

m

1 V 2 n − ( n o
)2 W , 2

= 0.

.


The substitution ¨ =  − 56.

=



 n ol

o =

,   + 

1 +1

,

=

 1

(mn o − n )( n o )−  +1 ,  +1

= ï –3 ! ( ï ñó ò – ñ )(ñó ò )–1 . The Legendre transformation  = ¨ g
, ¨2g

g = a [  (¨ )]−1 (¨2g
)3 .

ñóò ò ó

= ! ( ï )" ( ï ñó ò – ñ ). The substitution ¨ =  
−

ñóò ò ó

= ï –÷ –3 ñ ÷ ! ( ï ñ ó ò – ñ ). The transformation  = 1 a , ¨2g

g = ¨   (−¨2g
).


=

2

mn o
+ !n . (  + 1) n o


leads to an equation of the form 2.9.3.27:

¨ 
=   ( ) (¨ ). leads to a first-order separable equation: u

ñ óò ò ó

60.

= ï –1 ! (ñ )" ( ï ñó ò – ñ )ñó ò . The substitution ¨ ( ) =  
−

61.

ñ óò ò ó = ï –3 ! (ñ  ï )" (ï ñ óò

ñóò ò ó

= a ¨ g
− ¨



(  + 1)  +2 

 +  
− 

= ¨u a leads to an autonomous equation of the form 2.9.4.36:

leads to a first-order separable equation: ¨
=  ( ) (¨ ).



– ñ ). The transformation  = 1 a , = ¨u a leads to an autonomous equation of the form 2.9.4.36: ¨2g

g =  (¨ ) (−¨2g
).

Page 481

482 62.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 3 ñ óò ñóò ò ó

+ 2 ï 2 (ñó ò )2 = ! ( ï The contact transformation

m

ñóò

+ ñ ).


+ ,

=

2

n =


,

n o
=  ,

n ol

o =



where n = n (m ), leads to a linear equation:  (m ) n l o o =n . Inverse transformation:  = n o
, 63.

ñóò ò ó

=

ñ !IO ï ñ óò ï 2 ñ

=m

64.

n , n o


66.

n



 =

, ( n o
)2

1 2n − . n o
n ol

o ( n o
)3

¨ 
=  (¨ )+ ¨ − ¨ 2 . 
 leads to a first-order separable equation:  u

ï ñ óò ï ï ƒ ÷ ú –2 + ñ ! ( ñ )3  " O ñ

ñ óò ò ó = ñ (ñóò ) – ï –1 ñ óò 2

The transformation H =   + , ¨ H ( ¨ + B )¨ ‘
=  ( H ) (¨ ). 65.


=

.

P

The substitution ¨ ( ) =  –1

−

1

 

 + 2 


=  
 

P

.

leads to a first-order separable equation:

¡

= ù (ù – 1) ï –2 ñ + ! ( ï )" ( ïƒ÷ ñó ò – ù ïƒ÷ –1 ñ ). ¨ 
= The substitution ¨ =   
 − / B   −1 leads to a first-order separable equation: u ¨    ( ) ( ).

ñóò ò ó

ï 2 ñ óò ò ó

+ ô ï ñó ò – ôñ = ï D +2 !  ï D +1 ñó ò – ï The contact transformation (  ≠ −1):

m

= 
+ 



n = 

,

+1


−   ,

D¦ñ 

.

n o
= 

+1

,

n ol

o =

(  + 1)  +2 

 +  
− 

 2

leads to an autonomous equation of the form 2.9.1.1:  ( n ) n olo =  + 1. Inverse transformation:

 = (n o
)  67.

68.

=

,

 1 (mn o
− n )( n o
)−  +1 ,  +1

= – ï –1 ñó ò + ï –2 ! ( ïƒ÷ Df] )" ( ï ñó ò ). The transformation H =      , ¨ =  
 H (  ¨ + B )¨2‘
=  ( H ) (¨ ).

ñóò ò ó ñ óò ò ó

= ñ –1 (ñ ó ò )2 + ñ! The transformation H ( ¨ +  )¨2‘
=  ( H )

¡

69.

1 +1

(

ó D ñ ú )" (ñ óò ¦ñ

 H =   + , ¨ ¨( ).

ó

(ñó ò + ñ )(ñó ò ò ó + ñó ò ) =  –2 ! (  The contact transformation

ó ñóò

). = 



=

mn o
+ !n . (  + 1) n o


leads to a first-order separable equation:

leads to a first-order separable equation:

).

  −2 n ol

o = −

,  + 


= −n . where n = n (m ), leads to a linear equation:  (m ) n ol o m

=


 ,

n = 
+ ,

 n o
=  − ,

Inverse transformation:

70.



= mn o
.  = − ln n o
, = n − mn o , ñóò ò ó – ñóò =  2ó !   ó ñóò –  ó ñ   ñóò ò ó – ñ  .   1 b . The substitution ¨ = 
− leads to a first-order equation: ¨u
=  2  (  ¨ )(¨2
+ ¨ ).

Page 482

483

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2 b . The contact transformation




=  (  − ), m

n = ( 
)2 −



leads to a linear equation: n l o o

71.

72.

73.

74.

2

,

−2



 − 


 − 

= 2  (m ).

= ! ( ï )" (ñó ò sinh ï – ñ cosh ï ) + ñ . ¨ 
= The substitution ¨ = 
sinh  − cosh  leads to a first-order separable equation: p ¨ sinh 3 ( ) ( ). = ! ( ï )" (ñó ò cosh ï – ñ sinh ï ) + ñ . ¨ 
= The substitution ¨ = 
cosh  − sinh  leads to a first-order separable equation: p ¨ cosh 3 ( ) ( ).

ñóò ò ó

= ! ( ï )" (ñó ò sin ï – ñ cos ï ) – ñ . The substitution ¨ = 
sin  − cos  sin 3 ( ) (¨ ).

ñóò ò ó

¨ 
= leads to a first-order separable equation: p

= ! ( ï )" (ñ ó ò cos ï + ñ sin ï ) – ñ . The substitution ¨ = 
cos  + sin  cos 3 ( ) (¨ ).

ñ óò ò ó

leads to a first-order separable equation: ¨ 
=

7

+ 12 "Ió ò ñóò = ! (ñ ) %  ñóò "  , " = " (ï ). The substitution ¨ ( ) = 
£ leads to a first-order separable equation: ¨*¨u
=  ( ) % (¨ ).

"Iñóò ò ó

76.

!  ôñóò – õ ï  ñ óò ò ó

+  2 ôñó ò – 3 õÿñ   2 ôñó ò ñóò ò ó – õ  = 0, The contact transformation 2

n = 2  ( 
)3 − 3  ,



+ 3n leads to a linear equation:  (m ) n ol o Inverse transformation:  =



9

1 ( n o
)2 − m ,



=

= !  ñó ò 2 + ôñ  . The substitution ¨ ( ) = (
 )2 + 

ñóò ò ó

= ! (ñ )"  ñó ò 2 + 2 ôñ  – ô . The substitution ¨ ( ) = (
 )2 +2 

n o
= 3 
,

≠ 0.

n ol

o =

3






 − 

2

= 0.

2 1 ( n o
)3 − n , 81  3

ñóò ò ó



õ

3

=  ( 
)2 − ' ,

m

78.

n ol

o = 2 

ñóò ò ó

75.

77.

 n o
= 2 − 
,


=

1
n o , 3



 =

3 ˜n ol

o

. 2 !n o
n ol

o − 9

leads to a first-order separable equation: ¨u
= 2  (¨ ) +  .



leads to a first-order separable equation: ¨u
= 2  ( ) (¨ ).



ñ óò ò ó = !  ñóò – 2ô ï ñ óò

79.

+ 2 ôñ  . ¨ The substitution = + 1   2 leads to an autonomous equation of the form 2.9.4.78: ¨2

 =  (¨2
2 + 2  ¨ ) −  . 2

80.

ñóò ò ó

2

ó

– ñó ò =  2 !  ñó ò 2 – ñ 2   The contact transformation

m

 =  ( 
− ),

ñóò ò ó

–ñ  .

n = ( 
)2 −

2

,

 n o
= 2 − 
,

n ol

o = 2 



leads to an autonomous equation of the form 2.9.1.1: n l o o = 2  ( n ).

−2

 

 − 
 

 −

Page 483

484 81.

82.

83.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñ óò ò ó

ñ óò

=u ! 

ï ñ óò 2 + ôñ

. ¨ 
The substitution ( ) =  (  )2 +  +

1 2

leads to a first-order separable equation: ¨u
= 2  (¨ ) +  .



ñóò ò ó + 12 ñ óò =  –ó !   ó ñ óò 2 + ôñ

.  
2 ¨ The substitution ( ) =  (  ) +  leads to a first-order separable equation: ¨u
= 2  (¨ )+  .  !  ô (ñóò ) – õ ï  ñóò ò ó = V ôƒü (ñóò ) +1 – õ ( ü + 1)ñ±W V ôƒü (ñóò ) –1 ñóò ò ó – õW . The contact transformation (  ≠ 0, , ≠ −1)

m =  ( 
)  − ' ,

n = , ( 
) 

+1

n o
= ( , + 1) 
,

−  ( , + 1) ,

n ol

o =

( , + 1) 

 , ( 
)  −1 

 − 



= ( , + 1) n . leads to a linear equation:  (m ) n ol o Inverse transformation:  = 84.

85.

86.

r ñ óò ò ó

+

1 2

r òó ñ óò

 (n o
)  m − ,   ( , + 1)  =u ! 

r ñ óò 2 + ôñ 

r

,

The substitution ¨ ( ) = Y ( )(
 )2 +  2  (¨ ) +  .

r ñóò ò ó

+

1 2

r òóñóò

= ! ( ñ )" 

r ñóò 2 + 2ôñ 

The substitution ¨ ( ) = Y ( )(
 )2 + 2  2  ( ) (¨ ).

ñ óò ò ó

=

ï ÷ ñ ú (ñ óò ) 2÷ ÷ ++ú ú

+3 +2

H (

, ( n o
)   ( , + 1) 

=

),



r (ï

=

+1 +2

−

n

 ( , + 1)

,


=

n o
.

, +1

).

leads to a first-order separated equation: ¨u
=



–ô ,

r

=

r (ï

).

leads to a first-order separable equation: ¨u
=



= (ï

ñ óò

– ñ )(ñ

÷ +1 óò )– ÷ +ú +2 .

The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ leads to an equation of the form 2.9.4.25: a ¨ B +1 ¨ g

g = 1 ¨
2  +1−   ¨
¨ ¨a Q  (a  ¨ ) S ( g )  +1 (a g − )  , where  = − B + + 2 ,  = −¡ .

¡

2.9.5. Equations Not Solved for Second Derivative 1.

! (ï

)(ñó ò ò ó – ô )2 = (ñó ò )2 – 2 ôñ + õ .

Differentiating with respect to  , we obtain ( 

 −  )(2 




 +  


 − 2 
−  
) = 0.

Equating the second factor to zero and making the transformation ˆ = X

(1)

 ¨ 
£ M  , =  , we

arrive at a second-order constant coefficient linear equation of the form 2.1.9.1:

¨ Šª

Š − ¨ = 1   
, 2

(2)

whose right-hand side is to be expressed in terms of ˆ . Substituting the solution of equation (2) into the original one, we obtain a relation connecting integration constants. 1  Equating the first factor in (1) to zero, we find the singular solution: =  ( + L )2 + . 2 2

Page 484

485

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2.

! (ï

)(ñó ò ò ó – ôñ )2 = (ñó ò )2 – ôñ 2 + õ . Differentiating with respect to  , we obtain ( 

 − 

)[2  ( 


 − 


) +  
( 

 −  ) − 2 
] = 0.

Equating the second factor to zero, we arrive at a third-order linear equation: 2  ( 


 − 


) +  
( 

 −  ) − 2 
= 0.

Equating the first factor to zero, one can find the singular solution.

ï

3.

= ! (ñ ó ò ò ó ). ¨ 
). The substitution ¨ ( ) = 
leads to an equation of the form 1.8.1.7:  =  (p

4.

ñ = ! (ñóò ò ó ). The substitution ¨ ( ) = 12 (
 )2 leads to an equation of the form 1.8.1.8:

5.

ñ =ô ï

=  (¨p
).



+õ ï

+ ö + ! (ñó ò ò ó ). ¨ 

 +2  ). The substitution ¨ = −  2 − ' − ( leads to an equation of the form 2.9.5.4: ¨ =  (p 2

ï ñóò

6.

= ñ + ô (ñó ò )2 + õÿñó ò + ö + ! (ñó ò ò ó ). The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ (
 = a , 

 = 1 ¨2g

g ) leads to an equation of the form 2.9.5.5: ¨ = a 2 + 'a + ( +  (1 ¨2g

g ).

7.

! (ñ óò ò ó

8.

)+ Solution:

ï ñóò ò ó =

= 1 2

ñóò

L 1

. 2

+   ( L

1)

+L

2.

! (ñ óò ò ó

+ ñ ) = (ñó ò ò ó )2 + (ñó ò )2 . Differentiating with respect to  , we obtain

[ 
( 

 + ) − 2 

 ]( 


 + 
) = 0.

From the equation 


 + 
 = 0, it follows that: = # sin( + L

1)

+L

2,

where

#

2

=  (L

2 ).

Equating the expression in square brackets to zero, we arrive at the singular solution in parametric form: [2 −  

 (‡ )] ‡
 =ù M , = ‡ − 12   (‡ ). 2 V
Å 4  ( ‡ ) −   ( ‡ )W 9. 10.

ñóò

= ñ! (ñó ò ò ó¦ñ ). The transformation a =

ññóò ò ó

= (ñó ò )2 + 1 b . Solution:

2

, ¨ = (
 )2 leads to an equation of the form 1.8.1.11: ¨ = aª 2 (¨2g
).

!  ñóò ò ó¦ñ 

. =L

2 b . Solution:

exp( L

=L

where the constants L 3 b . Solutions:

1

1

2

 )+

sin( L

3

 ( L 22 ) exp(− L 2  ). 4 L 1 L 22

 )+L

2

cos( L

3

 ),

L 2 , and L 3 are related by the constraint ( L = A" £ −  (0) + L . 1,

2 1

+L

2 2)

L

2 3

+  (− L

2 3)

= 0.

Page 485

486 11. 12. 13.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ

=

ï ! (ï 3 ñ óò ò ó

).

The transformation  = 1 a ,

ï ñóò

– ñ = ! ( ïƒ÷

ï ñóò

– ñ = ! ( 

ï ñóò

– ñ = ! (ln ï

ï ñóò

– ñ = ! (sin ï

ñ„óò ò ó

ï ñ óò

– ñ = ! (r

),

¨ g

g ). = ¨u a leads to an equation of the form 2.9.5.4: ¨ =  (I

ñóò ò ó ). This is a special case of equation 2.9.5.16 with Y =   . ó ñóò ò ó

).

ñóò ò ó

).



This is a special case of equation 2.9.5.16 with Y =   . 14.

This is a special case of equation 2.9.5.16 with Y = ln  . 15.

).

This is a special case of equation 2.9.5.16 with Y = sin  . 16.

ñ óò ò ó

r

=

r (ï

).

  , ¨ =  
 − Y M Z

The transformation ˆ = X

¨ =  (2 ¨ Š
).

leads to an equation of the form 1.8.1.8:

2.9.6. Equations of General Form 2.9.6-1. Equations containing arbitrary functions of two variables. 1. 2.

3.

ñóò ò ó

=

H (ï , ñ óò

).

ñ óò ò ó

=

H (ñ , ñ ó ò

).

ñóò ò ó

=

H (ô ï

õÿñ , ñ óò

¨ 
=  ( , ¨ ). The substitution ¨ ( ) = 
leads to a first-order equation: u Autonomous equation. The substitution ¨ ( ) = "
 leads to a first-order equation: ¨*¨u
=   ( , ¨ ). +

).

The substitution  ¨ =  +  4.

ñóò ò ó

=

1 ñ H O ï , ñ óò ï w

P

 leads to an equation of the form 2.9.6.2: ¨ 

 = O ¨ , ¨ 
−

hP

.

.

Homogeneous equation. This is a special case of equation 2.9.6.6 with , = 1. 5.

ô ï  ï

+ õÿñ + ö , ñó ò . P + õÿñ + ö + ñ + 1 b . For ý − ˜ƒ = 0, the substitution  ¨ =  +  the form 2.9.6.2.

ñóò ò ó

=

1

ô ï

HwO

+ ( leads to an autonomous equation of

2 b . For ý − ˜ƒ ≠ 0, the transformation

H =  −  0, where 

0

and

0

¨ = − , 0

are the constants determined by the linear algebraic system of equations



0

+

0

+ ( = 0,

ƒ…

0

+ý

0

+ = = 0,

Page 486

487

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

leads to a homogeneous equation of the form 2.9.6.4:

¨ ¨ '‘

‘ = 1 ` O , ¨ ‘
, H H P

6.

7.

ñóò ò ó

=

ï –2 H (ï – ñ , ï 1– ñ óò

where

` (ˆ , ‡ ) =

1  + 'ˆ  O ,‡ .  + 'ˆ ƒ +± ý ˆ P

).

Generalized homogeneous equation. The transformation a = ln  , ¨ =  −  ¨ g
+ , ( , − 1)¨ =  (¨ , 3 ¨ g
+ , ¨ ). equation of the form 2.9.6.2: ¨ug

g + (2 , − 1)3

ñ óò ò ó

=

ñ ï ÷ ú ï 2H O ñ

ï

,

ñ óò ñ

P

.

Generalized homogeneous equation. The transformation H =   + , ¨ ‘ first-order equation: H ( ¨ + B )¨
=  ( H , ¨ ) + ¨ − ¨ 2 . 8. 9.

10.

ñóò ò ó

=

ô 2ñ

ñ óò ò ó

= (ô

ñóò ò ó

=w H O

+

ï

+ ôñ ).

+ ô )ñ +

H (ï , ñ óò

–

ñ ï

12. 13. 14. 15. 16. 17.

18.

–ô

ñóò ò ó

=

ñóò ò ó

=

H (ï , ï ñ óò

¨ 
= −¨ + " ( , ¨ ). leads to a first-order equation:  u 

– ñ ).

¨ 
= " ( , ¨ ). leads to a first-order equation: u

ï –2 H (ñ , ï ñ óò

– ñ ). The substitution ¨ ( ) =  
−

leads to a first-order equation: ( + ¨ )¨u
=  ( , ¨ ).

+ ( ô + 1)ñ ó ò = H ( ï , ï ñ ó ò

+ ôñ ).

The substitution ¨ =  
+ 

ï 2 ñ óò ò ó

= 2ñ +

ï 2 ñ óò ò ó

= ô ( ô + 1)ñ +

H ( ï , ï ñ óò

óò

H (ï , ï „ñ óò

H (ï , ñ óò


+ 

= 2 ôññ

The substitution ¨ = 
− 

2

ñóò ò ó

=

).

ó

ó

 –D H ( D ñ

,

The substitution ¨ =  ¨2

 − 2  ¨2
+  2 ¨ = 

ñóò ò ó

=

ñH ( D ó ñ ú

,

¨ 
=  ( , ¨ ). leads to a first-order equation: u

leads to a first-order equation:  ¨ 
= 2¨ +  ( , ¨ ).

ñ óò ò ó

+

+ ôñ ). ¨ 
= (  + 1)¨ +  ( , ¨ ). leads to a first-order equation:  u

– ôñ 2 ).

ó

D ñóò 



+ ñ ).

The substitution ¨ =  
+ The substitution ¨ = 

).

¨ 
= −  ¨ +  ( , ¨ ). leads to a first-order equation: u

The substitution ¨ ( ) =  
−

ï ñ óò ò ó

ï ñ

.

P

The substitution ¨ ( ) = 
− 11.

¨ 
=  ¨ +  ( , ¨ ). leads to a first-order equation: u

The substitution ¨ = 
− 

ï , ñ óò

=  
  leads to a

¡

H (ï , ñ óò

The substitution ¨ = 
+  2 2

leads to an

¨ 
=  ( , ¨ ). leads to a first-order equation: u

leads to a second-order autonomous equation of the form 2.9.6.2: ¨ 
−  ¨ ). (¨ , 2

ñóò ¦ñ

).

Equation invariant under “translation–dilatation” transformation. The transformation  H =   + , ¨ = 
  leads to a first-order equation: H ( ¨ +  )¨u‘
=  ( H , ¨ ) − ¨ 2. See also ¡ Paragraph 0.3.2-7.

Page 487

488 19.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=

ï –2 H (ƒï ÷  Df] , ï ñ óò

).

Equation invariant under “dilatation–translation” transformation. The transformation H =      , ¨ =  
 leads to a first-order equation: H (  ¨ + B )¨u‘
=  ( H , ¨ ) + ¨ . See also Paragraph 0.3.2-7. 20.

21.

22.

23.

24.

25.

ñ óò ò ó

=

 2fD ] H (ï  Df] ,  –fD ] ñ óò

ñ óò ò ó

=

ô ] ñ óò

ñóò ò ó

= (

ñ óò ò ó

=

H (ï , ñ óò

sinh ï – ñ cosh ï ) + ñ .

ñóò ò ó

=

H (ï , ñ óò

cosh ï – ñ sinh ï ) + ñ .

ñ óò ò ó

=

ï –2 H (ô ñ

ñ óò ò ó

=

ñH (ô ï

+ õ ln ñ ,

ñóò ò ó

=

H (ï , ñ óò

sin ï – ñ cos ï ) – ñ .

ñóò ò ó

=

H (ï , ñ óò

cos ï + ñ sin ï ) – ñ .

ñóò ò ó

= (r

ñ óò ò ó

=

+

H (ï , ñ óò

– ô

27.

28.

29.

30.

]

).

The substitution ¨ = 
−    leads to a first-order equation: ¨ 
=  ( , ¨ ). 2

ó

+

ó )ñ



+

H  ï , ñ óò

 The substitution ¨ = 
− 

–

 leads to a first-order equation: ¨ 
= −  ¨ +  ( , ¨ ).

+ õ ln ï ,

ï ñ óò

).

=  
 leads to a first-order equation: (  ¨ +  )¨u‘
=

+  ln  , ¨

ñ óò ¦ñ

).

‘ = 
 leads to a first-order equation: (  ¨ +  )¨
=

The transformation H =  +  ln , ¨  (H , ¨ ) − ¨ 2.

¨ 
=  ( , ¨ ) sin  . The substitution ¨ = 
sin  − cos  leads to a first-order equation: u ¨ 
=  ( , ¨ ) cos  . The substitution ¨ = 
cos  + sin  leads to a first-order equation: u 2

+

r òó )ñ

+

H (ï , ñ óò


The substitution ¨ =  − Y

r ò òó ó r ñ

+w H O

ï , ñ óò

–

–

ñ óò ò ó

=

r ñ óò ]

+

r ó

+

r ñ

r óò r ñ P

H (ï , ñ óò

r

),

=

r (ï

).


leads to a first-order equation: ¨  = − Y ¨ +  ( , ¨ ).

r

,

=

r (ï

).

Y
 ¨
leads to a first-order equation: ¨  = − +  ( , ¨ ).

Y

32.

.

¨ 
=  ( , ¨ ) cosh  . The substitution ¨ = 
cosh  − sinh  leads to a first-order equation: u



Y  The substitution ¨ =  − 31.

óñ

¨ 
=  ( , ¨ ) sinh  . The substitution ¨ = 
sinh  − cosh  leads to a first-order equation: u

The transformation H =   (H , ¨ ) + ¨ . 26.

).

The transformation H =     , ¨ =  −   
 leads to a first-order equation: ( !H ¨ + 1)¨u‘
=  (H , ¨ ) −  ¨ 2.

– r ),

r

=

r (ï , ñ

Y

).

The substitution ¨ = 
− Y ( , ) leads to a first-order equation: ¨ 
=  ( , ¨ ).

! 2 ñ óò ò ó

+

! !„óò ñóò

=

ž (ñ , „! ñóò

The substitution ¨ ( ) = 

),

!

= ! ( ï ).


leads to a first-order equation: ¨*¨u
= ` ( , ¨ ). 

Page 488

489

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

33.

ñ óò ò ó

=

H (ï , ñ

). Let  ≠ Y ( ) + j ( ), i.e., the equation is nonlinear. Then its order can be reduced by one if the right-hand side of the equation has the following form:

˜ J «E¦±` (‡ ) + X V 1 / 


 (‡ + ) +  1 J 2 

 « 2 

−3 2

 ( , ) = 

−1

W t  § , M

(1)

where

˜ = X  −3 J 2 « −1  , ‡ =  −1 J 2 « −1 − ˜ ;  , M P M ` = ` (‡ ),  =  ( ), and = ( ) are arbitrary functions, and , is˜ an arbitrary constant. The integral in (1) can always be expressed in terms of « and . The following cases are −1

« = exp O, X 

possible:

1 b . For  


 ≠ 0,

 ( , ) = 

J «ç` (‡ ) + 1  4

−3 2

2 b . For  = 

 ( , ) = 

2

−2

V 2 / 

 − (  
)2 W

−2

+ 12 

 2  
−  
+ 2 ,  + , 2 

+ ' + ( ,  
≠ −2 , ,  
≠ 23 , ,

J «ç` (‡ )+ 1  2

−3 2

−2

 2  
−  
+2 ,  +  , 2 + 14 ¢  

J «

−3 2

˜

,

J «

−3 2

˜

.

¢ = 4!( −  2.

where

3 b . For  = ý − 2 ,! , −2

 ( , ) =  2 3

4 b . For  =

ú

[` ( +


) +   + 2 , ],

ú

where

= − Xv

−1

M

 .

,! + ý ,

 ( , ) = `  

−’  +

−2

−2

  
+ 23 ,  + 89 , 2 ’ ,

where

In all these cases, the transformation

a = Xv

−1

 , M

‡ =

J «

−1 2

−1

−

’ = Xv

−3

M

 .

˜

leads to the autonomous equation ‡ g

g + 2 ,!‡ g
+ , 2 ‡ = ` (‡ ), which is reducible, with the aid of the substitution H (‡ ) = ‡ g
, to an Abel equation: HH 
+2 ,H + , 2 ‡ = ` (‡ ) (see Subsection 1.3.1). If , = 0, the solution of the original equation for case 1 b is as follows:

X

‡ £ 2 i (M ‡ ) + L

1

=A X

M 



+L

2,

where

i (‡ ) = X ` (‡ ) ‡ . M

If , = 0, the solution of the original equation for case 2 b is given by: 2X 34.

ñóò ò ó

ü

M

‡

8 i (‡ ) − ¢C‡

ó

2

+L

1

=A X



2

M



+ ' + (

+L

2,

where

i (‡ ) = X ` (‡ ) ‡ . M

+ ôñó ò + õÿñ =  H  ñó ò ¦ñ , ñóò ò ó¦ñ  .  ¨ 1 b . The substitution = ž leads to an autonomous equation:

¨ 

 + (2 Œ +  )¨ 
+ ( Π

2





+ Œ +  )¨ =   ¨  + Œ ¨ , ¨   + 2 Œ ¨  + Œ 2 ¨  .

2 b . Particular solution: = , ´ , where , is a root of the algebraic (transcendental) equation , ( Œ 2 + Œ +  ) =  ( , Œ , , Œ 2 ).

Page 489

490 35.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ô 1 (ñ ó ò ò ó

)2 + ô

2

ñ óò ñ óò ò ó

 = ž  J 2 ¨

The substitution 36.

ñóò ò ó

=

ñó'ò H

1

 ñóò ¦ñ ,

H  ï ñ óò ò ó , ï 2 ñ óò ò ó

–ï

óò

+ ô 4 (ñ

)2 + ô

5

ññ óò

+

ô 6ñ

2

=

ó

  HÊ ñ óò ¦ñ , ñ óò ò ó ¦ñ 

.

leads to an autonomous equation.

ñóò ò ó¦ñ

 = ž  ¨

The substitution 37.

ô 3 ññ óò ò ó

+

 + ñH

2

 ñóò ¦ñ ,

ó

ñóò ò ó¦ñ

 +  H

3

 ñóò ¦ñ ,

ñóò ò ó¦ñ

3

are related by the constraint

.

leads to an autonomous equation. + ñ  = 0.

ñóò

Solution: =L

1

 ln  + L 2  + L 3 ,

where L 2 is an arbitrary constant and the constants L  ( L 1 , L 3 ) = 0. 38.

H (ñ óò ò ó¦ñ , ñ ñóò ò ó

=L

where the constants L

L 2 , and L

1,

2 b . Solution:

1

where the constants L

L 2 , and L

1,

exp( L

1

3

3

 )+L

2

exp(− L

3

 ),

are related by the constraint  ( L

3

=L

H (ñ 3 ñ óò ò ó , ñ ñ óò ò ó

and L

– ñó ò 2 ) = 0.

1 b . Solution:

39.

1

cos( L

3

 )+L

2

sin( L

3

2 3,

4L

1

L 2L

2 3)

= 0.

 ),

are related by the constraint   − L

L

2 1+

−L

2 2,

2 3 , −(

L

2 2)

L

2 3

 = 0.

óò 2 ) = 0.

+ñ

Solutions: =A where the constants L 40.

Hv”Èñ

+

ñ óò 2 ï , ñ óò ò ó

+

L 2 , and L

1,

ñ óò

2ñ

–ñ

óò ò ó

óò 3

3

ü

L 1

2

+ 2L

2

 + L 3,

are related by the constraint  ( L

1

L

3

L 1 ) = 0.

= 0.

•

Solutions can be found from the relation ( − L 1 )2 = 2 L 2 ( − # ) + L 22 , where  ( L The question of whether there are other solutions calls for further investigation. 41.

HwOiñ

ñ óò ò ó ñ óò

+

ôñ óò , ñ D

ñ óò ò ó ñ óò

+1

1,

# ) = 0.

= 0.

P

A solution of this equation is any function that solves the first-order separable equation:


= L where the constants L 42.

HwO

ñ óò ò ó ñ óò

+ ñó ò ,

]

1

ñ óò ò ó ñ óò

and L

P

2

1

−

 +L , 2

are related by the constraint  ( L

2, −

L 1 ) = 0.

= 0.

A solution of this equation is any function that solves the first-order separable equation:


= L 1  − + L 2 , where the constants L

1

and L

2

are related by the constraint  ( L

2, −

L 1 ) = 0.

Page 490

491

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2.9.6-2. Equations containing arbitrary functions of three variables. 43.

H (ñ óò ò ó , ï ñ óò ò ó

ï 2 ñ óò ò ó

– ñó ò ,

– 2ï

ñóò

+ 2ñ ) = 0.

Solution:

=L

where the constants L

^_ 44.

L 2 , and L

1,

+L

2

 + L 3,

are related by the constraint  (2 L

3

H  ñóò ò ó , ï ñ óò ò ó

=L

H  ï 3 ñ óò ò ó , ï ñ óò ò ó

1,

L 2 , and L

3

ï 2 ñ óò ò ó

+ 2ñó ò ,

2



1

+L

2

 + L 3,

are related by the constraint  (2 L

+ï

ñóò =L

where the constants L

HÊ ï D +2 ñ óò ò ó , ï ñ óò ò ó

L 2 , and L

1,

3

=L 1,

H  ñ 3 ñ óò ò ó , ñ ñóò ò ó

L 2 , and L

3

+ï

Solution:

48.

49.

50.

1,

L 2 , and L

3

=L

– ñ , ñó ò ò ó – ñ , ñó ò ò ó – ñó ò  = 0.

H  ï , ñ óò

+

ôñ , ñ óò ò ó

–ô

2

Hv” ï

–

ñ óò 2 + 1 óò ñ óò ò ó ñ , ñ

+

L 2, 4 L 1 L

3−

L

2 2 ) = 0.



3

,

1

ô ï ñ óò

3, 2

L 1 , − L 2 ) = 0.

– ôñ  = 0.

 − + L 2  + L 3.

1



2

1, (

 +1) L 2, − L 3  = 0.

 = 0.

+ 2L

2

 + L 3, 1

L

3

−L

2 2,

L 1 , − L 2 ) = 0.

leads to a first-order equation:  ( , ¨ , ¨u
+ ¨ , ¨2
) = 0.

ñ , ñ óò ò ó

The substitution ¨ = 
+ 

L

are related by the constraint  ( L

H  ï , ñ óò

The substitution ¨ = 
−

+

ñóò 2 – ññóò 2

where the constants L

+

2

are related by the constraint    (  +1) L

ï ññóò ò ó

+ ñó ò 2 ,

 +L

are related by the constraint  (2 L

Solution: The constants L

1

óò , ï 2 ñ óò ò ó

+ ( ô + 1)ñ

1, −

– ñ  = 0.

Solution:

47.

L 2 , 2 L 3 ) = 0.

– ñó ò , 2ññó ò ò ó – (ñó ò )2  = 0.

where the constants L

46.

1, −

Reference: E. L. Ince (1964).

Solution:

45.

2



1

+ ôñó ò  = 0.

leads to a first-order equation:  ( , ¨ , ¨u
−  ¨ , ¨2
) = 0.

ñ óò 2 + 1 (ñ óò 2 + 1)3 ) 2 • ñ óò ò ó , ñ óò ò ó

= 0.

It is known that all functions of the form ( − L 1 )2 + ( − L 2 )2 = # 2 , where # = # ( L 1 , L 2 ) is determined from the algebraic (transcendental) equation  ( L 1 , L 2 , # ) = 0, are solutions of the original equation. 51.

ó

H   ñóò ò ó , ñ óò ò ó

+ ñó ò ,

ñ

–ï

ñóò

– ( ï + 1)ñó ò ò ó  = 0.

Solution: where the constants L

=L 1,

L 2 , and L

3

1



−



+L

2

 + L 3,

are related by the constraint  ( L

1,

L 2 , L 3 ) = 0.

Page 491

492 52.

SECOND-ORDER DIFFERENTIAL EQUATIONS

H O ñ óò ò ó ¦ñ , ñ ñ óò ò ó

óò

– (ñ

Solution:

=L

where the constants L 53.

H  ñóò ò ó

–ñ ,

ñóò ò ó

L 2 , and L

1,

1

HÊ ï ñ óò ò ó , ï 2 ñ óò ò ó

1,

–ï

ñ óò

L 2 , and L

where the constants L –ñ

Solution:

ñ óò

+ñ ,

1,

H  ñóò ò ó

+ñ ,

ñóò ò ó

L 2 , and L

ñ óò ò ó ï ñ óò ò ó , ñ óò ñ óò

where the constants L 56.

1,

exp(− L

3

 P

L 2 , and L

2

cosh  + L

= 0.

 ),

3

3

óò

)2

=L

1

1,

HÊ ñ , ï ñ óò 2, ï ñ óò ò ó

+

1 2

HÊ ñ , r ñ óò 2, r ñ óò ò ó

+

1 2

L 2 , and L

3

1

2L

2 3,

1)

= 0.

3, −

L 1, L

2 1

−L

2 2)

= 0.

1,

L 3, L

1

+L

2)

= 0.

= 0.

•

exp( L

2

 ) + L 3,

are related by the constraint  ( L

=L

L 2L

 ln  + L 2  + L 3 ,

1

ñ óò ò ó

1

3,

are related by the constraint  ( L (ñ

4L

2 3,

2, −

L 2 L 3 , − ln L 1 ) = 0.

3, −

L 1, L

sin ï – ñó ò cos ï , (ñó ò ò ó )2 + (ñó ò )2  = 0.

where the constants L

58.

2

ñ óò ò ó ¦ñ

ln ï  = 0.

óò ò ó ï

–ñ

– ln

Solution:

57.

 )+L

ü

are related by the constraint  (− L

3

=L

ñ óò ò ó ,  ñ óò ñ óò

3

sinh  + L

1

Solution:

Hv”

exp( L

 exp  – ï

are related by the constraint  ( L

3

=L

where the constants L

55.

ñ¦ñ óò ò ó ü

sinh ï – ñó ò cosh ï , (ñó ò ò ó )2 – (ñó ò )2  = 0.

Solution:

54.

óò

)2 , Ññ + ñ

sin  + L

2

cos  + L

3,

are related by the constraint  ( L

2 1

+L

2 2)

= 0.

ñ óò  = 0. ¨ The substitution ( ) =  (
 )2 leads to a first-order equation:   , ¨ , 12 ¨2
 = 0.  r òó ñ óò 

r

= 0,

=

r (ï

).

The substitution ¨ ( ) = Y ( )(
 )2 leads to a first-order equation:   , ¨ , 12 ¨2
 = 0. 

2.9.7. Some Transformations 1.

ñóò ò ó

+

ñóò ò ó

=

1 ñ ï –3 w H O ï , ï

P

= 0.

The transformation ˆ = 1  , ¨ =   leads to the equation ¨pŠª

Š +  (ˆ , ¨ ) = 0. 2.

3.

ù (ù

+ 1) ï

–2

ñ

+ï

3

The transformation ˆ =  

÷ H ( ï 2÷

2 +1

öï

+1

,

, ¨ = 

ïƒ÷ ñ

).

leads to the equation (2B + 1)2 ¨2Šª

Š =  (ˆ , ¨ ).

+ý ñ , ï = 0. +õ ô +õ P (˜ + leads to the equation ¨3Šª

Š + ¢ The transformation ˆ = M , ¨ =  +    +  where ¢ =  − ˜( .

ñóò ò ó

+ (ô

ï

+ õ )–3 HwO

ô ï

−2

 (ˆ , ¨ ) = 0,

M

Page 492

493

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

4.

ï 2 ñ óò ò ó

+ ô ï ñó ò + õÿñ + H ( ï , ñ ) = 0. Á The transformation  = ˆ , = ˆ  ¨ , where the parameters @ and “ are found from the simultaneous algebraic equations 2“ + 1 + (  − 1) @ = 0,

2

“

+ (  − 1)“…@ + @

2

= 0,

leads to an equation of the form

¨ ªŠ

Š + @ 2 ˆ −  5.

= ù (ù + 1) ï –2 ñ + ï 3÷ The transformation ˆ =    2

ñóò ò ó

ó

ñóò ò ó

H ( ô ï 2÷ +1

ó

+1

ó

7.

ñóò ò ó

=

ñ

+

ó

+ ý )3

Hv”

ñ óò ò ó

ñ

ï )v H ”

+ sinh–3 (

2

=



ô 2 ó ö 2 ó

+

+ý

ï

coth(

ñ óò ò ó

2

=

ñ

+ cosh–3 (

ï )v H ”

ï

tanh(

ï 2 ñ óò ò ó

1 4

+

ñ

+

ó

  ñ ö  2 ó + ý • 

,

7 ï Hv”ô

ln ï + õ ,

12.

ñ

.

cosh( Π )

ñ

.

¨ ªŠ

Š = Œ leads to the equation 3

ï

cosh(

7 ï •

)•

−2

 (ˆ , ¨ ).

.

leads to the equation ¨ ªŠ

Š = Œ

−2

 (ˆ , ¨ ).

= 0.

¨ ªŠ

Š +  leads to the equation 3

−2

 (ˆ , ¨ ) = 0.

ô ï ï

–ô ñ , . ï +1 | 2 – 1| •  −  ü ¨ ¨ ªŠ

Š =  (ˆ , ¨ ) + ¨ . The transformation ˆ = ln , = leads to the equation 43 2  +1 | − 1|

|ï

2

ñ óò ò ó

) ñóò ò ó

3 2

– 1|

+

ñ

2

=v H ” ln

+ sin–3 (

ï )H

” cot(

ï

ü

),

The transformation ˆ = cot( Œ  ), ¨ = 13.

)•

sinh( Π )

The transformation ˆ =  ln  +  , ¨ = £  11.

ï

sinh(

),

The transformation ˆ = tanh( Œ  ), ¨ = 10.

õ

ñ

),

The transformation ˆ = coth( Œ  ), ¨ = 9.

leads to the equation ¨3Šª

Š = (2 Œ )−2  (ˆ , ¨ ).

  2 +  ¨  , = 2  leads to the equation  (  2 + (  + M M ¨ Šª

Š = (2 ¢eŒ )−2  (ˆ , ¨ ), where ¢ =  − ˜( . M

The transformation ˆ =

8.

). leads to the equation  2 (2B + 1)2 ¨2Šª

Š =  (ˆ , ¨ ).

+ , ¨ = 

= 2 ñ +  3  H ( ô 2  + õ ,  ñ ).   The transformation ˆ =   2  +  , ¨ = 

 3 ( ö  2 ó

ïƒ÷ ñ

+õ ,

6.

2

 (ˆ Á , ˆ  ¨ ) = 0.

−2

ñ óò ò ó

+

2

ñ

+ cos–3 (

ï )H

” tan(

The transformation ˆ = tan( Œ  ), ¨

ï

), =

ñ ï

sin(

)•

sin( Π )

ñ cos(

ï

cos( Π )

= 0. leads to the equation ¨3Šª

Š + Œ

)•

−2

 (ˆ , ¨ ) = 0.

= 0.

¨ ªŠ

Š + Œ leads to the equation 3

−2

 (ˆ , ¨ ) = 0.

Page 493

494 14.

SECOND-ORDER DIFFERENTIAL EQUATIONS sin( ï + ô ) ñ = 0. , sin( ï + õ ) sin( ï + õ ) • sin( Œ  +  ) ¨ The transformation ˆ = , = leads to the equation sin( Œ  +  ) sin( Œ  +  )

ñóò ò ó

2

+

ñ

+ sin–3 (

ï

+ õ ) Hv”

¨ Šª

Š + [ Œ sin(  −  )]−2  (ˆ , ¨ ) = 0. 15.

(ï

2

) ñóò ò ó

+v H ” arctan ï + õ ,

3 2

+ 1)

ñ

7 ï

The transformation ˆ = arctan  +  , ¨ = £  16.

(ï

2

) ñ óò ò ó

3 2

+ 1)

+

H ”

arccot ï + õ ,

7 ï

ñ 2

2

ñóò ò ó

+

H (ï , ñ

) = 0.

The transformation  = Y ( H ),

+1

2

leads to the equation ¨ ªŠ

Š + ¨ +  (ˆ , ¨ ) = 0. = 0.

+1•

The transformation ˆ = arccot  +  , ¨ = £  17.

= 0.

+1•

2

+1

leads to the equation ¨3Šª

Š + ¨ +  (ˆ , ¨ ) = 0.

= ¨ £ !Y ‘
leads to the equation

'
‘
‘'
‘ 3 Y ‘'

‘ 2 ¨ ¨ '‘

‘ + ¬ 1 Y − O ® + ‘ 2 Y
4 Y ‘
P

−2

3J 2 ( !Y ‘
)   Y , ¨ £ !Y ‘
 = 0.

‘ The sign of the parameter  must coincide with that of the derivative Y
. 18.

+ ! ( ï , ñ )(ñó ò )3 + " ( ï , ñ )(ñó ò )2 = 0. Taking to be the independent variable, we obtain the following equation with respect to  =  ( ): 

− ( , )
−  ( , ) = 0.

ñóò ò ó



19.

H ( ï , ñ , „ñ óò , ñ óò ò ó



) = 0. Applying the Legendre transformation  = ¨ug
, = a ¨2g
− ¨ , where ¨ = ¨ (a ), and using the relations 
= a and 

 = 1 ¨2g

g , we arrive at the equation 1 O ¨ g
, a ¨ g
− ¨ , a , ¨

= 0. gg P Given a solution of the original equation, the corresponding solution of the transformed equation is written in parametric form as:

a = 
,

¨ =  
− ,

where

= ( ).

Page 494

Chapter 3

Third-Order Differential Equations 3.1. Linear Equations 3.1.1. Preliminary Remarks 1 b . A homogeneous linear equation of the third order has the general form

 3 ( ) 


 +  2 ( ) 

 +  1 ( ) 
+  0 ( ) = 0. Let

0

=

0(

(1)

 ) be a nontrivial particular solution of this equation. The substitution =

0(

 ) X H ( )  M

leads to a second-order linear equation:



3 0

H

+ (3 


+

2 0)

3 0

H
+ (3 



+ 2

3 0


+

1 0)

2 0

H = 0,

(2)

where the prime denotes differentiation with respect to  . 2 b . Let 1 = 1 ( ) and 2 = 2 ( ) be two nontrivial linearly independent particular solutions of equation (1). Then the general solution of this equation can be written in the form: =L

+L

1 1

2 2

+L

where

j = exp O − X

O

3

 

X

2

2 3

1



M P



j M

 −

1

X


−
 1 2

1 2

2

−2

j

 , M P

(3)

.

For specific equations described below in 3.1.2–3.1.9, often only particular solutions will be given, and the general solution can be obtained by formula (3). 3 b . A nonhomogeneous linear equation of the third-order has the form

 3 ( ) 


 +  2 ( ) 

 +  1 ( ) 
+  0 ( ) = ( ).

(4)

Let 1 = 1 ( ) and 2 = 2 ( ) be two linearly independent particular solutions of the corresponding homogeneous equation (1). Then the general solution of equation (4) is defined by formula (3) with:

j =¢

1  Ž O 1+ X L 3

−2 −

4 b . The substitution



= H exp O −

derivative is absent: where Y  =   

H


+  − Y 2
− 13 Y 3

3

¢  Ž

1 X 3 2 2

+Y



 , M P 

2 3

1

where

 =X

 

2 3

M

 , ¢ =


−
. 1 2

1 2

 reduces equation (1) to a form from which the second M P

 H
+  − 13 Y 2

− 13 Y 1 Y

2

+

2 27

Y

3 2

+Y

0

 H = 0,

( , = 0, 1, 2). 495

Page 495

496

THIRD-ORDER DIFFERENTIAL EQUATIONS

3.1.2. Equations Containing Power Functions 3.1.2-1. Equations of the form  3 ( )


 +  0 ( ) = ( ). 1.

ñóò ò ó ò ó

+

Ė

= 0.

Solution:

a

L

=

+L

 + L 3 L 1 exp(− ,! ) + L 1

2

where , = Π1 J 3 . 2.

ñóò ò ó ò ó

+

Ė

ô ï

=

2

=¨ +

Solution:

+õ 1

¨2


 + Œ ¨ = 0.

3.

ô ï ñ

ñ óò ò ó ò ó

=

ñ óò ò ó ò ó

+ (ô

ñóò ò ó ò ó

+

ñ óò ò ó ò ó

+ (3 ô

ñóò ò ó ò ó

=

Œ

ï

2 2

exp 

1 2

,!  cos 

2

3 2

,!  + L

3

exp 

1 2

,!  sin 

2

if Π= 0, 3 2

,! 

if Œ ≠ 0,

+ö .

( 

2

+ ' + ( ), where ¨ is the general solution of the equation 3.1.2.1:

+õ .

This is a special case of equation 5.1.2.3 with B = 3. 4.

ï

+ õ )ñ = 0.

For  = 0, this is an equation of the form 3.1.2.1. For  ≠ 0, the substitution ˆ =  +  leads to an equation of the form 3.1.2.3: 
Šª
Šª
Š + ˆ = 0. 5.

ô ï 3ñ

õï

=

.

The substitution ˆ =  6.

7.

2

ï

–ô

3

2

ï 3 )ñ

leads to an equation of the form 3.1.2.126: 2ˆ "
Šª
Šª
Š +3
Šª
Š + 14 ˆ

= 14  .

= 0.

Integrating, we obtain a second-order nonhomogeneous linear equation: 

 +  
+ (  2  2 −  ) = L exp  12  2  (see 2.1.2.31 for the solution of the corresponding homogeneous equation).

ô ï _ ñ

.

This is a special case of equation 5.1.2.4 with B = 3.

1 b . For ý = −9, −7, −6, −9 2, −3, −3 2, 1, and 3, see equations 3.1.2.17, 3.1.2.14, 3.1.2.11, 3.1.2.19, 3.1.2.10, 3.1.2.18, 3.1.2.3, and 3.1.2.5, respectively. 2 b . The transformation  = a

8.

9.

ñóò ò ó ò ó

+ [ô

ï ÷

– 3ô

3 3

2

ù ï 2÷

–1

, = ‡"a

leads to an equation of similar form: ‡


 = − a −:

‡ . : ¨ : 3 b . For ý ≠ −3, the transformation ˆ =  ( +3) J 3 , =  J 3 leads to an equation of the form 3.1.2.69: ˆ 3 ¨2Šª

Šª
Š + (1 − @ 2 )ˆ ¨2Š
+ ( @ 2 − 1 − @ 3 ˆ 3 )¨ = 0, where @ = : 3+3 . −1

+

ôù (ù

−2

– 1) ïƒ÷

–2

−6

]ñ = 0.

   +1   +1 Particular solution: 0 = exp O − . The substitution = exp O − X H ( )  leads v B +1 P B +1 P M


to a second-order linear equation of the form 2.1.2.47: H  −3   H  +(3  2  2  −3   B/  −1 ) H = 0.

ï ñóò ò ó ò ó

– ô 2(ô

ï

+ 3)ñ = 0.





Particular solution: 0 =    . The substitution =    X H ( )  leads to a second-order M equation of the form 2.1.2.108: "H 

 + 3(  + 1)H 
+ 3  (  + 2) H = 0.

Page 496

497

3.1. LINEAR EQUATIONS

10.

11.

ï 3 ñ óò ò ó ò ó

= ô (ô

– 1)ñ .

2

This is a special case of equation 3.1.2.175. Solution: =  (L 1   1 + L 2 where B 1 and B 2 are roots of the quadratic equation B + B +  2 − 1 = 0.

ï 6 ñ óò ò ó ò ó

=

ôñ

+

õï

2

( ï – ô )3 ( ï – õ )3 ñó ò ò ó ò

13.

(ô

−2

ï

2

+

õï

+ ö )3 ñ

óò ò ó ò ó

üIñ

=

ï 7 ñ óò ò ó ò ó

=

ôñ

+

õï

3

ï 7 ñ óò ò ó ò ó

ï

+ (ô

ï 9 ñ óò ò ó ò ó

+ (ô



= a

+ õ )ñ = 0.

ï )ñ

– 3ô

3

= a

2 2

ï 9 ñ óò ò ó ò ó

=

ôñ



=

ï 9 ) 2 ñ óò ò ó ò ó

=

ôñ

.

ôñ

.

−1

2

+ ' + (

leads to a constant coefficient

−1

,

= ¨ a

−2

leads to a linear equation of the form 3.1.2.3:

−1

,

= ¨ a

−2

leads to a linear equation of the form 3.1.2.4:

,

= ¨ a

−2

leads to a linear equation of the form 3.1.2.6:

=¨ a

−2

−1

.

The transformation  = a

ï 3 ) 2 ñ óò ò ó ò ó

2

= 0.

The transformation  = a ¨2g

g
g + (3  2 a −  3 a 3 )¨ = 0.

18.

  ),

.

.

The transformation  ¨2g

g
g − ( 'a +  )¨ = 0.

17.

3

– ö^ñ = 0, ô ≠õ.  − ¨ The transformation a = ln û , = leads to a constant coefficient linear equation:  −  ûû (  −  )2 û ¨ 2 ¨


2 ¨

2 ¨
(  −  )3 ( g g g − 3 g g + 2 gû ) − ( û = 0.

The transformation  ¨2g

g
g + a ¨ +  = 0.

16.

+L

leads to a constant coefficient linear equation of the

, ¨ = M + '   + (  linear equation: ¨3Šª

Šª
Š + (4 !( −  2 )¨2Š
= , ¨ .

15.

2

ó

The transformation ˆ = X

14.

 

.

The transformation  = a −1 , = ¨ a form 3.1.2.2: ¨3g

g
g +  ¨ +  = 0.

12.

2

,

¨ g

g
g + a 3 ¨ = 0. leads to an equation of the form 3.1.2.5: u

This is a special case of equation 5.1.2.8 with B = 1. 19.

This is a special case of equation 5.1.2.9 with B = 1. 3.1.2-2. Equations of the form  3 ( )


 +  1 ( )
 +  0 ( ) = ( ). 20.

21.

ñ óò ò ó ò ó

+

ôƒõÿñ óò

ñóò ò ó ò ó

+

ô ï ñóò

+ô

2

ï

(3 –

õ

–ô

ï 2 )ñ

= 0.

Integrating, we obtain a second-order nonhomogeneous linear equation:

 +  
 + (  2  2 +  −  ) = L exp  12  2  (see 2.1.2.31 for the solution of the corresponding homogeneous equation).

Solution:

+ ôùñ = 0, ù = 1, 2, 3, uuu = ‡ (  −1), where ‡ is the solution of the second-order linear equation ‡

 + ‡ = L .

Page 497

498 22.

23.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

+

ñóò ò ó ò ó

+

ô ï ñ óò

– 2 ôñ = 0.

ô ï ñóò

+ õ (ô

The substitution ¨ ¨2

 +  ¨ = 0.


− 2 leads to a second-order linear equation of the form 2.1.2.2:

=

ï

õ 2 )ñ

+

24.

25.

26.

= 0.



Particular solution: 0 =   . The substitution ¨ = 
+  equation of the form 2.1.2.12: ¨u

 −  ¨2
+ (  +  2 )¨ = 0. −

ñóò ò ó ò ó

+

ô ï ñóò

ï

+ ( ôƒõ

+

ô

+

õ 3 )ñ

leads to a second-order linear

= 0.

Integrating yields a second-order linear equation: "

 −  
 + (  +  2 ) = L  2.1.2.108 for the solution of the corresponding homogeneous equation with L = 0).

ñóò ò ó ò ó

ï

+ (ô

−



(see

+ õ )ñó ò + ôñ = 0.

Integrating yields a second-order nonhomogeneous linear equation: 

 + (  +  ) = L 2.1.2.2 for the solution of the corresponding homogeneous equation).

ñ óò ò ó ò ó



ï

+ (ô

+ õ )ñ

óò

– ôñ = 0.

(see




 −  +  (  +  )2 leads to a second-order linear equation of the form 2.1.2.67: žˆ H Šª

Š + 3 H Š
+  −2 ˆ 2 H = 0. Particular solution:

27.

ñóò ò ó ò ó

+ (ô

ñ óò ò ó ò ó

+ (2 ô

ñ óò ò ó ò ó

+ (ô

ï

0

=  +  . The transformation ˆ =  +  , H =

+ õ )ñó ò + 3 ôñ = 0.

The substitution   ˆ =  +  leads to a linear equation of the form 3.1.2.21 with B = 3: 
Šª
Šª
Š + ˆ 
Š + 3  = 0. 28.

ï

+ õ )ñ

óò

+

ôñ

= 0. 1 2

The substitution   ˆ =  + 
Šª
Šª
Š + 2 ˆ 
Š +  = 0. 29.

30.

31.

32.

ï

–

õ 2 )ñ ó ò

+

ôƒõ ï ñ

= 0.

The substitution ¨ = 
+  ¨2

 −  ¨2
+  ¨ = 0.

ñóò ò ó ò ó

ï

+ (ô

–

õ 2 )ñ óò

leads to a second-order linear equation of the form 2.1.2.108:

ï

+ ô (õ

 leads to a linear equation of the form 3.1.2.48 with B = 1:

+ 1)ñ = 0.

Integrating yields a second-order linear equation: "

 −  
 +  for the solution of the corresponding homogeneous equation with L

ñóò ò ó ò ó

+ (ô

ï

ñóò ò ó ò ó

+ (ô

ï

+ õ )ñó ò + ö ( ô

ï

+

õ

= L  − = 0).

(see 2.1.2.108

+ ö 2 )ñ = 0.

The substitution ¨ = 
+ ( leads to a second-order linear equation of the form 2.1.2.12: ¨2

 − ( ¨2
+ (  +  + ( 2 )¨ = 0. + õ )ñó ò + ö

Particular solution:

0

ï (ö 2 ï

2

+

ô ï

+

õ

– 3 ö )ñ = 0.

= exp  − 21 (˜ 2  . The substitution

= exp  − 12 (˜ 2  X H ( )  leads to

a second-order linear equation of the form 2.1.2.31: H 

 − 3 (˜"H 
+ (3 ( 2  33.



ñóò ò ó ò ó

+

ô ï 2 ñ óò

+ô

ï ñ

2

M

+  +  − 3 ( ) H = 0.

= 0.

This is a special case of equation 3.1.2.47 with B = 1.

Page 498

499

3.1. LINEAR EQUATIONS

34.

35.

36.

37.

38.

39.

40.

41.

ô ï 2 ñ óò

ñóò ò ó ò ó

+

ñóò ò ó ò ó

–ô

ñóò ò ó ò ó

+

ñóò ò ó ò ó

+ ( ô – 1) õ

ñóò ò ó ò ó

+ (ô

ï

ñóò ò ó ò ó

+ (ô

ï

ï ñ

– 2ô

= 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.7: ¨2

 +  2 ¨ = 0. 2

ï 2 ñ óò

+ô

ï ñ

2

= 0.

Integrating yields a second-order linear equation: "

 +  
 −  = L exp  12  2  (see 2.1.2.108 for the solution of the corresponding homogeneous equation with L = 0).

ô ï 2 ñ óò

ï

+ õ (ô

2

+

õ 2 )ñ

= 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.31: ¨ 

 −  ¨ 
+ (  2 +  2 )¨ = 0. 2

ï 2 ñ óò

+

õ 2 ï (ƒô õ ï

2

+ 2 ô + 1)ñ = 0.

Integrating, we obtain a second-order nonhomogeneous linear equation:

 − ' 
 + (  2  2 +  ) = L exp  − 12 ' 2  (see 2.1.2.31 for the solution of the corresponding homogeneous equation). 2

+ õ )ñó ò + 2 ô

ï ñ

= 0.

Integrating yields a second-order nonhomogeneous linear equation:

 + (  (see 2.1.2.4 for the solution of the corresponding homogeneous equation). 2

–

õ 2 )ñ óò

+

ô ï

õ ï )ñ

(2 –

= 0.

Integrating yields a second-order linear equation: "

 +  
 +  2 = L  the solution of the corresponding homogeneous equation with L = 0).

ñóò ò ó ò ó

+ (ô

ñóò ò ó ò ó

– (3 õ

ï

2

+ õ )ñó ò + ö ( ô

ï

2

õ

+

+

ö 2 )ñ



+ ) =L

(see 2.1.2.13 for

= 0.

The substitution ¨ = 
+ ( leads to a second-order linear equation of the form 2.1.2.13: ¨ 

 − ( ¨ 
+ (  2 +  + ( 2 )¨ = 0. 2

ï

2

+ ô + 3 õ )ñó ò + 2 õ

ï (õ 2 ï

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0:

ñóò ò ó ò ó

ï

+ (ô

2

+

õï

ü ï [(ô

+ ö )ñó ò +

2 1

– ô )ñ = 0.

= exp 

1 2

'

2

+ £   ,

 ' 2  cos   £ |  |  ,

1 1 = exp 2

3 b . Particular solutions with  = 0: 42.



2

1

+

= exp 

ü 2 )ï

2

+

1 2

õï

' 2  ,

2

2

= exp  2 = exp

=  exp 

1 2

1 2 1 2

'

2

− £   .

 ' 2  sin   £ |  |  .

' 2  .

+ ö – 3 ü ]ñ = 0.

= exp  − 21 ,! 2  v X H ( )  leads M


to a second-order equation of the form 2.1.2.31: H  − 3 ,!"H  + [(  + 3 , 2 ) 2 + ' + ( − 3 , ] H = 0.

Particular solution:

43.

ñóò ò ó ò ó

+ (ô

ñóò ò ó ò ó

+

ï

4

+

0

õ ï )ñ óò

= exp  − 12 ,! 2  . The substitution

– 2( ô

ï

3

+ õ )ñ = 0.

This is a special case of equation 3.1.2.49 with B = 2. 44.

45.

ô ƒï ÷ ñ óò

– 2ô

ïƒ÷ –1 ñ

= 0.

The substitution ¨ =  
− 2 ¨2

 +   ¨ = 0.

leads to a second-order linear equation of the form 2.1.2.7:

+ ô ï ÷ ñ ó ò + ôù ï ÷ –1 ñ = 0. Integrating yields a second-order nonhomogeneous linear equation:

 +   2.1.2.7 for the solution of the corresponding homogeneous equation).

ñ óò ò ó ò ó

=L

(see

Page 499

500 46.

47.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

ñóò ò ó ò ó

+

ô ïƒú

+1

ø

49.

50.

52.

ñ

ô ï 2÷ ñ óò

−1

ôù ï 2÷ –1 ñ

+

=L

1

ø



= 0.

= 0.

ø

Á 2 (‡ ) + L 2 

Á (‡ ) n Á (‡ ) + L 3 "n Á 2 (‡ ), where @ =

Á (‡ ) and n Á (‡ ) are the Bessel functions.

ñ óò ò ó ò ó

+ 2ô

ï ÷ ñ óò

+ ôù

ñóò ò ó ò ó

+ (ô

ï 2÷

+

õ ïƒ÷

–1

ñ óò ò ó ò ó

+ (ô

ï 2÷

+

õï ÷

–1

= L 1¨

ï ÷ –1 ñ

£ Ñ  +1 1 , ‡ = ; 2(B + 1) 2(B + 1)

= 0.

¨ 2 ¨ ¨ Solution: 2 + L 3 2 . Here, 1 and 2 are a fundamental set of solutions of a second-order linear equation of the form 2.1.2.7: 2¨p

 +   ¨ = 0. 2 1

+ L 2¨ 1¨

ï 2÷

)ñó ò – 2( ô

–1

+

õ ïƒ÷

–2

)ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.10: ¨2

 + (  2  + '  −1 )¨ = 0.

óò

)ñ

– ö [( ô + ö 2 ) ï

3

÷

+ ( õ + 3 ö^ù ) ï

÷

2 –1

+ ù (ù – 1) ï

÷

–2

]ñ = 0.

(˜  +1 (˜  +1 . The substitution = exp O XvH ( )  leads to B +1 P B +1 P M


 2 2  a second-order equation of the form 2.1.2.47: H  +3 (˜ H  +[(  +3 ( ) +(  +3 (˜B )  −1] H = 0. Particular solution:

51.

+ 3) ïƒú

+ ô (þ

The substitution  = a , = ¨ a −2 leads to an equation of the form 3.1.2.45 with B = − − 5: ¡ ¨2g

g
g + a − + −5 ¨2g
−  ( + 5)a − + −6 ¨ = 0. ¡

Solution:

48.

ñóò

ï ñóò ò ó ò ó

+

ôñóò

ï ñ óò ò ó ò ó

+

ô ï ñ óò

0

+ õ (õ

2

ï

= exp O

+ ô )ñ = 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.108:  ¨2

 − ' ¨2
+ (  2  +  )¨ = 0. – [õ (ô +

õ 2)ï 



53.

54.

55.

ï ñóò ò ó ò ó

+ (õ – ô

2

0

ï )ñ óò

. The substitution



+ ôƒõÿñ = 0.

The substitution ¨ = 
+   ¨2

 −  ¨2
+  ¨ = 0.

ï ñóò ò ó ò ó

+ (ô

ï

2

ï ñóò ò ó ò ó

+ (ô

ï

3

= 

 X H (

)  leads to a second-order M


linear equation of the form 2.1.2.108: "  H  + 3( ' + 1)H  + [(  + 3  2 ) + 6  ] H = 0. Particular solution:

= 

+ ô + 3 õ 2 ]ñ = 0.

leads to a second-order linear equation of the form 2.1.2.108:

+

õ ï )ñ óò

– 2( ô

ï

+

õ ï )ñ óò

– 2( ô

ï

+ õ )ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.2: ¨ 

 + (  +  )¨ = 0. 2

+ õ )ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.4: ¨2

 + (  2 +  )¨ = 0.

56.

(ô

ï

+ õ )ñ„ó ò ò ó ò

57.

(ô

ï

+ 2)ñ„ó ò ò ó ò

ó

+ ö^ñó ò + ü ( ôƒü

2

ï ñóò

3

The substitution ¨ = (  +  )¨ 

 − , (  + 

ó

–ô

3

Particular solutions:

ï

+

ñ

= 0.

õü

2

+ ö )ñ = 0.


+ , leads to a second-order linear equation of the form 2.1.2.108: )¨ 
+ ( , 2  + , 2 + ( )¨ = 0. + 2ô 1

2

= ,

2

 =  − .

Page 500

501

3.1. LINEAR EQUATIONS

58.

ï

( ô„ö

+

õ^ö

– ô )ñ„ó ò ò ó ò

ó

Particular solutions: (ô

60.

(ô ï

61.

(ô

ï

59.

+ õ )ñ

óò ò ó ò ó

1

ï

+ (ö

64.

65.

66.

67.

+ õ )ñ„ó ò ò ó ò ó + [( ö – ôƒü ) ï 2

= 0.

+ õ )ñ

óò ò ó ò ó

ï

– (ô

ï 2 ñ óò ò ó ò ó

3 3

ï 2 ñ óò ò ó ò ó

+ (ô

ï

ï 2 ñ óò ò ó ò ó

+ (ô

ï

2

– 3ô

1

ï 2ñ

– 6ñó ò + ô

ý

–õ

ü ]ñ óò

+

õ 3 )ñ ó ò

ï

+ ü (ö

+ ý )ñ = 0.


+ , leads to a second-order linear equation of the form 2.1.2.108: )¨2
+ ( (˜ + )¨ = 0. M

The substitution ¨ = (  +  )¨2

 − , (  + 

ï

+

M

2

2





=

,

ï

+ 2õ

= ¨

ï

ï (ô 2 ï

+ ôƒõ

2

– 3ô –

õ 2 )ñ

= 0.

= exp  − 21  2  .

2

= 0.

¨ 


 + 6 2 2 ¨ 

 + leads to an equation of the form 3.1.2.173:  3 2

2

+

õï

–þ

+

õï

+ ö )ñó ò – ü [( ô +

2

–þ

)ñ

óò

+ (þ

ï

– 1)(ô

+ õ )ñ = 0.

The substitution ¨ =  
+ ( − 1) leads to a second-order linear equation of the form ¡ 2.1.2.108:  ¨ 

 − ( + 1)¨ 
+ (  +  )¨ = 0. ¡ 2

The substitution ¨ = 
− ,  2 ¨2

 + ,! 2 ¨2
+ [(  + , 2 )

ï 2 ñ óò ò ó ò ó

+ (ô

ï ÷

ï 2 ñ óò ò ó ò ó

+ (ô

ïƒ÷

ï 2 ñ óò ò ó ò ó

– 3ô

ï ÷

–

õ

2

+1

–

õ

– õ )ñ

2

ü 2 )ï

+ ô ( õ – 1) ï

óò

2

+

õï

+ ö ]ñ = 0.

leads to a second-order linear equation of the form 2.1.2.135: + ' + ( ]¨ = 0.

÷ –1 ñ

= 0.

The substitution ¨ =  
+ (  − 1) leads to a second-order equation of the form 2.1.2.67:  ¨ 

 − (  + 1)¨ 
+   −1 ¨ = 0. 2

– õ )ñó ò + ô ( õ – 1) ïƒ÷

ñ

= 0.

The substitution ¨ =  
+ (  − 1) leads to a second-order linear equation of the form 2.1.2.67:  ¨3

 − (  + 1)¨2
+   ¨ = 0. +1

(ù + ô

ï ÷

+1

)ñ

óò

ï ÷ (ù

+ô

ï (ô ï

+ õ )ñ„ó ò ò ó ò

ó

+ ï (ö

ï

+ ý )ñó ò – 2( ö

–ù

2

+ 2ô

2

ï 2÷

   +1 , 1 = exp O B +1 P   +1 . 1 =  , 2 = 

2 b . Particular solutions with B = −1:

69.

ñ

+ s[( ô s2 + ö ) ï + õ s2 + ý ]ñ = 0.

óò

1 b . Particular solutions with B ≠ −1:

68.

3

v . 2 = 

The substitution ¨ = 
+ ° leads to a second-order linear equation of the form 2.1.2.108: ¨ 
+ [( !° 2 + ( ) + ˜° 2 + ]¨ = 0. (  +  )¨2

 − ° (  +  )2

The substitution (  3 − 12)¨ + 2  = 0. 63.

+ õ )ñó ò + ô„ö

=  +  ,

+ ý )ñ

Particular solutions: 62.

ï

– ö 3 (ô

ï

+2

)ñ = 0.

2

=  exp O

  +1 . B +1 P

+ ý )ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.108: (  +  )¨ 

 + ( (˜ + )¨ = 0. M

ï 3 ñ óò ò ó ò ó

+ (1 – ô 2 ) ï

ñóò

+ (õ

ï

3

+ô

2

– 1)ñ = 0.

For  = A 1, we have a constant coefficient equation of the form 3.1.2.1. For  = 0, we obtain the Euler equation 3.1.2.175.

Page 501

502

THIRD-ORDER DIFFERENTIAL EQUATIONS 1 b . If  ≠ 0 and  is a positive integer greater than 1, then the solution is: = where Œ 1 , Œ 2 , and Œ degree ≤ 3(  − 1).

3

1−

 ½

3



=1

.

L  exp(− Œ   )  ( ),

are roots of the cubic equation Œ

3

.

=  and  ( ) are polynomials of

2 b . Denote the solution of the original equation for arbitrary (including complex)  by  . Then the following recurrence relation holds:



+3

=   + (2  + 3)

−1



 − (  + 1)(2 + 3)(

−2


 −

−3

 ),

(1)

where the prime denotes differentiation with respect to  .  Since the functions 1 1 =  −  , corresponding to three values of Œ determined by the 3 equation Œ =  , form a fundamental set of solutions, formula (1) makes it possible to find  all  for any integer values of B not divisible by 3. In particular, 2 = ( −1 + Œ )  −  , where 3 Œ =. 70.

71.

72.

73.

74.

75.

76.

77.

ï 3 ñ óò ò ó ò ó

+ (4 ï 3 + ô ï )ñ ó ò – ôñ = 0. = L 1  ø Á 2 ( ) + L 2  ø Á ( ) n Á ( ) + L Solution: Bessel functions; 4 @ 2 = 1 −  .

3

"n Á 2 ( ), where ø Á ( ) and n Á ( ) are the

ï 3 ñ óò ò ó ò ó

+ ï [ ô ï 2 + 3 õ (1 – õ )]ñ ó ò + 2 õ ( ô ï 2 + õ 2 – 1)ñ = 0. 1 b . Particular solutions with  > 0: 1 = ' sin  £   , 2 = ' cos  £   .

2 b . Particular solutions with  < 0:

1

3 b . Particular solutions with  = 0:

1

ï 3 ñ óò ò ó ò ó

= ' exp  − £ −   ,

= ' ,

2

= '

+1

2

= ' exp   £ −   .

.

+ ï ( ô ï 2 + õ ï + ö )ñó ò + ( ü – 1)( ô ï 2 + õ ï + ö + ü 2 + ü )ñ = 0. The substitution ¨ =  
+ ( , − 1) leads to a second-order linear equation of the form 2.1.2.131:  2 ¨2

 − ( , + 1) ¨2
+ (  2 + ' + ( + , 2 + , )¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ô ïƒ÷ ñó ò + ( õ – 1)(ô ïw÷ –1 + õ 2 + õ )ñ = 0. The substitution ¨ =  
+ (  − 1) leads to a second-order linear equation of the form 2.1.2.132:  2 ¨2

 − (  + 1) ¨2
+ (   −1 +  2 +  )¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ï ( ô ïƒ÷ + õ )ñó ò – 2( ô ïƒ÷ + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.118:  2 ¨2

 + (   +  )¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ï ( ô ïƒ÷ + õ – ö )ñó ò + ( ö – 1)(ô ïw÷ + õ + ö 2 )ñ = 0. The substitution ¨ =  
+ ( ( − 1) leads to a second-order linear equation of the form 2.1.2.132:  2 ¨2

 − ( ( + 1) ¨2
+ (   +  + ( 2 )¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ( ô ï 2÷ + 1 – ù 2 ) ï ñó ò + [ õ ï 3÷ + ô (ù – 1) ï 2÷ + ù 2 – 1]ñ = 0. The transformation ˆ =   B , H =   −1 leads to a constant coefficient linear equation: H Šª

Šª
Š + !H Š
+ ˜H = 0.

ï 2(ô ï

+ õ )ñó ò ò ó ò ó + ( ö ï – õÿþ 2 – õÿþ )ñó ò + (þ – 1)(ö + ôþ 2 + ôþ )ñ = 0. The substitution ¨ =  
+ ( − 1) leads to a second-order linear equation of the form ¡ 2.1.2.172:  (  +  )¨3

 − ( + 1)(  +  )¨3
+ ( ( +  2 +  )¨ = 0.

¡

¡

¡

Page 502

503

3.1. LINEAR EQUATIONS

78.

79.

ï (ô ï

2

õï

+

+ ö )ñó ò ò ó ò

ñóò

– 2ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.179: (  2 + ' + ( )¨2

 + ¨ = 0.

ï 5 ñ óò ò ó ò ó

= ô (ï

ï 6 ñ óò ò ó ò ó

+

ñóò

– 2ñ ).

a

 2VL  2VL

=

Solution: 80.

+ï

ó

ô ï 2 ñ óò

1

+L

2

exp  £    + L

£ −   + L 1 + L 2 cos 

+ (õ – 2ô

ï )ñ

3

exp  − £    W

£ −   W 3 sin 

if  > 0, if  < 0.

= 0.

The transformation  = a , = ¨ a −2 leads to a constant coefficient linear equation of the form 3.1.2.82 with  2 = 0: ¨2g

g
g +  ¨2g
−  ¨ = 0. −1

3.1.2-3. Equations of the form  3 ( )


 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ). 81.

ñóò ò ó ò ó

+ 3 ôñó ò ò ó + 3 ô

 =  − (L

Solution: 82.

ñ óò ò ó ò ó

2

ñóò

+

1+L

ô 3ñ

= 0.

2 2  + L 3  ).

ô 2 ñ óò ò ó + ô 1 ñ óò + ô 0 ñ = 0. A third-order constant coefficient linear equation. . Denote ( Œ ) = Œ 3 +  2 Œ 2 +  1 Œ +  0 . . 1 b . Let the characteristic polynomial ( Œ ) be factorizable: . ( Œ ) = ( Œ − Œ )( Œ − Œ )( Œ − Œ ), where Œ , Œ , and Œ are real numbers. 1 2 3 1 2 3   L 1   + L 2    + L 3    if all the roots Œ  are different, +

1

=

Solution:

2 b . Let

.

=L

84.

ôñóò ò ó

1

+ (õ

ï

ñóò ò ó ò ó

+

ñóò ò ó ò ó

+ 3 ôñó ò ò ó + 2( õ

2

1+L 1+L

( Œ ) = ( Œ − Œ 1 )( Œ

Solution: 83.

(L (L



1



2

3

1 + L 3  2  )   2 + L 3  2 )  1

if Œ if Œ

3

+ 2  1 Π+  0 ), where 

+

−



1



+ ö )ñó ò + ( ôƒõ

(L

ï

2

cos “/ + L

3

2 1

1 1

=Œ =Œ

86.

87.

2

≠ Œ 3, = Œ 3.

<  0.

sin “/ ), where “ = Å  0 −  21 .

+ ô„ö + õ )ñ = 0.

 Integrating yields a second-order linear equation: "

 + ( ' + ( ) = L  −  (see 2.1.2.2 for the solution of the corresponding homogeneous equation with L = 0).

ï

ï

+ ô 2 )ñó ò + õ (2 ô

+ 1)ñ = 0.

This is a special case of equation 3.1.2.113 with B = 0 and 85.

2

ñóò ò ó ò ó

+

ôñóò ò ó

+ (õ

ï

2

+

öï

+ ý )ñó ò + ô ( õ

The substitution ¨ = 
+  ¨2

 + ( ' 2 + (˜ + )¨ = 0. M

ñ óò ò ó ò ó

+

ôñ óò ò ó

+

õ ï ÷ ñ óò

+

ñóò ò ó ò ó

+ 3ô

ï ñóò ò ó

+ 3ô

ï ñóò

2 2

2

+

öï

= 1.

+ ý )ñ = 0.

leads to a second-order linear equation of the form 2.1.2.6:

ôƒõ ï ÷ ñ

The substitution ¨ = 
+  ¨2

 + '  ¨ = 0.

ï

¡

= 0.

leads to a second-order linear equation of the form 2.1.2.7: + (ô

3

ï

3

+ õ )ñ = 0.

The substitution = ¨ exp  − 12  2  leads to a constant coefficient linear equation of the form 3.1.2.82 with  2 = 0: ¨2


 − 3  ¨2
+  ¨ = 0.

Page 503

504 88.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

ô ï ñóò ò ó

ï

+ ( ôƒõ

Particular solutions: 89.

90.

ñ óò ò ó ò ó

ï ñ óò ò ó

+ 3ô

–

=

−

2 2

+

1

ï

+ (2 ô





ô

ï ñ óò ò ó

+ 3ô

2 1

+ L 2¨ 1¨

õ 2 )ñ óò

+

,

=

−

óò

+

2

+ õ )ñ

ôƒõÿñ

= 0.



 X

exp  2 ' − 12    2

ôƒõ ï ñ

ï

+ (2 ô

¨ 2 ¨ 2 + L 3 2 . Here,

ï

+ 2õ

2 2

ñóò ò ó ò ó

ï ñóò ò ó

+ 3ô

ï

+ 3( ô

óò

+ ô )ñ

ï

+ õ (2 ô

ï ñóò ò ó

+ 3ô

+ [2( ô

2

+ õ )ï

3 3

   ¨ 2

The substitution = exp  ¨2


 + [(  − 3  2 ) + ( ]¨ = 0.

ñóò ò ó ò ó

ï

+ ô )ñó ò + ( ô

2 2

− 12

92.

+ 1)ñ = 0.

2

2

+

õï

94.

ñ óò ò ó ò ó

ï

+ (ô

+ õ )ñ

óò ò ó

leads to a linear equation of the form 3.1.2.4:

+ ô ]ñó ò + 2 õ

ï (ô ï

+ 1)ñ = 0.

2

96.

ñ óò ò ó ò ó

+ ( ôƒõ

ï

+

ô

ñóò ò ó ò ó

ï

óò ò ó

+ õ )ñ

+

óò

+ ö )ñ

ôƒõ 2 ï ñ óò

= ,

1

+

õ^ö^ñ

= 0.

=

2

– ôƒõ −





ñ

2

−





.

ï

+ 1)ñ = 0.

Integrating yields a second-order linear equation: "

 +  
 + (˜ = L  for the solution of the corresponding homogeneous equation with L = 0).

ñóò ò ó ò ó

ï

+ (ô

+

õ

+ ö )ñó ò ò ó + ( ô„ö

ñóò ò ó ò ó

ï

+ (ô

+ ,

õ^ö 2

+ s)ñ„ó ò + s( ô = 

 2

ï

2

−





(see 2.1.2.28

+ õ )ñ = 0.

, where Œ

+ 2 ô )ñó ò + ô [( ö – ôƒõ ) ï

+ õ )ñó ò ò ó + ( ö

ï

 1

ï

(see 2.1.2.28 for

= 0.

+ õ )ñó ò ò ó + [( ôƒõ + ö ) ï + ô ]ñ„ó ò + ö ( õ

+ (ô

Particular solutions: 1 =  equation Œ 2 + (Œ + ° = 0. 97.

ô

+

= 2.

¡

Integrating yields a second-order linear equation: "

 +  
 + ( = L  the solution of the corresponding homogeneous equation with L = 0).

Particular solutions: 95.

ï

+ ( ôƒõ

= 1.

¡

+ ö )ñ = 0.

This is a special case of equation 3.1.2.113 with B = 1 and 93.

 .

= 0.

This is a special case of equation 3.1.2.113 with B = 1 and 91.

M

¨ Solution: 1 and 2 are linearly independent solutions ¨ p

of a second-order equation of the form 2.1.2.28:   +  ¨2
+ 14  ¨ = 0.

ñ óò ò ó ò ó

= L 1¨

ô

+

1

and Œ

2

are roots of the quadratic

+ õ ]ñ = 0.

 − 21  2  . The substitution = exp  − 12  2  X H ( )  leads to a M   ) H 
+[  2  2 +( ( −2  ) −  ]H = 0. second-order linear equation of the form 2.1.2.31: H 

 +(  −2  Particular solution:

98.

99.

ñóò ò ó ò ó

+ (ô

ï

ñóò ò ó ò ó

+ (ô

ï

0 = exp

+ õ )ñó ò ò ó + ( ö

ï

+ ý )ñó ò + [ ô„ö

ï

2

+ ( ô„ý +

õ^ö )ï

+

ö

+ õ^ý ]ñ = 0.

Integrating yields a second-order linear equation: "

 + ( (˜ + ) = L exp  − 12  2 − '  (see M 2.1.2.2 for the solution of the corresponding homogeneous equation with L = 0). + õ )ñó ò ò ó + ( 

ï

2

+

ï

+ )ñó ò – ü [ 

ï

2

+ ( ôƒü +  ) ï +

ü

2

+

õü

+ ]ñ = 0.

The substitution ¨ = 
− , leads to a second-order linear equation of the form 2.1.2.31: ¨2

 + (  +  + , )¨2
+ [ ƒ… 2 + ( , + ý ) + , 2 + , + = ]¨ = 0.

Page 504

505

3.1. LINEAR EQUATIONS

100.

– ï 2 ñó ò ò ó + ( ô + õ – 1)ï ñ„ó ò – ôƒõÿñ = 0. The following three series, converging for any  , make up a fundamental set of solutions:

ñóò ò ó ò ó

1

2

3

101.

102.

103.

½

=1

 (  − 3)(  − 3)  ( − 3B + 3)(  − 3B + 3) 3   , (3B )!

=1

(  − 1)(  − 1)(  − 4)(  − 4)  (  − 3B + 2)(  − 3B + 2) 3   (3B + 1)!

=1+ ÷

½



= + ÷ =





2

2

½

(  − 2)(  − 2)(  − 5)(  − 5)  (  − 3B + 1)(  − 3B + 1) 3   (3B + 2)!

+ ÷



=1

105.

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

–

õÿñóò

= 0.

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

ïƒ÷ –1 ñ óò

– 2ô

1

.

2

2

= cos   £ −   ,

ïƒ÷ –2 ñ

+ 2ô

= ,

= exp(− £  ),

= exp( £  ).

2

= sin   £ −   .

= 0.

2

= .

+ ô ï ÷ ñ ó ò ò ó + õ ï ÷ –1 ñ ó ò – 2( ô + õ ) ï ÷ –2 ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.45: ¨2

 +   ¨2
+ (  +  )  −1 ¨ = 0.

ñ óò ò ó ò ó ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

+

ôƒõ ƒï ÷ ñ óò

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

– (ô

1

ïƒ÷

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

= exp  –1

õ 2 (ô ïƒ÷

õ ï 2 )ñ óò

–

– õ )ñ = 0.

'  cos 

− 12

ïƒ÷

1

=



−

õ ïƒú ñóò 0

ôù ïƒ÷

+

–

= .



,

–1 2

õ ïƒú

–1

ñ

3 2

2

'  ,

−



 X

+1

= sin 

õ 2 )ñ óò

–

=

2

õ ï (ô ïƒ÷

+

£  , 1 1 = cos  2 

+ ( ôƒõ

+ ô ïƒ÷ ñó ò ò ó + Particular solution:

ñóò ò ó ò ó

+

2

Particular solutions: 108.

ïƒ÷ ñ

– ôƒõ

Particular solutions: 107.

+2

+ ô ïƒ÷ ñó ò ò ó – 2 ô ïƒ÷ –2 ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.45: ¨ 

 +   ¨ 
+   −1 ¨ = 0.

Particular solutions: 106.

,

ñóò ò ó ò ó

Particular solutions: 104.

+1

+

2

= exp  − 21 '  sin 

2

3 2

'  .

+ 3)ñ = 0. 1 2

 2£   .

ôƒõÿù ïƒ÷ –1 ñ

exp O 2 ' −



= 0.

B +1

 

+1

P3M

 .

= 0.

109.

+ ô ïƒ÷ ñó ò ò ó + õ ïƒú ñó ò + õ ïƒú –1 ( ô ïƒ÷ +1 + þ )ñ = 0. ¨ 
+   ¨ = 0. The substitution ¨ = 

 + ' +2 leads to a first-order linear equation: u

110.

ñóò ò ó ò ó

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

Particular solutions: 111.

ïƒ÷

– õ (2 ô 1

=



+ 3 õ )ñó ò +



,

2

= 

õ 2 (ô ïƒ÷ 



+ 2 õ )ñ = 0.

.

+ ô ïƒ÷ ñó ò ò ó + ( ôƒõ ïƒ÷ – õ 2 + ö )ñó ò + ö ( ô ïƒ÷ – õ )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ 1 and Œ equation Œ 2 + Œ + ( = 0.

ñóò ò ó ò ó

2

are roots of the quadratic

Page 505

506 112.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

+ (õ

ïƒú

Particular solution: 113.

114.

ïƒ÷ ñóò ò ó

ï ÷

+ 3ô

+ (2 ô

ñóò ò ó ò ó

= ( ïƒ÷ – ô )ñó ò ò ó + ( ô

ïƒú

+ 2õ

2 2

L 2¨ 1¨

ïƒ÷

ñóò ò ó ò ó

+ (ô

ïƒ÷



1

,

ïƒ÷

+ õ )ñó ò ò ó + ( ô„ö

ñóò ò ó ò ó + (ô ïƒ÷

¡

– õ )ñó ò ò ó + ö

Particular solution: 117.

ñóò ò ó ò ó

0

2

õ^ö

+

 1

119.

ñ óò ò ó ò ó

+ (ô

ï ÷

+ ö )ñ

ñ óò ò ó ò ó

+ (ô

ï ÷

+

óò ò ó

123.

ïƒ÷ +ú

)ñó ò + õ (2 ô

+þ

ïƒú

–1

)ñ = 0.

õ ƒï ÷ ñ

. , where Œ

2

+þ

)ñó ò + (þ

 2

= 

2



= 

+ö

and Œ

ïƒú )ñ

1

are roots of the quadratic

2

ïƒ÷

+ ö 2 )( ô

, where Œ

ïƒ÷

– õ ( ôƒõ

1

+

and Œ

õ

– ö )ñ = 0.

are roots of the quadratic

2

= 0.

.

ïƒ÷

õ ïƒú )ñ óò

+

ï ÷

ñóò ò ó ò ó

+ (ô

ïƒ÷

+

ñóò ò ó ò ó

+ ( ôƒõ

ïƒ÷

õ ï )ñ óò ò ó

+

ñóò ò ó ò ó

+ (ô

ïƒ÷

+

+ õ (ô 1

ïƒú ñ

+ ôƒõ

÷

+ ( ôù + õ ) ï

ô ïƒ÷

+ ( ôƒõ

v 1 = 

–1 1

ï ÷

+1

+ 2)ñ

= exp  − 12 ' 2  ,

−

ïƒ÷

+1

óò

–1

]ñ

= 0.

óò

ï ÷

+ õ [ö

2

–1

+ (ù – 1) ï

÷

–2

]ñ = 0.

v X 2 = 

= ,

2

=

−

2





+ ö^ñó ò + ö ( ô

= 0.

= exp  − 12 ' 2  X exp 

+

õ

– ö 2 )ñó ò + ö ( ôƒõ exp  2 (˜ − 12 '  

−

,

ôƒõ ï ÷ ñ

+

õ^ö ï

+

+ õ )ñó ò ò ó + ôƒõ

õ ïƒú )ñ óò ò ó

2

ïƒ÷ ñóò

– ôƒõ

2

ïƒ÷ –1 ñ

ïƒ÷

+

õ ïƒú )ñ

= cos   £ (  ,

2 b . Particular solutions with ( < 0:

1

= exp  − £ − (  ,

+ (ô

ïƒ÷

+

õ ïƒú )ñ óò ò ó

ï ñóò ò ó ò ó

+ 3ñó ò ò ó +

The substitution ¨ ¨2


 +  ¨ = 0.

1

ô ï ñ

=

+ ( ôƒõ

=

= 0.

−

v

,

ïƒ÷ +ú 2

=

M

+1

' 2   . M – ô„ö

ïƒ÷

+ õ )ñ = 0.

 .

= 0.

= 0.

1

ñóò ò ó ò ó

ïƒ÷

1 2

.

1 b . Particular solutions with ( > 0:

Particular solutions: 124.



+ [ ô„ö

õ ï )ñ ó ò ò ó

Particular solutions: 122.

–1

= 0.

 Integrating yields a second-order linear equation: "

 +   
 + '  −1 = L  − v (see 2.1.2.45 for the solution of the corresponding homogeneous equation with L = 0).

Particular solutions: 121.



−  . 0 = 

Particular solutions: 120.

ïƒú ñóò

=

+ ( ïƒ÷ + ô )ñó ò ò ó + ( ô

Particular solution: 118.

ïƒ÷

– õ )ñó ò +

Particular solutions: 1 =  , equation Π2 + (Π+ + ( 2 = 0. 116.

+ ôù

õ ïƒú )ñ

¨ 2 ¨ ¨ Solution: 2 + L 3 2 . Here, 1 and 2 form a fundamental set of solutions ¨ 2
u ¨

1   + 2 ' +2¨ = 0. of the second-order linear equation:  +  2 1+

Particular solutions: 1 =  equation Π2 + Π+  = 0. 115.

+

v . 0 = 

ñóò ò ó ò ó

= L 1¨

ïƒ÷

– ö 2 )ñó ò – ö ( ô„ö

+ −

õ^ö ïƒú

v X

+

= sin   £ (  .

2

2

= exp   £ − (  .

õÿþ ƒï ú –1 – ö 2 )ñ óò ï ÷ +ú – ô„ö ƒï ÷ + ö ( ôƒõ ƒ

exp O 2 (˜ −

' + ¡

+1

+ 1 P3M

+

õÿþ ïƒú

–1

)ñ = 0.

 .

leads to a constant coefficient linear equation of the form 3.1.2.1:

Page 506

507

3.1. LINEAR EQUATIONS

125.

ï ñ óò ò ó ò ó

óò ò ó

– 3ùñ

=  3

Solution:

ï ñ

+ô +2

O

¨2


 +  ¨ = 0.

126. 2 ï

ñ óò ò ó ò ó

óò ò ó

+ 3ñ

+ô

½

= 0,



1 2

ï ñ

M M

 O

3P

=õ ,



= 0, 1, 2,

uuu

, where ¨ is the general solution of equation 3.1.2.1:

P

2

ô

 ‘

4

ù ¨

≠ 0.

H  LlÁ X  ¡ £ , where Œ 1 , Œ 2 , and Œ 3 are roots of the cubic equation M 3 0 2H +  Á =1 2 Œ 3 +  = 0; Œ 4 = − F for  > 0 and Œ 4 = + F for  < 0. In addition, the constants L*Á are related by the constraint £  ( L 1 + L 2 + L 3 + L 4 ) +  = 0, and the integrals are taken along =

Solution:

straight lines.

127.

128.

129.

130.

ï ñóò ò ó ò ó

+ 3ñó ò ò ó +

ô ï 2ñ

=õ .

ï ñóò ò ó ò ó

+ 3ñó ò ò ó +

ô ï 4ñ

=

õï

ï ñ óò ò ó ò ó

+

ôñ óò ò ó

ôƒõÿñ óò

+

õ 3ï ñ

ï ñóò ò ó ò ó

+ ( ô + õ )ñó ò ò ó – ï

The substitution ¨ = 

¨ 


 +  ¨ =  . leads to an equation of the form 3.1.2.3: I

The substitution ¨ =  +

.

¨ 


 +  3 ¨ = ' . leads to an equation of the form 3.1.2.5: I

ñóò

– ôñ = 0,

Solution:

½

= where = 1 = −1, ý ý 3 = −1. 131.

132.

1

==

134.

135.

2

+

ôñóò ò ó

+ ( õ – ö 2)ï

ï ñ óò ò ó ò ó

+

ôñ óò ò ó

+ [( ö –

3

Á

= 0, ý

ï ñóò ò ó ò ó

LlÁ X

=1 2

ñóò

ô

;

:

¡

¡

|a | 

> 0.

|a − 1|(  −2) J 2 

−1 2

= 1; for  > 0, =

3

= 1 and ý

õ ï )ñ

= 0.

+ ö (ô –

õ ï )ñ

– ö ( ô„ö +

õ

> 0,

−

3

g M

a,

= + F ; for  < 0, =

3

= −F

and

The substitution ¨ = 
− ( leads to a second-order linear equation of the form 2.1.2.108:  ¨2

 + ( (˜ +  )¨2
+ ( ' + ! ( )¨ = 0.

Particular solutions: Œ 2 + Œ + ( = 0. 133.

= 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.108:  ¨2

 + (  − ' )¨2
+  2  ¨ = 0.

ï ñóò ò ó ò ó

+

ï ñóò ò ó ò ó

+ (ô

ôñóò ò ó

+

1

õ 2)ï

= 

õ ƒï ÷ ñ óò

+ ôƒõ ]ñ

 1

,

+ õ )ñó ò ò ó – ô

2

The substitution ¨ = 
+   ¨2

 +  ¨2
−  ¨ = 0.

ï ñóò ò ó ò ó

+ (ô

ï

+ õ )ñó ò ò ó +

= 

 2

, where Œ

+ õ ( ô + ù – 1) ïw÷

The substitution ¨ = 

 + ' 

ï

2

óò

õÿñ

ö ï ñóò

−1

–1

ñ

1

= 0. and Œ

2

are roots of the quadratic equation

= 0.

leads to a first-order linear equation:  ¨ 
+  ¨ = 0.

= 0. leads to a second-order linear equation of the form 2.1.2.108: – ö^ñ = 0.

The substitution ¨ =  
− leads to a second-order linear equation of the form 2.1.2.108:  ¨2

 + (  +  − 1)¨3
+ (˜ ¨ = 0.

Page 507

508 136.

137.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï ñ óò ò ó ò ó

+ ( ô ï + 3)ñ ó ò ò ó + ( õ ï + 2 ô )ñ ó ò + ( ö ï + õ )ñ = 0. The substitution ¨ =  leads to a constant coefficient linear equation of the form 3.1.2.82: ¨2


 +  ¨2

 +  ¨2
+ ( ¨ = 0.

ï ñóò ò ó ò ó

+ (ô

Solution: 138. 139.

ï

−1

=

ï ñóò ò ó ò ó

+ 2)ñó ò + õ [ õ ( ô – õ ) ï + ô ]ñ = 0.

ï

+ 3)ñó ò ò ó + ô ( õ

2 2 V L 1  (  −  )  + L 2  −   J 2 cos  3 '  + L 3  −   J 2 sin  3 '  W . 2 2

+ [ ô ( õ + 1) ï + õ ]ñ„ó ò ò ó + ô 2 õ ï ñó ò – ô  Particular solutions: 1 =  , 2 =  −  .

2

õÿñ

= 0.

ï ñóò ò ó ò ó

– ( ï + 2 ô )ñó ò ò ó – ( ï – 2 ô – 1)ñ„ó ò + ( ï – 1)ñ = 0.  Solution: = L 1  +   +1 [ L 2 ®  +1 ( )+ L 3 ;  +1 ( )], where ®  ( ) and Bessel functions.

140. 2 ï

;  (

) are the modified

– 4( ï + ô – 1)ñ„ó ò ò ó + (2 ï + 6 ô – 5)ñ„ó ò + (1 – 2 ô )ñ = 0.   Solution: = L 1  +    J 2 [ L 2 ®  ( 2) + L 3 ;  ( 2)], where ®  ( H ) and modified Bessel functions.

ñóò ò ó ò ó

141. 2 ï ñó ò ò ó ò ó + 3(2 ô Solution:

ï

ï

+ ü )ñ„ó ò ò ó + 6( õ =

.

½

4

Á

+

ôƒü )ñ óò

 ‘ . LlÁÑX  ¡  [ ( H )]( 

J

−2) 2

0

=1

ï

+ (2 ö

+ 3õ

L

H , M

ü )ñ

= −L

4

ü

= 0,

1

−L

2

;  (H

) are the

> 0. −L

3,

where ( H ) = H 3 + 3 !H 2 + 3 ˜H + ( ; Œ 1 , Œ 2 , and Œ 3 are roots of this polynomial, which are assumed to be different; Œ 4 = − F for  > 0 and Œ 4 = + F for  < 0. 142.

143. 144.

145.

146.

+ ( ô ï + õ )ñó ò ò ó + [( ô„ö + s – ö 2 ) ï + õ^ö ]ñó ò + s[( ô – ö ) ï + õ ]ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + (Œ + ° = 0.

ï ñóò ò ó ò ó ï ñóò ò ó ò ó

+ ( ô ï 2 + õ + 2)ñó ò ò ó – ôƒõ ( õ + 1)ñ = 0. This is a special case of equation 3.1.2.145 with B = 2.

ï ñóò ò ó ò ó

+ ( ô ï 2 + õ )ñó ò ò ó + 4 ô ï ñó ò + 2 ôñ = 0. Integrating the equation twice, we arrive at a first-order linear equation: 
 + (  L 1 + L 2 .

+ ( ô ïƒ÷ + õ + 2)ñó ò ò ó – ôƒõ ( õ + 1) ïƒ÷ –2 ñ = 0. The substitution ¨ =  
+  leads to a second-order linear equation of the form 2.1.2.45: ¨ 

 +   −1 ¨ 
−  (  + 1)  −2 ¨ = 0.

ï ñóò ò ó ò ó

+ (ô

ïƒ÷

+ 3)ñó ò ò ó + (2 ô

ï ñóò ò ó ò ó

+ (ô

ïƒ÷

+1

ï ñóò ò ó ò ó

+ (ô

ïƒ÷

1

=

−1

ïƒ÷

1

=

−1

+ 3)ñó ò ò ó + ( ôƒõ

Particular solutions:

1

=

−1

–1

+

õ ï )ñ óò

+ õ (ô

cos  £   ,

+ 3)ñó ò ò ó + ô ( õ

Particular solutions: 148.

+  − 2) =

ï ñóò ò ó ò ó

Particular solutions: 147.

2

ï

+ 2) ïƒ÷

2

,

+ 2ô 2

=

−1

ïƒ÷

–1



–

  .

−1 −

2

+ 1)ñ = 0.

sin  £   .

ñóò + 2 õ (ôƒõ ïƒ÷ 3

exp  − 21 '  cos 

ïƒ÷

=

ïƒ÷

'  ,

õ 2 ï )ñ óò

+1 2

+

ô ïƒ÷

=

+ õ (ô

–

−1

ïƒ÷

õ 2 ï )ñ

= 0.

exp  − 21 '  sin 

–1

2

3 2

'  .

– õ )ñ = 0.

Page 508

509

3.1. LINEAR EQUATIONS

ï

+ õ )ñ„ó ò ò ó ò

149. ( ô

ó

+ [ õ ( ô + 1)ï +

Particular solutions:

1

= ,

õ

2 2

+ 1]ñó ò ò ó + =





−

õ 2 ï ñ óò

–

õ 2ñ

= 0.

.

150. ( ô ï + õ )ñ ó ò ò ó ò ó + ü ( ô ï + õ )ñ ó ò ò ó + ( ö ï + ý )ñ ó ò + ü ( ö ï + ý )ñ = 0. The substitution ¨ = 
+ , leads to a second-order linear equation of the form 2.1.2.108: (  +  )¨2

 + ( (˜ + )¨ = 0. 151.

M

+ [( ô + ö ) ï + õ + ý ]ñ„ó ò + ( + 2 )[( ö – ô ) ï + ý – õ ]ñ = 0. Particular solutions: 1 = exp( ° 1  ), 2 = exp( ° 2  ), where ° 1 and ° 2 are roots of the quadratic equation ° 2 + “±° + Œ + “ 2 = 0.

ñ óò ò ó ò ó ( ô ï + õ )„ ï + õ )„ñ óò ò ó ò ó

152. ( ô

+ ( ö ï + ý ) ñ óò ò ó

ï + ý )ñ óò ò ó – ü

+ (ö

 , 1 = 

Particular solutions:

[(3 ôƒü + 2 ö ) ï + 3 õ

+ 2 ý ]ñ„ó ò + ü 2 [(2 ôƒü + ö ) ï + 2 õ

ü

 . 2 =  

ü + ý ]ñ

= 0.

153. ( ô ï + õ )ñ ó ò ò ó ò ó + ( ö ï + ý )ñ ó ò ò ó + s ï ( ô ï + õ )ñ ó ò + s[ ö ï 2 + ( ô + ý ) ï + õ ]ñ = 0. The substitution ¨ = 

 + ˜°  leads to a first-order linear equation: (  +  )¨u
+( (˜ + )¨ = 0. 154.

(1 – ï )ñ ó ò ò ó ò ó

+ ï (ô ï

– 2 ô + 1)ñ ó ò ò ó + (– ô ï 2

= ,

Particular solutions:

1

(ô ï

+1

2

2



+ 2 ô – 1)ñ ó ò + 2 ô ( ï

M

– 1)ñ = 0.

= .

155. ( ô ï + õ )ñ„ó ò ò ó ò ó + ( ö ï + ý )ñó ò ò ó + s ïƒ÷ ( ô ï + õ )ñó ò + s ïƒ÷ –1 [ ö ï 2 + ( ôù + ý ) ï + õÿù ]ñ = 0. The substitution ¨ = 

 + °˜  leads to a first-order linear equation: (  +  )¨u
+( (˜ + )¨ = 0. 156.

–1)ñ„ó ò ò ó ò ó + ï ( ƒ ô õ ïƒ÷

Particular solutions: 157.

158.

159.

ï 2 ñ óò ò ó ò ó

161.

 . 2 = 

2

= ,

+(2 õ ïƒ÷ – ô 2 õ ïƒ÷

+2

+ ô ) ñ óò +2ôƒõ (ô ï 2

M

–1)ïw÷ ñ

= 0.

+ 3 ï ñ ó ò ò ó – 3ñ ó ò + ô ï 2 ñ + õ = 0. = (¨u  )
 , where the function ¨ = ¨ ( ) satisfies a constant coefficient linear Solution: equation of the form 3.1.2.2: ¨u


 +  ¨ =  .

ï 2 ñ óò ò ó ò ó

+ 6 ï ñó ò ò ó + 6ñó ò + ô ï 2 ñ = õ . The substitution ¨ =  2 leads to a constant coefficient linear equation of the form 3.1.2.2: ¨2


 +  ¨ =  .

ï 2 ñ óò ò ó ò ó

– 3(ù + þ Solution: =

160.

1

–2 õ ïƒ÷ – ô 2 ) ñ óò ò ó

)ï

»

ñ óò ò ó

−1



+ 3ù (3þ

( > − 3“ − 1)

óò

+ 1)ñ

+»

–ï

2

Á

−1

( > − 3 @ − 2)

ñ ½

3



=1

where the    are three roots of the cubic equation  

3

=0

ï 2 ñ óò ò ó ò ó

+ 6 ï ñ ó ò ò ó + 6ñ The substitution ¨ = 

ï 2 ñ óò ò ó ò ó

– 2(ù + 1)ï Solution: = where

.

™

L

óò

2

ñ óò ò ó

+L L 1 (  1

=0

þ ,ù

= 0,

 L   Ÿ ¼ ,

= 1, 2, 3,

uuu

> = M ,  M

= 1.

+ ô ï 3ñ = õ . ¨ 


 +  ¨ =  . leads to a linear equation of the form 3.1.2.3: p – (ô 2 2

ï

2

– 6ù )ñ

óò

+ 2ô

+ L 3  2  +1  − 4B + 2) + L 2 



4

2 .

ï ñ

ù

= 0,

= 1, 2, 3,

uuu

if  = 0,  2  Æ ( ) if  ≠ 0, ( ) + L 3  −

( ) and Æ ( ) are some polynomials of degree ≤ 2B + 2.

Page 509

510 162.

163.

164.

165.

166.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó ò ó

+ 3 ï ñó ò ò ó + (4 ô 2 ï 2 D + 1 – 4 ô 2 õ 2 )ñó ò + 4 ô Solution: = L 1 ø 2 (  ) + L 2 ø (  ) n (  ) + L 3 n    Bessel functions.

168.

169.

170.



= 0.

(  ), where

2

ø  (H

) and n ( H ) are the



ï 2 ñ óò ò ó ò ó

+ ô ï 2 ñó ò ò ó + ( õ ï + ö )ñó ò + ô ( õ ï + ö )ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.111:  2 ¨2

 + ( ' + ( )¨ = 0. + ô ï 2 ñ ó ò ò ó + ( õ ï ÷ + ö )ñ ó ò + ô ( õ ï ÷ + ö )ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.118:  2 ¨ 

 + ( '  + ( )¨ = 0.

ï 2 ñ óò ò ó ò ó ï 2 ñ óò ò ó ò ó

– ( ï + ô ) ï ñó ò ò ó + ô (2 ï + 1)ñ„ó ò – ô ( ï + 1)ñ = 0.  Solution: = L 1  +  (  +1) J 2 V L 2 ø  +1  2 £     + L 3 n  +1  2 £   W , where ø  ( H ) and n  ( H ) are the Bessel functions.

ï 2 ñ óò ò ó ò ó

– (ï

2

– 2 ï )ñó ò ò ó – ( ï



Solution: = L 1 Bessel functions. 167.

ï D ñ

3 2 –1

ï 2 ñ óò ò ó ò ó

+£  VL

– 2 ï ( ï – 1)ñ„ó ò ò ó + ( ï

2 2

+ô

2

®  (

2

– 14 )ñó ò + ( ï

)+L

– 2ï +

  Solution: = L 1 + £   J 2 V L modified Bessel functions.

2

®  (

3

1 4

;  (  )W

2

– 2ï +

ô

2

– 14 )ñ = 0.

, where ®  ( ) and

– ô 2 )ñó ò + ( ô

2

;  (

) are the modified

– 14 )ñ = 0.

2) + L

3 ;  (  2)W , where ®  ( H ) and ;  ( H ) are the

ï 2 ñ óò ò ó ò ó

– 3( ï – ô ) ï ñó ò ò ó + [2 ï 2 + 4( õ – ô ) ï + ô (2 ô – 1)]ñ„ó ò – 2 õ (2 ï – 2 ô + 1)ñ = 0. Solution: = L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 and ¨ 2 are a fundamental set of solutions of a second-order linear equation of the form 2.1.2.108:  ¨p

 + (  −  )¨2
+  ¨ = 0.

ï 2 ñ óò ò ó ò ó

+ ï [( ô + ö ) ï + õ ]ñ„ó ò ò ó The substitution ¨ = 
+ ( with B = 1:  2 ¨2

 +  (  + 

+ [( ô„ö +  ) ï 2 + ( õ^ö +  ) ï + ]ñ„ó ò + ö (  ï 2 +  ï + )ñ = 0. leads to a second-order linear equation of the form 2.1.2.146 )¨2
+ ( ƒ… 2 + ý± + = )¨ = 0.

+ ( ô ï ÷ +1 + õ ï )ñ ó ò ò ó + [ ô ( õ – 2) ï Particular solutions: 1 =  + 1 , 2 =  + equation 2 + (  − 3) + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

¡

÷

+ ö ]ñ ó ò + ô ( ö – õ + 2) ï ÷ –1 ñ = 0. 2 , where 1 and 2 are roots of the quadratic

¡

¡

¡

171. 2 ï ( ï – 1)ñ„ó ò ò ó ò ó + 3(2 ï – 1)ñ„ó ò ò ó + (2 ô ï + õ )ñó ò + ôñ = 0. Solution: = L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 and ¨ 2 form a fundamental set of solutions of the equation 2 ( − 1)¨3

 + (2 − 1)¨3
+  12  + 14  − 12  ¨ = 0, which is reduced, by means of the substitution  = cos2 ˆ , to the Mathieu equation 2.1.6.29: 2¨uŠª

Š = (  +  − 2 +  cos 2ˆ )¨ . 172. ( ô 2 ï 2 + ô 1 ï + ô 0 )ñó ò ò ó ò ó + ( õ 1 ï + õ 0 )ñó ò ò ó + ( ö 1 ï + ö 0 )ñó ò – þ ö 1 ñ = 0. Here, ( 1 ≠ 0 and is a positive integer. A solution of this equation is a polynomial of ¡ degree that can represented as follows:

¡

= where c

½+ 

O− =0

1 

( 1P

= M , ´ ®

M



,ž + ®´ −+

−1

[( 

2

+  1  +  0)c

3

+ (  1  +  0)c

2

 + ( 0c ].  + ,

Á Á =  +1 with @ ≠ −1. @ +1

Page 510

511

3.1. LINEAR EQUATIONS

173.

174.

175.

176.

177. 178.

ï 3 ñ óò ò ó ò ó

+ 6ï

2

ñóò ò ó

+ (ô

ï

3

– 12)ñ + 2 õ = 0.

Solution: = (¨u  2 )
 , where ¨ = ¨ ( ) satisfies a constant coefficient linear equation of the form 3.1.2.2: ¨ 


 +  ¨ =  .

ï 3 ñ óò ò ó ò ó

+ ô ï 2 ñ ó ò ò ó + õ ï ñ ó ò + ( ô – 2) õÿñ = 0. This is a special case of equation 3.1.2.175. Solution: = L 1  B 1 and B 2 are roots of the quadratic equation B 2 − B +  = 0.

180.

181.

182.

183. 184.

 + L 2   1 + L 3   2 , where

ï 3 ñ óò ò ó ò ó

+ ô ï 2 ñó ò ò ó + õ ï ñó ò + ö^ñ = 0. The Euler equation. The substitution a = ln | | leads to a constant coefficient linear equation of the form 3.1.2.82: g

g
g + (  − 3)
g
g + (2 −  +  ) g
+ ( = 0.

ï 3 ñ óò ò ó ò ó

+ 3 ô ï 2 ñó ò ò ó + 3 ô ( ô – 1)ï ñ„ó ò + [ õ ï 3 + ô ( ô – 1)(ô – 2)]ñ = 0. The substitution ¨ =   leads to a constant coefficient linear equation of the form 3.1.2.1: ¨2


 +  ¨ = 0.

ï 3 ñ óò ò ó ò ó

+ 3 ô ï 2 ñó ò ò ó + 3 ô ( ô – 1)ï ñ„ó ò + [ õ ïƒ÷ + ô ( ô – 1)(ô – 2)]ñ = 0. The substitution ¨ =   leads to an equation of the form 3.1.2.7: ¨u


 + ' 

ï 3 ñ óò ò ó ò ó

+ 3(1 – ô ) ï

2

ñóò ò ó

+ ï [4 õ

Solution: = L 1   ø Á 2 ( ' v ) + L are the Bessel functions. 179.

2−

ö ï 2}

2 2

2

 

ø

2 2

+ ( ô ï 2 + õ )ñó ò ò ó + 2(2 ô – 9) ï ñ„ó ò + 2( ô – 6)ñ = 0. Integrating the equation twice, we arrive at a first-order linear equation:  L 1 + L 2 .

+ ï 2 ( ô ï + õ )ñó ò ò ó + ö ï Particular solutions: 1 =   1 , B 2 − B + ( = 0.

ï 3 ñ óò ò ó ò ó

+ ô ïƒ÷ ñó ò ò ó + õ ï ñó ò + õ ( ô Particular solutions: 1 =  + 1 , equation 2 − +  = 0.

+ 2 ôƒõ ï + ö )ñ ó ò + ( ô 2 õ ï 2 + õ^ö – 2 ö )ñ = 0. −    2 , where B 1 and B 2 are roots of the quadratic 2=

2

ïƒ÷ 2

– 2)ñ = 0. =  + 2 , where

–2

¡

+ ï 2 ( ô ïƒ÷

ï 3 ñ óò ò ó ò ó

+

+ õ )ñó ò ò ó + ï ( ô ïƒ÷ + õ – 1)ñó ò + ( ô Particular solutions: 1 = cos(ln  ), 2 = sin(ln  ).

ï 2 (ô ïƒ÷


 +[(  −6) 2 +  ] =

+ ö ( ô ï + õ – 2)ñ = 0.  2 , where B 1 and B 2 are roots of the quadratic equation 2=

+ ï 2 (2 ô ï + õ )ñ ó ò ò ó + ï ( ô 2 ï  Particular solutions: 1 =  −    1 , equation B 2 − B + ( = 0.

¡

3

ñóò

ï 3 ñ óò ò ó ò ó

ï 3 ñ óò ò ó ò ó

¨ = 0.

ö + 3ô (ô – 1)]ñ„óò + [4 õ 2 ö 2 ( ö – ô ) ï 2 } + ô (4 b 2 ö 2 – ô 2 )]ñ = 0. Á ( ' v ) n Á ( ' v ) + L 3   n Á 2 ( ' v ), where ø Á (‡ ) and n Á (‡ ) + 1 – 4b

ï 3 ñ óò ò ó ò ó ï 3 ñ óò ò ó ò ó

−3

õ + ö + 1)ñóò ò ó ï + [  ï 2÷ + ( ô„ö +  ) ï ÷ +

+ +

1

¡

ïƒ÷

õ^ö ]ñ óò

and +

õ

¡

2

are roots of the quadratic

– 3)ñ = 0.

+ ( ö – 1)(

ï 2÷

+

ï ÷

+ )ñ = 0.

The substitution ¨ =  
+ ( ( − 1) leads to a second-order linear equation of the form ¨ 
+ ( ƒ… 2  + ý±  + = )¨ = 0. 2.1.2.146:  2 ¨2

 +  (   +  )2

185. ( ô ï + õ ) ï 3 ñ ó ò ò ó ò ó + ( ö Particular solutions: B 2 − B + ° = 0.

ï

+ ý )ï 2 ñ  1, 1=

+ s( ô ï + õ ) ï  2 , where B 2=

óò ò ó

+ s[( ö – 2 ô ) ï + ý – 2 õ ]ñ = 0. 1 and B 2 are roots of the quadratic equation

ñ óò

Page 511

512 186. 187.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 6 ñ óò ò ó ò ó

+ 6 ï 5 ñó ò ò ó – ôñ + 2 õ ï = 0. The substitution  = a −1 leads to an equation of the form 3.1.2.62: a 2 
g
g
g −6
g + a

ï 2 (ƒï ÷ + ô )ñ óò ò ó ò ó + ï ( õ ï +1 +2ù ïƒ÷ + ö ï )ñ óò ò ó

2

−2 'a = 0.

+[2 õ ü ï +1 + ù (ù –1) ïƒ÷ ]ñó ò + õ ü ( ü –1) ï ñ = 0. Integrating the equation twice, we arrive at a first-order linear equation: (  +  )
 +( '  + ( ) = L 1 + L 2 .

3.1.3. Equations Containing Exponential Functions 3.1.3-1. Equations with exponential functions. 1.

ñóò ò ó ò ó

ó (ô 2  2 ó

– ô

ó

+ 3 ô



 

2



)ñ = 0.

2.

ñóò ò ó ò ó

+

ô ó ñóò

0

óñ

+ ô

=

Œ

. The substitution

P

õ  ó

= exp O



 



X H ( )  leads to P  M  
Œ

2 2 a second-order linear equation of the form 2.1.3.27: H  +3    H  +(3    +3 Π ) H = 0. Particular solution:

= exp O

+

.





Integrating yields a second-order linear equation:  +    = ' “ −1   + L for the solution of the corresponding homogeneous equation with  = L = 0).





(see 2.1.3.1

3.

ñóò ò ó ò ó

+ ô ñó ò + õ ( ô + õ 2 )ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.3.10: ¨2

 −  ¨2
+ (    +  2 )¨ = 0.

ó

ó

4.

ñóò ò ó ò ó

ó

ó

+ ô ñó ò + [ ô ( – õ )  – õ 3 ]ñ = 0.   Integrating yields a second-order linear equation: "

 +  
 + (   +  2 ) = L   2.1.3.10 for the solution of the corresponding homogeneous equation with L = 0).

(see

5.

ñóò ò ó ò ó

+ ( ô – õ 2 )ñó ò + ôƒõ ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.3.10: ¨2

 −  ¨2
+    ¨ = 0.

ó

6.

ñóò ò ó ò ó

+ ( ô – õ 2 )ñó ò + ô ( – õ )  ñ = 0.   Integrating yields a second-order linear equation: "

 +  
 +   = L   (see 2.1.3.10 for the solution of the corresponding homogeneous equation with L = 0).

ó

7.

ñóò ò ó ò ó

+ ( ô + õ )ñó ò The substitution ¨ = ¨2

 − ( ¨2
+ (    + 

ó

8.

ñóò ò ó ò ó

+ ( ô 2  + õ )ñó ò – ö ( ô 2  + õ + ö 2 )ñ = 0. The substitution ¨ = 
− ( leads to a second-order linear equation of the form 2.1.3.27: ¨2

 + ( ¨2
+ (   2   +    + ( 2 )¨ = 0.

9.

ñóò ò ó ò ó

– 3 ô

ó

ó

ó

ó

ó (ô  ó

+ )ñó ò +

Particular solutions: 10.

ñ óò ò ó ò ó

– (3 ô

ó

+ ö ( ô + õ + ö 2 )ñ = 0. 
+ ( leads to a second-order linear equation of the form 2.1.3.10: + ( 2 )¨ = 0.

 

2 2

ó

ó

1

= exp O

+ 3 ô



ó

ó

ô ó (2ô 2  2 ó Œ



 

+ õ )ñ

óò



P

,

+ ô



2

2

–

=  exp O

)ñ = 0.

ó (2ô 2  2 ó 

 



P

.

– 2õ –

2

)ñ = 0.

   −  £  , 2 = exp O   P Œ Œ    £ 2 b . Particular solutions with  < 0: 1 = exp O   cos   −   , 2 = exp O  Œ P Œ 1 b . Particular solutions with  > 0:

1

= exp O



Œ



 

+ £ 

 P

P

.

sin   £ −   .

Page 512

513

3.1. LINEAR EQUATIONS

11.

12.

13.

ñ óò ò ó ò ó

+

ôñ óò ò ó

+

õ  ó ñ óò

ó

+ ôƒõ

 ñ

The substitution ¨ = 
+  ¨2

 +    ¨ = 0.

ñóò ò ó ò ó

+

ôñóò ò ó

+ ( õ

ñóò ò ó ò ó

+

ôñóò ò ó

+ ( õ

ó

= 0.

leads to a second-order linear equation of the form 2.1.3.1:

ó

+ ö )ñó ò + [ õ ( ô + ) 

+ ô„ö ]ñ = 0.

  Integrating yields a second-order linear equation: "

 + (   + ( ) = L  −  (see 2.1.3.2 for the solution of the corresponding homogeneous equation with L = 0). 2



ó

ö ó )ñ óò

+

The substitution ¨ = 
+  ¨2

 + (   2   + (   )¨ = 0.

+ ô ( õ

2

ó



+

ö ó )ñ

= 0.

leads to a second-order linear equation of the form 2.1.3.27:

14.

ñóò ò ó ò ó

+

ô ó ñóò ò ó

– õ 2 ( ô

+ õ )ñ = 0. The substitution ¨ = 
−  leads to a second-order linear equation of the form 2.1.3.27: ¨2

 + (    +  )¨2
+  (    +  )¨ = 0.

15.

ñ óò ò ó ò ó

+

ô  ó ñ óò ò ó

– õÿñ

16.

17.

óò

2

= exp   £   .

2 b . Particular solutions with  < 0:

2

= sin  £ −   .

+

+ ôƒõ

ó ñóò

= cos  £ −   ,

ñóò ò ó ò ó

ô ó ñóò ò ó

1

+

õ 3ñ

= 0.

ñóò ò ó ò ó

+

ô ó ñóò ò ó

+ ôƒõ

ó ñóò

+

õ 2 (ô  ó

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.3.27: ¨2

 + (    −  )¨2
+  2 ¨ = 0.

ñóò ò ó ò ó

+

ô ó ñóò ò ó

1

ñ óò ò ó ò ó

+

ô  ó ñ óò ò ó

ñóò ò ó ò ó + ô ó ñ óò ò ó ñ óò ò ó ò ó

+

ô  ó ñ óò ò ó

0

23.

ñóò ò ó ò ó

+ ( ô

ñóò ò ó ò ó

+ ( ô

ó



=

ó

+ ( õ

=

+ ( ôƒõ

+ 3 õ )ñó ò +



,

2

– ö 2 )ñ

óò

v

ó

–

õ

1

=

+ õ )ñó ò ò ó – ôƒõ



3 2

'  ,

õ 2 (ô  ó .

– ö ( ô„ö



ó

+

+ ö )ñó ò + ö ( ô

2

 =  : 1 ,

−

= 



2

.

+ [ô ( õ + )

Particular solutions: 22.

ó

– õ (2 ô 1

– õ )ñ = 0.

= exp  − 12 '  cos 

Particular solutions: 1 equation ý 2 + 'ý + ( = 0. 21.

= 0.

= exp  − £   ,

Particular solution: 20.

 ñ

1

Particular solutions: 19.

ó

– ôƒõ

1 b . Particular solutions with  > 0:

Particular solutions: 18.

ó

 2



õ 2 ]ñ ó ò

–

,

2

ó

 ñ

=

−



 X

= exp  − 21 '  sin 

2

3 2

'  .

+ 2 õ )ñ = 0.

õ ó )ñ ó

+ ôƒõ

= 0.

– õ )ñ = 0.

 =  : 2 , where ý

2

ó 

2

and ý

1

ó

 ñ

exp O 2 ' −

= 0.

Œ



 



2

P3M

are roots of the quadratic

 .

= 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.3.27: ¨2

 +    ¨2
−    ¨ = 0.

ó

+ õ )ñó ò ò ó – ö 2 ( ô

ó

+

õ

+ ö )ñ = 0.

The substitution ¨ = 
− ( leads to a second-order linear equation of the form 2.1.3.27: ¨2

 + (    +  + ( )¨2
+ ( (    +  + ( )¨ = 0.

Page 513

514

THIRD-ORDER DIFFERENTIAL EQUATIONS

ó

24.

+ ( ô  + õ )ñ ó ò ò ó The substitution ¨ = ¨2

 + (    +  − ( )¨2


25.

ñóò ò ó ò ó

26.

ñ óò ò ó ò ó

ó

ó

+ ö ( ô  + õ )ñ ó ò + ö 3 ñ = 0. 
+ ( leads to a second-order linear equation of the form 2.1.3.27: + ( 2 ¨ = 0.

ó

+ ( õD + 2 ô )ñó ò ò ó – ô ( õD  Particular solutions: 1 =   ,

ñ óò ò ó ò ó = (  ó

2

ó – ô )ñ ó ò ò ó + ( ô 

Particular solutions: 1 equation ý 2 + ý +  = 0.

+ ô )ñó ò – 2 ô 3 ñ = 0.  =  − +   .

ó

– õ )ñ ó ò + õ  ñ .   : 1 =  , 2 =  : 2 , where ý

and ý

1

are roots of the quadratic

27.

ñ óò ò ó ò ó

+ ( ô  + õ )ñ ó ò ò ó The substitution ¨ = ¨2

 + (    +  + ° )¨2


ó

28.

ñóò ò ó ò ó

ó

+ ( ö  + ý )ñ ó ò – s[( ô s + ö )   + õ s + ý + s2 ]ñ = 0. 
− ° leads to a second-order linear equation of the form 2.1.3.27:  + [( !° + ( )  + ˜° + + ° 2 ]¨ = 0.

+ ( ô + õ )ñó ò ò ó The substitution ¨ = ¨2

 + (    +  + ° )¨2


+ ( ö 2  + ý )ñó ò – s( ö 2  + ô s  + õ s + ý + s2 )ñ = 0. 
− ° leads to a second-order linear equation of the form 2.1.3.27:   + ( (  2  + !°  + ˜° + + ° 2 )¨ = 0.

29.

ñóò ò ó ò ó

ó

+ ( ô

ó

2

ó

M

ó

ó

+ õ )ñó ò ò ó + ( ö

ó

ó

,

ó

M

ó

+ ý )ñó ò – ü [ ü ( ô + ü )  2  + ( ô + 3 ü/ + õ ü + ö ) -



Particular solution: 0 = exp O   . The substitution ΠP to a second-order linear equation of the form 2.1.3.27. 30.

ñ óò ò ó ò ó

+ (2 ô

ó + õ )ñ ó ò ò ó



Particular solutions: 31.

ñóò ò ó ò ó

ó

+ ( ô

1

0

ó

ó (ô   ó





= exp O −  

– õ )ñó ò ò ó +

Particular solution:

ñóò ò ó ò ó

+ ô

Œ

ö  ó ñ óò

=





.



,

P

ó

õ  ó )ñ óò

+ ô





ó [ô ( õ

=  exp O −  

2

– õ ( ôƒõ

ó

óò

+ 2 õ + 3 )ñ

= exp O

+ ö

Œ

ó )ñ 

ó



P

Œ

ó ,

2

+

 



+ 2 )



+

õ

+ ý ]ñ = 0.

XZH ( )  leads M P

ó + õ

+

2

]ñ = 0.

.

= 0.

32.

+ (  + ô )ñó ò ò ó + ( ô +  Particular solution: 0 =  −  .

33.

ñóò ò ó ò ó

+ ( ô + õ  )ñó ò ò ó + ö^ñó ò + ö ( ô + õ  )ñ = 0. ¨ 
+ (    +     )¨ = 0. The substitution ¨ = 

 + ( leads to a first-order linear equation: u

34.

ñóò ò ó ò ó

ó

+ ( ô

ó

ó

+

õ ó )ñ óò ò ó

−v  , 1 = 

Particular solutions:

ó

ó

 

( + )

ó

 ñ

= 0.

ó

ó – ö 2 ]ñóò ó + ö [ ôƒõ (  + )

+ õ ( ö + b ) 

ó

ó

+ ô ( õ  + 2 )ñó ò ò ó – [ ôƒõ (  + ) + ]ñó ò – 2 ô

  Particular solutions: 1 =   , 2 =  −  +  “ .

36.

( ô + õ )ñ ó ò ò ó ò ó – ô Particular solution:

ó

ó ( õ^öD

óñ

0

– ô„ö

  −v  X exp O 2 (˜ −  Á  . 2 =  @ P M

35.

37.

ñóò ò ó ò ó

+ [ ôƒõ

+ ôƒõ

3

ó

 ñ

ó

+

õb ó ]ñ

= 0.

= 0.

= 0.  =  + .

+ ô + ö )ñó ò ò ó ò ó – ( õ^ö 3 D  Particular solutions: 1 =  v ,

ó

+ ô 3 + ö 3 )ñó ò + −  + . 2 = 

ô„ö (ô

2

– ö 2 )ñ = 0.

Page 514

515

3.1. LINEAR EQUATIONS

38.

( ô



ó

óò ò ó ò ó

+ õ )ñ

+ ( ö

ó 

óò ò ó

+ ý )ñ

+ ü ( ô

ó 

óò

+ õ )ñ

+ ü ( ö

ó 

+ ý )ñ = 0.

£ ,  , 2 = sin   £ ,  . 1 = cos   £ − ,  , 2 = exp   £ − ,  . 1 = exp  − 

1 b . Particular solutions with , > 0: 2 b . Particular solutions with , < 0:

3.1.3-2. Equations with power and exponential functions. 39.

ñóò ò ó ò ó

+

ô ó ñóò

+

õ ï (ô  ó

Particular solution: 40.

ñóò ò ó ò ó

ï

+ (ô

+ õ ) 

Particular solution: 41.

ñóò ò ó ò ó

ï

+ (ô

+ õ ) 

42.

43.

ñóò ò ó ò ó + ô ó ñ óò ò ó

= exp 

0

ó ñóò

+

= 0.

óñ

– 2 ô

= 0. 2

= (  +  ) .

0

ô ï  ó ñóò ò ó

'  . 2

óñ

õ ƒï ÷ ñ óò

õ ïƒ÷

+

+ (õ

ï

2

– ô

ó )ñ óò

+

ô ï 2   ó ñóò ò ó

Particular solutions: 45.

ñóò ò ó ò ó

+ (ô

ï

+

ñóò ò ó ò ó

+ ( ôƒõ

õ ó )ñ óò ò ó

ï  ó

ñóò ò ó ò ó

+

ô ïƒ÷ ( õ  ó

48.

ñ óò ò ó ò ó

+ (ô

ï ÷

– 2 õ

50.

ï ñóò ò ó ò ó

+

ôñóò ò ó

ï ñóò ò ó ò ó

+

ô ï  ó ñóò

  , 2





=  ,

ó )ñ ó ò ò ó

+ ï ( õ

–

ó

= exp 

ô 2 õ ï  ó ñóò ïƒ÷  ó

2

=  −



õ  ó (2ô ï ÷

= exp O

1

2

Œ



 

óñ

 P

,

+ ö )ñó ò + [ õ (

2

õ ó ñ

+ )ñó ò – 2 ô

+ Π.

= 0.

3

ïƒ÷ ñ

ó

– õ  + 3 )ñ ó ò ó + õ [ ô ïƒ÷ ( õ 2

ï

= 0.

 2  X exp  12   2   . M

− 21 –ô

 . 2 =  −

= ,

1

Particular solutions: 49.

− 12

+ 2 )ñó ò ò ó – ( ôƒõ

Particular solutions:

= 0.

+ 2)ñó ò + ôƒõ

+ ô )ñó ò ò ó +

1

óñ

= .

ï  ó

= exp 

õ ó

+

+ 2 ô

2

2

2

+ ô (õ

1

Particular solutions: 47.

= ,

1

Particular solutions: 46.

ï  ó ñóò

– 2ô

+ 3)ñ = 0.

£   , 2 = sin  1  2 £   . 1 = cos   2 1 2£ −   , 2 = exp  − 12  2 £ −   . 1 = exp  − 2  1 2

2 b . Particular solutions with  < 0:

ñóò ò ó ò ó

+ ù )ñ = 0.

õ ï (ô ï 2   ó

+

1 b . Particular solutions with  > 0:

44.

ï  ó

(ô

–1

¨ 
+    ¨ = 0. leads to a first-order linear equation: u

The substitution ¨ = 

 + ' 

ñóò ò ó ò ó

– 3 õ )ñ = 0.

2

=  +  .

ó ñóò

+

õ 2ï

− 12

– ô

0

Particular solution:

+

=  exp O

+ ô ) 

ó

Œ



 



P

ó

= 0.

– ) + 2 õ

ó

–

2

]ñ = 0.

.

+ ô„ö ]ñ = 0.

 ¨ 
+  ¨ = 0. The substitution ¨ = 

 + (   + ( ) leads to a first-order linear equation:  u – 2 ô

The substitution ¨ =  ¨2

 +    ¨ = 0.

óñ

= 0.


− 2 leads to a second-order linear equation of the form 2.1.3.1:

Page 515

516 51.

52.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ó

ï ñ óò ò ó ò ó

ó

ï ñ óò ò ó ò ó

+ (ô

ï ó

54.

55.

56. 57.

óò ò ó

+ 3)ñ

Particular solutions: 53.

ó

= (   – ô ï )ñ ó ò ò ó + ( ô  – õ ï )ñ ó ò + õ  ñ .   Particular solutions: 1 =  : 1 , 2 =  : 2 , where ý equation ý 2 + ý +  = 0.

1

ó

ï 2 ñ óò ò ó ò ó

ï

+ ô (õ −1

=

ó

 ñ óò

+ 2) 

+ õ ( ôƒõ

2

exp  − 21 '  cos 

ó

3 2

and ý

1

ï ó

'  ,

2

+

ô  ó −1

=

are roots of the quadratic

2

–õ

2

ï )ñ

= 0.

exp  − 21 '  sin 

ó

2

3 2

'  .

+ ï ( ô ï  + õ )ñó ò ò ó + [ ô ( õ – 2)ï - + ö ]ñó ò + ô ( ö – õ + 2) - ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 3)B + ( −  + 2 = 0.

ó

ï 3 ñ óò ò ó ò ó

ó

+ õ ï 2  ñó ò ò ó + ô ï ñó ò + ô ( õ – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B equation B 2 − B +  = 0.

ó

ï 3 ñ óò ò ó ò ó

and B

1

ó

2

are roots of the quadratic

ó

+ ï 2 ( ô + õ )ñó ò ò ó + ï ( ôƒõ + ö – õ )ñó ò + ö ( ô – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B + ( = 0.

ó

( ô + õ ï )ñó ò ò ó ò ó – ô Particular solution:

ó

( ô

+

õ ï 2 )ñ óò ò ó ò ó

ó

58.

( ô ï  + õ )ñó ò ò ó ò ó + Particular solution:

59.

(ô

+ õ )ñó ò ò ó ò

0

– ô

Particular solution:

ï 2 ó

óñ

ó

0

õÿñ

+

Particular solution:

= 0.  =   + ' .

óñ

= 0.

= 



+ ' 2 .

= 0. − . 0 =  +  

õÿñ

= 0. 0

= 

2

+ 

−



.

3.1.4. Equations Containing Hyperbolic Functions 3.1.4-1. Equations with hyperbolic sine. 1.

2.

+ ôñó ò ò ó + õ sinh2 ï ñó ò + ôƒõ sinh2 ï ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.4.1: ¨2

 +  sinh2  ¨ = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

sinh ÷ (

ô

+

ï )ñ óò ò ó

+

õÿñóò

+ ôƒõ sinh ÷ (

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ï )ñ

= 0.

= cos   £   ,

2

= exp  − £ −   ,

= sin   £   . 2

= exp   £ −   .

3.

+ ô sinh ÷ ( ï )ñó ò ò ó + õ ïƒú ñó ò + õ ïƒú –1 [ ô ï sinh ÷ ( ï ) + þ ]ñ = 0. + leads to a first-order linear equation: ¨u
+  sinh ( Œ  )¨ = 0. The substitution ¨ = 

 + ' *

4.

ñ óò ò ó ò ó

ñóò ò ó ò ó

+

ô

sinh ÷ (

ï )ñ ó ò ò ó

Particular solutions:

1

+ ôƒõ sinh ÷ (

= exp 

− 12

ï )ñ ó ò 2

'  cos 

+ 3 2

õ 2 [ô

'  ,

sinh÷ ( 2

ï

) – õ ]ñ = 0.

= exp  − 21 '  sin 

2

3 2

'  .

Page 516

517

3.1. LINEAR EQUATIONS

5.

ñóò ò ó ò ó

+

sinh ÷ (

ô

ï )ñ óò ò ó

Particular solutions: 6.

7.

1

10.

ñóò ò ó ò ó

+

ô ï

sinh ÷

ï ñóò ò ó

ñ óò ò ó ò ó

+

ô ï

2

sinh ÷ (

12.

13.

14.

16.

1

18.

– 2ô

ï )ñ óò

= ,

2

õ ï (ô ï

+

2

sinh ÷

ï

ñ óò ò ó ò ó

+ ( ô sinh ÷

ï

+

õ ï )ñ ó ò ò ó

sinh ÷ (

1 2

= .

2

+ 2 ô sinh ÷ (

ï )ñ ó ò

2

1

ï

+ õ (ô

) + 2 õ ]ñ = 0.

+ 3)ñ = 0.

£  . 2 = sin  

2

ï

ï

– õ 2 )ñ ó ò + ö ( ô sinh ÷ ï – õ )ñ = 0. 2 = exp( Œ 2  ), where Œ 1 and Œ 2 are roots of the

ö

£  , 1 = cos   1 2

sinh ÷ (

õ 2[ô

.

– ô sinh ÷

2

) + 3 õ ]ñó ò +

ï )ñ

= (sinh ÷ ï – ô )ñó ò ò ó + ( ô sinh ÷ ï – õ )ñó ò + õ sinh ÷ ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

= 0.

sinh ÷

= exp  − 12 ' 2  ,

ï

+ 2)ñ

+ ôƒõ sinh ÷

óò

ï ñ

= exp  − 12 ' 2  X exp 

2

and Œ

1

1 2

ï ñ óò ò ó ò ó

are roots of the

2

= 0.

' 2   . M

+ ï ( ô sinh2 ï + õ )ñ ó ò – 2( ô sinh2 ï + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.4.1: ¨2

 + (  sinh2  +  )¨ = 0.

+ ( ô ï 2 sinh ÷ ï + õ ï )ñó ò ò ó + [ ô ( õ – 2) ï sinh ÷ Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

¡

¡

¡

¡

ï

+ ö ]ñó ò + ô ( ö – õ + 2) sinh ÷ ï ñ = 0. 1 and 2 are roots of the quadratic

¡

+ ï 2 ( ô sinh÷ ï + õ )ñó ò ò ó + ï ( ôƒõ sinh ÷ ï + ö – õ )ñó ò + ö ( ô sinh ÷ ï – 2)ñ = 0. Particular solutions: 1 =  + 1 , 2 =  + 2 , where 1 and 2 are roots of the quadratic ¡ ¡ equation 2 + (  − 1) + ( = 0.

ï 3 ñ óò ò ó ò ó

sinh ÷ ï ñó ò ò ó ò ó

+ ôñó ò ò ó + ôƒõÿñó ò +

sinh ÷

ï ñóò ò ó ò ó

1

= exp 

+ ôñó ò ò ó + õ sinh ÷

– õ sinh ÷

õ 2(ô

− 12

'  cos 

ï ñóò

+

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

sinh ÷

ï ñóò ò ó ò ó

+ ôñó ò ò ó – õ (2 ô + 3 õ sinh ÷

Particular solutions:

17.

ï



= 

2

ï 

ñóò ò ó ò ó

Particular solutions: 15.

,

+ (õ

ï )ñ ó ò ò ó

Particular solutions: 11.



=

+ ô sinh ÷ ï ñ ó ò ò ó + ( ôƒõ sinh÷ ï + Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

Particular solutions: 9.



ñ óò ò ó ò ó

Particular solutions: 8.

– õ [2 ô sinh ÷ (

1

=





,

2

= 

2

3 2

ôƒõÿñ

ï )ñ

= 0.

'  ,

2

= exp  − 21 '  sin 

= 0.

= cos   £   ,

2

= exp  − £ −   ,





ï )ñ óò

+

õ 2 (ô

2

3 2

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ sinh ÷

ï )ñ

= 0.

.

sinh ÷ ï ñó ò ò ó ò ó + ñó ò ò ó + [( õ – ô 2 ) sinh÷ ï + ô ]ñó ò + õ (1 – ô sinh ÷ ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0. sinh ÷ (

ï )ñ óò ò ó ò ó

+ô

Particular solutions:

ï 2 ñ óò ò ó 1

ï ñóò

– 2ô

= ,

2

2

+ 2 ôñ = 0.

= .

Page 517

518 19. 20.

THIRD-ORDER DIFFERENTIAL EQUATIONS sinh ÷ ï ñó ò ò ó ò ó + ( ô sinh ÷ ï + ô Particular solutions: 1 =  ,

sinh ÷

ï ñóò ò ó ò ó

sinh ÷

ï

+ (ô

Particular solutions: sinh ÷

ï

21.

ï ñ óò ò ó ò ó

1

ï

3

22.

sinh ÷

ï ñóò ò ó ò ó

1

Particular solutions:

ï

3

23.

sinh ÷

ï ñóò ò ó ò ó

1

25.

sinh ÷ ï ñó ò ò ó ò ó + ô Particular solutions:

ï

3

ï

3

sinh ÷

ï ñóò ò ó ò ó

ï

+ ï )ñ

−1

=

+ï

Particular solutions:

1

2

=

−1

+ (ô

ï ñóò

–ô

=

−2

,

= 0.

ï )ñ óò

+

ôñ

sinh ÷

ï

=

−1

2

= .

+ 2)ñ

ï ñóò

+ ô (sinh ÷

óò

+ 2(2 sinh ÷

ï

– ô )ñ = 0.

+ 6(2 sinh ÷

ï

– ô )ñ = 0.

= .

+ ï ( ô – sinh ÷ ï )ñó ò + ( ô – 3 sinh ÷ = cos(ln  ), 2 = sin(ln  ). + ô )ñó ò ò ó + ï [ ô – ( õ + 1) sinh ÷

ï

=

2 −

,

2

2

=

 2   . M

1 2

ï

+ ï )ñ = 0.

sin  £   .

3

2

= 0.

ï

2

2

,

ñ

2

= exp  − 21  2  X exp 

ï ñóò

– 6 ï sinh ÷

(sinh÷ 1

2

2

cos  £   ,

ï 2 ñ óò ò ó 1

óò ò ó

– 2 ï sinh ÷

ô ï 2 ñ óò ò ó

+

Particular solutions: 24.

= exp  − 12  2  ,

ô ï 2 ñ óò ò ó

+

+ 1)ñó ò ò ó + ô −  . 2 = 

+ 1)ñó ò ò ó + ô ( ï + 2 sinh ÷

ï

+ (3 sinh ÷

Particular solutions:

ï

ï )ñ

ï ]ñ óò

.

= 0. + õ (2 sinh ÷

ï

– ô )ñ = 0.

3.1.4-2. Equations with hyperbolic cosine. 26.

27.

28.

+ ôñó ò ò ó + õ cosh(2 ï )ñ„ó ò + ôƒõ cosh(2 ï )ñ = 0. leads to a second-order linear equation of the form 2.1.4.9: The substitution ¨ = 
+  ¨2

 +  cosh(2 )¨ = 0.

ñóò ò ó ò ó

+ ôñó ò ò ó + õ cosh2 ï ñó ò + ôƒõ cosh2 ï ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.4.10: ¨2

 +  cosh2  ¨ = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

cosh÷ (

ô

+

ï )ñ óò ò ó

õÿñóò

+

+ ôƒõ cosh÷ (

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ï )ñ

= 0.

= cos   £   ,

2

= exp  − £ −   ,

= sin   £   . 2

= exp   £ −   .

29.

+ ô cosh÷ ( ï )ñó ò ò ó + õ ïƒú ñó ò + õ ïƒú –1 [ ô ï cosh÷ ( ï ) + þ ]ñ = 0. + leads to a first-order linear equation: ¨u
+  cosh ( Œ  )¨ = 0. The substitution ¨ = 

 + ' *

30.

ñóò ò ó ò ó

ñóò ò ó ò ó

cosh÷ (

ô

+

ï )ñ óò ò ó

Particular solutions: 31.

ñ óò ò ó ò ó

+

ô

cosh÷ (

ï )ñ ó ò ò ó

Particular solutions: 32.

1

1

+ ôƒõ cosh÷ (

= exp 

− 12

'  cos 

– õ [2 ô cosh ÷ ( =





,

2

= 

+ ô cosh÷ ï ñó ò ò ó + ( ôƒõ cosh÷ ï + Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

ñóò ò ó ò ó

cosh÷ (

ï )ñ óò + õ 2 [ô 2 3



ö



ï

2

'  ,

) + 3 õ ]ñ

2

óò

= exp 

+

õ 2 [ô

ï

) – õ ]ñ = 0.

− 21

'  sin 

cosh÷ (

ï

2

3 2

'  .

) + 2 õ ]ñ = 0.

. – õ 2 )ñó ò + ö ( ô cosh÷ ï – õ )ñ = 0. 2 = exp( Œ 2  ), where Œ 1 and Œ 2 are roots of the

Page 518

519

3.1. LINEAR EQUATIONS

33.

ñóò ò ó ò ó

+

ô ï

cosh÷

ï ñóò ò ó

Particular solutions: 34.

ñóò ò ó ò ó

+

ô ï

2

cosh÷ (

36.

38.

39.

40.

41.

42.

44.

46.

= ,

2

cosh÷

£  . 2 = sin  

2

cosh÷ (

ï

2

1 2

2

= .

+ 3)ñ = 0.

2

+ 2 ô cosh÷ (

ï )ñ óò

ï ï )ñ

= (cosh÷ ï – ô )ñ ó ò ò ó + ( ô cosh÷ ï – õ )ñ ó ò + õ cosh÷ ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó

+ ( ô cosh÷

ï

õ ï )ñ óò ò ó

+

1

cosh÷

ï

+ õ (ô

= exp  − 12 ' 2  ,

ï

+ 2)ñó ò +

ôƒõ

cosh÷

ï ñ

= exp  − 12 ' 2  X exp 

2

= 0.

and Œ

1

1 2

ï ñóò ò ó ò ó

are roots of the

2

= 0.

' 2   . M

+ ï [ ô cosh(2 ï ) + õ ]ñ„ó ò – 2[ ô cosh(2 ï ) + õ ]ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.4.9: ¨2

 + [  cosh(2 ) +  ]¨ = 0.

ï ñóò ò ó ò ó

+ ï ( ô cosh2 ï + õ )ñó ò – 2( ô cosh2 ï + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.4.10: ¨2

 + (  cosh2  +  )¨ = 0.

= (cosh÷ ï – ô ï )ñó ò ò ó + ( ô cosh÷ Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ +  = 0.

ï ñóò ò ó ò ó

– õ ï )ñó ò + õ cosh÷ ï = exp( Œ 2  ), where Œ

ï

2

+ ( ô ï 2 cosh÷ ï + õ ï )ñó ò ò ó + [ ô ( õ – 2) ï cosh ÷ Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

¡

¡

¡

¡

+ ï 2 ( ô cosh÷ Particular solutions: equation 2 + (  − 1)

ï 3 ñ óò ò ó ò ó

cosh ÷ ï ñó ò ò ó ò ó cosh ÷

ï ñóò ò ó ò ó

ï

+õ 1 = +(

1

= exp 

+ ôñó ò ò ó + õ cosh÷

– õ cosh÷

õ 2 (ô

− 12

'  cos 

ï ñóò

2 b . Particular solutions with  < 0:

1

1

=



,

2

= 

3 2

ï )ñ

'  ,

1

and Œ

are roots of the

2

+ ö ]ñó ò + ô ( ö – õ + 2) cosh÷ ï ñ = 0. 1 and 2 are roots of the quadratic

¡

= 0. 2

= exp  − 21 '  sin 

= 0.

= cos   £   ,

2

= exp  − £ −   ,

+ ôñó ò ò ó – õ (2 ô + 3 õ cosh ÷



2

ôƒõÿñ

+ 1

ï ñóò ò ó ò ó

.

óò ò ó

1 b . Particular solutions with  > 0: cosh ÷

ï

ñ

+ ï ( ôƒõ cosh÷ ï + ö – õ )ñ ó ò + ö ( ô cosh÷ ï – 2)ñ = 0.  + 1 , 2 =  + 2 , where 1 and 2 are roots of the quadratic ¡ ¡ = 0. )ñ

+ ôñó ò ò ó + ôƒõÿñó ò +

Particular solutions:

45.

– 2ô

õ ï (ô ï

+

ñ óò ò ó ò ó

Particular solutions:

43.

ï )ñ óò

£  , 1 = cos  

1

Particular solutions: 37.

– ô cosh÷

2

1 2

ï )ñ óò ò ó

Particular solutions:

35.

ï

+ (õ





ï )ñ óò

+

õ 2 (ô

2

3 2

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ cosh ÷

ï )ñ

= 0.

.

cosh ÷ ï ñó ò ò ó ò ó + ñó ò ò ó + [( õ – ô 2 ) cosh÷ ï + ô ]ñó ò + õ (1 – ô cosh ÷ ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ 1 and Œ 2 are roots of the quadratic 2 equation Œ + Œ +  = 0. cosh ÷ (

ï )ñ óò ò ó ò ó

+ô

Particular solutions:

ï 2 ñ óò ò ó 1

ï ñóò

– 2ô

= ,

2

2

+ 2 ôñ = 0.

= .

Page 519

520 47.

THIRD-ORDER DIFFERENTIAL EQUATIONS cosh÷

ï ñóò ò ó ò ó

+ ( ô cosh÷

Particular solutions: 48.

cosh ÷

ï ñóò ò ó ò ó

ï

+ (ô

1

cosh÷

ï

ï ñóò ò ó ò ó

1

ï

3

50.

cosh÷

ï ñ óò ò ó ò ó

+ô

1

ï

3

cosh÷

ï ñ óò ò ó ò ó

+ô

1

ï

3

52.

cosh÷

ï ñ óò ò ó ò ó

+ô

1

ï

3

cosh÷

ï ñ óò ò ó ò ó

+ï

ï

= =

2

+ ï )ñó ò ò ó + ( ô – 2 ï cosh÷ −1

,

2

= .

– 6 ï cosh÷ −2

,

2

ï ñ óò

=

= 0.

ï )ñ óò

+ ôñ = 0.

= exp  − 21   X exp 

−1

=

ï

1 2

 2   . M

+ 2)ñó ò + ô (cosh÷

ï

sin  £   .

+ 2(2 cosh ÷

ï

– ô )ñ = 0.

+ 6(2 cosh ÷

ï

– ô )ñ = 0.

= .

ï

1

ñ

+ ï )ñ = 0.

3

(cosh÷

2 −

2

2

1 = cos  ln |  |  ,

Particular solutions:

–ô

2

2

ï ñ óò

+ ï ( ô – cosh ÷

2

ï ñóò

cosh÷

ï

cos  £   ,

−1

ï 2 ñ óò ò ó

Particular solutions: 53.

= exp  − 12   ,

=

2

−  . 2 = 

2

ï 2 ñ óò ò ó

Particular solutions:

+ 1)ñó ò ò ó + ô

+ 1)ñó ò ò ó + ô ( ï + 2 cosh ÷

ï

ï 2 ñ óò ò ó

Particular solutions: 51.

= ,

+ (3 cosh ÷

Particular solutions:

ô ï

+

cosh÷

Particular solutions: 49.

ï

+ ô )ñ

óò ò ó

,

=

2

ï )ñ ó ò

+ ( ô – 3 cosh÷

2 = sin  ln |  |  .

+ ï [ ô – ( õ + 1) cosh ÷

2

ï )ñ

ï ]ñ ó ò

= 0. + õ (2 cosh ÷

ï

– ô )ñ = 0.

.

3.1.4-3. Equations with hyperbolic sine and cosine. 54.

ñóò ò ó ò ó

+ [ ô sinh(2 ï ) + õ ]ñ„ó ò +

cosh(2 ï )ñ = 0.

ñ óò ò ó ò ó

+ [ ô sinh(2 ï ) + õ ]ñ

ñóò ò ó ò ó

+ [ ô cosh(2 ï ) + õ ]ñ„ó ò +

ñóò ò ó ò ó

+ [ ô cosh(2 ï ) + õ ]ñ„ó ò + 2 ô sinh(2 ï ) ñ = 0.

ñóò ò ó ò ó

+ ( õ cosh ï – ô

ô

This is a special case of equation 3.1.9.26 with  ( ) = 12 [  sinh(2 ) +  ]. 55.

56.

57.

58.

óò

+ 2 ô cosh(2 ï ) ñ = 0.

Integrating yields a second-order linear equation: "

 + [  sinh(2 ) +  ] = L .

ô

sinh(2 ï )ñ = 0.

Solution: = L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 and ¨ 2 form a fundamental set of solutions of the modified Mathieu equation 2.1.4.9: 4 ¨u

 + [  cosh(2 ) +  ]¨ = 0. Integrating yields a second-order linear equation: "

 + [  cosh(2 ) +  ] = L (see 2.1.4.9 for the solution of the corresponding modified homogeneous Mathieu equation with L = 0). 2

)ñó ò + õ (sinh ï – ô cosh ï )ñ = 0.

This is a special case of equation 3.1.9.29 with  ( ) =  cosh  . 3.1.4-4. Equations with hyperbolic tangent. 59.

ñ óò ò ó ò ó

–ô

3

tanh( ô

ï )ñ

= 0.

= cosh(  ) XZH ( )  leads to a M

second-order linear equation of the form 2.1.4.43: H  + 3  tanh(  ) H 
+ 3  2 H = 0. Particular solution:

0

= cosh(  ). The substitution

Page 520

521

3.1. LINEAR EQUATIONS

60.

ñ óò ò ó ò ó

=

ñóò ò ó ò ó

– 3ô

+ (1 – ô ) tanh ï

ôñ óò

ñ

.

This is a special case of equation 3.1.9.30 with  ( ) =  and ( ) = cosh  . 61.

2

ñóò

+ 2ô

3

Particular solutions: 62. 63.

64.

tanh÷

1

67.

ô

ñóò ò ó ò ó

+

ôñóò ò ó

+ [ õ tanh(

ñóò ò ó ò ó

+

ôñóò ò ó

– [2 ô tanh(

ï

ñ óò ò ó ò ó

– tanh ï

ñ óò ò ó

1

ï

) + 3 ]ñ„ó ò +

= cosh( Π ),

óò

– ôñ

ñ

ñóò ò ó ò ó

+

ô

tanh÷ (

ï )ñ óò ò ó

+

ñóò ò ó ò ó

+

ô

tanh÷ (

ï )ñ óò ò ó

+ ôƒõ tanh÷ (

õ ïƒú ñóò

The substitution ¨ = 

 + ' +

ñóò ò ó ò ó

tanh÷ (

ô

+

1

ñóò ò ó ò ó

+

tanh÷ (

ô

ï )ñ óò ò ó



ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó



,

ñóò ò ó ò ó + ô

tanh÷

1

= 

2

ñóò ò ó ò ó

+

ô ï

tanh÷

1

ï ñóò ò ó



73.

ñ óò ò ó ò ó

+

ô ï

2

tanh÷ (

ñ óò ò ó ò ó

= (tanh ÷

ï

= cosh  ,

2

+ (õ

ï

2

ï

+1

= cosh  ,

ï )ñ ó ò ò ó

Particular solutions: 74.

ï 

– ô )ñ

1

= ,

óò ò ó

) – ô ]ñ = 0.

tanh÷ (

ï

[ô

ï

3

cosh  .

3

cosh  .

)+þ

]ñ = 0.

2

'  ,

3 2

– 2ô

)+ö –

2

ïƒ÷

=  cosh  .

õ 2 ]ñ óò

ï )ñ óò

2

'  sin 

tanh÷ (

ï

+

ï )ñ ó ò

– õ )ñ 2

+ ö [ ô tanh ÷ ( 1

3 2

'  .

) + 2 õ ]ñ = 0.

óò

ï

) – õ ]ñ = 0.

and ý

õ ï (ô ï

+2

2

tanh÷

ï –ô

tanh÷

ï

2

are roots of the

1 2

ï

+ 2 tanh ï )ñ = 0.

+ 3)ñ = 0.

£  . 2 = sin  

= .

ï

) – õ ]ñ = 0.

− 21

(2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0.

=  cosh  .

2

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0.

õ 2 [ô

+ 3)ñó ò + (2 ô tanh÷

tanh÷ (

+ ( ô tanh ÷

= exp 

2

ï

= exp(ý 2  ), where ý

2

2

ï

tanh÷ (

) + 3 õ ]ñó ò +

ï

– ô tanh÷

1 2

õ 2[ô

+

.

£  , 1 = cos  

Particular solutions:

£ − ) + L 2 sin( 

tanh ï + 3)ñ„ó ò + [ ô

– (2 ô tanh÷

ï ñóò ò ó

2

'  cos 

− 12

+ [ ôƒõ tanh ÷ (

ïƒ÷

– (2 ô

–1

exp( £  ) + L

2

ï )ñ óò

– õ [2 ô tanh ÷ ( =

ï

) + 2 tanh(

leads to a first-order linear equation: ¨ 
+  tanh ( Œ  )¨ = 0.

= exp 

ï )ñ óò ò ó 1

õ ïƒú

+

ï

[2 ô tanh2 (

= 0.

2 b . Solution for  < 0:

Particular solutions: 72.

) + ö ]ñ = 0.

=  cosh( Π ).

2

+ ô tanh ï

2

= L 1 exp(− £  ) + L = L 1 cos( £ −  ) + L

Particular solutions: 71.

ï

) + ö ]ñ„ó ò + ô [ õ tanh(

Particular solutions: 1 = exp(ý 1  ), quadratic equation ý 2 + 'ý + ( = 0. 70.

.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.4.22: ¨2

 + [  tanh( Œ  ) + ( ]¨ = 0.

Particular solutions: 69.

ï )ñ

This is a special case of equation 3.1.9.30 with  ( ) =  tanh   and ( ) = cosh  .

Particular solutions: 68.

=  cosh(  ).

2

+ tanh ï (1 – ô tanh ÷

ï ñ óò

=

1 b . Solution for  > 0:

66.

= 0.

= cosh(  ),

ñ óò ò ó ò ó

Particular solutions: 65.

ï )ñ

tanh( ô

2

+ 2 ô tanh÷ (

+ õ tanh÷

ï ñ

ï )ñ

= 0.

.

= exp( Œ 2  ), where Œ

1

and Œ

2

are roots of the

Page 521

522 75.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

+ ( ô tanh÷

ï

õ ï )ñ ó ò ò ó

+

Particular solutions: 76.

ñóò ò ó ò ó

ïƒ÷

+ [ô

1

ñóò ò ó ò ó

1

ñóò ò ó ò ó

+ ( ô tanh÷

1

80.

81.

82.

83.

ï 2 ñ óò ò ó ò ó

+ (ô

ï

2

1

tanh÷

ï

2

=





,

ï 3 ñ óò ò ó ò ó

+

tanh÷

ï 2(ô

tanh ÷

ï ñóò ò ó ò ó

+

ôñóò ò ó

ï

óò ò ó

+ õ )ñ

+ ôƒõÿñó ò + = exp 

1

tanh ÷

+ õ tanh÷

ï ñóò ò ó ò ó

+

ôñóò ò ó

+ ï ( ôƒõ tanh ÷

ï

2

– õ tanh ÷

õ 2 (ô

2

'  cos 

ï ñóò

3 2

and

1

'  ,

+ ôƒõÿñ = 0.

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

+

ôñóò ò ó

tanh ÷

ï ñóò ò ó ò ó

– õ (2 ô + 3 õ tanh÷ 1





=

,

tanh ÷ ( tanh ÷ tanh ÷

ï ñ óò ò ó ò ó

ï

tanh÷

= 

+ ñó ò ò ó + [( õ – ô 2 ) tanh÷

ï )ñ óò ò ó ò ó

ï ñ óò ò ó ò ó

2

ô ï 2 ñ óò ò ó

+

1

+ ( ô tanh÷ + (ô

ï

ï ñóò ò ó ò ó

1

ï

1

,

ô ï

+

ï

2

ï ñóò 2

+ 1)ñ

óò ò ó

=

óò ò ó

2

= 0.

are roots of the quadratic

+ ö ( ô tanh ÷

¡

ï ñ

ï

– 2)ñ = 0.

are roots of the quadratic

2

3 2

'  .

= sin  £   .

2

2

= exp  £ −   .

+ 2 õ tanh ÷

õ 2 (ô



, where Œ

+

ô 2 ï ñ óò

and Œ

1

–ô

+ ô ( ï + 2 tanh ÷

+ ï )ñó ò ò ó + ( ô

−1

+ 1)ñ = 0.

ï )ñ

= 0.

2

ï )ñ

= 0.

are roots of the quadratic

+ 2 ôñ = 0.

= exp  − 12  2  ,

ï

2

= . + 1)ñ

2

+ ô ]ñó ò + õ (1 – ô tanh÷

= 

2

+

ï

.

−  . 2 = 

= ,

+ (3 tanh÷

Particular solutions:



– 2ô

= ,

tanh÷ 1

1

ï



ï )ñ óò 

+ ô tanh÷

ï

= exp  − 21 '  sin 

2

= cos  £   ,

ï ñóò ò ó ò ó

= 0.

= 0.

1

tanh ÷

(1 – õ tanh ï ) + 1]ñ = 0.

+1

¡

óò

and

1

¡

ï )ñ

' 2   . M

+ ö ]ñó ò + ô ( ö – õ + 2) tanh ÷

ï

+ ö – õ )ñ

=  + 2 , where

1 2

= 0.

ô 2 ƒï ÷ –1 ñ

+

– 1]ñó ò + õ (– ôƒõ tanh ÷

1 b . Particular solutions with  > 0:

Particular solutions: 88.

ïƒ÷ ñóò

+ [ ô ( õ – 2) ï tanh ÷

− 12

Particular solutions:

Particular solutions: 87.

2

= cosh  .

2

Particular solutions: 1 =  + 1 , equation 2 + (  − 1) + ( = 0. ¡ ¡

Particular solutions: 86.

]ñó ò ò ó – ô

ï

ï ñ

ïƒ÷

– õ )ñó ò ò ó

ï )ñ óò ò ó

+õ

– 1) ïƒ÷ – 1]ñó ò + õ [ ô

2

= cosh( Π ).

ï

tanh÷

ôƒõ

+

= exp  − 12 ' 2  X exp 

2

–1

– 1) tanh ÷

2

Particular solutions: 1 =  equation Π2 + Π+  = 0.

85.

ïƒ÷

óò

+ 2)ñ

Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0. ¡ ¡

Particular solutions: 84.

= ,

ï

= cosh  .

2

– 1) – ô

+ [ô ( õ

Particular solutions: 79.

,

– ôƒõ tanh÷

ï

+1





=

ï )(ô ïw÷

+ [ tanh(

Particular solutions: 78.

= exp  − 12 ' 2  ,

(tanh ï – õ ) – õ ]ñ„ó ò ò ó + [ ô ( õ

Particular solutions: 77.

tanh ÷

ï

+ õ (ô

cos   £   ,

2

ñ

= 0.

ï )ñ ó ò

+ ôñ = 0.

= exp  − 21  2  X exp 

2

tanh ÷

ï 2

=

−1

ï

1 2

 2   . M

+ 2)ñó ò + ô (tanh÷

sin   £   .

ï

+ ï )ñ = 0.

Page 522

523

3.1. LINEAR EQUATIONS

ï

3

89.

tanh÷

ï ñóò ò ó ò ó

+

Particular solutions:

ï

3

90.

tanh÷

ï ñ óò ò ó ò ó

+

1

ï

3

tanh÷

ï ñ óò ò ó ò ó

+

1

ï

3

tanh÷

ï ñ óò ò ó ò ó

=

−2

,

2

= .

2

– 6 ï tanh ÷ ,

3

ï ñóò

+ 2(2 tanh÷

ï

– ô )ñ = 0.

ï ñ óò

+ 6(2 tanh÷

ï

– ô )ñ = 0.

= .

2

+ ï ( ô – tanh÷

ô ï 2 ñ óò ò ó

1 = cos  ln |  |  ,

Particular solutions: 92.

=

−1

ô ï 2 ñ óò ò ó

Particular solutions: 91.

– 2 ï tanh ÷

ô ï 2 ñ óò ò ó

+ï

2

(tanh÷

Particular solutions:

1

2 −

=

ï

ï )ñ

2 = sin  ln |  |  .

+ ï [ ô – ( õ + 1) tanh÷

+ ô )ñ

óò ò ó

,

=

2

+ ( ô – 3 tanh÷

ï )ñ ó ò

2

ï ]ñ ó ò + õ

= 0. (2 tanh÷

ï

– ô )ñ = 0.

.

3.1.4-5. Equations with hyperbolic cotangent. 93.

ñ óò ò ó ò ó

–ô

3

ï )ñ

coth( ô

= 0.

= sinh(  ) XZH ( )  leads to a M

second-order linear equation of the form 2.1.4.52: H  + 3  coth(  ) H 
+ 3  2 H = 0. Particular solution:

94.

ñóò ò ó ò ó

=

ñ óò ò ó ò ó

– 3ô

0

= sinh(  ). The substitution

+ (1 – ô ) coth ï

ôñóò

ñ

.

This is a special case of equation 3.1.9.30 with  ( ) =  and ( ) = sinh  . 95.

2

ñ óò

+ 2ô

3

Particular solutions: 96.

97.

98.

coth÷

ï ñóò

1

ô

ñ óò ò ó ò ó

+

ôñ óò ò ó

+ [ õ coth(

ñ óò ò ó ò ó

+

ôñ óò ò ó

– [2 ô coth(

ï )ñ

.

ï

óò

) + ö ]ñ

+ ô [ õ coth(

ï

) + ö ]ñ = 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.4.44: ¨2

 + [  coth( Œ  ) + ( ]¨ = 0.

ñóò ò ó ò ó

– coth ï

ñóò ò ó

1

ñóò ò ó ò ó

ï

2

coth ï

ô

ñ

2

+

[2 ô coth2 (

ï

ï

) + 2 coth(

) – ô ]ñ = 0.

=  sinh( Π ). = 0.

= L 1 exp(− £  ) + L 2 exp( £  ) + L 3 sinh  . = L 1 cos( £ −  ) + L 2 sin( £ −  ) + L 3 sinh  .

+ ( ô coth ï – ôƒõ – õ )ñ„ó ò ò ó + ( ôƒõ 1

ñóò ò ó ò ó

+

ô

coth÷ (

ï )ñ óò ò ó

ñóò ò ó ò ó

+

ô

coth÷ (

ï )ñ óò ò ó

=





,

1

2

+

õ ïƒú ñóò

+

ôƒõ

The substitution ¨ = 

 + ' +

Particular solutions:

óò

) + 3 ]ñ

= sinh( Π ),

– ôñó ò +

Particular solutions:

102.

=  sinh(  ).

This is a special case of equation 3.1.9.30 with  ( ) =  coth   and ( ) = sinh  .

2 b . Solution for  < 0:

101.

2

+ coth ï (1 – ô coth÷

=

1 b . Solution for  > 0:

100.

= 0.

= sinh( ),

ñóò ò ó ò ó

Particular solutions: 99.

ï )ñ

coth( ô

= exp 

2

– ô – 1)ñó ò + õ (– ôƒõ coth ï + ô + 1)ñ = 0.

= sinh  . +

õ ïƒú

–1

[ô

ï

coth÷ (

ï

)+þ

]ñ = 0.

leads to a first-order linear equation: ¨ 
+  coth ( Œ  )¨ = 0.

coth÷ ( − 12

ï )ñ óò

'  cos 

2

+ 3 2

õ 2 [ô

'  ,

coth÷ ( 2

ï

) – õ ]ñ = 0.

= exp  − 21 '  sin 

2

3 2

'  .

Page 523

524 103.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

coth÷ (

ô

Particular solutions: 104.

ñóò ò ó ò ó

+

ô ï

coth÷

ï ñóò ò ó

Particular solutions: 105.

ñóò ò ó ò ó

+

ô ï

2

coth÷ (

107. 108.

110.

111.

112.



=

+ ô ï ÷ ñ ó ò ò ó – (2 ô Particular solutions:

,

ï

+ (õ

1 2

1

ï

+

+ [ coth( ï )( ô Particular solutions:

1

¡

¡

¡

¡

+ ï 2 ( ô coth÷ Particular solutions: equation 2 + (  − 1)

ï 3 ñ óò ò ó ò ó

coth ÷ ï ñó ò ò ó ò ó

+

ôñóò ò ó

Particular solutions:

113. coth ÷

ï ñóò ò ó ò ó

+

ôñóò ò ó

ï

+1

ï

+ õ (ô

ï

1

2

coth÷

ôƒõÿñóò

+

= exp 

− 12

+ õ coth÷

'  cos 

ï ñóò

2

ïƒ÷ ñóò

Particular solutions:

=

ï





,

2

2

3 2

ï )ñ

ï ñ

+ ö ]ñ 1 and

1 2

= 0.

' 2   . M

ô 2 ƒï ÷ –1 ñ óò ¡

+ 2 coth ï )ñ = 0.

ï

= 0.

+ ô ( ö – õ + 2) coth÷ ï ñ = 0. 2 are roots of the quadratic

= 0.

'  ,

2

= cos  £   ,

= exp  − 21 '  sin 

= 

2

= exp  − £ −   ,

– õ (2 ô + 3 õ coth÷ 1

– ô coth÷

ï

+

+ ôƒõÿñ = 0. 1

ôñóò ò ó

+2

= exp  − 12 ' 2  X exp 

2

– õ coth÷

õ 2 (ô

2 b . Particular solutions with  < 0: +

= 0.

)ñó ò ò ó + ï ( ôƒõ coth÷ ï + ö – õ )ñó ò + ö ( ô coth÷ ï – 2)ñ = 0.  + 1 , 2 =  + 2 , where 1 and 2 are roots of the quadratic ¡ ¡ = 0.

1

ï ñóò ò ó ò ó

ï )ñ

+ 2)ñó ò + ôƒõ coth÷

ï

– 1) – ô ïƒ÷ –1 ]ñó ò ò ó – ô =  , 2 = sinh( Œ  ).

1 b . Particular solutions with  > 0: 114. coth ÷

+ 3)ñ = 0.

2

+ 3)ñó ò + (2 ô coth ÷ =  sinh  .

= exp  − 12 ' 2  ,

+õ 1 = +( +

ï

) + 2 õ ]ñ = 0.

coth ï + 3)ñ ó ò + [ ô ï ÷ (2 coth2 ï – 1) + 2 coth ï ]ñ = 0. = sinh  , 2 =  sinh  .

+ ( ô ï 2 coth÷ ï + õ ï )ñ ó ò ò ó + [ ô ( õ – 2) ï coth ÷ Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

coth÷

+ 2 ô coth÷ (

ïw÷

ñóò ò ó ò ó

2

ï

= .

2

õ ï )ñ óò ò ó 1

1 2

ï )ñ óò

2

+ ô coth÷ ï ñó ò ò ó – (2 ô coth÷ Particular solutions: 1 = sinh  , + ( ô coth÷

õ ï (ô ï

+

coth÷ (

õ 2 [ô

£  . 2 = sin  

coth÷ (

ï

ñóò ò ó ò ó ñóò ò ó ò ó

ï )ñ óò

2

– 2ô

= ,

) + 3 õ ]ñó ò +

.

– ô coth÷

2

ï ÷

ñ óò ò ó ò ó



= 

2

ï 

£  , 1 = cos  

1

Particular solutions: 109.

1



ï )ñ óò ò ó

Particular solutions: 106.

– õ [2 ô coth÷ (

ï )ñ óò ò ó





ï )ñ óò

+

õ 2(ô

2

3 2

'  .

= sin  £   . 2

= exp  £ −   .

+ 2 õ coth÷

ï )ñ

= 0.

.

115. coth ÷ ï ñ ó ò ò ó ò ó + ñ ó ò ò ó + [( õ – ô 2 ) coth÷ ï + ô ]ñ ó ò + õ (1 – ô coth ÷ ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0. 116. coth ÷ (

ï )ñ óò ò ó ò ó

+

ô ï 2 ñ óò ò ó

Particular solutions:

1

ï ñóò

– 2ô

= ,

117. coth ÷ ï ñó ò ò ó ò ó + ( ô coth÷ ï + ô Particular solutions: 1 =  ,

ï

2

+ 2 ôñ = 0.

2

= .

2

+ 1)ñó ò ò ó + ô  =  − .

2

ï ñóò

–ô

2

ñ

= 0.

Page 524

525

3.1. LINEAR EQUATIONS

118. coth ÷

ï ñ óò ò ó ò ó

ï

+ (ô

coth÷

Particular solutions: coth÷

ï

119.

1

ï

3

coth÷

ï ñóò ò ó ò ó

+ô

1

ï

3

121.

coth÷

ï ñóò ò ó ò ó

+ô

1

ï

3

coth÷

ï ñóò ò ó ò ó

+ô

1

ï

3

coth÷

ï ñ óò ò ó ò ó

+ï

=

−2

1 2

1

ï ñóò

= .

2

,

2

3

ï ñóò

= .

2

+ ï ( ô – coth ÷

= cos  ln   ,

(coth÷

Particular solutions:

,

2

– 6 ï coth÷

ï 2 ñ óò ò ó

Particular solutions: 123.

=

ï 2 ñ óò ò ó

Particular solutions: 122.

– 2 ï coth÷ −1

ï

,

2

+ ôñ = 0.

−1

=

=

 2   . M

+ 2(2 coth÷

ï

– ô )ñ = 0.

+ 6(2 coth÷

ï

– ô )ñ = 0.

+ ( ô – 3 coth÷

ï )ñ óò

+ ï )ñ = 0.

ï

sin   £   .

+ ï [ ô – ( õ + 1) coth÷

2

1 2

+ 2)ñó ò + ô (coth÷

ï

ï )ñ

= sin  ln   .

2

óò ò ó

+ ô )ñ

2 −

=

coth÷

ï

cos   £   ,

−1

=

ï )ñ ó ò

= exp  − 21  2  X exp 

2

+ ï )ñó ò ò ó + ( ô

ï

ï 2 ñ óò ò ó

Particular solutions:

+ ô ( ï + 2 coth÷

óò ò ó

+ 1)ñ

= exp  − 12  2  ,

+ (3 coth÷

ï ñóò ò ó ò ó

Particular solutions: 120.

ï

ï ]ñ ó ò

.

= 0. + õ (2 coth ÷

ï

– ô )ñ = 0.

3.1.5. Equations Containing Logarithmic Functions 3.1.5-1. Equations with logarithmic functions. 1.

2.

ñóò ò ó ò ó

ln ÷ (

ô

+

ï )ñ óò ò ó

5.

6.

+

ln÷ (

ôƒõ

1

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ó

ln ÷

ô

+

ï ñóò ò ó

+

ñóò ò ó ò ó

+

ô

ln ÷ (

ï )ñ óò ò ó

ln ÷

ï ñóò

= exp 

− 12

ôƒõ 1

Particular solutions: 4.

õÿñóò

1 b . Particular solutions with  > 0:

Particular solutions: 3.

+

+

1

=



,

ï

ï

+ õ )ñó ò ò ó +

ö“ñóò

ï

õ 2 [ô





2

ï

+ õ )ñ = 0.

= exp  − £ − (  ,

ln ÷

ï ñ óò ò ó ò ó

+

ôñ óò ò ó

Particular solutions:

+

– õ )ñó ò + õ ln ÷

1

2

+

õ 2 (ô

– õ ln÷

= exp 

− 12

'  cos 

ôƒõÿñ óò

'  .

) + 2 õ ]ñ = 0.

+ ö ( ô ln÷ ï – õ )ñ = 0. = exp( Œ 2  ), where Œ 1 and Œ

1

ï

ï

3 2

õ 2 )ñ óò

2 b . Particular solutions with ( < 0: – ô )ñó ò ò ó + ( ô ln÷

2

.

= cos  £ (  ,

ï

= exp  £ −   .

ln÷ (

1

= (ln÷

2

= exp  − 21 '  sin 

1 b . Particular solutions with ( > 0:

ñóò ò ó ò ó

= sin   £   .

– õ )ñ = 0.

) + 3 õ ]ñó ò +

3 2

+ ö ( ô ln÷

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0.

7.

2

2

+ ô ln ÷ ï ñó ò ò ó + ( ôƒõ ln÷ ï + ö – Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0. + ( ô ln÷

ln ÷

'  ,

ñóò ò ó ò ó ñóò ò ó ò ó

2

= exp  − £ −   ,

õ 2 (ô

= 

2

= 0.

= cos   £   ,

'  cos 

– õ [2 ô ln÷ (



ï )ñ

2

ï ñ

= sin  £ (  . 2

= exp   £ − (  .

.

= exp( Œ 2  ), where Œ

2

ï )ñ 3 2

1

and Œ

= 0.

'  ,

2

are roots of the

2

= exp  − 21 '  sin 

2

2

are roots of the

3 2

'  .

Page 525

526 8.

THIRD-ORDER DIFFERENTIAL EQUATIONS ln ÷

ï ñóò ò ó ò ó

+ ñó ò ò ó + [( ô –

õ

2

) ln÷

Particular solutions: 1 =  equation Œ 2 + Œ +  = 0. 9.

10.

ln ÷

ï ñ óò ò ó ò ó





=

Particular solutions:

1

ln ÷ (

+ õ ln÷ (

ï )ñ óò ò ó ò ó

+

+ õ ]ñó ò + ô (1 – õ ln ÷

ï

,

ôñóò ò ó

,

2

 2

= 

2

– õ (2 ô + 3 õ ln ÷

ôñ óò ò ó

+

 1

ï )ñ ó ò

ï )ñ óò

+





= 

, where Œ

1

ï )ñ

and Œ

+ 2 õ ln÷

õ 2 (ô

= 0. 2

ï )ñ

are roots of the quadratic = 0.

.

ôƒõÿñ

+

= 0.

1 b . Particular solutions with  > 0:

1

= cos   £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

= sin   £   .

2

2

= exp   £ −   .

3.1.5-2. Equations with power and logarithmic functions. 11.

ñóò ò ó ò ó

+ õ ) ln ÷ (

ï

+ (ô

Particular solution: 12.

ñ óò ò ó ò ó

Particular solution: 13.

– ô ln÷ (

ï )ñ ó ò

– 2 ô ln÷ (

+

ô

ln ÷ (

ï )ñ óò ò ó

+

+

ô ï

ln÷

ï ñóò ò ó

+ (õ

ñóò ò ó ò ó ñóò ò ó ò ó

ï ñ óò ò ó

ï )ñ

= 0.

= (  +  ) .

0

ln ÷

+

= 0.

2

ô

ñ óò ò ó ò ó

ï )ñ

=  +  .

0

+ õ ) ln ÷ (

ï

+ (ô

ï )ñ óò

ï

+ (õ

+ ö )ñ

óò

+ ( ôƒõ

ln÷

ï

ï

ln÷

ô„ö

+

ï

+ õ )ñ = 0.





Integrating yields a second-order linear equation:  + ( ' + ( ) = L exp O − lX ln   M P (see 2.1.2.2 for the solution of the corresponding homogeneous equation with L = 0). 14.

15.

ñóò ò ó ò ó

+ ( ô ln÷

ï

+

18.

ln÷

ï

ñóò ò ó ò ó

+ (ô

ñóò ò ó ò ó

+ ( ôƒõ

ï

20.

–1

ï

ln÷ (

ï

+

õ ï (ô ï

2

[ô

2

– ô ln÷

ï )ñ óò

£  , 1 = cos  

1

1 2

)+þ

]ñ = 0.

ln÷

+ 3)ñ = 0.

ln÷

ï

ï

2

+ 2)ñó ò + ôƒõ ln÷

= exp  − 12 '  , 2

ï

ï

£  . 2 = sin   1 2

2

+ õ (ô

+ õ )ñó ò ò ó + ô ( õ

2

– 1) ln ÷

ï ñ

= 0.

= exp  − 12 '  X exp  2

ï ñóò

– ôƒõ ln ÷

ï ñ

' 2   . M

1 2

= 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.5.13: ¨2

 +  ln  ¨2
−  ln   ¨ = 0.

ï

ln÷

+ ô ln÷

ï

Particular solutions: 19.

ï

õ ï )ñ óò ò ó

Particular solutions: 17.

õ ïƒú

+

¨ 
+  ln ( Œ  )¨ = 0. The substitution ¨ = 

 + ' +* leads to a first-order linear equation: u Particular solutions:

16.

õ ïƒú ñóò

ñóò ò ó ò ó

+

ô ï

2

ln÷ (

1

ï )ñ óò ò ó

ï

+ õ )ñó ò ò ó +

= , – 2ô

2

=

−

ln÷ (

ï





ôƒõ 2 ï

ï )ñ óò

= ,

ï ñóò ò ó ò ó

ï

ln2 ï + 1)ñó ò – ôƒõ ( õ

ô ï ñóò ò ó

– õ (õ

The substitution ¨ = 
+   ¨2

 − (  2  ln2  +  )¨ = 0.

– ôƒõ

2

ln ÷

ï ñ

= 0.

+ 2 ô ln÷ (

ï )ñ

= 0.

2

1

+

ï ñóò

.

Particular solutions:

2

ln÷

= .

ï

ln2 ï + 1)ñ = 0.

leads to a second-order linear equation of the form 2.1.5.3:

Page 526

527

3.1. LINEAR EQUATIONS

21.

ï ñ óò ò ó ò ó

+

ln÷ (

ô

ï )ñ ó ò ò ó

õ ï ñ óò

+

+ ôƒõ ln÷ (

23.

2 b . Particular solutions with  < 0:

ï ñ óò ò ó ò ó

õ 2ï

+

ln ï

ô ï

ñ óò ò ó

+ ( ôƒõ =

Particular solutions:

1

= (ln÷

ï )ñ óò ò ó

ï ñóò ò ó ò ó

ï

–ô

ï −





ln ï –

ï ñóò ò ó ò ó

ln ÷

ï

+ (ô



–

õ ï )ñ óò

,

+ ( ô ln÷

Particular solutions: 25.

ï ñóò ò ó ò ó

ln ÷

ï

+ (ô

ï

1

ï

ï ñóò ò ó ò ó

1

ï 2 ñ óò ò ó ò ó

,

2

=

+ [ ô ( õ – ln ï ) ïw÷ + 2]ñó ò ò ó + ô

Particular solutions: 27.

=

−1

ln÷ (

ô

+

1

ï )ñ óò ò ó

= , +

2

+

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0: 28.

29.

30.

31.

32.

33.

ï

+ õ ln÷

õ ï )ñ óò

+

2

ï

=

−1

M

 .

ï ñ

.

ïƒ÷ –1 ñ óò

–ô

ln ÷ (

ï

–

ïƒ÷ –2 ñ

ï )ñ

ln ÷

ï

ï

1

and Œ

2

are roots of the

+ 1)ñ = 0.

sin   £   .



  .

−1 −

ôƒõ

= 0.



+ õ (ô

+ 2 ô ln ÷

= ln  −  + 1.

õ ï 2 ñ óò



+2 )

= exp( Œ 2  ), where Œ

2

ln÷

ï

ôƒõÿñ

−

cos   £   ,

+ 3)ñó ò ò ó + ( ôƒõ

Particular solutions: 26.

=

−1

+

   ( 2 =   XZ

+ 3)ñó ò ò ó + (2 ô ln ÷

ï

óò

+ ô )ñ

−

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 24.

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp   £ −   . 1 = exp  − 

1 b . Particular solutions with  > 0:

22.

ï )ñ

õ 2 ï )ñ óò

+ õ ( ô ln÷

ï

– õ )ñ = 0.

= 0.

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp   £ −   . 1 = exp  − 

ï 2 ñ óò ò ó ò ó

+ ï 2 ( ô ln ï + õ )ñ ó ò ò ó + 2 ô ï ñ ó ò – ôñ = 0. Integrating the equation twice, we arrive at a first-order linear equation:
 + (  ln  +  ) = L 1 + L 2 .

ï 2 ñ óò ò ó ò ó

ï [ô ï

– 3ô

ln2 (

ï

ï

+ ô [2 ô

2 2

1

ï 2 ñ óò ò ó ò ó

+

ln ï +

õ ï )ñ óò ò ó

ï 2 ñ óò ò ó ò ó

+ (ô

ln÷

+

Q

+ 2ï ( õ

ï

ï

ln3 (

= exp lX ln( Π )  S ,

Particular solutions:

ï 2(ô

óò

) + 1]ñ

2

M

) + 1]ñ = 0.

=  exp lX ln( Π )  S .

Q

M

+ ô )ñó ò – ôñ = 0.

Integrating the equation twice, we arrive at a first-order linear equation:
 + (  ln  + ' ) = L 1 + L 2 .

ï

2

ï

õ ï )ñ óò ò ó

+ [ ô ( õ – 2) ï ln ÷

ï

Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0. ¡ ¡

+ ö ]ñó ò + ô ( ö – 1

and

¡

2

õ

+ 2) ln ÷

ï ñ

= 0.

are roots of the quadratic

ï 3 ñ óò ò ó ò ó

+ ï 2 ( ô ln ï + õ )ñ ó ò ò ó + 2 ô ï ñ ó ò – ôñ = 0. Integrating the equation twice, we obtain a first-order linear equation: 
 + (  ln  +  − 2) = L 1 + L 2 .

ï 3 ñ óò ò ó ò ó

+

ô

ln÷ (

ï )ñ ó ò ò ó

+

õ ï 3 ñ óò

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0:

+

ôƒõ

ln ÷ (

ï )ñ

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp   £ −   . 1 = exp  − 

Page 527

528 34.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 3 ñ óò ò ó ò ó

ô ï

+

2

ln÷

Particular solutions: 35.

36.

37.

38.

ï 3 ñ óò ò ó ò ó

ô ï

+

2

ln÷

−2

=

Particular solutions:

1

ï 3 ñ óò ò ó ò ó

+

ï 2(ô

ln ï +

õ ï )ñ óò ò ó

ï 3 ñ óò ò ó ò ó

+

ï 2(ô

ln÷

+ õ )ñ

+ 2(2 – ô ln ÷

ñóò ,

– 6ï

ï ñ óò ò ó

2

,

2

= 0.

ï )ñ

= 0.

2

= . + 6(2 – ô ln ÷

ñ óò

ï )ñ

3

= .

+ 2ï ( õ

ï

óò

+ ô )ñ

– ôñ = 0.

Integrating the equation twice, we obtain a first-order linear equation: 
 +(  ln  + ' −2) = L 1 + L 2 .

ï

óò ò ó

+ ï ( ôƒõ ln÷

Particular solutions: 1 =  + 1 , equation 2 + (  − 1) + ( = 0. ¡ ¡

ln ÷

ï ñ óò ò ó ò ó

ln ÷

ï

+ (ô

ï

ln ÷

ï ñóò ò ó ò ó

+ ( ô ln÷

ln ÷ (

ï )ñ ó ò ò ó ò ó

+

+ 1)ñ

ï

+ô

ï

1

= ,

ô ï 2 ñ óò ò ó

– 2ô

Particular solutions:

ï

= ,

1

2

+ ô ( ï + 2 ln÷

óò ò ó

2

+ 1)ñó ò ò ó + ô

2

 . 2 =  −

ï ñ óò 2

ö

+

– õ )ñ

=  + 2 , where

= exp  − 12  2  ,

1

Particular solutions: 40.

−1

=

1

Particular solutions: 39.

– 2ï

ï ñóò ò ó

¡

ï )ñ ó ò

óò

+ ö ( ô ln ÷

1

and

¡

2

–ô

2

– 2)ñ = 0.

are roots of the quadratic

+ ôñ = 0.

= exp  − 21  2  X exp 

ï ñóò

ï

ñ

1 2

 2   . M

= 0.

+ 2 ôñ = 0.

=  2.

3.1.6. Equations Containing Trigonometric Functions 3.1.6-1. Equations with sine. 1.

2.

3.

4.

5.

ñ óò ò ó ò ó

+

ô

sin ï

ñóò ò ó ò ó

+

ô

sin2 ï

ñ óò ò ó ò ó

+ [ ô sin(

ñ óò ò ó ò ó

+

– õ ( ô sin ï +

ñ óò

õ 2 )ñ

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a second-order linear equation of the form 2.1.6.2: ¨2

 +   sin  + 3  2  ¨ = 0. 4

ñóò

– õ ( ô sin2 ï +

õ 2 )ñ

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a second-order linear equation of the form 2.1.6.3: ¨ 

 +  sin2  + 3  2  ¨ = 0. 4

ï

óò

) + õ ]ñ

– ö [ ô sin(

ï

) + õ + ö 2 ]ñ = 0.

 The substitution ¨ =  v J 2(
 − ( ) leads to a second-order linear equation of the form 2.1.6.2: ¨2

 + V  sin( Œ  ) +  + 3 ( 2 W ¨ = 0. 4

ôñ óò ò ó

+ [ õ sin(

ï

) + ö ]ñ

The substitution ¨ = 
+  ¨2

 + [  sin( Œ  ) + ( ]¨ = 0.

ñ óò ò ó ò ó

+

ôñ óò ò ó

+ õ sin2 (

ï )ñ ó ò

The substitution ¨ = 
+  ¨2

 +  sin2 ( Œ  ) ¨ = 0.

óò

+ ô [ õ sin(

ï

) + ö ]ñ = 0.

leads to a second-order linear equation of the form 2.1.6.2: + ôƒõ sin2 (

ï )ñ

= 0.

leads to a second-order linear equation of the form 2.1.6.3:

Page 528

529

3.1. LINEAR EQUATIONS

6.

7.

ñ óò ò ó ò ó

sin ÷ (

ô

+

ï )ñ ó ò ò ó

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ñ óò ò ó ò ó

sin ÷ (

ô

+

+ ôƒõ sin÷ (

ï )ñ ó ò ò ó

Particular solutions: 8.

ñóò ò ó ò ó

+

sin ÷ (

ô

1

10. 11.

= exp 

1





=

ï

+ ô sin ÷„ï ñó ò ò ó + ( ôƒõ sin ÷„ï + ö – Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

13.

18.

19.

sin÷ (

õ 2 [ô

ï

2

3 2

'  .

) + 2 õ ]ñ = 0.

. + ö ( ô sin÷„ï – õ )ñ = 0. = exp( Œ 2  ), where Œ 1 and Œ

õ 2 )ñ óò 2

2

are roots of the

+ ( ô sin÷„ï + õ )ñó ò ò ó + ö^ñó ò + ö ( ô sin÷„ï + õ )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ (  . = exp  − £ − (  ,

1

= exp   £ − (  .

2

= (sin ÷„ï – ô )ñó ò ò ó + ( ô sin ÷„ï – õ )ñó ò + õ sin ÷„ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

+ ( ô sin÷„ï +

ñ óò ò ó ò ó

+

ô ï

sin ÷

õ ï )ñ óò ò ó 1

ï ñ óò ò ó

ñóò ò ó ò ó

+ ( ôƒõ

ï

sin÷„ï +

ñóò ò ó ò ó

+

ô ï

sin ÷ (

2

Particular solutions: 17.

= exp  − 21 '  sin 

2

ñóò ò ó ò ó

Particular solutions: 16.



) – õ ]ñ = 0.

+ ô sin ÷ ( ï )ñó ò ò ó + õ ïƒú ñó ò + õ ïƒú –1 [ ô ï sin÷ ( ï ) + þ ]ñ = 0. The substitution ¨ = 

 + '  +2 leads to a first-order linear equation: ¨u
+  sin ( Œ  ) ¨ = 0.

ï ñ óò ò ó ò ó

ï

+ õ (ô

= exp 

+ (õ

ï

2

− 12

sin ÷„ï + 2)ñó ò + ôƒõ sin÷„ï

'  , 2

– ô sin÷

2

ï )ñ ó ò

£  , 1 = cos   1 2

Particular solutions: 15.

2

ï

ñóò ò ó ò ó

Particular solutions: 14.



'  ,

= exp   £   .

= sin   £ −   .

2

sin ÷ (

) + 3 õ ]ñó ò +

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0: 12.

2

= cos   £ −   ,

= 

2

= 0.

= exp  − £   ,

'  cos 

− 12

,

ï )ñ

ï )ñ ó ò + õ 2 [ ô 2 3

– õ [2 ô sin ÷ (

ï )ñ óò ò ó

Particular solutions: 9.

– ôƒõ sin ÷ (

óò

– õÿñ

2

sin ÷„ï + õ )ñó ò ò ó +

ô 1

= ,

ï )ñ óò ò ó 1

– 2ô

= ,

2

ï

2

=

−





sin÷ ( 2

= exp  +

− 12

õ ï (ô ï

ôƒõ 2 ï

sin ÷„ï

and Œ

2

are roots of the

= 0.

'  X exp  12 '  2   . M 2

2

sin÷

ï

£  . 2 = sin   1 2

ñ

1

+ 3)ñ = 0.

2

ñóò

– ôƒõ

+ 2 ô sin÷ (

ï )ñ

2

sin ÷„ï

ñ

= 0.

.

ï )ñ óò

= 0.

= .

+ ï [ ô sin( ï ) + õ ]ñ ó ò – 2[ ô sin( ï ) + õ ]ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.6.2: ¨2

 + [  sin( Œ  ) +  ]¨ = 0.

= (sin÷ ï – ô ï )ñ ó ò ò ó + ( ô sin÷ ï – õ ï )ñ ó ò + õ sin ÷ ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ï ñ óò ò ó ò ó ï ñóò ò ó ò ó

+ (ô

ï

sin ÷„ï + 3)ñó ò ò ó + (2 ô sin ÷„ï +

Particular solutions:

1

=

−1

cos   £   ,

õ ï )ñ óò

2

=

−1

+ õ (ô

ï

1

and Œ

2

are roots of the

sin ÷„ï + 1)ñ = 0.

sin   £   .

Page 529

530 20.

21.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó ò ó

= (sin ÷„ï – ô ï 2 )ñó ò ò ó + ( ô sin ÷„ï – õ ï 2 )ñó ò + õ sin÷„ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ 1 and Œ quadratic equation Œ 2 + Œ +  = 0.

ï 3 ñ óò ò ó ò ó

+ ô ï 2 sin ÷ ( ï )ñó ò ò ó + õ ï Particular solutions: 1 =  + 1 , equation 2 − +  = 0.

¡

22.

23.

25.

27.

sin ÷

ï ñ óò ò ó ò ó

+

ôñ óò ò ó

sin ÷„ï

ñóò ò ó ò ó

+

29.

ôñóò ò ó

1

¡

= exp 

– õ sin ÷

õ 2 (ô ñóò

+ ôƒõÿñ = 0.

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ó

+

ï )ñ

'  cos  1

sin ÷„ï

2

− 12

1 b . Particular solutions with  > 0:

¡

3 2

1

=





,

2

'  ,

2

= cos   £   ,

= exp  − 21 '  sin 

2

= exp  − £ −   ,

– õ (2 ô + 3 õ sin ÷„ï )ñó ò +

ôñóò ò ó

= 0.





= 

2

3 2

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ sin ÷„ï )ñ = 0.

õ 2 (ô

.

sin ÷„ï ñó ò ò ó ò ó + ñó ò ò ó + [( õ – ô 2 ) sin ÷„ï + ô ]ñó ò + õ (1 – ô sin ÷„ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic 2 equation Œ + Œ +  = 0. sin ÷ (

ï )ñ óò ò ó ò ó

ô ï 2 ñ óò ò ó

+

1

ï ñóò

– 2ô

= ,

2

+ 2 ôñ = 0.

=  2.

sin ÷„ï ñó ò ò ó ò ó + ( ô sin ÷„ï + ô ï + 1)ñó ò ò ó +  Particular solutions: 1 =  , 2 =  −  . sin ÷„ï

ï

ñóò ò ó ò ó

sin ÷ï

30.

ï

+ (ô

ñóò ò ó ò ó

ï

3

31.

sin ÷„ï

ñóò ò ó ò ó

1

ï

3

sin ÷„ï

ñóò ò ó ò ó

1

1

sin ÷„ï ñó ò ò ó ò ó + ô ï Particular solutions: 3

sin ÷„ï

ñóò ò ó ò ó

+

=

ô ï 2 ñ óò ò ó

+

3

ï

=

ô ï 2 ñ óò ò ó

+

Particular solutions:

ï

= exp  − 12  2  ,

+ (3 sin ÷„ï + ï )ñó ò ò ó + ( ô

Particular solutions: 32.

ô 2 ï ñ óò

–ô

2

ñ

= 0.

sin÷„ï + 1)ñó ò ò ó + ô ( ï + 2 sin ÷„ï )ñó ò + ôñ = 0.

Particular solutions:

34.

+

+ õ sin ÷„ï

Particular solutions:

33.

óò

+ ôƒõÿñ

Particular solutions:

28.

2

sin2 ï ñó ò ò ó ò ó + ô sin2 ï ñó ò ò ó + õÿñó ò + ôƒõÿñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.6.21: sin2  ¨ 

 +  ¨ = 0.

Particular solutions:

26.

+ õ [ ô sin ÷ ( ï ) – 2]ñ = 0. =  + 2 , where 1 and 2 are roots of the quadratic

ñóò

¡

Particular solutions:

24.

are roots of the

2

ï

2

Particular solutions:

1 2

=

−1

,

2

,

2

= exp  − 21  2  X exp 

2

=

−1

 2   . M

sin   £   .

+ 2(2 sin ÷„ï – ô )ñ = 0.

ñóò

+ 6(2 sin ÷„ï – ô )ñ = 0.

=  3.

1 2

sin ÷„ï + 2)ñó ò + ô (sin÷„ï + ï )ñ = 0.

ñóò

=  2.

– 6 ï sin ÷„ï −2

ï

cos   £   ,

– 2 ï sin ÷„ï −1

2

1

+ ï ( ô – sin ÷„ï )ñó ò + ( ô – 3 sin ÷„ï )ñ = 0. = cos(ln  ), 2 = sin(ln  ).

1

=

ñóò ò ó

(sin÷„ï + ô )ñó ò ò ó + ï [ ô – ( õ + 1) sin ÷„ï ]ñó ò + õ (2 sin ÷„ï – ô )ñ = 0. −

2

,

2

=

2

.

Page 530

531

3.1. LINEAR EQUATIONS

3.1.6-2. Equations with cosine. 35.

36.

37. 38.

cos(2 ï )ñ

ñ óò ò ó ò ó

+

ñóò ò ó ò ó

+ [ ô cos(

ñ óò ò ó ò ó

+

ôñ óò ò ó

ñóò ò ó ò ó

+

ô

ô

– õ [ ô cos(2 ï ) +

óò

õ 2 ]ñ

ï

+ ( õ cos 2 ï + ö )ñ

The substitution ¨ = 
+  cos÷ (

ï )ñ óò ò ó

ñ óò ò ó ò ó

cos÷ (

ô

+

+ ô ( õ cos 2 ï + ö )ñ = 0.

óò

+ ôƒõ cos÷ (

õÿñóò

ñ óò ò ó ò ó

cos÷ (

ô

+

ñóò ò ó ò ó

ï )ñ ó ò

1





=

,

2

ï

= 



ö

–

+ ( ôƒõ cos÷„ï +

ñóò ò ó

Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

43.

44.

cos÷ (

ï )ñ ó ò ò ó

õ ï ú ñ óò

ñ óò ò ó ò ó

+

ñóò ò ó ò ó

+ ( ô cos÷„ï + õ )ñó ò ò ó +

ô

+

3 2

'  ,

) + 3 õ ]ñ

óò

2

ï

2

= exp  − 21 '  sin  cos÷ (

õ 2 [ô

+

) – õ ]ñ = 0.

ï

3 2

'  .

) + 2 õ ]ñ = 0.

. + ö ( ô cos÷„ï – õ )ñ = 0.

õ 2 )ñ óò

õï ú

+

cos÷ (

= exp( Œ 2  ), where Œ

–1

[ô

cos÷ (

ï

ï

)+þ

1

and Œ

are roots of the

2

]ñ = 0.

+ ö ( ô cos÷„ï + õ )ñ = 0.

ö^ñóò

1 b . Particular solutions with ( > 0:

1

= cos  £ (  ,

2 b . Particular solutions with ( < 0:

1

= exp  − £ − (  ,

ñ óò ò ó ò ó

= (cos÷

ï

– ô )ñ

+ ( ô cos÷

óò ò ó

ñóò ò ó ò ó

+ ( ô cos÷„ï +

õ ï )ñ óò ò ó

ñóò ò ó ò ó

+

ô ï

cos÷„ï

ñóò ò ó ò ó

+ ( ôƒõ

ï

1

ñóò ò ó

Particular solutions: 47.

2

¨ 
+  cos ( Œ  ) ¨ = 0. The substitution ¨ = 

 + ' +2 leads to a first-order linear equation: u

Particular solutions: 46.



2

ï

– õ )ñ

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 45.

= 0.

õ 2[ô

+

'  cos 

− 12

– õ [2 ô cos÷ (

ï )ñ ó ò ò ó

cos÷„ï

ô

+

= exp 

ï )ñ

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp  £ −   . 1 = exp  − 

+ ôƒõ cos÷ ( 1

Particular solutions:

42.

+ ö 2 ]ñ = 0.

¨ 

 +(  cos 2 + ( )¨ = 0. leads to the Mathieu equation 2.1.6.29: u

+

ï )ñ ó ò ò ó

Particular solutions:

41.

õ

)+

 The transformation ˆ = 12 Œ  , ¨ =  v J 2 (
 − ( ) leads to the Mathieu equation 2.1.6.29: ¨2Šª

Š + 4 Œ −2 V  cos(2ˆ ) +  + 3 ( 2 W ¨ = 0. 4

2 b . Particular solutions with  < 0:

40.

ï

) + õ ]ñ„ó ò – ö [ ô cos(

1 b . Particular solutions with  > 0:

39.

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a Mathieu equation of the form 2.1.6.29: ¨2

 + V  cos(2 ) + 3  2 W ¨ = 0. 4

cos÷„ï +

Particular solutions:

ï

+ õ (ô

2

ï

2

+ õ cos÷

óò

2

 2£   ,

= cos 

ô

cos÷„ï + õ )ñó ò ò ó + ôƒõ

1

= ,

=

= exp   £ − (  .

.

−





1

ñ

2

õ ï (ô ï

= sin  2

ï

1 2

2

and Œ

2

are roots of the

= 0.

= exp  − 12 ' 2  X exp 

1

2

2

= exp( Œ 2  ), where Œ

– ô cos÷„ï )ñó ò + 1 2

ï ñ

= sin  £ (  .

cos÷„ï + 2)ñó ò + ôƒõ cos÷„ï

= exp  − 12 ' 2  ,

+ (õ

2

1 2

' 2   . M

cos÷ï + 3)ñ = 0.

 2£   .

cos÷„ï

ñóò

– ôƒõ

2

cos÷„ï

ñ

= 0.

.

Page 531

532 48.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

ô ï

cos÷ (

2

ï )ñ óò ò ó

Particular solutions: 49. 50.

51.

52.

53.

1

ï ñóò ò ó ò ó

cos÷ (

ï

– 2ô

= ,

2

2

55.

ï ñóò ò ó ò ó

+ ï ( ô cos2 ï + õ )ñó ò – 2( ô cos2 ï + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.6.30: ¨2

 + (  cos2  +  )¨ = 0.

= (cos÷ ï – ô ï )ñ ó ò ò ó + ( ô cos÷ ï – õ ï )ñ ó ò + õ cos÷ ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ï ñ óò ò ó ò ó ï ñóò ò ó ò ó

+ (ô

ï

cos÷„ï + 3)ñó ò ò ó + (2 ô cos÷ï + 1

ï 2 ñ óò ò ó ò ó

ï 2 )ñ óò ò ó

= (cos÷„ï – ô

cos   £   ,

−1

=

Particular solutions:

+ ô ï 2 cos÷ ( ï )ñó ò ò ó + õ Particular solutions: 1 =  + 1 , equation 2 − +  = 0.

cos2 ï

¡

¡

cos÷„ï

58.

cos÷„ï

ñóò ò ó ò ó

+ ô cos2 ï

ñóò ò ó ò ó

+ ôñó ò ò ó + ôƒõÿñó ò +

ñóò ò ó ò ó

60.

1

õ 2 (ô

sin   £   .

ï 2 )ñ óò

+ õ cos÷„ï

ñ

= exp( Œ 2  ), where Œ

. 1

2

3 2





,

2

'  ,

2

= cos   £   ,

= 

2





õ 2 (ô

Šª

Šª
Š +  sin2 ˆ

sin 2 ˆ

= exp  − 21 '  sin 

= exp  − £ −   ,

+ ôñó ò ò ó – õ (2 ô + 3 õ cos÷„ï )ñó ò + =

are roots of the

2

¡

+ ôƒõÿñ = 0. 1

1

and Œ

– õ cos÷„ï )ñ = 0.

= exp  − 12 '  cos 

ñóò

are roots of the

2

cos÷„ï + 1)ñ = 0.

¡

2 b . Particular solutions with  < 0:

ñóò ò ó ò ó

ï

and Œ

+ õ [ ô cos÷ ( ï ) – 2]ñ = 0. =  + 2 , where 1 and 2 are roots of the quadratic

1

2

3 2

Šª

Š +

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ cos÷„ï )ñ = 0.

.

cos÷„ï ñó ò ò ó ò ó + ñó ò ò ó + [( õ – ô 2 ) cos÷„ï + ô ]ñó ò + õ (1 – ô cos÷„ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0. cos÷ (

ï )ñ óò ò ó ò ó

+ô

ï 2 ñ óò ò ó

Particular solutions: 61.

2

−1

+ õ (ô

1

ñóò ò ó + õÿñóò + ôƒõÿñ = 0. z + 2 leads to an equation of the form 3.1.6.22:

+ ôñó ò ò ó + õ cos÷„ï

Particular solutions: 59.

ï ñóò

=

2

2

1 b . Particular solutions with  > 0: cos÷„ï

õ ï )ñ óò

+ ( ô cos÷„ï – õ

ï 3 ñ óò ò ó ò ó

Particular solutions: 57.

= 0.

+ ï ( ô cos 2 ï + õ )ñ„ó ò – 2( ô cos 2ï + õ )ñ = 0. ¨ 

 +(  cos 2 +  )¨ = 0. The substitution ¨ =  
−2 leads to the Mathieu equation 2.1.6.29: u

The substitution  = ˆ  
Š +  = 0. 56.

ï )ñ

= .

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 54.

+ 2 ô cos÷ (

ï )ñ óò

cos÷„ï

1

– 2ô

= ,

ï ñóò

2

+ 2 ôñ = 0.

=  2.

+ ( ô cos÷„ï + ô ï + 1)ñó ò ò ó + ô  Particular solutions: 1 =  , 2 =  −  .

ñóò ò ó ò ó

2

ï ñóò

–ô

2

ñ

= 0.

Page 532

533

3.1. LINEAR EQUATIONS

62.

cos÷

ï ñ óò ò ó ò ó

cos÷

ï

+ (ô

Particular solutions:

ï

cos÷„ï

63.

ñóò ò ó ò ó

1

ï

3

cos÷„ï

ñóò ò ó ò ó

1

ï

3

cos÷„ï

ñóò ò ó ò ó

1

ï

cos÷„ï ñó ò ò ó ò ó + ô ï Particular solutions:

ï

3

3

67.

cos÷

ï ñ óò ò ó ò ó

+

= =

ï

2

Particular solutions:

1 2

=

−1

,

ï

,

+

ôñ

= 0.

= exp  − 21  2  X exp 

2

=

−1

1 2

 2   . M

sin   £   .

ñóò

+ 2(2 cos ÷„ï – ô )ñ = 0.

ñóò

+ 6(2 cos ÷„ï – ô )ñ = 0.

=  3.

2

ï )ñ ó ò

cos÷„ï + 2)ñó ò + ô (cos÷„ï + ï )ñ = 0.

=  2.

2

– 6 ï cos÷„ï −2

2

cos   £   ,

−1

– 2 ï cos÷„ï

ô ï 2 ñ óò ò ó

+

Particular solutions: 66.

+ ô ( ï + 2 cos ÷

óò ò ó

= exp  − 12  2  ,

ô ï 2 ñ óò ò ó

+

Particular solutions: 65.

+ 1)ñ

+ (3 cos÷„ï + ï )ñó ò ò ó + ( ô

Particular solutions: 64.

ï

+ ï ( ô – cos÷„ï )ñó ò + ( ô – 3 cos÷„ï )ñ = 0. = cos(ln | |), 2 = sin(ln | |).

ñóò ò ó 1

(cos÷ 1

ï

= | |

2 −

+ ï [ ô – ( õ + 1) cos ÷

óò ò ó

+ ô )ñ

,

2

= | |

2

.

ï ]ñ ó ò

+ õ (2 cos÷

ï

– ô )ñ = 0.

3.1.6-3. Equations with sine and cosine. 68.

69.

70.

71. 72.

73.

+ [ ô sin( ï ) + õ ]ñ ó ò + ô cos( ï ) ñ = 0. Integrating yields a second-order linear equation: "

 + [  sin( Π ) +  ] = L the solution of the corresponding homogeneous equation with L = 0).

ñ óò ò ó ò ó

+ [ ô sin( ï ) – õ 2 ]ñ ó ò + ô [ cos( ï ) – õ sin( ï )]ñ = 0.  By integrating and substituting ¨ =   J 2 , we obtain a second-order nonhomogeneous  3 ¨

 + V  sin( Œ  ) − 14  2 W ¨ = L  3  J 2 (see 2.1.6.2 for the solution of the linear equation: corresponding homogeneous equation).

ñ óò ò ó ò ó

+ [ ô cos(2ï ) + õ ]ñ„ó ò – ô sin(2 ï ) ñ = 0. Solution: = L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 , ¨ 2 is a fundamental set of solutions of the Mathieu equation 2.1.6.29: 4¨u

 + [  cos(2 ) +  ]¨ = 0.

ñóò ò ó ò ó

+ [ ô cos(2ï ) + õ ]ñ ó ò – 2 ô sin(2 ï ) ñ = 0. Integrating yields a nonhomogeneous Mathieu equation: "

 + [  cos(2 ) +  ] = L .

ñ óò ò ó ò ó

+ [ ô cos(2ï ) – õ 2 ]ñó ò – ô [ õ cos(2 ï ) + 2 sin(2 ï )]ñ = 0.  By integrating and substituting ¨ =   J 2 , we obtain a nonhomogeneous Mathieu equation:  ¨2

 + [  cos(2 ) − 1  2 ]¨ = L  3  J 2 . 4

ñóò ò ó ò ó ñóò ò ó ò ó

– 3 ô [ ô sin2 ( õ

ï

Particular solutions: 74. 75.

(see 2.1.6.2 for

) + õ cos( õ 1

ï )]„ñ óò

= exp −



Q 

+ ô sin( õ

cos( ' )S ,

ï )[õ

2

2

+ 2ô

=  exp −

2



Q 

sin2 ( õ

ï )]ñ

= 0.

cos( ' )S .

sin( ï )ñ„ó ò ò ó ò ó + õÿñó ò ò ó + 3 ô 2 sin( ï )ñó ò + 2 ô 3 cos( ï )ñ = 0. This is a special case of equation 3.1.9.105 with  ( ) = 0.

ô

sin( ï )ñ ó ò ò ó ò ó + [ ô + (2 + 1) cos( ï )]ñ ó ò ò ó – ( 2 + 2 ) sin( This is a special case of equation 3.1.9.106 with  ( ) = 0.

ï )ñ ó ò

–

2

cos(

ï )ñ

= 0.

Page 533

534 76.

THIRD-ORDER DIFFERENTIAL EQUATIONS sin2 ï

+ 3 sin ï cos ï

ñóò ò ó ò ó

+ [cos 2 ï + 4 b ( b + 1) sin2 ï ]ñó ò + 2 b ( b + 1) sin 2 ï

ñ„óò ò ó

ñ

= 0.

Solution: =L +L 2 1 2+L Here, 1 , 2 form a fundamental set of solutions of the Legendre equation 2.1.2.154, with the argument  of the functions 1 and 2 substituted by cos  . 2 1 1

2 3 2.

3.1.6-4. Equations with tangent. 77.

78.

ñóò ò ó ò ó

+

ñ óò ò ó ò ó

–ô

ô

3

tan( ô

ï )ñ

= 0.

tan( ô

ï )ñ

= 0.

Integrating yields a second-order nonhomogeneous linear equation:
Šª
Š + tan ˆ 
Š − = L cos ˆ , where ˆ =   (see 2.1.6.53 for the solution of the corresponding homogeneous equation). 3

Particular solution: 0 = cos(  ). The substitution = cos(  ) XZH ( )  leads to a secondM order linear equation of the form 2.1.6.53: H Šª

Š − 3 tan ˆtH Š
− 3 H = 0, where ˆ =  . 79.

ñ óò ò ó ò ó

+ 3ô

2

ñ óò

+ 2ô

3

tan( ô

Particular solutions: 80.

ñ óò ò ó ò ó

ôñ óò

+

1

ï )ñ

= 0.

= cos(  ),

+ ( ô – 1) tan ï

ñ

2

=  cos(  ).

= 0.

Particular solution: 0 = cos  . The substitution linear equation: H 

 − 3 tan 2H 
+ (  − 3) H = 0. 81.

ñ óò ò ó ò ó

ôñ óò

+

+ (ô –

2

) tan(

ï )ñ

= cos pXvH ( )  leads to a second-order

M

= 0.

= cos( Π ) XZH ( )  leads to a secondM

Š ª Š order linear equation of the form 2.1.6.53: H − 3 tan ˆtH Š
+ ( Œ −2 − 3) H = 0, where ˆ = Œ  . Particular solution:

82.

83.

84.

ô

tan2 ï

ñóò ò ó ò ó

+

ñóò ò ó ò ó

+ [ ô tan2 (

ñóò ò ó ò ó

+

0

= cos( Π ). The substitution

– õ ( ô tan2 ï +

ñóò

õ 2 )ñ

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a second-order linear equation of the form 2.1.6.51: ¨3

 +   tan2  + 34  2  ¨ = 0.

ï

) + õ ]ñó ò – ö [ ô tan2 (

ï

)+

õ

+ ö 2 ]ñ = 0.

 The transformation ˆ = Œ  , ¨ =  v J 2 ( 
− ( ) leads to an equation of the form 2.1.6.51: ¨2Šª

Š + Œ −2 (  tan2 ˆ +  + 3 ( 2 )¨ = 0. 4 tan÷„ï

ô

ñóò

+ tan ï ( ô tan÷„ï – 1)ñ = 0.

Particular solution: 0 = cos  . The substitution = cos  X linear equation: H 

 − 3 tan 2H 
+ (  tan  − 3) H = 0. 85.

86.

ñóò ò ó ò ó

+

ôñóò ò ó

+ ( õ tan2 ï + ö )ñó ò + ô ( õ tan2 ï + ö )ñ = 0.

The substitution ¨ = 
+  ¨2

 + (  tan2  + ( )¨ = 0.

ñóò ò ó ò ó

+

ôñóò ò ó

H ( )  leads to a second-order M

leads to a second-order linear equation of the form 2.1.6.51:

+ [3 + 2 ô tan(

Particular solutions:

1

ï )]„ñ óò

= cos( Π ),

2

+

2

{ ô [1 + 2 tan2 (

=  cos( Π ).

ï

)] + 2 tan(

ï )}ñ

= 0.

Page 534

535

3.1. LINEAR EQUATIONS

87.

ñ óò ò ó ò ó

+

ï )ñ ó ò ò ó

tan(

2 b . Solution for  < 0:

ñóò ò ó ò ó

ô

+

ï )„ñ óò ò ó

tan(

+

Particular solution:

ï )ñ

– ô tan(

= 0.

= L 1 exp  − £   + L = L 1 cos   £ −   + L

1 b . Solution for  > 0:

88.

óò

– ôñ

õÿñóò

+ ( ô +

õ

2

–

2

exp   £   + L

3

cos( Π ).

2

sin   £ −   + L

3

cos( Π ).

ï )ñ

) tan(

= 0.

H

= cos( Π ). The transformation  =

0

= cos( Œ  ) X ¨

,

Œ to a second-order linear equation of the form 2.1.6.131: ¨p‘'

‘ +  Œ  Œ −2 − 3 − 2 Œ −1 tan2 H  ¨ = 0. 89.

ñ óò ò ó ò ó

tan÷ (

ô

+

ï )ñ ó ò ò ó

+

õÿñ óò

tan÷ (

ôƒõ

+

2 b . Particular solutions with  < 0:

ñ óò ò ó ò ó

tan÷ (

ô

+

ï )ñ ó ò ò ó

Particular solutions: 91.

ñóò ò ó ò ó

tan÷ (

ô

+

1

ï )ñ óò ò ó

Particular solutions: 92.

ñóò ò ó ò ó

tan÷„ï

ô

+

+

tan÷ (

ôƒõ

= exp 

ï )ñ ó ò

– õ [2 ô tan÷ ( 1



=



,

ï

= 

2

+ ( ôƒõ tan÷„ï +

ñóò ò ó

ö

Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0. 93. 94.

95.

tan÷ (

ô

ï )ñ óò ò ó

õ ïƒú ñóò



3 2

'  ,

2

) + 3 õ ]ñó ò +

ï

= exp  − 21 '  sin  tan÷ (

õ 2 [ô

= exp( Œ 2  ), where Œ

2

õ ïƒú

[ô

ï

tan÷ (

ï

+ ( ô tan÷„ï + õ )ñó ò ò ó + ö^ñó ò + ö ( ô tan÷„ï + õ )ñ = 0.

+

ï

2

3 2

'  .

) + 2 õ ]ñ = 0.

.

ñóò ò ó ò ó

+

) – õ ]ñ = 0.

–1

)+þ

1

and Œ

2

are roots of the

]ñ = 0.

¨ 
+  tan ( Œ  ) ¨ = 0. The substitution ¨ = 

 + ' +2 leads to a first-order linear equation: u 1 b . Particular solutions with ( > 0:

1

= cos   £ (  ,

2 b . Particular solutions with ( < 0:

1

= exp  − £ − (  ,

ñóò ò ó ò ó ñ óò ò ó ò ó ñóò ò ó ò ó ñóò ò ó ò ó ñóò ò ó ò ó

2

= (tan÷„ï – ô )ñó ò ò ó + ( ô tan÷„ï – õ )ñó ò + õ tan÷„ï

+ ( ô tan÷

ï

+

+

ô ï

tan÷„ï

õ ï )ñ ó ò ò ó 1

ñóò ò ó

+ ( ôƒõ

ï

+

ô ï

2

tan÷ (

Particular solutions:

ô 1

ï

2

= cos 

ï

tan÷

ï 2

 2£   ,

= , – 2ô

= ,

2

ï 2

=

−





= exp   £ − (  .

.

ôƒõ

+

tan÷

1

ï ñ

2

õ ï (ô ï

= sin  2

ï

1 2

2

and Œ

2

are roots of the

= 0. 1 2

' 2   . M

tan÷„ï + 3)ñ = 0.

 2£   .

tan÷„ï

ñóò

– ôƒõ

+ 2 ô tan÷ (

ï )ñ

2

tan÷„ï

ñ

= 0.

.

tan÷ ( 2

ñ

2

= exp  − 12 ' 2  X exp 

– ô tan÷„ï )ñó ò + 1 2

óò

+ 2)ñ

tan÷„ï + õ )ñó ò ò ó + ôƒõ

ï )ñ óò ò ó 1

+ õ (ô

= sin   £ (  .

= exp( Œ 2  ), where Œ

2

= exp  − 12 ' 2  ,

+ (õ 1

tan÷„ï +

Particular solutions: 99.

õ 2 [ô

+

Particular solutions: 98.



tan÷ (

+

ñóò ò ó ò ó

Particular solutions: 97.

= 0.

– õ 2 )ñó ò + ö ( ô tan÷„ï – õ )ñ = 0.

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 96.

2

'  cos 

− 12

M

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp   £ −   . 1 = exp  − 

1 b . Particular solutions with  > 0:

90.

ï )ñ

 leads − 3  tan H ¨2‘
+

−1

ï )ñ óò

= 0.

= .

Page 535

536

THIRD-ORDER DIFFERENTIAL EQUATIONS

100.

– [ õ ( ô + tan ï ) ïw÷ + ô ]ñó ò ò ó + [ õ ( ô 2 + 1) ïƒ÷ + 1]ñó ò + ô [ õ ( ô tan ï – 1) ïw÷ – 1]ñ = 0.  Particular solutions: 1 =   , 2 = cos  .

101.

ñóò ò ó ò ó

ñóò ò ó ò ó

– ( ôƒõ tan÷„ï + õ tan÷

Particular solutions: 102. 103.

104.

105.

106.

+ [ tan( ï )( ô Particular solutions:

ñóò ò ó ò ó

1

ïw÷

1

ï ñ óò ò ó ò ó

ï

+1

+ ô )ñó ò ò ó + [ õ ( ô

 =  ,

= cos  .

2

+ 1) + ô ïƒ÷ –1 ]ñó ò ò ó – ô =  , 2 = cos( Œ  ).

2

ïƒ÷ ñóò

+ ô

2

ïƒ÷ –1 ñ

= (tan÷„ï – ô ï )ñó ò ò ó + ( ô tan÷„ï – õ ï )ñó ò + õ tan÷„ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ï ñóò ò ó ò ó ï ñóò ò ó ò ó

tan÷„ï + 3)ñó ò ò ó + (2 ô tan÷„ï +

ï

+ (ô

−1

=

Particular solutions:

1

ï 2 ñ óò ò ó ò ó

ï 2)ñ óò ò ó

= (tan÷„ï – ô

ï 3 ñ óò ò ó ò ó

+

ô ï

tan÷ (

2

ï )ñ óò ò ó

¡

tan ÷ ï ñ ó ò ò ó ò ó

¡

+

ôñ óò ò ó

2

+ ( ô tan÷„ï –

õ ï 2 )ñ óò

+

õ ï ñóò

=  + 1,

Particular solutions: 109. tan ÷„ï

ñóò ò ó ò ó

+

ôñóò ò ó

+

ôƒõÿñ óò 1

+

2

− 12

= exp 

+ õ tan÷„ï

=  + 2 , where

+

ôƒõÿñ 1

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ó

+

ôñóò ò ó

Particular solutions: 111. tan ÷„ï

ñóò ò ó ò ó

= 0.

ï )ñ óò ò ó ò ó

ñóò ò ó ò ó

3 2

1

=





,

2

= 

ï

ñ

. 1

and Œ

are roots of the

2

) – 2]ñ = 0. and

1

¡

¡

2

are roots of the quadratic

= 0.

'  ,

= exp  − 21 '  sin 

2

= 0.

õ 2 (ô

are roots of the

2

tan÷„ï + 1)ñ = 0.

ï

+ õ tan÷„ï

2

= exp  − £ −   ,



and Œ

sin   £   .

= cos   £   ,



1

2

3 2

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ tan÷„ï )ñ = 0.

.

+ ñó ò ò ó + [( õ – ô 2 ) tan÷„ï + ô ]ñó ò + õ (1 – ô tan÷„ï )ñ = 0.

+

ô ï 2 ñ óò ò ó

Particular solutions: 113. tan ÷„ï

ï )ñ

– õ (2 ô + 3 õ tan ÷„ï )ñó ò +

Particular solutions: 1 =   equation Π2 + Π+  = 0.

112. tan ÷ (

2

'  cos 

ñóò

−1

+ õ [ ô tan÷ (

– õ tan÷

õ 2 (ô

=

+ õ (ô

= exp( Œ 2  ), where Œ

2

1 b . Particular solutions with  > 0: 110. tan ÷„ï

õ ï )ñ óò

cos   £   ,

Particular solutions: 1 equation 2 − +  = 0.

108.

+ 1) tan÷„ï + 1]ñó ò + ô ( ôƒõ tan÷ +1 ï – õ tan÷„ï – 1)ñ = 0.

+ ï ( ô tan2 ï + õ )ñ ó ò – 2( ô tan2 ï + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.6.51: ¨2

 + (  tan2  +  )¨ = 0.

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 107.

2

1

= ,

+ ( ô tan÷„ï +

Particular solutions:

1

– 2ô

ô ï

= ,

1



,

ï ñóò 2

=  

2

2



, where Œ

1

and Œ

2

are roots of the quadratic

+ 2 ôñ = 0.

2

= .

+ 1)ñó ò ò ó + ô

 . 2 =  −

2

ï ñóò

–ô

2

ñ

= 0.

Page 536

537

3.1. LINEAR EQUATIONS

114. tan ÷

ï ñ óò ò ó ò ó

tan÷

ï

+ (ô

Particular solutions:

ï

tan÷„ï

115.

1

ï

3

tan÷ï

ñóò ò ó ò ó

1

ï

3

tan÷ï

ñóò ò ó ò ó

1

ï

3

tan÷

ï ñ óò ò ó ò ó

1

ï

3

tan÷ï

ñóò ò ó ò ó

+

– 2 ï tan÷„ï

ï

1 2

−1

=

2

−2

=

ï

,

2

ñóò

,

2

ï )ñ ó ò

+

ôñ

= 0.

= exp  − 21  2  X exp 

1 2

 2   . M

tan÷„ï + 2)ñó ò + ô (tan÷„ï + ï )ñ = 0. 2

=

−1

sin   £   .

+ 2(2 tan ÷„ï – ô )ñ = 0.

=  2.

– 6 ï tan÷„ï

ñóò

+ 6(2 tan ÷„ï – ô )ñ = 0.

=  3.

+ ï ( ô – tan÷

ô ï 2 ñ óò ò ó

+

Particular solutions: 119.

+ ô ( ï + 2 tan÷

cos   £   ,

−1

=

ô ï 2 ñ óò ò ó

+

Particular solutions: 118.

óò ò ó

= exp  − 12  2  ,

ô ï 2 ñ óò ò ó

+

Particular solutions: 117.

+ 1)ñ

+ (3 tan ÷„ï + ï )ñó ò ò ó + ( ô

ñóò ò ó ò ó

Particular solutions: 116.

ï

= cos(ln  ),

ï )ñ ó ò

+ ( ô – 3 tan ÷

= sin(ln  ).

2

ï )ñ

= 0.

(tan÷„ï + ô )ñó ò ò ó + ï [ ô – ( õ + 1) tan ÷„ï ]ñó ò + õ (2 tan÷„ï – ô )ñ = 0.

Particular solutions:

1

−

=

2

,

2

=

2

.

3.1.6-5. Equations with cotangent. 120.

ñóò ò ó ò ó

ô

3

+

cot( ô

ï )ñ

= 0.

The substitution  = a + 121.

ñ óò ò ó ò ó

–ô

3

cot( ô

ï )ñ

ñóò ò ó ò ó

+ 3ô

2

ñóò

– 2ô

3

ñ óò ò ó ò ó

ôñ óò

+

125.

126.

cot2 ï

ñóò ò ó ò ó

+

ô

ñóò ò ó ò ó

+

ôñóò ò ó

ñóò

tan( a ) = 0.

leads to a linear equation of the form 3.1.6.77: "
g
g
g + 

3

tan( a ) = 0.

cot( ô 1

ï )ñ

0

= 0.

= sin(  ),

ñ

2

=  sin(  ).

= 0.

= sin  .

– õ ( ô cot2 ï +

õ 2 )ñ

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a second-order linear equation of the form 3 ¨

2 2.1.6.81:  +   cot  + 34  2  ¨ = 0. + ( õ cot2 ï + ö )ñó ò + ô ( õ cot2 ï + ö )ñ = 0.

The substitution ¨ = 
+  ¨2

 + (  cot2  + ( )¨ = 0.

ñóò ò ó ò ó

+

cot÷„ï

ô

ñóò

Particular solution: 127.

3

+ (1 – ô ) cot ï

Particular solution: 124.



2

Particular solutions: 123.

leads to a linear equation of the form 3.1.6.78: "
g
g
g − 

= 0.

The substitution  = a + 122.



2

ñóò ò ó ò ó

+

ôñóò ò ó

leads to a second-order linear equation of the form 2.1.6.81:

+ cot ï (1 – ô cot÷„ï )ñ = 0. 0

= sin  .

+ [3 – 2 ô cot(

Particular solutions:

1

ï )]„ñ óò

= sin( Π ),

2

+

2

{ ô [1 + 2 cot2 (

=  sin( Π ).

ï

)] – 2 cot(

ï )}ñ

= 0.

Page 537

538 128.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

–

ï )ñ óò ò ó

cot(

1 b . Solution for  > 0:

– ôñ

2 b . Solution for  < 0: 129.

ñóò ò ó ò ó

cot÷ (

ô

+

ï )ñ óò ò ó

Particular solutions: 130.

ñóò ò ó ò ó

cot÷ (

ô

+

1

ï )ñ óò ò ó

Particular solutions: 131.

ñóò ò ó ò ó

cot÷„ï

ô

+

+

óò

cot÷ (

ôƒõ

ï )ñ óò ï





=

,

2

= 

cot÷ (

ï )ñ óò ò ó

õ ïƒú ñóò

exp  £   + L

3

sin( Π ).

2

sin   £ −   + L

3

sin( Π ).

3 2

cot÷ (

'  ,





ï

) – õ ]ñ = 0.

2

= exp  − 21 '  sin 

2

) + 3 õ ]ñó ò +

ï

cot÷ (

õ 2 [ô

3 2

'  .

) + 2 õ ]ñ = 0.

. + ö ( ô cot÷ï – õ )ñ = 0.

õ 2 )ñ óò

Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

= exp( Œ 2  ), where Œ

2

õ ïƒú

cot÷ (

ï

and Œ

are roots of the

2

ï

ñóò ò ó ò ó

+

) + þ ]ñ = 0. The substitution ¨ = 

 + ' +* leads to a first-order linear equation: ¨u
+  cot ( Œ  ) ¨ = 0.

133.

ñóò ò ó ò ó

+ ( ô cot÷„ï + õ )ñó ò ò ó +

+

+

–1

+ ö ( ô cot÷„ï + õ )ñ = 0.

ö^ñóò

1 b . Particular solutions with ( > 0:

1

= cos   £ (  ,

2 b . Particular solutions with ( < 0:

1

= exp  − £ − (  ,

ñóò ò ó ò ó ñóò ò ó ò ó

+

ô ï

cot÷„ï

ñóò ò ó

Particular solutions: 136.

ñóò ò ó ò ó

+ ( ôƒõ

ï

1

ñ óò ò ó ò ó

+

ô ï

cot÷ (

2

1

ô ï ñóò ò ó ò ó

140.

 2£   ,

1

2

ï

– 2ô

= ,

2

1

= ,

=

−





cot÷ ( 2

2

= sin 

2

ôƒõ 2 ï

ñ

= exp   £ − (  .

.

1 2

2

1

and Œ

2

are roots of the

cot÷„ï + 3)ñ = 0.

 2£   .

cot÷„ï

ñóò

– ôƒõ

+ 2 ô cot÷ (

ï )ñ

cot÷„ï

2

ñ

= 0.

.

ï )ñ ó ò

= .

ï )]„ñ óò ò ó

õ ï (ô ï

2

–

= sin( Π ).

ï ñóò

2

+

ï ñóò ò ó ò ó

+ ï ( ô cot2 ï + õ )ñó ò – 2( ô cot2 ï + õ )ñ = 0.

ï ñóò ò ó ò ó

+ (ô

ñ

= 0.

= 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.6.81: ¨2

 + (  cot2  +  )¨ = 0.

ï

cot÷„ï + 3)ñó ò ò ó + (2 ô cot÷„ï +

Particular solutions: 141.

= ,

+ [1 – ( ô + 1) ï cot(

Particular solutions: 139.

1 2

2

= exp( Œ 2  ), where Œ

– ô cot÷„ï )ñó ò +

2

= cos 

ï )ñ ó ò ò ó

Particular solutions: 138.

ï

cot÷„ï + ô cot÷„ï + õ )ñó ò ò ó +

Particular solutions: 137.

+ (õ

2

= sin   £ (  .

2

= (cot÷„ï – ô )ñó ò ò ó + ( ô cot÷„ï – õ )ñó ò + õ cot÷„ï

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 135.

[ô

1

132.

134.

ô

2

2

õ 2 [ô

+

= exp  − 12 '  cos 

+ ( ôƒõ cot÷„ï + ö –

ñóò ò ó

= 0.

= L 1 exp  − £   + L = L 1 cos   £ −   + L

– õ [2 ô cot÷ ( 1

ï )ñ

+ ô cot(

ï 2 ñ óò ò ó ò ó

= (cot÷„ï – ô

1

=

−1

ï 2)ñ óò ò ó

õ ï )ñ óò

cos   £   ,

2

+ ( ô cot÷ï –

õ ï 2 )ñ óò

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0.

2

=

−1

+ õ (ô

ï

cot÷„ï + 1)ñ = 0.

sin   £   . + õ cot÷„ï

ñ

= exp( Œ 2  ), where Œ

. 1

and Œ

2

are roots of the

Page 538

539

3.1. LINEAR EQUATIONS

ï 3 ñ óò ò ó ò ó

142.

+

ô ï

cot÷ (

2

ï )ñ óò ò ó

+

=  + 1,

Particular solutions: 1 equation 2 − +  = 0.

¡

143.

cot÷ ï ñ ó ò ò ó ò ó

¡

óò ò ó

+ ôñ

+

Particular solutions: 144. cot÷

ï ñ óò ò ó ò ó

1

+

– õ cot÷

õ 2 (ô

ï ñ óò

ï )ñ

2

'  cos 

3 2

ï

) – 2]ñ = 0. and

1

¡

'  ,

2

+ ôƒõÿñ = 0.

= cos  £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

+ ôñó ò ò ó – õ (2 ô + 3 õ cot÷„ï )ñó ò +

Particular solutions: 146. cot÷

ï ñ óò ò ó ò ó

=





,

Particular solutions: 1 =  equation Π2 + Π+  = 0.

147. cot÷ (

ï )ñ óò ò ó ò ó

ô ï 2 ñ óò ò ó

+

Particular solutions: 148. cot÷„ï

ñóò ò ó ò ó

1

149. cot÷„ï

ñóò ò ó ò ó

+ (ô

ï

1

ï

cot÷„ï

ñóò ò ó ò ó

1

ï

3

cot÷„ï

ñóò ò ó ò ó

+ô

1

ï

3

cot÷„ï

ñóò ò ó ò ó

+ô

1

ï

3

cot÷

ï ñ óò ò ó ò ó

+ô

1

3

cot÷

ï ñ óò ò ó ò ó

+ï

=

2

1

= =

  ,

óò

1

−1

ï

,

2

= sin  £   .

,

2

ï

+ õ (1 – ô cot÷

, where Œ

=

−

2

,

ï )ñ

and Œ

1

= 0. 2

are roots of the quadratic

–ô

2

ñ

= 0. = 0.

  X exp  12   2   . M

cot÷„ï + 2)ñó ò + ô (cot÷„ï + ï )ñ = 0. 2

=

−1

sin   £   .

+ 6(2 cot÷„ï – ô )ñ = 0.

ï )ñ ó ò

+ ( ô – 3 cot÷

= sin(ln  ).

2

+ ï [ ô – ( õ + 1) cot÷

óò ò ó

ôñ

2

ñóò

=  3.

2

= exp 

− 21

+ 2(2 cot÷„ï – ô )ñ = 0.

+ ï ( ô – cot÷ + ô )ñ

= exp   £ −   .

2

ñóò

=  2.

– 6 ï cot÷„ï −2

2

cos   £   ,

= cos(ln  ),

(cot÷

Particular solutions:

−1

'  .

+ 2 õ cot÷„ï )ñ = 0.

õ 2 (ô

ô 2 ï ñ óò

2

– 2 ï cot÷„ï

ï 2 ñ óò ò ó

Particular solutions:

ï

−  . 2 = 

= exp 

ï 2 ñ óò ò ó

Particular solutions: 153.

=  2.

− 12

3 2

+ 2 ôñ = 0.

+ 1)ñó ò ò ó +

= ,

ï 2 ñ óò ò ó

Particular solutions: 152.

2

 2

= 

2

2

.

+ ô ]ñ

+ (3 cot÷„ï + ï )ñó ò ò ó + ( ô

Particular solutions: 151.

,

2

cot÷„ï + 1)ñó ò ò ó + ô ( ï + 2 cot÷„ï )ñó ò +

Particular solutions: 150.

= ,

ô ï

ï

ï ñóò

– 2ô

+ ( ô cot÷„ï +

Particular solutions:

 1





= 

2

+ [( õ – ô 2 ) cot÷

óò ò ó

+ñ

1

are roots of the quadratic

= exp  − 21 '  sin 

1

ñóò ò ó ò ó

2

¡

= 0.

1 b . Particular solutions with  > 0: 145. cot÷„ï

154.

=  + 2 , where

2

− 12

= exp 

+ õ cot÷

óò ò ó

+ ôñ

ôƒõÿñ óò

+ õ [ ô cot÷ (

õ ï ñóò

=

2

.

ï )ñ ï ]ñ ó ò

= 0. + õ (2 cot÷

ï

– ô )ñ = 0.

3.1.7. Equations Containing Inverse Trigonometric Functions 1.

ñ óò ò ó ò ó

+

ôñ óò ò ó

+

õÿñ óò

+

ö^ñ

= arcsin 

ï

.

This is a special case of equation 5.1.6.26.

Page 539

540 2.

THIRD-ORDER DIFFERENTIAL EQUATIONS + arcsin ï ñ ó ò ò ó + ôñ ó ò + ô arcsin ï ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   ,

ñ óò ò ó ò ó

2 b . Particular solutions with  < 0: The substitution ¨ = 

 + 

1

= sin   £   .

2

= exp  − £ −   ,

= exp   £ −   .

2

¨ 
+ arcsin   ¨ = 0. leads to a first-order linear equation: u

3.

+ arcsin¦ï ñó ò ò ó + ô ïƒ÷ ñó ò + ô ïƒ÷ –1 ( ï arcsin³ï + ù )ñ = 0. ¨ 
+ arcsin   ¨ = 0. The substitution ¨ = 

 +   leads to a first-order linear equation: u

4.

ñóò ò ó ò ó

ñóò ò ó ò ó

+ arcsin¦ï

ñóò ò ó

6.

ñóò

ô 2 2(arcsin¦ï 3

ô

1

= exp  − 12   cos 

Particular solutions: 5.

arcsin ³ï

+

+

  ,

2

– ô )ñ = 0. 2

2

= exp  − 12   sin 

  .

3 2

+ arcsin¦ï ñó ò ò ó – ô (2 arcsin ¦ï + 3 ô )ñó ò + ô 2 (arcsin¦ï + 2 ô )ñ = 0.   Particular solutions: 1 =   , 2 =    .

ñóò ò ó ò ó

+ arcsin¦ï ñó ò ò ó + ( ô arcsin ¦ï + õ – ô 2 )ñó ò + õ (arcsin³ï – ô )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó

7.

= (arcsin¦ï – ô )ñó ò ò ó + ( ô arcsin ³ï – õ )ñó ò + õ arcsin¦ï ñ . The substitution ¨ = 

 +  
+  leads to a first-order linear equation: ¨ 
= arcsin   ¨ .

8.

ñóò ò ó ò ó

ñóò ò ó ò ó

+

ï

arcsin ³ï

ñóò ò ó

Particular solutions: 9.

ñóò ò ó ò ó

ô ï )ñ óò ò ó

Particular solutions: 10.

+ ï 2 arcsin³ï Solution: =L

11. 12.

1

 + L 2 2 + L

3

2

 2£   ,

2

= sin 

1 2

arcsin ³ï + 3)ñ = 0.

2

 2£   .

−3

j M

= exp  − 21   X exp  2

2

– 2 ï arcsin¦ï

X 

ô ï (ï

+ ô ( ï arcsin¦ï + 2)ñó ò + ô arcsin¦ï

= exp  − 12   ,

ñóò ò ó O

1 2

– arcsin³ï )ñó ò +

2

1

ñóò ò ó ò ó

2

= cos 

1

+ (arcsin ³ï +

ï

+ (ô

+ 2 arcsin ¦ï

ñóò

 − X 

−2

j

+ ( ô ï arcsin  ï + arcsin ï + ô )ñ ó ò ò ó + ô  Particular solutions: 1 =  , 2 =  −  .

ñ óò ò ó ò ó

 , M P 2

ï

ñ

arcsin 

+ arccos¦ï ñó ò ò ó + ôñó ò + ô arccos¦ï ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   ,

The substitution ¨ = 

 + 

1

1 2

= 0.

 2   . M

= 0.

where j = exp O − X 

ñóò ò ó ò ó

2 b . Particular solutions with  < 0:

ñ

2

= exp  − £ −   ,

ï ñ óò

–ô

2

2

arcsin

arcsin  

ï ñ

 . M P

= 0.

= sin   £   . 2

= exp  £ −   .

leads to a first-order linear equation: ¨ 
+ arccos   ¨ = 0.

13.

+ arccos¦ï ñó ò ò ó + ô ïƒ÷ ñó ò + ô ïƒ÷ –1 ( ï arccos¦ï + ù )ñ = 0. The substitution ¨ = 

 +   leads to a first-order linear equation: ¨ 
+ arccos   ¨ = 0.

14.

ñóò ò ó ò ó

ñóò ò ó ò ó

+ arccos¦ï

ñóò ò ó

Particular solutions: 15.

+ 1

ô

arccos³ï

= exp 

− 12

ñóò

+

  cos 

ô 2 2 (arccos³ï 3 2

+ arccos ï ñ ó ò ò ó – ô (2 arccos ï + 3 ô )ñ   Particular solutions: 1 =   , 2 =    .

ñ óò ò ó ò ó

  ,

óò

+

ô

2

2

– ô )ñ = 0. = exp  − 12   sin 

(arccos

ï

2

3 2

  .

+ 2 ô )ñ = 0.

Page 540

541

3.1. LINEAR EQUATIONS

16.

+ arccos ï ñ ó ò ò ó + ( ô arccos  Particular solutions: 1 =  1 , equation Œ 2 + Œ +  = 0.

ñ óò ò ó ò ó

ï 2

+ õ – ô 2 )ñ ó ò + õ (arccos  =  2 , where Œ 1 and Œ

ï 2

– ô )ñ = 0. are roots of the quadratic

17.

= (arccos¦ï – ô )ñó ò ò ó + ( ô arccos¦ï – õ )ñó ò + õ arccos¦ï ñ . The substitution ¨ = 

 +  
+  leads to a first-order linear equation: ¨ 
= arccos   ¨ .

18.

ñóò ò ó ò ó

ñóò ò ó ò ó

ï

arccos³ï

+

ñóò ò ó

Particular solutions: 19.

ñóò ò ó ò ó

1

+ (arccos³ï +

+ ï 2 arccos Solution: =L

21. 22.

1

 + L 2 2 + L

3

2

1

O

2

1 2

= cos 

– arccos¦ï )ñó ò +

 2£   ,

− 12

  ,

−3

j M

−2

ñ

= 0.

  X exp   2   . M 2

1 2

ï ñ

= 0.

 , where j = exp O − Xç 2 arccos    . M P M P

j

+ ( ô ï arccos¦ï + arccos¦ï + ô )ñó ò ò ó + ô  Particular solutions: 1 =  , 2 =  −  .

2

ï

arccos¦ï

+ arctan³ï ñó ò ò ó + ôñó ò + ô arctan³ï ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   ,

ñóò ò ó ò ó

1

arccos¦ï + 3)ñ = 0. arccos¦ï

ô

+ 2 arccos

ñóò ò ó ò ó

2

= exp  − £ −   ,

ñóò

–ô

2

arccos¦ï

ñ

= 0.

= sin   £   . 2

= exp  £ −   .

The substitution ¨ = 

 + 

¨ 
+ arctan   ¨ = 0. leads to a first-order linear equation: u

ñ óò ò ó ò ó

óò

23.

+ arctan ï ñ ó ò ò ó + ô ï ÷ ñ The substitution ¨ = 

 +  

24.

ñóò ò ó ò ó

+ arctan³ï

ñóò ò ó

Particular solutions:

26.

− 21

= exp 

2

ï ñ óò

 − 3X\

2 b . Particular solutions with  < 0:

25.

2

 2£   .

1 2

= sin 

2

2

– 2 ï arccos

Xe

ô ï (ï

+ ô ( ï arccos³ï + 2)ñó ò +

= exp 

ï ñ óò ò ó

ñ óò ò ó ò ó

ï

ô ï )ñ óò ò ó

Particular solutions: 20.

+ (ô

+ 1

ô

+ ô ï ÷ –1 ( ï arctan ï + ù )ñ = 0. ¨ 
+ arctan   ¨ = 0. leads to a first-order linear equation: u

arctan¦ï

ñóò

+ ô 2 (arctan³ï – ô )ñ = 0.

2

= exp  − 12   cos 

3 2

  ,

+ arctan³ï ñó ò ò ó – ô (2 arctan ³ï + 3 ô )ñó ò +   Particular solutions: 1 =   , 2 =    .

ñóò ò ó ò ó

ô

= exp  − 12   sin 

2 2

2

3 2

  .

(arctan¦ï + 2 ô )ñ = 0.

+ arctan³ï ñó ò ò ó + ( ô arctan³ï + õ – ô 2 )ñó ò + õ (arctan¦ï – ô )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó

27.

= (arctan ï – ô )ñ ó ò ò ó + ( ô arctan ï – õ )ñ ó ò + õ arctan ï ñ . ¨ 
= arctan   ¨ . The substitution ¨ = 

 +  
 +  leads to a first-order linear equation: u

28.

ñóò ò ó ò ó

ñ óò ò ó ò ó

+

ï

arctan³ï

ñóò ò ó

Particular solutions: 29.

ñ óò ò ó ò ó

+ (arctan

ï

+

Particular solutions:

1

+ (ô

2

= cos 

ô ï )ñ ó ò ò ó 1

ï

1 2

– arctan¦ï )ñó ò + ô

 2£   ,

+ ô ( ï arctan

= exp  − 12  2  ,

= sin 

2

2

ï

+ 2)ñ

ï (ï

2

arctan³ï + 3)ñ = 0.

 2£   .

1 2

óò

+

ô

arctan

= exp  − 21  2  X exp 

ï ñ 1 2

= 0.

 2   . M

Page 541

542 30.

THIRD-ORDER DIFFERENTIAL EQUATIONS

=L 31. 32.

ï ñ óò ò ó

+ ï 2 arctan Solution:

ñ óò ò ó ò ó

1

 + L 2 2 + L

3

2

O

+ ( ô ï arctan Particular solutions:

ñ óò ò ó ò ó

ï

– 2 ï arctan

XZ

−3

j

ï ñ óò −2

 − 3XZ M

+ arctan ï + ô )ñ −  . 1 =  , 2 = 

ï ñ

+ 2 arctan

j

 , M P

óò ò ó

+

ô 2ï

where j = exp O − XÈ

+ arccot ï ñ ó ò ò ó + ôñ ó ò + ô arccot ï ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   , 1

ï ñ óò

arctan

ñ óò ò ó ò ó

2 b . Particular solutions with  < 0:

= 0.

–ô

2

2

arctan 

arctan

ï ñ

 . M P

= 0.

= sin   £   .

2

= exp  − £ −   ,

2

= exp   £ −   .

¨ 
+ arccot   ¨ = 0. leads to a first-order linear equation: u

The substitution ¨ = 

 +  33.

+ arccot³ï ñó ò ò ó + ô ïƒ÷ ñó ò + ô ïƒ÷ –1 ( ï arccot³ï + ù )ñ = 0. ¨ 
+ arccot   ¨ = 0. The substitution ¨ = 

 +   leads to a first-order linear equation: u

34.

ñ óò ò ó ò ó

ñóò ò ó ò ó

+ arccot

ï ñ óò ò ó

Particular solutions: 35. 36.

ï ñ óò

+ ô arccot 1

= exp 

− 12

+

  cos 

ô 2 2(arccot ï 3 2

  ,

– ô )ñ = 0.

2

= exp  − 12   sin 

2

3 2

+ arccot³ï ñó ò ò ó – ô (2 arccot¦ï + 3 ô )ñó ò + ô 2 (arccot³ï + 2 ô )ñ = 0.   Particular solutions: 1 =   , 2 =    .

  .

ñóò ò ó ò ó

+ arccot³ï ñó ò ò ó + ( ô arccot¦ï + õ – ô 2 )ñó ò + õ (arccot¦ï – ô )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic 2 equation Œ + Œ +  = 0.

ñóò ò ó ò ó

37.

= (arccot³ï – ô )ñó ò ò ó + ( ô arccot³ï – õ )ñó ò + õ arccot³ï ñ . The substitution ¨ = 

 +  
+  leads to a first-order linear equation: ¨ 
= arccot   ¨ .

38.

ñóò ò ó ò ó

ñóò ò ó ò ó

+

ï

arccot¦ï

ñóò ò ó

Particular solutions: 39.

1

+ (arccot¦ï + ô

ñóò ò ó ò ó

+ ï 2 arccot¦ï Solution:

ñóò ò ó ò ó =L

41. 42.

1

 + L 2 2 + L

O

2

2

– arccot³ï )ñó ò +

 2£   ,

= cos 

1 2

= exp 

− 12

  , 2

– 2 ï arccot¦ï

Xç

2

ô ï (ï 1 2

= sin 

2

arccot¦ï + 3)ñ = 0.

 2£   .

+ ô ( ï arccot¦ï + 2)ñó ò + ô arccot³ï

−3

j M

2

= exp 

 − 3Xe

−2

− 21

j

ñ

= 0.

  X exp   2   . M 2

+ 2 arccot¦ï

ñóò

+ ( ô ï arccot¦ï + arccot¦ï + ô )ñó ò ò ó +  Particular solutions: 1 =  , 2 =  −  .

ñóò ò ó ò ó

1 2

ñ

= 0.

 , where j = exp O − Xç 2 arccot   . M P M P

ô 2ï

arccot¦ï

ñóò

–ô

2

arccot³ï

ñ

= 0.

ï ñóò ò ó ò ó

+ ( ô ï 2 + õ )ñó ò ò ó + 4 ô ï ñó ò + 2 ôñ = arcsin ¦ï . Integrating the equation twice, we arrive at a first-order linear equation:

 
+ (  43.

1

ñóò ò ó 3

ï

ï )ñ óò ò ó

Particular solutions: 40.

+ (ô

ï ñóò ò ó ò ó

2

+  − 2) = L

1+L

 2 + X  ( − a ) arcsin  a a , M 0

+ ( ï arcsin¦ï + 3)ñó ò ò ó + (2 arcsin ¦ï + ô

Particular solutions:

1

=

−1

cos   £   ,

2

=

ï )ñ óò −1



0

is any number.

+ ô ( ï arcsin ¦ï + 1)ñ = 0.

sin   £   .

Page 542

543

3.1. LINEAR EQUATIONS

44.

ï ñ óò ò ó ò ó

ï

+ ( ï arccos

+ 3)ñ

Particular solutions: 45.

ï ñóò ò ó ò ó

1

ï ñóò ò ó ò ó

1

1

ï 3 ñ óò ò ó ò ó


+ ( 

3

ï

2

+

2

51.

arcsin³ï

53.

ï 3 ñ óò ò ó ò ó

ï

2

+

1

arcsin³ï

1

ï 3 ñ óò ò ó ò ó

+ ï 2 arcsin³ï Particular solutions:

ï 3 ñ óò ò ó ò ó

ï

2

+

56.

58.

cos   £   ,

2

=

−1

cos   £   ,

2

=

+ ô ( ï arctan¦ï + 1)ñ = 0.

ï )ñ óò

+ ô ( ï arccot¦ï + 1)ñ = 0.

−1

−1

sin  £   .

+ 1)ñ = 0.

sin   £   . sin   £   .

=

 2 + X  ( − a ) arcsin a a , M 0

1+L

– 2ï −1

ñóò

,

2

– 6ï −2

ñóò

,

2



0

is any number.

+ 2(2 – arcsin ³ï )ñ = 0.

=  2.

+ 6(2 – arcsin ³ï )ñ = 0.

=  3.

1

+ ï (arcsin¦ï – 1)ñó ò + (arcsin¦ï – 3)ñ = 0. = cos(ln  ), 2 = sin(ln  ).

1

=

(arcsin³ï + 1)ñó ò ò ó + ï (arcsin ¦ï – ô – 1)ñó ò – ô (arcsin¦ï – 2)ñ = 0.

ï 3 ñ óò ò ó ò ó

−

2 

,

2

=

2 

.

+ ï 2 (arcsin³ï + ô )ñó ò ò ó + ï ( ô arcsin ³ï + õ – ô )ñó ò + õ (arcsin ³ï – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

ï 3 ñ óò ò ó ò ó

ï

2

+

ï ñ óò ò ó

arccos

ï 3 ñ óò ò ó ò ó

ï

2

+

arccos¦ï

ï 3 ñ óò ò ó ò ó

+ ï 2 arccos¦ï Particular solutions:

ï 3 ñ óò ò ó ò ó

ï

2

+

1

=

ñóò ò ó 1

=

– 2ï

−1

,

– 6ï −2

,

+ 2(2 – arccos

ñ óò 2

ñóò 2

2

ï )ñ

= 0.

= .

+ 6(2 – arccos¦ï )ñ = 0.

=  3.

1

+ ï (arccos³ï – 1)ñó ò + (arccos³ï – 3)ñ = 0. = cos(ln  ), 2 = sin(ln  ).

1

=

ñóò ò ó

(arccos³ï + 1)ñó ò ò ó + ï (arccos¦ï – ô – 1)ñó ò – ô (arccos³ï – 2)ñ = 0.

Particular solutions: 57.

−1

ô ï )ñ óò

−1

ñóò ò ó

Particular solutions: 55.

=

ñóò ò ó

Particular solutions: 54.

=

ñóò ò ó

Particular solutions: 52.

=

+ ) =L

Particular solutions: 50.

=

+ ô ( ï arccos

+ [( ô + 6) ï 2 + õ ]ñó ò ò ó + 2(2 ô + 3) ï ñ„ó ò + 2 ôñ = arcsin ¦ï . Integrating the equation twice, we arrive at a first-order linear equation:

Particular solutions: 49.

2

ï

ô ï )ñ ó ò

ï 3 ñ óò ò ó ò ó

 48.

cos  £   ,

+

+ ( ï arccot³ï + 3)ñó ò ò ó + (2 arccot¦ï + ô

Particular solutions: 47.

−1

ï

+ (2 arccos 

+ ( ï arctan³ï + 3)ñó ò ò ó + (2 arctan³ï +

Particular solutions: 46.

=

óò ò ó

ï 3 ñ óò ò ó ò ó

−

2 

,

2

=

2 

.

+ ï 2 (arccos ï + ô )ñ ó ò ò ó + ï ( ô arccos ï + õ – ô )ñ ó ò + õ (arccos ï – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

ï 3 ñ óò ò ó ò ó

+

ï

2

arctan³ï

Particular solutions:

ñóò ò ó 1

=

– 2ï

−1

,

ñóò 2

+ 2(2 – arctan¦ï )ñ = 0.

=  2.

Page 543

544 59.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 3 ñ óò ò ó ò ó

ï

2

+

ï ñ óò ò ó

arctan

Particular solutions: 60. 61.

1

ï 3 ñ óò ò ó ò ó

+ ï 2 arctan³ï Particular solutions:

ï 3 ñ óò ò ó ò ó

ï

2

+

63.

ï

1

(arctan

66.

ï 3 ñ óò ò ó ò ó

= 0.

+ ï (arctan

óò ò ó

2  2 = 

,

ï

óò

– ô – 1)ñ

ï

– ô (arctan

– 2)ñ = 0.

.

+ ï 2 (arctan¦ï + ô )ñó ò ò ó + ï ( ô arctan¦ï + õ – ô )ñó ò + õ (arctan¦ï – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

ï 3 ñ óò ò ó ò ó

ï

2

+

arccot¦ï

ï 3 ñ óò ò ó ò ó

ï

2

+

1

1

+ ï 2 arccot¦ï Particular solutions: +

ï

2

(arccot

−1

=

ï ñ óò ò ó

arccot

ï 3 ñ óò ò ó ò ó ï 3 ñ óò ò ó ò ó

– 2ï

ñóò ò ó

ñóò

,

2

– 6ï

=

−2

ñ óò

,

2

+ 2(2 – arccot³ï )ñ = 0.

=  2.

+ 6(2 – arccot 3

= .

ï )ñ

= 0.

+ ï (arccot¦ï – 1)ñó ò + (arccot³ï – 3)ñ = 0. = cos(ln  ), 2 = sin(ln  ).

ñóò ò ó ï

1

+ 1)ñ

ï 3 ñ óò ò ó ò ó

+ ï (arccot

óò ò ó

2 −  1 = 

Particular solutions: 67.

= .

2

2 −  1 = 

Particular solutions: 65.

,

3

ï )ñ

+ ï (arctan³ï – 1)ñó ò + (arctan³ï – 3)ñ = 0. = cos(ln  ), 2 = sin(ln  ).

+ 1)ñ

Particular solutions: 64.

=

−2

+ 6(2 – arctan

ñ óò

ñóò ò ó

Particular solutions: 62.

– 6ï

2  2 = 

,

ï

– ô – 1)ñ

óò

– ô (arccot

ï

– 2)ñ = 0.

.

+ ï 2 (arccot¦ï + ô )ñó ò ò ó + ï ( ô arccot¦ï + õ – ô )ñó ò + õ (arccot¦ï – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

3.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions 1. 2. 3. 4. 5. 6.

ó

= tan ï ñ + ô (ñó ò + tan ï Particular solution: 0 = cos  .

ñóò ò ó ò ó

ó

ñ

).

ó

+ ô ñó ò ò ó + (2 ô tan ï + 3)ñ„ó ò + [ ô Particular solutions: 1 = cos  , 2 =  cos  .

ñóò ò ó ò ó

ó

ó

+ ô ñó ò ò ó + (3 – 2 ô cot ï )ñó ò + [ ô Particular solutions: 1 = sin  , 2 =  sin  .

ñóò ò ó ò ó

cosh÷ ï ñ ó ò ò ó + (2 ô cosh÷ Particular solutions: 1 = cos  ,

ï

ñ óò ò ó ò ó + ô

tan ï + 3)ñ 2 =  cos  .

óò

ó

ó

(2 tan2 ï + 1) + 2 tan ï ]ñ = 0. (2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

+ [ ô cosh÷

cosh÷ ï ñó ò ò ó + (3 – 2 ô cosh ÷ ï cot ï )ñó ò + [ ô cosh÷ Particular solutions: 1 = sin  , 2 =  sin  .

ï

ñóò ò ó ò ó + ô

ï

sinh÷ ï ñó ò ò ó + (2 ô sinh ÷ Particular solutions: 1 = cos  ,

ñóò ò ó ò ó + ô

ï

tan ï + 3)ñ„ó ò + [ ô sinh ÷ 2 =  cos  .

(2 tan2 ï + 1) + 2 tan ï ]ñ = 0.

ï

(2 cot2 ï + 1) – 2 cot ï ]ñ = 0. (2 tan2 ï + 1) + 2 tan ï ]ñ = 0.

Page 544

545

3.1. LINEAR EQUATIONS

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

+ ô sinh÷ ï ñó ò ò ó + (3 – 2 ô sinh ÷ ï cot ï )ñó ò + [ ô sinh÷ Particular solutions: 1 = sin  , 2 =  sin  . tanh÷ ï ñó ò ò ó + (2 ô tanh ÷ Particular solutions: 1 = cos  ,

ï

ñóò ò ó ò ó + ô

(2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

ï

ñóò ò ó ò ó

tan ï + 3)ñ„ó ò + [ ô tanh÷ 2 =  cos  .

tanh÷ ï ñó ò ò ó + (3 – 2 ô tanh ÷ ï cot ï )ñó ò + [ ô tanh÷ Particular solutions: 1 = sin  , 2 =  sin  .

ï

ñóò ò ó ò ó + ô

coth÷ ï ñó ò ò ó + (2 ô coth÷ Particular solutions: 1 = cos  ,

ï

ñóò ò ó ò ó + ô

tan ï + 3)ñ„ó ò + [ ô coth÷ 2 =  cos  .

+ ô coth÷ ï ñ ó ò ò ó + (3 – 2 ô coth ÷ ï cot ï )ñ Particular solutions: 1 = sin  , 2 =  sin  .

ñ óò ò ó ò ó

óò

+ [ ô coth÷

+ ô ln÷ ï ñó ò ò ó – (2 ô ln ÷ ï tanh ï + 3)ñ„ó ò + [ ô ln÷ Particular solutions: 1 = cosh  , 2 =  cosh  . + ô ln ÷ ï ñó ò ò ó – (2 ô ln÷ ï coth ï + 3)ñ„ó ò + [ ô ln÷ Particular solutions: 1 = sinh  , 2 =  sinh  .

ï

ñóò ò ó ò ó

+ ô ln ÷ ï ñó ò ò ó + (2 ô ln ÷ ï tan ï + 3)ñ„ó ò + [ ô ln÷ Particular solutions: 1 = cos  , 2 =  cos  .

ï

ñóò ò ó ò ó

+ ô ln ÷ ï ñó ò ò ó + (3 – 2 ô ln ÷ Particular solutions: 1 = sin  ,

ñóò ò ó ò ó

ï

cot ï )ñó ò + [ ô ln÷ 2 =  sin  .

ï

ï

(2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

ï

(2 tan2 ï + 1) + 2 tan ï ]ñ = 0. (2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

(2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0.

ï

ñóò ò ó ò ó

(2 tan2 ï + 1) + 2 tan ï ]ñ = 0.

ï

(2 coth2 ï – 1) + 2 coth ï ]ñ = 0. (2 tan2 ï + 1) + 2 tan ï ]ñ = 0. (2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

cos÷ ï ñ ó ò ò ó – (2 ô cos÷ ï tanh ï + 3)ñ ó ò + [ ô cos÷ Particular solutions: 1 = cosh  , 2 =  cosh  .

(2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0.

ï

ñ óò ò ó ò ó + ô

cos÷„ï ñó ò ò ó – (2 ô cos÷„ï coth ï + 3)ñ„ó ò + [ ô cos÷„ï (2 coth2 ï – 1) + 2 coth ï ]ñ = 0. Particular solutions: 1 = sinh  , 2 =  sinh  .

ñóò ò ó ò ó + ô

sin ÷„ï ñó ò ò ó – (2 ô sin ÷„ï tanh ï + 3)ñ„ó ò + [ ô sin ÷„ï (2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0. Particular solutions: 1 = cosh  , 2 =  cosh  .

ñóò ò ó ò ó + ô

+ ô sin÷„ï ñó ò ò ó – (2 ô sin ÷„ï coth ï + 3)ñ„ó ò + [ ô sin÷„ï (2 coth2 ï – 1) + 2 coth ï ]ñ = 0. Particular solutions: 1 = sinh  , 2 =  sinh  .

ñóò ò ó ò ó

tan÷„ï ñó ò ò ó – (2 ô tan÷„ï tanh ï + 3)ñ„ó ò + [ ô tan÷„ï (2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0. Particular solutions: 1 = cosh  , 2 =  cosh  .

ñóò ò ó ò ó + ô

tan÷ ï ñ ó ò ò ó – (2 ô tan÷ ï coth ï + 3)ñ ó ò + [ ô tan÷ Particular solutions: 1 = sinh  , 2 =  sinh  .

ï

ñ óò ò ó ò ó + ô

cot÷ ï ñ ó ò ò ó – (2 ô cot÷ ï tanh ï + 3)ñ ó ò + [ ô cot÷ Particular solutions: 1 = cosh  , 2 =  cosh  .

ñ óò ò ó ò ó + ô

ï

(2 coth2 ï – 1) + 2 coth ï ]ñ = 0.

(2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0.

+ ô cot÷„ï ñó ò ò ó – (2 ô cot÷„ï coth ï + 3)ñ„ó ò + [ ô cot÷ï (2 coth2 ï – 1) + 2 coth ï ]ñ = 0. Particular solutions: 1 = sinh  , 2 =  sinh  .

ñóò ò ó ò ó

Page 545

546 24. 25. 26. 27. 28.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ó

ó

+ ( õD + 2 ô ) cosh÷ ï ñó ò ò ó – ô ( õD cosh÷   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

ó

ó

+ ( õD + 2 ô ) sinh ÷ ï ñó ò ò ó – ô ( õD sinh ÷   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

ó

ï ï

ó

ó

+ ( õ D + 2 ô ) coth÷ ï ñ ó ò ò ó – ô ( õ D coth÷   Particular solutions: 1 =   , 2 =  −  +   .

ñ óò ò ó ò ó

ó

ï

+ ô )ñ

ó

+ ( õD + 2 ô ) ln÷ ï ñó ò ò ó – ô ( õD ln÷ ï + ô )ñó ò – 2 ô   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

ó

ó

ó

29.

+( ô ln ÷ ï –2 õ )ñó ò ò ó – õ (2 ô ln÷ ï – õ +3)ñó ò + õ   Particular solutions: 1 = exp(   ), 2 =  exp(   ).

30.

ñ óò ò ó ò ó

ñóò ò ó ò ó

+ ( ô cos÷

ï

ó )ñ ó ò ò ó

– 2 õ

Particular solutions:

1

ñ óò ò ó ò ó + ( õ D ó

ó (2ô

– õ



= exp(   ),

2

cos÷

ó

32.

ñóò ò ó ò ó

Particular solutions:

34.



= exp(   ),

ï ñ

ln÷

ï ( õ  ó

ó [ô

óò

– 2ô

ñóò ò ó ò ó

ó

+ ( ô tan÷„ï – 2 õ

2

ó

Particular solutions:

ó

ó

+ 1) + ô ]ñó ò ò ó – ô 1 =  , 2 = cos  .

ó )ñ óò ò ó

1

–

õ ó (2ô 

= exp(   ),

2

ó



+ ô [

ï  ó ñóò

+

ô ó ñ

ó

ó

+ [   (cot ï + ô ) + ô ]ñ ó ò ò ó + [( ô 2 + 1)   Particular solutions: 1 =  −  , 2 = sin  .

ñ óò ò ó ò ó

ó

+ [ ô – cot ï ( ô ï  Particular solutions: 1 =  ,

ñóò ò ó ò ó

ó

+ 1)]ñó ò ò ó – ô 2 = sin  .



ó

óò

+ 1]ñ

ï  ó ñóò

+

+ ô [

ô ó ñ

ï ñ ï ñ

= 0. = 0.

–1)+2 õ

ó

ï ñ

ó

–1]ñ = 0.

– 1]ñ = 0.

= 0.

ó

– 1) + 2 õ

ñ

– 1]ñ = 0.

= 0.

tan ï – 1) – 1]ñ = 0.

= 0.

tan÷„ï – õ + 3)ñó ò ó + õ [ ô tan÷ ï ( õ  =  exp(   ).

ó

ó

ó (ô 

= 0.

= 0.

sin÷„ï

3

óò

+ 1]ñ

+ ( õD + 2 ô ) tan÷„ï ñó ò ò ó – ô ( õD tan÷„ï + ô )ñó ò – 2 ô   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

cos÷

3

= 0.

– 1) + 2 õ

ó

ó

ñóò ò ó ò ó

ó

sin ÷„ï – õ + 3)ñó ò ó + õ [ ô sin ÷„ï ( õ  =  exp(   ).

– [   (tan ï + ô ) + ô ]ñ ó ò ò ó + [( ô 2 + 1)   Particular solutions: 1 =   , 2 = cos  .

ñ óò ò ó ò ó

36.

39.

ó (2ô

– õ

coth÷

ln÷

+ 2 ô ) sin ÷„ï ñó ò ò ó – ô ( õD sin ÷„ï + ô )ñó ò – 2 ô   Particular solutions: 1 =   , 2 =  −  +   .

+ [tan ï (ô ï - Particular solutions:

38.

1

ñóò ò ó ò ó + ( õD ó

35.

37.

ó )ñ óò ò ó

3

3

– õ + 3)ñ ó ò ó + õ [ ô cos÷„ï ( õ  =  exp(   ).

+ 2 ô ) cos÷ ï ñ ó ò ò ó – ô ( õ D cos÷ ï + ô )ñ   Particular solutions: 1 =   , 2 =  −  +   . + ( ô sin÷„ï – 2 õ

– 2ô

ï ñ ï ñ

tanh÷

3

ó

ï

31.

33.

óò

sinh ÷

3

+ ô )ñó ò – 2 ô

ï

cosh÷

3

+ ô )ñó ò – 2 ô

ó

+ ( õD + 2 ô ) tanh÷ ï ñó ò ò ó – ô ( õD tanh÷   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

+ ô )ñó ò – 2 ô

3



ó

– 1) + 2 õ

tan÷„ï

ó

ñ

ó

– 1]ñ = 0.

= 0.

(1 – ô cot ï ) + 1]ñ = 0.

= 0.

Page 546

547

3.1. LINEAR EQUATIONS

40.

ñ óò ò ó ò ó

+ ( ô cot÷

ï

– 2 õ

Particular solutions: 41. 42. 43.

ñóò ò ó ò ó + ( õD ó

ó )ñ ó ò ò ó 1

–



= exp(   ),

–[cosh÷ ï (tan ï + ô )+ ô ]ñ  Particular solutions: 1 =   ,

ñ óò ò ó ò ó

– õ + 3)ñ ó ò ó + õ [ ô cot÷„ï ( õ  =  exp(   ).

2

ó

óò ò ó

+[( ô 2 +1) cosh÷ 2 = cos  .

ñóò ò ó ò ó

ñ óò ò ó ò ó

+[sinh ÷

ï

–[tanh÷ +[tanh÷

−  , 1 = 

2

2

ï

 , 1 = 

2

(cot ï + ô )+ ô ]ñ„ó ò ò ó +[( ô

Particular solutions:

2

2

50.

ñ óò ò ó ò ó

ñóò ò ó ò ó

+ [ ô tan÷

ï

(tanh ï – õ ) – õ ]ñ

Particular solutions:

1





=

,

2

ñóò ò ó ò ó +[ô

+1]ñ

óò + ô

[sinh ÷

ï (ô

ï

+1]ñó ò + ô [sinh ÷

óò ò ó

+ [ô ( õ

= cosh  .

2

ï

(1– ô cot ï )+1]ñ = 0.

ï

(1– ô cot ï )+1]ñ = 0.

ï

+1]ñó ò + ô [tanh ÷

ï (ô

ï

+1]ñó ò + ô [tanh÷

ï

+1]ñó ò + ô [coth÷ +1]ñó ò + ô [coth÷

tan ï –1)–1]ñ = 0.

(1– ô cot ï )+1]ñ = 0.

ï ï

tan ï –1)–1]ñ = 0.

(tan ï –1)–1]ñ = 0. (1– ô cot ï )+1]ñ = 0.

– 1) tan÷ ï – 1]ñ ó ò + õ [ ô tan÷„ï (1 – õ tanh ï ) + 1]ñ = 0.

1

= exp  − £ − (  ,

2

= exp   £ − (  .

tan÷„ï (coth ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) tan÷„ï –1]ñó ò + õ [ ô tan÷„ï (1– õ coth ï )+1]ñ = 0. 1





=

,

2

= sinh  .

+ ( ô tan÷„ï + õ cothú ï )ñó ò ò ó + ö^ñó ò + ö ( ô tan÷„ï + õ cothú ï )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ (  .

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0: 54.

ï

tan ï –1)–1]ñ = 0.

+ ( ô tan÷„ï + õ tanhú ï )ñó ò ò ó + ö^ñó ò + ö ( ô tan÷„ï + õ tanhú ï )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ (  .

Particular solutions: 53.

ï

ï

– 1]ñ = 0.

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0: 52.

[cosh÷

ñóò ò ó ò ó

+[coth÷ ï (cot ï + ô )+ ô ]ñ„ó ò ò ó +[( ô 2 +1) coth÷  Particular solutions: 1 =  −  , 2 = sin  .

51.

óò + ô

= sin  .

ó

= 0.

+1]ñ

ï

+1) tanh ÷

49.

ñ ï (ô

+1) tanh ÷

–[coth÷ ï (tan ï + ô )+ ô ]ñ„ó ò ò ó +[( ô 2 +1) coth÷  Particular solutions: 1 =   , 2 = cos  .

cot÷ï

[cosh÷

= cos  .

2

−  , 1 = 

+1) sinh ÷

– 1) + 2 õ

óò + ô

= sin  .

(tan ï + ô )+ ô ]ñ„ó ò ò ó +[( ô

ï

Particular solutions:

ñóò ò ó ò ó

+[( ô 2 +1) sinh ÷ 2 = cos  .

(cot ï + ô )+ ô ]ñ„ó ò ò ó +[( ô

Particular solutions:

ñóò ò ó ò ó

óò ò ó

3

ó

+1]ñ

ï

+[cosh÷ ï (cot ï + ô )+ ô ]ñ ó ò ò ó +[( ô 2 +1) cosh ÷  Particular solutions: 1 =  −  , 2 = sin  .

45.

48.

ó

ï

ñ óò ò ó ò ó

–[sinh÷ ï (tan ï + ô )+ ô ]ñ  Particular solutions: 1 =   ,

47.

cot÷

+ 2 ô ) cot÷„ï ñó ò ò ó – ô ( õD cot÷„ï + ô )ñó ò – 2 ô   Particular solutions: 1 =   , 2 =  −  +   .

44.

46.

õ ó (2ô

ñóò ò ó ò ó +[ô

1

= exp  − £ − (  ,

2

= exp  £ − (  .

cot÷„ï (tanh ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) cot÷„ï –1]ñó ò + õ [ ô cot÷„ï (1– õ tanh ï )+1]ñ = 0.

Particular solutions:

1

=





,

2

= cosh  .

Page 547

548 55.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

cot÷

ô

+

tanhú

ï

ï ñ óò ò ó

– õÿñ

óò

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0: 56.

ñ óò ò ó ò ó

+[ ô cot÷

ï

(coth ï – õ )– õ ]ñ

Particular solutions: 57.

1





=

,

2

ï

ñóò ò ó ò ó

+[ ô ln÷

ñóò ò ó ò ó

ln ÷

ô

+

1

tanhú

ï

1





=

,

ï ñóò ò ó

2

–

= cosh  .

– ôƒõ ln ÷

õÿñóò

ñóò ò ó ò ó

+[ ô ln÷

Particular solutions:

1





=

,

2

2

ï

= exp   £   .

= sin   £ −   .

óò + õ [ô

cot÷

£  , 1 = exp  −  £ −  , 1 = cos 

ï

= exp   £ − (  .

2

ï ñ 2 2

= exp   £   .

= sin  £ −   .

–1]ñó ò + õ [ ô ln÷

1

= exp  − £ − (  ,

– [ln÷ ï (tan ï + ô ) + ô ]ñ ó ò ò ó + [( ô 2 + 1) ln÷  Particular solutions: 1 =   , 2 = cos  .

2

[ln÷

63.

+ [ln÷ ï (cot ï + ô ) + ô ]ñ„ó ò ò ó + [( ô 2 + 1) ln÷  Particular solutions: 1 =  −  , 2 = sin  .

ï

+ 1]ñó ò + ô [ln÷

64.

ñóò ò ó ò ó +[ô

ñóò ò ó ò ó

ñ óò ò ó ò ó

+

ô

cos÷

ï

tanhú

1





=

ï ñ óò ò ó

,

2

+

= cosh  .

õÿñ óò

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0:

ñóò ò ó ò ó +[ô

ï (ô ï

tan ï – 1) – 1]ñ = 0.

(1 – ô cot ï ) + 1]ñ = 0.

+

ôƒõ

cos÷

ï

tanhú

ï ñ

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp  £ −   . 1 = exp  − 

cos÷„ï (coth ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) cos÷„ï –1]ñó ò + õ [ ô cos÷„ï (1– õ coth ï )+1]ñ = 0.

Particular solutions:

1





=

,

2

= sinh  .

+ ( ô cos÷ ï + õ cothú ï )ñ ó ò ò ó + ö^ñ ó ò + ö ( ô cos÷ 1 b . Particular solutions with ( > 0: 1 = cos  £ (  ,

ñ óò ò ó ò ó

2 b . Particular solutions with ( < 0:

ñóò ò ó ò ó

óò + ô

cos÷„ï (tanh ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) cos÷„ï –1]ñó ò + õ [ ô cos÷ï (1– õ tanh ï )+1]ñ = 0.

Particular solutions:

68.

(1– õ coth ï )+1]ñ = 0.

= exp   £ − (  .

+ 1]ñ

67.

ï

+ ( ô ln÷ ï + õ cothú ï )ñó ò ò ó + ö^ñó ò + ö ( ô ln÷ ï + õ cothú ï )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos  £ (  , 2 = sin  £ (  .

ï

66.

(1– õ tanh ï )+1]ñ = 0.

ï

= 0.

ñ óò ò ó ò ó

65.

(1– õ coth ï )+1]ñ = 0.

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0:

62.

ï

+ õ cothú ï )ñ = 0. £ ( . 2 = sin  

ï

tanhú

= sinh  .

= 0.

–1]ñó ò + õ [ ô ln÷

ï

(coth ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) ln÷

ï

2

= exp  − £ − (  ,

(tanh ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) ln÷

ï

ï ñ

–1]ñ

+ ( ô cot÷ ï + õ cothú ï )ñ ó ò ò ó + ö^ñ ó ò + ö ( ô cot÷ 1 b . Particular solutions with ( > 0: 1 = cos   £ (  ,

2 b . Particular solutions with  < 0:

61.

–1) cot÷

= sinh  .

1 b . Particular solutions with  > 0:

60.

2

ñ óò ò ó ò ó

Particular solutions: 59.

tanhú

ï

£  , 1 = exp  −  £ −  , 1 = cos  

óò ò ó +[ô ( õ

2 b . Particular solutions with ( < 0: 58.

– ôƒõ cot÷

1

ï

+ õ cothú ï )ñ = 0. £ ( . 2 = sin 

= exp  − £ − (  ,

2

= exp   £ − (  .

+[ ô sin ÷„ï (tanh ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) sin ÷„ï –1]ñó ò + õ [ ô sin÷„ï (1– õ tanh ï )+1]ñ = 0.

Particular solutions:

1

=





,

2

= cosh  .

Page 548

549

3.1. LINEAR EQUATIONS

69.

ñ óò ò ó ò ó

+

ô

sin ÷

ï

tanhú

ï ñ óò ò ó

+

õÿñ óò

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0: 70.

ñóò ò ó ò ó +[ô

sin ÷

ôƒõ

tanhú

ï

1

=





,

2

= sinh  .

+ ( ô sin÷„ï + õ cothú ï )ñó ò ò ó + ö^ñó ò + ö ( ô sin÷„ï + õ cothú ï )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ (  .

ó

= exp  − £ − (  ,

ó

+ [ ô ï 2  ( õ – ln ï ) + 2]ñ„ó ò ò ó + ô ï  ñó ò – ô Particular solutions: 1 =  , 2 = ln  −  + 1.

73.

(   – 1)ñ ó ò ò ó ò ó – ( ô Particular solutions:

76. 77. 78. 79. 80. 81. 82. 83. 84.

ï ñóò ò ó ò ó

1

72.

75.

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp  £ −   . 1 = exp  − 

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0:

74.

ï ñ

sin÷„ï (coth ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) sin ÷„ï –1]ñó ò + õ [ ô sin÷„ï (1– õ coth ï )+1]ñ = 0.

Particular solutions: 71.

+

ó



ó

ó

+ tan ï )ñ ó ò ò ó + (   +  , 1 =  2 = cos  .

óñ

2

= exp   £ − (  . = 0.

+ ô ( ô tan ï – 

ô 2 )ñ ó ò

ô

cosh÷ ï ñó ò ò ó ò ó + [tan ï ( ô cosh ÷ ï + ï ) + 1]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = cos  .

ñóò

+ ñ = 0.

ô

cosh÷ ï ñó ò ò ó ò ó + [1 – cot ï ( ô cosh ÷ ï + ï )]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sin  .

ñóò

+ ñ = 0.

ô

sinh ÷ ï ñó ò ò ó ò ó + [tan ï ( ô sinh ÷ ï + ï ) + 1]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = cos  .

ñóò

+ ñ = 0.

ô

sinh ÷ ï ñó ò ò ó ò ó + [1 – cot ï ( ô sinh ÷ ï + ï )]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sin  .

ñóò

+ ñ = 0.

tanh÷ ï ñ ó ò ò ó ò ó + [tan ï ( ô tanh÷ ï + ï ) + 1]ñ Particular solutions: 1 =  , 2 = cos  .

–ï

ñ óò

+ ñ = 0.

tanh÷ ï ñó ò ò ó ò ó + [1 – cot ï ( ô tanh ÷ ï + ï )]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sin  .

ñóò

+ ñ = 0.

–ï

ñ óò

+ ñ = 0.

coth÷ ï ñó ò ò ó ò ó + [1 – cot ï ( ô coth÷ ï + ï )]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sin  .

ñóò

+ ñ = 0.

óò ò ó

ô ô

coth÷ ï ñ ó ò ò ó ò ó + [tan ï ( ô coth÷ ï + ï ) + 1]ñ Particular solutions: 1 =  , 2 = cos  .

ô

óò ò ó

ô

ln÷ ï ñ ó ò ò ó ò ó + [tanh ï ( ï – ô ln ÷ ï ) – 1]ñ Particular solutions: 1 =  , 2 = cosh  .

–ï

ñ óò

+ ñ = 0.

ln÷ ï ñó ò ò ó ò ó + [coth ï (ï – ô ln ÷ ï ) – 1]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sinh  .

ñóò

+ ñ = 0.

ô

óò ò ó

ô

ln÷ ï ñó ò ò ó ò ó + [tan ï ( ô ln ÷ ï + ï ) + 1]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = cos  .

ô

ñóò



ó )ñ

= 0.

+ ñ = 0.

Page 549

550

THIRD-ORDER DIFFERENTIAL EQUATIONS ln÷

ô

85.

ï ñ óò ò ó ò ó

+ [1 – cot ï ( ô ln ÷

Particular solutions: cos÷„ï

ô

86.

cos÷„ï

ô ô ï

cos÷„ï sin ÷„ï

ô

ñóò ò ó ò ó

sin ÷

ô

ï ñ óò ò ó ò ó

ô ï

sin ÷„ï tan÷

ô

ï ñ óò ò ó ò ó

tan÷„ï

ô ô ï

ñóò ò ó ò ó

tan÷„ï cot÷„ï

ô

cot÷„ï

ô

ñóò ò ó ò ó

ô ï

cot÷

= ,

2

= sinh  . ln ï +

2

õ ï 2 )ñ óò ò ó

1

= ,

2

1

= ,

2

1

= ,

ñóò

ñóò

+ ñ = 0.

ñ óò

+ ñ = 0.

= ln  −  + 1. = cosh  .

ï

) – 1]ñ

= sinh  .

2 2

+ï

ln ï +

óò ò ó

–ï

õ ï 2 )ñ óò ò ó

= ln  −  + 1.

ï ñóò

+

– ñ = 0.

– ñ = 0.

ñóò

+ ñ = 0.

1

1

= , = ,

2

2

+ (2 ô tan÷„ï – ï 1

= ,

2

ï

) – 1]ñ

= cosh  . = sinh  . ln ï +

2

óò ò ó

õ ï 2 )ñ óò ò ó

1

= ,

2

= cosh  .

+ [coth ï (ï – ô cot÷„ï ) – 1]ñó ò ò ó – ï

ï ñ óò ò ó ò ó

1

= ,

+ (2 ô cot÷

Particular solutions:

1

ï

= ,

2

–ï 2

+ï

ñóò

ñóò

+ ñ = 0.

ñóò

+ ñ = 0.

= ln  −  + 1.

+ [tanh ï ( ï – ô cot÷„ï ) – 1]ñó ò ò ó – ï

Particular solutions: 97.

1

+ ñ = 0.

+ [coth ï ( ï – ô tan ÷„ï ) – 1]ñó ò ò ó – ï

Particular solutions: 96.

2

ñóò

+ ñ = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

= ,

+ ñ = 0.

ñ óò

Particular solutions: 95.

1

ñóò

–ï

Particular solutions: 94.

+ ñ = 0.

= cosh  .

+ [tanh ï ( ï – ô tan÷

Particular solutions: 93.

2

+ (2 ô sin÷„ï – ï

ñóò ò ó ò ó

Particular solutions: 92.

= ,

+ [coth ï (ï – ô sin ÷

Particular solutions: 91.

= sin  .

ñ óò

+ [tanh ï ( ï – ô sin ÷„ï ) – 1]ñó ò ò ó – ï

Particular solutions: 90.

1

+ (2 ô cos÷„ï – ï

ñóò ò ó ò ó

Particular solutions: 89.

2

–ï

+ [coth ï ( ï – ô cos÷„ï ) – 1]ñó ò ò ó – ï

ñóò ò ó ò ó

Particular solutions: 88.

= ,

óò ò ó

+ [tanh ï ( ï – ô cos ÷„ï ) – 1]ñó ò ò ó – ï

ñóò ò ó ò ó

Particular solutions: 87.

1

+ ï )]ñ

ï

= sinh  .

2

ln ï +

õ ï 2 )ñ ó ò ò ó

= ln  −  + 1.

+

ï ñ óò

– ñ = 0.

– ñ = 0.

3.1.9. Equations Containing Arbitrary Functions ]

Notation:  =  ( ), = ( ), and % = % ( ) are arbitrary functions of  ;  ,  , ( , B , and Πare arbitrary parameters. 3.1.9-1. Equations of the form  3 ( )


 +  1 ( )
 +  0 ( ) = ( ). 1.

ñ óò ò ó ò ó

= ! ( ï )ñ .

The transformation  = a

−1

, = ‡"a

−2

leads to an equation of the same form: ‡ g

g
g = −a

−6

 (1 a )‡ .

Page 550

551

3.1. LINEAR EQUATIONS

2.

3.

ñ óò ò ó ò ó

ô ï öï

=I ! O

+õ + ý P (ö

ñ

ï

. + ý )6   +  ¨ ¨ ªŠ

Šª
Š = ¢ The transformation ˆ = , = leads to a simpler equation: u (˜ + ( (˜ + )2 M M where ¢ =  − ˜( .

!„ñóò ò ó ò ó

–

!„ó ò ò ó ò óñ

= 0.

!„ñ óò ò ó ò ó

+

! óò òó ò ó ñ

= X H

=" .

6.

ñóò ò ó ò ó

+

!„ñóò

– ( ô! + ô 3 )ñ = 0.

 Particular solution: 0 =   . The substitution ¨ 3 ¨

2 ¨
equation:  +   + (  +  2 )¨ = 0.

ô ï (!

+

Particular solution:

0

ñóò ò ó ò ó

+

!„ñóò

+

ô 2ï

2

M

= 
− 

8.

9.

ñ óò ò ó ò ó

= exp  − 12  2  . The substitution

óò + ô!„ñ = 0.  Particular solution: 0 =  −  . The substitution ¨


¨ ¨ ¨ equation:  −   +  = 0. ñóò ò ó ò ó

+ ( ! – ô 2 )ñ

+

ï !„ñóò ï

+ (ô

11.

ñóò ò ó ò ó

+ (ô

2

=

ï

− 3  ) H = 0.

leads

leads to a second-order linear


− 2 leads to a second-order linear

+ õ ) !„ñó ò – 2 ô!„ñ = 0.

+ (! – ô

ï )ñ óò

2 2

+ (! – ô

ï ÷ )ñ ó ò

2 2

= (  +  )
 − 2 

+

ô ï (!

0

= exp  − 12  2  . The substitution

– ô [ï

leads to a

– 3 ô )ñ = 0.

to a second-order linear equation: H 

 − 3 "H 
+ (2  2 

ñ óò ò ó ò ó

M

=  +  .

0

Particular solution:

12.

= exp  − 12  2  XZH ( ) 

Particular solution: 0 = (  +  )2 . The substitution ¨ second-order linear equation: ¨u

 + (  +  )  ¨ = 0.

ñóò ò ó ò ó

 +L .

+ õ ) !„ñó ò – ô!„ñ = 0.

Particular solution: 10.

M

leads to a second-order linear

= 
+ 

– 2 !„ñ = 0.

Particular solution: 0 =  2 . The substitution ¨ equation: ¨ 

 +   ¨ = 0.

ñóò ò ó ò ó

= Xw

– 3 ô )ñ = 0.

to a second-order linear equation: H 

 − 3 "H 
+ (  + 3  2  7.

 leads to a second-order linear



 −  

+  



Integrating yields a second-order linear equation:  5.

 (ˆ )¨ ,

M

Particular solution: 0 =  . The substitution equation:  H 

 + 3  
H 
+ 3  

 H = 0. 4.

−3

÷ !



+ 3 ôù

–1

− 3  +  ) H = 0.

+ ù (ù – 1) ï

÷

–2

]ñ = 0.

M

leads



  +1 XZH ( )  P

B + 1 P M leads to a second-order linear equation: H  + 3   H 
+ (2  2  2  + 3 B/  −1 +  ) H = 0.

13.

Particular solution:

0

ï ñ óò ò ó ò ó

ï

+

ï !„ñ óò

– [( ô

Particular solution:

= exp O

B +1

+ 1) ! + ô

 . 0 =  

3

ï

 

ï 2÷

2

= exp  − 12  2  Z X H ( ) 

+1

. The substitution

= exp O

+ 3 ô 2 ]ñ = 0.

Page 551

552 14.

15.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó ò ó

+ (ï

ï 2 ñ óò ò ó ò ó

+ [ï (ô

–ô

!

2

– ô )ñó ò + ( ô – 1) !„ñ = 0.

Particular solution: 0 =  1−  . The substitution ¨ =  linear equation:  ¨3

 − (  + 1)¨2
+  ¨ = 0.

ï

Particular solution: 16.

ï (ï

+ 1)ñ„ó ò ò ó ò

ï 3 ñ óò ò ó ò ó

+

= +

0

!„ñ

+ −1

= 0.

.

+ ï ( ! – ï – 3)ñ„ó ò – ( ï + 1) !„ñ = 0.

ó

Particular solution: 17.

óò

+ 1) ! – 6]ñ


+ (  − 1) leads to a second-order

ï !„ñóò



=  .

0

+ ( ô – 1)( ! + ô

2

+ ô )ñ = 0.

Particular solution: 0 =   . The substitution ¨ =  
+ (  − 1) leads to a second-order linear equation:  2 ¨2

 − (  + 1) ¨2
+ (  +  2 +  )¨ = 0. 1−

18.

ï 6 ñ óò ò ó ò ó

+

ï 2 „! ñóò

+ (ô

Particular solution: 19.

ñóò ò ó ò ó

+ (! – ô

 

2 2

ô!

– 2ï

! )ñ

ó (!

+ 3 ô

J . 0 =  

ó )ñ óò

– ô

0

= exp O

= 0.

ñóò ò ó ò ó

+ [(1 +

õD ó ) !

ñóò ò ó ò ó

=

ñ óò ò ó ò ó

=

ñóò ò ó ò ó

+



– ô 2 ]ñó ò +



ó

+

2

)ñ = 0.

    . The substitution = exp O   X H ( )  leads to Œ

P ΠM 
P 2 2      a second-order linear equation: H + 3   H + (  + 2    + 3 Π ) H = 0. Particular solution:

21.

+

2

Particular solution:

20.

3

ô!„ñ

  + . 0 =  −

!„ñóò

+ tanh ï (1 – ! )ñ .

!„ñ óò

+ coth ï (1 – ! )ñ .

!„ñóò

+ tan ï ( ! – 1)ñ = 0.

= 0.

This is a special case of equation 3.1.9.30 with ( ) = cosh  . 22.

This is a special case of equation 3.1.9.30 with ( ) = sinh  . 23.

Particular solution: 0 = cos  . The substitution linear equation: H 

 − 3 tan 2H 
+ (  − 3) H = 0. 24.

ñóò ò ó ò ó

+

!„ñóò

ñóò ò ó ò ó

+

!„ñóò

+

!„óò ñ

0

= sin  .

=" .

Integrating yields a second-order linear equation: 26.

27.

ñóò ò ó ò ó

+ 2 !„ñó ò +

!„ó¿ò ñ



 + 

+

!„óò ñóò

=X

M

 +L .

= 0.

= L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 and ¨ Solution: of the second-order linear equation: 2¨u

 +  ¨ = 0.

ñóò ò ó ò ó

H ( )  leads to a second-order M

+ cot ï (1 – ! )ñ = 0.

Particular solution: 25.

= cos  X

+ ! (2 !„ó ò –

!

2

)ñ = 0.

Integrating yields a second-order linear equation:

2

are linearly independent solutions



 +  
+ 

2

= L exp O X



 . M P

Page 552

553

3.1. LINEAR EQUATIONS

28.

ñóò ò ó ò ó

+ ( ô – 1) !

2

ñóò

– [ !„ó ò ò ó – (2 ô + 1) !

!ƒó ò

+ ô!

3

]ñ = 0.

Integrating yields a second-order equation:



 +  
+ ( 

2

−  
) = L exp O X



 . M P

29.

ñóò ò ó ò ó

+ ( ! – ô 2 )ñó ò + ( !„ó ò – ô! )ñ = 0. ¨ 
−  ¨ = 0. The substitution ¨ = 

 +  
 +  leads to a first-order linear equation: u

30.

ñóò ò ó ò ó

=

" óò ò ó ò ó ñ "

+ ! ( ï ) Oñó ò –


The substitution ¨ = 
−

" óò ñ " P

.

leads to a second-order linear equation.



3.1.9-2. Equations of the form  3 ( )


 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ). 31.

+ ôñó ò ò ó + õÿñó ò + ö^ñ = ! ( ï ). This is a special case of equation 5.1.6.26 with B = 3.

32.

ñóò ò ó ò ó

+ ôñó ò ò ó + !„ñó ò + ô!„ñ = 0. ¨ 

 +  ¨ = 0. The substitution ¨ = 
+  leads to a second-order linear equation: u

33.

ñóò ò ó ò ó

34.

ñóò ò ó ò ó

+ !„ñó ò ò ó – ô 2 ( ! + ô )ñ Particular solution: 0 =  equation: ¨3

 + (  +  )¨2


ñóò ò ó ò ó

+

!„ñóò ò ó

+

ôñóò

= 0.

  . The substitution ¨ = 
−  +  (  +  )¨ = 0.

+ ô!„ñ = 0.

leads to a second-order linear

1 b . Particular solutions with  > 0: 1 = cos  £   , 2 = sin  £  2 b . Particular solutions with  < 0: 1 = exp  − £ −   , 2 = exp   The substitution ¨ = 

 +  leads to a first-order linear equation:

35.

+ !„ñó ò ò ó + ô ïƒ÷ ñó ò + ô ïƒ÷ The substitution ¨ = 

 +    

36.

ñóò ò ó ò ó

+ !„ñó ò ò ó + ô!„ñó ò + ô The substitution ¨ = 
+ 

37.

ñóò ò ó ò ó

ñóò ò ó ò ó

+

!„ñóò ò ó

+

ô!„ñóò

Particular solutions:

3

ñ

–1

£ −  . ¨u
+  ¨ = 0.

( ï ! + ù )ñ = 0. leads to a first-order linear equation: ¨ 
+  ¨ = 0.

= 0. ¨ 

 +(  −  )2 ¨ 
+  2 ¨ = 0. leads to a second-order linear equation: u

+ ô 2 ( ! – ô )ñ = 0. 1

.

= exp  − 12   cos 

2

3 2

  ,

2

= exp  − 12   sin 

2

3 2

  .

38.

ñóò ò ó ò ó

+ !„ñó ò ò ó + "Iñó ò + % = 0. ¨ 

 +  2 ¨ 
+ ¨ + % = 0. The substitution ¨ = 
leads to a second-order linear equation: u

39.

ñóò ò ó ò ó

+ !„ñó ò ò ó – ô (2 ! + 3 ô )ñ„ó ò + ô 2 ( ! + 2 ô )ñ = 0.   Particular solutions: 1 =   , 2 =    .

40.

+ !„ñó ò ò ó + ï "Iñó ò – "Iñ = 0. ¨ 

 +(  −1)¨2
+  2 ¨ = 0. The substitution ¨ =  
− leads to a second-order equation:  u

41.

ñóò ò ó ò ó

ñóò ò ó ò ó

+ !„ñó ò ò ó + (" – ô 2 )ñó ò – ô ( ô! + " )ñ = 0.  Particular solution: 0 =   . The substitution ¨ 3 ¨

2 ¨
  + )¨ = 0. equation:  + (  +  )  + ( 

= 
− 

leads to a second-order linear

Page 553

554 42.

43.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+ !„ñó ò ò ó + ( ô! + õ – ô 2 )ñó ò + õ ( ! – ô )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó

+ ( ! – ô )ñó ò ò ó – ô 2 !„ñ Particular solution: 0 =   equation: ¨3

 +  ¨2
+  ¨



= 0. . The substitution ¨ = 0.

1

and Œ

= 
− 

2

are roots of the quadratic

leads to a second-order linear

44.

ñóò ò ó ò ó = ( ! – ô )ñóò ò ó + (ô! – õ )ñóò + õ!„ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0. The substitution ¨ = 

 +  
 +  leads to a first-order linear equation: ¨3
=  ¨ .

45.

ñóò ò ó ò ó

+ ( ! – ô )ñó ò ò ó + "Iñó ò – ô ( ô! + " )ñ = 0.  Particular solution: 0 =   .

46.

ñóò ò ó ò ó

+ ( ! + ô )ñó ò ò ó + ( ô! + " )ñó ò + ô"Iñ = 0.  Particular solution: 0 =  −  .

47.

ñóò ò ó ò ó

+

ï !„ñóò ò ó

ï

+ (ô

Particular solutions: 48.

ñ óò ò ó ò ó

ï

+ (ô

1

Particular solution: 49.

ñ óò ò ó ò ó

+ (! +

ô ï )ñ óò ò ó

0

ñ óò ò ó ò ó

+

ï 2 „! ñ óò ò ó

– 2ï

Particular solutions: Solution: = L 1  + L 51.

ñ óò ò ó ò ó

+ (! +

ô ï )ñ óò ò ó

!„ñ óò

=

2

+ ô (ï

Particular solutions: 50.

= cos 

+ï

óò ò ó

+ õ ) !„ñ

– ! )ñó ò + ô

2

1

 £  ,

2

óò

+

ô!„ñ

= exp  − 12  2  ,

2

!„ñ óò

1 2

 2£   .

= 0. = exp  − 21  2  X exp 

2

+ (" + 2 ô ) ñ

óò

M

+ ô [ï

 − 3XZ + (1 – ô

"

−2

j

− 12

53.

ñóò ò ó ò ó

ñóò ò ó ò ó

ï

2

+

õï

Particular solution:

2

ô 2 ï „! ñóò

–ô

−  . 2 = 

2

!„ñ

ï 2 ) ! ]ñ

= 0.

0

2

= 

ï

= 0.

+ ' + ( .

+ ï ( ï ! + " )ñó ò ò ó – "Iñó ò – 2 !„ñ = 0. Particular solution: 0 =  2 . The substitution ¨   ¨ = 0. equation: ¨3

 +  (  + )¨2
+ – ï (ô

leads to a second-order

+ ö ) !„ñó ò ò ó – 2 ô!„ñ = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

 2   . M

 , where j = exp O − Xw 2   . M P M P

Particular solution: 0 = exp    . The substitution ¨ = 
+  3 ¨

2 ¨
linear equation:  +   + ( −   )¨ = 0.

+ (ô

1 2

+ 2 !„ñ = 0.

=  , 2 =  2. 2 XÈ −3 j 2 + L 3 O  1

= sin 

– 2 !„ñ = 0. + 2)ñ

!

+ ( ô ï ! + ! + ô )ñó ò ò ó + Particular solutions: 1 =  ,

55.

+ 3)ñ = 0.

2

+ 2  +  .

52.

54.

1 2

ï (ï 2 !

=


− 2 leads to a second-order linear

+ õ ) !„ñó ò ò ó + ( õ – ô 2 ) !„ñó ò + 2 ô!„ñ = 0.

Particular solution:

0

=

2

+  + 12 ( 

2

−  ).

Page 554

555

3.1. LINEAR EQUATIONS

56.

– [(2 ï + ô ) ! + ( ï

ñóò ò ó ò ó

Particular solution: 57.

ï ñ óò ò ó ò ó

+ ï (ô

óò ò ó

+ 3ñ

ï

Particular solution: 58.

60.

2

64.

65.

ï ñóò ò ó ò ó

+ (ï

ï ñóò ò ó ò ó

+ (ï

ï ñ óò ò ó ò ó

+ (ï

ï

ï

– (ô

2

– 1) !„ñ = 0.

.

1

+ (ô

ï !

=

+L

2

ô 2 )ñ óò

  + X  ( − a )  (a ) a , M 0

ô ï )ñ óò

+ ô (ï

cos( £  ),

−1

− 21

exp 

2

=

2

! =

ï ! 2

  cos 

+ 2! – ô ,

ï

– [ô (ô

+ 2) !„ñó ò + ô ( ô

=

1

ï ñóò ò ó ò ó

−1

−1

1

óò ò ó

+ 3)ñ

!

2

ï )ñ ó ò

+ 2) ! + ( ô

ï



0

is any number.

+ 1)" ]ñ = 0.

+ 1)ñ = 0. −1

sin( £  ).

+

!

3 2

–ô

2

  ,

ï )ñ

2

=

= 0. −1

exp  − 12   sin 

2

3 2

  .

+ ô ( ! – ô )ñ = 0.

   .

−1 −

+ ( ï ! + 3)ñó ò ò ó + (2 ! + ô ïƒ÷ +1 )ñó ò + ô ïƒ÷ ( ï ! + ù + 1)ñ = 0. ¨ 

 +   2 ¨ 
+ The substitution ¨ =  leads to an equation of the form 3.1.9.35: ¨I


 +  2 ¨  −1  (  + B ) = 0.

ï ñóò ò ó ò ó

+ ( ï 2 ! + ô + 2)ñó ò ò ó – ô ( ô + 1) !„ñ = 0. Particular solution: 0 =  −  . The substitution ¨ =  equation: ¨ 

 +   ¨ 
− (  + 1)  ¨ = 0.

ï ñóò ò ó ò ó

+ [ï

2

ï

(ô

2

ï ñ óò ò ó ò ó

+ [ï (ô

ï

0

(ô

ï

– 1)ñ„ó ò ò ó ò

ó

ï 2 ñ óò ò ó ò ó

+

ï 2 ñ óò ò ó ò ó

+ ï [( ô

ï !„ñóò ò ó

+ ï [ï (ô

ï 2 ñ óò ò ó ò ó

+ ï (ï ! By integrating,  2 

 + (  − 2)

=  + 

– 1) ! + ï

2


+ 

leads to a second-order linear

+ 1) ! + 3]ñ„ó ò ò ó – 2 !„ñ = 0.

ï

0

ï

2

(ô

– 2) ! – ô 1

0

2

= ,

ï

−1

.

2

+ 1)" + 3]ñ

ï

=  + 

+ [ï (ô

Particular solution:

70.

=

+ 3)ñó ò ò ó + ( ô

!

Particular solution: 69.

=  + 

1

Particular solutions: 68.

−1

+ 3)ñó ò ò ó + (2 ! +

!

Particular solution: 67.

óò

+ 1) !„ñ

+ ï ( ! – 2 ô )ñó ò ò ó + ï (" +  Particular solution: 0 =    .

Particular solution: 66.

+  +  .

ï ñóò ò ó ò ó

Particular solutions:

63.

+ õ )" ]ñó ò ò ó + 2 !„ñó ò + 2"Iñ = 0.

+  − 2) = L

Particular solutions: 62.

2

ô ï

+ ( ô ï 2 + õ )ñó ò ò ó + 4 ô ï ñó ò + 2 ôñ = ! ( ï ). Integrating the equation twice, we arrive at a first-order linear equation:

Particular solutions: 61.

+

= 2

0

ï ñóò ò ó ò ó

 
+ (  59.

0

2

−1

óò ò ó

– 2 !„ñ

óò

– 2"Iñ = 0.

ï 2)!

+

ô 2]ñ óò

.

2

]ñó ò ò ó + [(2 – ô

 . 2 = 

2

+ 2ô (ô

ï

– 1) !„ñ = 0.

+ 1)" + 2 ! – 6]ñ„ó ò + "Iñ = 0.

= +

−1

.

+ 1) ! + 3]ñ„ó ò ò ó – 2 !„ñ = 0. 0

= +

−1

.

+ ô )ñ ó ò ò ó + [( ô – 2)ï ! + õ ]ñ ó ò + ( õ – ô + 2) !„ñ = 0. we obtain the second-order nonhomogeneous Euler equation 2.1.9.15: 
+ (  −  + 2) = L exp O − X   .

M P

Page 555

556 71.

72.

73. 74.

THIRD-ORDER DIFFERENTIAL EQUATIONS ( ô ï + õ )ï ñ óò ò ó ò ó + (  ï +  )ñ óò ò ó + ï ñ óò + ñ = ! . Integrating yields a second-order linear equation: (  +  ) ( + 2  − ƒ ) = X¯  + L .

ï 3 ñ óò ò ó ò ó

+ õ ï ñó ò + ö^ñ = ! ( ï ). Nonhomogeneous Euler equation. The substitution a = ln | | leads to an equation of the form g 3.1.9.31: g

g
g + (  − 3) g

g + (  −  + 2) g
+ ( =  (A  ).

ï 3 ñ óò ò ó ò ó

+ ( ô + 2) ï 2 ñó ò ò ó + ï !„ñó ò + ô!„ñ = 0. Particular solution: 0 =  −  .

ï 3 ñ óò ò ó ò ó

+ [( ô + 6) ï 2 + õ ]ñó ò ò ó + 2(2 ô + 3) ï ñ„ó ò + 2 ôñ = ! ( ï ). Integrating the equation twice, we arrive at a first-order linear equation: 3


+ ( 

−2

ï 3 ñ óò ò ó ò ó

 
+ (  +



−2 −1

ï 2 „! ñóò ò ó

ï 3 ñ óò ò ó ò ó

+

ï 2 „! ñóò ò ó

79.

81.

82.

83.

ï 3 ñ óò ò ó ò ó

1

+L

2

  + X  ( − a )  (a ) a , M 0



0

is any number.

1

ñóò

=

ñóò

=

1

+L

2

  + X  ( − a )a −2  0

−3

+ 2(2 – ! )ñ = 0. −1

,

2

 (a ) a , M 

0

is any number.

=  2.

+ 6(2 – ! )ñ = 0. −2

,

2

=  3.

+ ï 2 !„ñó ò ò ó + ï ( ! – 1)ñó ò + ( ! – 3)ñ = 0. Particular solutions: 1 = cos(ln | |), 2 = sin(ln | |).

ï 3 ñ óò ò ó ò ó

+

ï 2(!

ï 3 ñ óò ò ó ò ó

+ 1)ñó ò ò ó + ï ( ! – ô – 1)ñ„ó ò – ô ( ! – 2)ñ = 0. 1

=

−

2 

,

=

2

2 

.

+ ï 2 ( ! + ô )ñó ò ò ó + ï ( ô! + õ – ô )ñó ò + õ ( ! – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

ï 3 ñ óò ò ó ò ó

+ ï 2 ( ! + ô )ñó ò ò ó + ï [" + ( ô – 1) ! ]ñ„ó ò + ( ô – 2)"Iñ = 0. Particular solution: 0 =  2−  . The substitution ¨ =  
+ (  − 2) leads to a second-order linear equation:  2 ¨2

 +   ¨2
+ ¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ï 2 ( ! + 2 ô ï )ñ ó ò ò ó + ï (2 ô ï ! + ô 2 ï 2 + õ )ñ ó ò + ( ô 2 ï 2 ! + õ! – 2 õ )ñ = 0.   Particular solutions: 1 =  −    1 , 2 =  −    2 , where B 1 and B 2 are roots of the quadratic equation B 2 − B +  = 0.

ó

+ ô ñó ò ò ó – 3 2 ñó ò + 2 3 ñ = ! ( ï ). Integrating the equation twice, we arrive at a first-order linear equation:

ñóò ò ó ò ó

 − 84.

1

– 6ï

Particular solutions:

80.

+ ) =L

– 2ï

Particular solutions:

78.

+ ) =L

+ ï 2 ( õ ï 2 D +1 – 3 ô )ñó ò ò ó + 2 ô ( ô + 1)(2 ô + 1)ñ = ! ( ï ). Integrating the equation twice, we arrive at a first-order linear equation:

Particular solutions: 77.

2

ï 3 ñ óò ò ó ò ó 

76.

M

+ ô ï 2 ñ óò ò ó

 75.



 + [( ƒ − 2  ) + ý −  ] 
+

ñ óò ò ó ò ó


  + ( + 2 Œ  − ) = L

1+L

+ ( ! + ô )ñ ó ò ò ó + [ ô! + (1 + õ  Particular solution: 0 =  −  +  .

D

 2 + X  ( − a )  −  g  (a ) a , M 0

ó )" ] ñ ó ò



0

is any number.

+ ô"Iñ = 0.

Page 556

557

3.1. LINEAR EQUATIONS

85.

ñóò ò ó ò ó

ó

+ ( õD

 , 1 = 

Particular solutions: 86.

ñóò ò ó ò ó

+( ! –2 ô

ñóò ò ó ò ó

+ ô )ñó ò – 2 ô

+ ( ! – ô

= exp O

1

ó )ñ óò ò ó

3

−  +  . 2 = 

ó )ñ óò ò ó – ô  ó (2 ! – ô  ó

Particular solutions: 87.

ó !

+ 2 ô ) !„ñó ò ò ó – ô ( õD

Œ





  



=  exp O

2

ó )ñ óò

+ (" – 2 ô

= 0.

ó [(ô  ó

+3 )ñó ò + ô

,

P

!„ñ

Œ





 

ó

–

2

]ñ = 0.

.

P

ó [(ô  ó

– ô

– ) ! +2 ô

+ )! + " +

2





]ñ = 0.

X H ( )  leads to Z   . The substitution = exp O   Œ

P ΠP  M  
 2 2 a second-order linear equation: H  + (  + 2   ) H  + (2    + +    + Π ) H = 0. Particular solution:

88.

ñóò ò ó ò ó

– [( ô + ö +

õD ó ) !

ó

 ñóò ò ó ò ó

+ (2 

ó

– ô + ö ]ñó ò ò ó + [( ö

v , 1 = 

Particular solutions: 89.

= exp O

0

+ J



ó

–ô

2

2

−  + ( . 2 = 

+ )ñó ò ò ó + (

ó



2

+

õ^öD ó ) !

+ 2J C

– ô„ö ]ñó ò + ô„ö ( ô + ö ) !„ñ = 0.

ó )ñ óò 

+ J

2

ó

  ñ

= ! ( ï ).

Integrating the equation twice, we arrive at a first-order linear equation:

  90.

ñóò ò ó ò ó

+


  + (ý   + = ) = L

!„ñóò ò ó

1

+L

2

ï )("

+ "Iñó ò – [ #! + tanh(

Particular solution:

0

  + X  ( − a )  (a ) a , M 0 2

+

ñóò ò ó ò ó

+

!„ñóò ò ó

= cosh( Π ). The substitution

Particular solutions: 92.

ñ óò ò ó ò ó

+

!„ñ óò ò ó

1

ñ óò ò ó ò ó

1

ñóò ò ó ò ó ñóò ò ó ò ó ñóò ò ó ò ó

ï )(ï !

+ [ coth(

1

ï )(ï !

Particular solutions: 97.

ï ñ óò ò ó ò ó ñ óò ò ó ò ó

2

+ [( ô 2

+ [ï

2

+

!„ñ óò ò ó

= ,

Particular solution:

óò

2

= ,

1

= ,

2

óò ò ó 2

óò

– 1) ! – 1]ñ

2

M

leads to a

+ 2 Œ" tanh( Œ  )] H = 0.

) – 1] ! + 2 tanh( ) – 1] ! + 2 coth(

ï )}ñ

ï )}ñ

= 0. = 0.

+ ô [(1 – ô tanh ï ) ! + 1]ñ = 0.

– 1) ! – 1]ñ„ó ò + ô [(1 – ô coth ï ) ! + 1]ñ = 0.

= sinh  .

ï !„ñóò

+

2

!„ñ

= 0.

ï !„ñóò

+

2

!„ñ

= 0.

!„ñ

= 0.

2

= cosh( Π ). 2

= sinh( Π ). +

ï !„ñ óò

–

ï )("

–

= ln  −  + 1.

+ [ #! + tan( 0

ï

{[2 coth2 (

= cosh  . 2

ï

{[2 tanh2 (

=  sinh( Π ).

– 1) – ! ]ñ„ó ò ò ó –

( ô – ln ï ) ! + 2]ñ + "Iñ

2

2

2

+

– 1) – ! ]ñ„ó ò ò ó –

1

Particular solutions: 98.

óò ò ó

 , 1 = 

Particular solutions: 96.

= sinh( Π ),

+ [(coth ï – ô ) ! – ô ]ñ„ó ò ò ó + [( ô + [ tanh(

óò

2

is any number.

= cosh( Π ) XZH ( ) 

=  cosh( Π ).

2

) + 3 ]ñ

 , 1 = 

Particular solutions: 95.

ï

+ [(tanh ï – ô ) ! – ô ]ñ

Particular solutions: 94.

) + 3 ]ñ„ó ò +

= cosh( Π ),

– [2 ! coth(

Particular solutions: 93.

ï

– [2 ! tanh(

0

)]ñ = 0.

second-order linear equation: H 

 + [  + 3 Πtanh( Π )] H 
+ [ + 3 Π91.



2

)]ñ = 0.

= cos( Π ). The substitution

= cos( Π ) XZH ( )  leads to a second-

order linear equation: H 

 + [  − 3 Œ tan( Œ  )] H 
+ [ − 3 Œ

2

M

− 2 Œ" tan( Œ  )] H = 0.

Page 557

558 99.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

!„ñóò ò ó

Particular solutions: 100.

ñóò ò ó ò ó

+

!„ñóò ò ó

 , 1 = 

−  , 1 = 

ñóò ò ó ò ó

+ [! +

ï )(ï !

tan(

ñóò ò ó ò ó

+ [! –

ï )(ï !

cot(

Particular solutions: 105.

ô

ï )„ñ óò ò ó ò ó

sin(

+

õÿñóò ò ó

2

+ 3 ô

2

– 2 cot(

= 0.

{[1 + 2 cot2 (

ï )}ñ

= 0.

+ 1) ! + 1]ñ„ó ò + ô [(1 – ô cot ï ) ! + 1]ñ = 0.

2

= sin  . 2

ï !„ñóò

!„ñ

= 0.

!„ñ

= 0.

+

2

+

2

= cos( Π ).

+ 1)]ñ„ó ò ò ó –

= ,

1

ï )] !

ï )}ñ

= cos  .

2

2

+ 2 tan(

=  sin( Π ).

+ 1)]ñ„ó ò ò ó –

= ,

1

ï )] !

{[1 + 2 tan2 (

+ 1) ! + 1]ñ„ó ò + ô [( ô tan ï – 1) ! – 1]ñ = 0.

2

+ [(cot ï + ô ) ! + ô ]ñ„ó ò ò ó + [( ô

ñóò ò ó ò ó

Particular solutions: 104.

2

2

+

2

– [( ô + tan ï ) ! + ô ]ñ„ó ò ò ó + [( ô

ñóò ò ó ò ó

Particular solutions: 103.

= sin( Π ),

1

2

=  cos( Π ).

2

ï )]„ñ óò

+ [3 – 2 ! cot(

Particular solutions: 102.

) + 3 ]ñ„ó ò +

= cos( Π ),

1

Particular solutions: 101.

ï

+ [2 ! tan(

2

ï !„ñóò

= sin( Π ).

sin(

ï )ñ óò

+ 2 ô

3

ï )ñ

cos(

= ! (ï ).

Integrating the equation twice, we arrive at a first-order linear equation:

 sin( Π ) 
+ [  − 2 Œ cos( Œ  )] = L 106. sin(

ï )ñ ó ò ò ó ò ó

1

+L

ï )]ñ óò ò ó

2

+ [ ô + (2 + 1) cos(

  + X  ( − a )  (a ) a , M 0

–(

2

+ 2 ) sin(

ï )ñ ó ò

–



0

2

cos(

is any number.

ï )ñ

= ! (ï ).

Integrating the equation twice, we arrive at a first-order linear equation: sin( Π ) 
+ [  + cos( Œ  )] = L 107. ( ! – 1)ñ

óò ò ó ò ó

– [ ô! +

Particular solutions: 108.

ñ óò ò ó ò ó

+

!„ñ óò ò ó

+ "Iñ

óò

ï )]ñ óò ò ó

tan(

 , 1 =  + ( !" + "

2

ó ò )ñ

+L

2

+(

2

1

  + X  ( − a )  (a ) a , M 0

!

+

ô 2 )ñ ó ò

110.

ñóò ò ó ò ó

+ 3 !„ñó ò ò ó + ( !„ó ò + 2 ! 2 1

+ L 2¨ 1¨

) – #! ]ñ = 0.

= 0.



 +

= L exp O − Xv

 . M P

+ 2" )ñó ò + (2 !" + "Ió ò )ñ = 0.

+ ( ! + " )ñó ò ò ó + ( !„ó ò +

!"

+ % )ñó ò + ( %„òó + "/% )ñ = 0.



 +  
+ %

Integrating yields a second-order linear equation: 111.

ï

is any number.

¨ 2 ¨ ¨ Solution: 2 + L 3 2 . Here, 1 , 2 is a fundamental set of solutions of the u ¨

2 ¨
1 second-order linear equation:  +   + 2 ¨ = 0.

ñóò ò ó ò ó

= L 1¨

2

+ ô [ ô tan(

0

= cos( Π ).

Integrating yields a second-order linear equation: 109.



ñ óò ò ó ò ó

+ ( ! + " )ñ

óò ò ó

+ (2"

óò

The substitution ¨ = 
+

+

!"

+ % )ñ

óò

+ ("

óò ò ó

+

!" óò

= L exp O − Xw

 . M P

+ "/% )ñ = 0.

¨ 

 +  2 ¨ 
+ % ¨ = 0. leads to a second-order linear equation: u

Page 558

3.2. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò˜Í Ò

=

 Ë Ÿ Ï¢¡

(Ï

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

559

3.2. Equations of the Form í ïwî²î²ïwî ï = † ì  í  ( í­ï î ) ( í­ïwî²î ï ) S 3.2.1. Classification Table Table 28 presents below all solvable equations whose solutions are outlined in Subsections 3.2.2– 3.2.4. Two-parameter families (in the space of the parameters ƒ , ý , = , and > ), one-parameter families, and isolated points are represented in a consecutive fashion. Equations are arranged in accordance with the growth of > , the growth of = (for identical > ), the growth of ý (for identical > and = ), and the growth of ƒ (for identical > , = , and ý ). The number of the equation sought is indicated in the last column in this table. In Subsections 3.2.2–3.2.4, the value of the insignificant parameter # is in many cases defined in the form of a function of two (one) auxiliary coefficients  and  ,

# = Y ( ,  )

(1)

and the corresponding solutions are represented in parametric form,

 =  1 (U , L 1 , L 2 , L 3 ,  ),

=  2 (U , L

1,

L 2 , L 3 ,  ),

(2)

where U is the parameter, L 1 , L 2 , and L 3 are arbitrary constants, and  1 and  2 are some functions. Having fixed the auxiliary coefficient sign  > 0 (or  > 0), one should express the coefficient  in terms of both # and  with the help of

 = j ( # ,  ). Substituting this formula into (2) yields a solution of the equation under consideration (where the specific numerical value of the coefficient  can be chosen arbitrarily). The case  < 0 (or  < 0), which may lead to a branch of the solution or to a different domain of definition of the variables  and in (2), should be considered in a similar manner. TABLE 28 Solvable equations of the form 


 = #* 9

>

=

arbitrary

arbitrary

arbitrary ( > ≠ 2) = + 4ý + 5 = + 2ý + 3 3= + 7 2(= + 2) 3= + 7 2(= + 2) arbitrary ( > ≠ 1, 2) arbitrary ( > ≠ 2) 3ý + 4 2ý + 3 arbitrary ( > ≠ 32 )

arbitrary (= ≠ −1) arbitrary (= ≠ −1)

: ( 
); ( 

 ) < ƒ

Equation

0

arbitrary

3.2.4.15*

0

0

3.2.4.1

arbitrary (ý ≠ −1)

0

3.2.4.174

arbitrary (= ≠ −2)

− 12

0

3.2.4.10

arbitrary (= ≠ −2)

1

0

3.2.4.7

−1

−1

0

3.2.4.175

−1

0

0

3.2.4.11

0

arbitrary (ý ≠ − 23 )

0

3.2.4.8

0

− 12

0

3.2.4.87

ý

* Given are formulas of reduction to the generalized Emden–Fowler equation.

Page 559

560

THIRD-ORDER DIFFERENTIAL EQUATIONS TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 ) <

>

=

ƒ ý

Equation

1

arbitrary (ý ≠ −1)

0

3.2.4.2

1

−1

0

3.2.4.13

1

1

0

3.2.4.4

3

arbitrary (ý ≠ − 23 )

0

3.2.4.9

−1

3

− 75

0

3.2.4.168

−1

3 arbitrary (= ≠ −1)

0

0

3.2.4.164

0

0

3.2.4.3

0

arbitrary

− 41 (= + 5)

0

3.2.4.171

0

−2ý − 5

arbitrary (ý ≠ −2)

0

3.2.4.5

0

−13

1

0

3.2.4.153

0

−13

3

0

3.2.4.155

0

−7

0

0

3.2.4.141

0

−7

1

0

3.2.4.145

0

−4

− 12

0

3.2.4.127

0

−4

0

0

3.2.4.123

0

−3

−2

0

3.2.4.95

0

−3

−1

0

3.2.4.30

0

−3

0

0

3.2.4.26

0

−3

1

0

3.2.4.91

0

− 73 − 73 − 73 − 73 − 73 − 73 − 73 − 73 − 95 − 95

− 10 3 − 73 − 43 − 56 − 12

0

3.2.4.76

0

3.2.4.42

0

3.2.4.52

0

3.2.4.133

0

3.2.4.131

0

0

3.2.4.48

1

0

3.2.4.38

2

0

3.2.4.70

− 13 5

0

3.2.4.64

1

0

3.2.4.60

arbitrary ( > ≠ 1) arbitrary ( > ≠ 1) arbitrary ( > ≠ 2) 3ý + 4 2ý + 3

0

0 0 0 0 0 0 0 0 0

Page 560

3.2. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò˜Í Ò

=

 Ë Ÿ Ï¢¡

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

561

TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 ) <

>

= ý

0

−1

−2

0

3.2.4.22

0

−1

0

0

3.2.4.18

0

0

0

3.2.2.2

0

0

3

3.2.3.3

0

0

− 72 − 72 − 52

0

3.2.2.3

0

0

− 52

1

3.2.3.4

0

0

−2

0

3.2.2.6

0

0

− 43

− 43

3.2.3.5

0

0

− 43

0

3.2.2.4

0

0

− 32

3.2.3.7

0

0

0

3.2.2.8

− 53

3.2.3.6

ƒ

Equation

0

0

− 54 − 54 − 76

0

0

− 76

0

3.2.2.5

0

0

−3

3.2.3.8

0

0

− 12 − 12

− 32

3.2.3.9

0

0

− 12

0

3.2.2.7

0

0

0

arbitrary

3.2.3.1

0

0

0

0

3.2.2.1

0

0

1

arbitrary

3.2.3.2

0

1

1

0

3.2.4.35

0

2

− 72

0

3.2.4.165

0

2

0

3.2.4.161

0

3

0 arbitrary (ý ≠ −2)

0

3.2.4.85

0

3

−2

0

3.2.4.82

0

5

−5

0

3.2.4.105

0

5

− 20 7

0

3.2.4.117

0

5

− 15 7

0

3.2.4.111

0

5

0

0

3.2.4.101

1 2

0

− 52

0

3.2.4.74

1 2

3

− 15 8

0

3.2.4.114

1 2

3

20 − 13

0

3.2.4.120

Page 561

562

THIRD-ORDER DIFFERENTIAL EQUATIONS TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 ) <

= > 1 2 1 2 2 3 4 5

ƒ ý

Equation

3

− 54

0

3.2.4.108

3

0

0

3.2.4.104

0

− 76

0

3.2.4.157

−4 arbitrary (= ≠ 1)

− 12

0

3.2.4.137

−1

0

3.2.4.140

1

−3

− 12

0

3.2.4.32

1

−3

1

0

3.2.4.24

1

−1

0

3.2.4.177

1

1

−1 arbitrary (ý ≠ −1)

0

3.2.4.14

1

1

−1

0

3.2.4.17

1

1

1

0

3.2.4.21

3

0

3.2.4.159

0

3.2.4.151

0

3.2.4.109

0

3.2.4.57

3

− 34 − 12 − 23 − 12 − 12

0

3.2.4.148

3

0

0

3.2.4.144

− 94

1

0

3.2.4.66

0

− 13

0

3.2.4.169

0

1

0

3.2.4.62

0

− 52

0

3.2.4.80

0

0

3.2.4.121

0

− 23 − 72

0

3.2.4.68

−7

1

0

3.2.4.54

− 52 − 13 7 − 13

1

0

3.2.4.45

1

0

3.2.4.78

1

0

3.2.4.72

0

1

0

3.2.4.40

1

1

0

3.2.4.50

3

0

0

3.2.4.126

3

1

0

3.2.4.130

1

8 7 8 7 6 5 5 4 5 4 5 4 9 7 9 7 9 7 13 10 27 20 18 13 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5

3 0 −4

Page 562

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

=

 Ë Ÿ Ï¢¡

(Ï

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

563

TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 ) <

> 7 5 10 7 22 15 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 23 15 11 7 8 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 21 13 33 20 17 10 12 7 12 7 12 7 7 4 7 4

=

ƒ ý

Equation

11

1

0

3.2.4.135

0

− 52

0

3.2.4.43

0

− 23

0

3.2.4.115

= − 1)

0

3.2.4.173

−3

− 12

0

3.2.4.100

−3

1

0

3.2.4.97

0

−2

0

3.2.4.99

0

− 12

0

3.2.4.84

0

1

0

3.2.4.93

1

1

0

3.2.4.28

3

−2

0

3.2.4.98

3

− 12

0

3.2.4.94

3

0

0

3.2.4.29

− 13

− 12

0

3.2.4.116

−4

− 12

0

3.2.4.47

1

1

0

3.2.4.125

3

−4

0

3.2.4.55

3

0

3.2.4.46

0

3.2.4.79

0

3.2.4.73

3

− 74 − 10 7 − 23 − 12

0

3.2.4.41

3

0

0

3.2.4.51

3

1

0

3.2.4.129

3

5

0

3.2.4.136

−6

0

3.2.4.69

0

3.2.4.122

0

3.2.4.81

0

3.2.4.170

0

3.2.4.67

0

3.2.4.63

0

− 12 − 12 − 12 − 12 − 13 8 − 12 − 52

0

3.2.4.56

0

1

0

3.2.4.147

arbitrary

3 3

− 13 −4 1 3

3 3

1 2(

Page 563

564

THIRD-ORDER DIFFERENTIAL EQUATIONS TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 ) <

> 7 4 9 5 13 7 13 7

2

=

ƒ ý

Equation

1

1

0

3.2.4.143

− 13

− 12

0

3.2.4.110

− 12

1

0

3.2.4.160

0 arbitrary (= ≠ −1)

1

0

3.2.4.152

0

0

3.2.4.12

2

−1

arbitrary (ý ≠ 0)

0

3.2.4.139

2

−1

−1

0

3.2.4.176

2

−1

0

0

3.2.4.16

2

0

−2

0

3.2.4.33

2

3

−2

0

3.2.4.25

2

3

0

0

3.2.4.19

11 5 7 3 5 2 5 2 5 2 5 2 5 2

0

− 52

0

3.2.4.138

− 43

0

3.2.4.158

−4

− 12 − 12

0

3.2.4.75

− 11 4

1

0

3.2.4.113

27 − 13 − 32

1

0

3.2.4.119

1

0

3.2.4.107

1 arbitrary (= ≠ −3)

1

0

3.2.4.103

1

0

3.2.4.86

3

−2ý − 5

arbitrary (ý ≠ −2)

0

3.2.4.6

3

arbitrary

−= − 2

0

3.2.4.172

3

−9

2

0

3.2.4.106

3

−6

0

3.2.4.59

3

−6

0

3.2.4.166

3

− 17 3

− 12 1 2 − 53

0

3.2.4.77

3

− 33 7 21 −5

2

0

3.2.4.118

− 75

0

3.2.4.65

− 12 − 53

0

3.2.4.37

0

3.2.4.44

2

0

3.2.4.112

3

3 3

−4

3

− 11 3 23 −7

3

Page 564

3.2. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò˜Í Ò

=

 Ë Ÿ Ï¢¡

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

565

TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 ) <

>

= ý

3

−3

−2

0

3.2.4.96

3

−3

−1

0

3.2.4.23

3

−3

− 21

0

3.2.4.90

3

−3

1

0

3.2.4.83

3

− 35

− 35

0

3.2.4.53

3

− 21 − 21

0

3.2.4.149

3

− 35 − 34

0

3.2.4.150

3

−1

−2

0

3.2.4.31

3

− 32

0

3.2.4.134

3

0

0

3.2.4.128

3

0

0

3.2.4.132

3

0

− 35 − 25 − 35 − 21

0

3.2.4.89

3

0

1

−3

3.2.4.88

3

1

−4

0

3.2.4.142

3

1

− 25

0

3.2.4.124

3

1

−2

0

3.2.4.27

3

1

− 35

0

3.2.4.49

3

1

−1

0

3.2.4.20

3

1

0

3.2.4.34

3

1

− 21 1 2

0

3.2.4.162

3

1

2

0

3.2.4.102

3

3

−7

0

3.2.4.154

3

3

−4

0

3.2.4.146

3

3

−2

0

3.2.4.92

3

3

0

3.2.4.39

3

3

0

3.2.4.61

3

3

− 35 − 57 − 21

0

3.2.4.58

3

3

0

0

3.2.4.36

3

5

− 35

0

3.2.4.71

3

7

−7

0

3.2.4.156

4

− 59

1

0

3.2.4.167

4

1

1

0

3.2.4.163

ƒ

Equation

Page 565

566

THIRD-ORDER DIFFERENTIAL EQUATIONS

3.2.2. Equations of the Form 1. 2.

ñ óò ò ó ò ó =
. Solution: = 16 * # 

3

+L

−3 J 2 X V L 1  2 ‡ * + L 2  − ‡ * sin  £ 3 . U  W U + L 3, M −1 =  L 12 V L 1  2 ‡ * + L 2  − ‡ * sin  £ 3 . U  W , 3 1

 J . .

−3 9 2 3

=
ñ –5 ) 2 . Solution in parametric form: 7 1

X (U 3 − 3U + L 2 )−3 J

2

=
ñ –4 ) 3 . Solution in parametric form:

ñóò ò ó ò ó

7 1

 = L where ² =

A (4U ü

−1

Xv²

®

= L

6 3 1(

U − 3U + L 2 )−1 ,

(2UÈ®·Tç² )2 U + L

= XÈU ²

− 1),

3

U + L 3, M

M

−1

U + L 2 , # = A 18  M

=
ñ –7 ) 6 . Solution in parametric form:

9 1 (2

= L

3,

# = −6

where

 J .

−3 7 2

UÈ®·Te² )3 ,

 J .

−3 7 3

ñóò ò ó ò ó

where ² =

]

 + L 0.

ñóò ò ó ò ó

 = L

5.

1

=
ñ –7 ) 2 . Solution in parametric form:

where # = −8

4.

+L

2

ñ óò ò ó ò ó

 = L

3.



2

â$äãÑãÑäã ä = M âJ¥

ü

 = L

13 1

A (4U

− 1),

3

−1

X ²

®

(2UÈ®·Te² )−5 J

=X U ²

−1

M

2

M

U + L 3,

= L

U + L 2 , # = T 18 

18 1 (2

UÈ®·Tç² )−3 ,

 J .

−3 13 6

In the solutions of equations 6 and 7, the following notation is used:

:

L

ø

J U ) + L 2 n 1 J 3 (U ) for the upper sign, L 1 ® 1 J 3 (U ) + L 2 ; 1 J 3 (U ) for the lower sign, where ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; 1 J 3 (U ) are the modified Bessel =

™

1 1 3(

functions. 6.

ñóò ò ó ò ó

=
ñ –2 . Solution in parametric form:

 = L 7.

1

:

−2

M

U + L 3,

= L

=
ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

 = L 8.

−1

X U

1

X

:

M

U + L 3,

= L

1

U

J :

−2 3

−2

: U J 2,

2 2 3 1

,

where

where

# =A

# =T

4 −3 3 . 3





 J

4 −3 3 2 . 3



=
ñ –5 ) 4 . This is a special case of equation 3.2.4.171 with = = 0.

ñ óò ò ó ò ó

Page 566

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

=

 Ë Ÿ Ï¢¡

(Ï

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

567

â$äãÑãÑäã ä = Mx槦!âJ¥

3.2.3. Equations of the Form For ƒ = 0, see Subsection 3.2.2. 1.

ñ óò ò ó ò ó

=
Solution:

ï ^

. = #3 ( ) + L

2

3.

if ƒ ≠ −1, −2, −3;

( ƒ + 1)( ƒ + 2)( ƒ + 3) 1 2 ln | | − 34  2 2 − ln | | +  1 2 ln |  |








if ƒ = −1; if ƒ = −2; if ƒ = −3.

=
ï 3 ñ –7 ) 2 . Solution in parametric form:

1

3 1

J

−3 2

OXv

U +L M

−1 3

P

4 −1 1

= L

,



J

−3 2

OXv

 J . .

 2 ‡ * + L 2  − ‡ * sin  £ 3 . U  , # = 8 

M

U +L

−2 3

P

,

−6 9 2 3

=
ï ñ –5 ) 2 . Solution in parametric form:

ñ óò ò ó ò ó

= L where # = 6 

7 1

X (U Q

8 3 1(

U

3

− 3U + L

− 3U + L

2)

2)

−1

 J .

Q

J

−3 2

X (U

M 3

−1

U + L 3S − 3U + L

,

2)

J

−3 2

M

U + L 3S

−2

,

−4 7 2

In the solutions of equations 5 and 6, the following notation is used:

² =

A (4U ü

3

− 1),

®

−1

= XÈU ²

=
ï –4 ) 3 ñ –4 ) 3 . Solution in parametric form:

M

U + L 2.

ñ óò ò ó ò ó

 = L

7 1

Q

Xv²

−1

where # = T 18  6.

+3

ñ óò ò ó ò ó

 = L

5.

 + L 0 , where  9

=
ï ^ ñ . See equation 3.1.2.7.

where  = L

]





1

ñóò ò ó ò ó

 = L

4.

+L

2






 ( ) =

2.



(2UÈ®·Tç² )2 U + L

M

J  J .

3

S

−1

,

= L

5 1 (2

UÈ®·Tç² )3 Z X ² Q

−1

(2UÈ®·TC² )2 U + L

M

3

S

−2

,

−5 3 7 3

=
ï –5 ) 3 ñ –7 ) 6 . Solution in parametric form:

ñ óò ò ó ò ó

13 1

 = L = L where # = T 18 

Q

8 1 (2

J  J .

XZ²

−1

(2UÈ®·Tç² )−5 J

UÈ®·Tç² )−3 v X ² Q

−1

2

M

U + L 3S

(2UÈ®lTç² )−5 J

−1

2

M

,

U + L 3S

−2

,

−4 3 13 6

Page 567

568 7.

THIRD-ORDER DIFFERENTIAL EQUATIONS =
ï –3 ) 2 ñ –5 ) 4 . Solution in parametric form:

ñóò ò ó ò ó

 = L

3 1

O X U

where H = L 8.

2

J H

J  J

−1 2 −1 2 3 4

+ 14 U

2

−1

U +L M

3

P

2 1

= L

,

+ 4 $IU 1 J 2 ,  = exp O X H

J

−1 2

=
ï –3 ñ –1 ) 2 . Solution in parametric form:

J H

J  J

−1 2 −1 2 3 4

*O X U

U , # = 12 C $  M P

M

J  J .

−2

U +L

3

P

,

−3 2 9 4

ñóò ò ó ò ó

 =L where

:

O X

−1

U +L M

:

3

P

= 'U 2 J

,

:

3

2

O X

: M

−2

U +L

3

P

,

ø

J U ) + L 2 n 1 J 3 (U ) for the upper sign, L 1 ® 1 J 3 (U ) + L 2 ; 1 J 3 (U ) for the lower sign, ( U ) and n ( U ) are the Bessel functions, and ® 1 J 3 (U ) and ; 1 J 3 (U ) are the modified Bessel 1J 3 1J 3 # =A

ø

1

 J

4 3 2 , 3

L

™

=

1 1 3(

functions. 9.

=
ï –3 ) 2 ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

.

 = L

.

3

.

exp O 2 X

U , M P

 = L

3

.

2

.

exp O 2 X

U . M P

Here, , L 2 ) is the general solution of the second Painlev´e transcendent: .

= A"U . = + 2(. U ,3L, 1and # = A 14  −3 J 2  3 J 2 .

*+*

3.2.4. Equations with |¨ | + | © | ≠ 0 1.

ñóò ò ó ò ó

=
(ñó ò )` (ñó ò ò ó ) ª , Solution in parametric form:

 = L 1; + < where # = A 2.

−1

XçU

J O 1 AIU ;

= +1 2< 2−>

−2

 = L 1: + < X where # = A

]

ý +1 2< 1−>

<

< P

1 −2

≠ 2. = L 1; +2 <

U + L 3, M

−3

XCO 1 AIU ;

+1 2

< P

1 −2

M

U + L 2,

; <.

+2 −3 1− −

Q





X¥ 1 AIU :

 <

S

≠ –1,

1 +1 1−



≠ 1.

< M

 : <.

J

−1 2

U + L 2S

M

U + L 3,

= L

<

2 −2 1

U ,

−1 2 −2 − −

In the solutions of equations 3–10, the following notation is used:

² = £ 1 ACU + 3.

 ;

S

≠ –1, +1 2

−1 2

=
ñ _ ñ ó ò (ñ ó ò ò ó ) ª , Solution in parametric form:

ñ óò ò ó ò ó



+1 ,

« = X (1 ACU +

=
(ñó ò )` , ≠ –1. Solution in parametric form:

)

ñóò ò ó ò ó

 = L 1+ X U where

¡

=

= −1 2

, # = A3¡

+ 1 2+  4

J ²

−1

−2 −2

+ .

−1 2



J

+1 −1 2

M

M

U + L 2,

U + L 3,

 = ²u« − U .

= L 1+

−1

« ,

Page 568

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

4.

ñ óò ò ó ò ó

=


ññ óò (ñ óò ò ó )ª

S

,

 Ë Ÿ Ï¢¡

=

(Ï

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

569

≠ 2.

Solution in parametric form:

 = L 13 + =

where 5.

¡

+1

1 , # = −8  2  ¡ > −2

ñ óò ò ó ò ó =
ñ _ (ñ óò

)

_

–2 –5



,

J ²

−1 2

Xv«

−1

U + L 3, M

2 1

= L

+

+2

² ,

1J + 2 2 S . Q ( + 1) 

−3

A

¡

≠ –2.

Solution in parametric form:

 = L 1+ = −ý − 3, #

where 6.

ñóò ò ó ò ó

¡ =


ñ _ (ñ óò

)–2_

−3

J «

1 −2 4 (−1)

=A

−3 2

−1

U + L 3, M

+ ( + 1)  2 + ¡

(ñó ò ò ó )3 ,

–5

J ²

−1 2

XZU





−2 3−

+

−2

+2

U «

2 1

= L

−1

«

,

+ .

≠ –2.

Solution in parametric form:

 = L 1+ = ý , # = T 2(

where 7.

ñóò ò ó ò ó

¡ =

+3

+ 1) 

¡

`

3 +7 +4


ñ (ñ óò )` (ñ óò ò ó

) 2`

J ²

−1

+

+3

−3 2

XZ« −2

+

−2



,



U + L 3, M

= L

2 1

+

−1

,

.

≠ –2.

Solution in parametric form:

 = L 1+

ñóò ò ó ò ó

+

+

+2

+ J 2²

−

XZ«

−1

M

U + L 3,

2(= + 2) 2  −1 ( + 1)  + =− , # =− ¡ A ¡ S = +1 +2 Q 2

where 8.

2

¡ =


ñ _ (ñ óò ò ó

2 +2

¡

_

3 +4 ) 2 +3

_



,

= L

4 1

 , 1

4 2 + A S . Q ( + 1)( + 2) 

¡

¡

≠ –3  2.

Solution in parametric form:

 = L 1+ =−

where 9.

ñóò ò ó ò ó

¡ =

ý

ý +1


ñ _ (ñ óò

2

+2

+

XÈU + «

−1

, # =(

¡

_

+

−2

−1

+ 3)

3 +4 +3

)3 (ñó ò ò ó ) 2_

−



,

+

 +

−1

²

U + L 3, M

2 2 + S 2 Q ( + 1) 

A

+1

+

( +1)2 1

= L 1 +3

U +

+1

+

+2)

«

−

+

−1

,

.

¡

≠ –3  2.

Solution in parametric form:

 = L 1+ where

¡

=−

2

+

+

−1

XZ« +

+1



J ²

−1 2

2ý + 3 , # = ¡  2 ý +1 ( + 2)3

¡

−

−1

2

+

+

M

+1 +1

U + L 3, Q

(

A ¡

= L

+

( −1)( 1

« +

+2

,

1

+ 1)( + 2)  + ¡ S . 4 2

Page 569

570 10.

THIRD-ORDER DIFFERENTIAL EQUATIONS

`

3 +7 +4

=
ñ –1 ) 2 (ñó ò )` (ñó ò ò ó ) 2` Solution in parametric form:

ñóò ò ó ò ó

 = L 1+ where 11.

=

¡

2

+2

+

−7



,

X U

+

−1 2

≠ –2.

²

−1

1−= 2( + 3) 1 J 2 A , # =− ¡  1+= +1 Q (

¡

ñóò ò ó ò ó

=
(ñó ò )–1 (ñó ò ò ó ) ª , Solution in parametric form:

S

≠ 2.

+ «

+3 2



¡

12.

=
( ñ ó ò )` ( ñ ó ò ò ó )2 , Solution in parametric form: 1− 1+



13.

;

ñ óò ò ó ò ó

=
ñ –1 ñ ó ò (ñ ó ò ò ó ) ª , Solution in parametric form:

 = L

1

where # = 14.

X U

1

1 (T 1)− <  2 < 1−>

1

XÈU

where # = T (ý + 1) 

1− 1+

< exp(T U 2 ) U + L ® 2 < M

3− 1−



: ¬XZU

−1−



2 1 +3

, .

X U

1

3− 1+

;

; exp(T U 2 ) U + L 3 , M

≠ 1.

−2 1−

:

+ 1)2  + S 2 2

(

A ¡ Q

−8 1

< exp(T 2U 2 ) U + L , XÈU < −2 3 M

2 1

= L

1− 1+

: .

J

−1 2

M

U + L 3,

= L

U + L 3,

= 'U

2 1 exp(

T"U 2 ),

<.

=
ñ _ ñó ò ñóò ò ó ,  ≠ –1. Solution in parametric form:

ñóò ò ó ò ó

 =L

15.

S

XÈU exp(T"U 2 ) ¬XÈU

2 +1

= L

≠ –1.

; exp(T U 2 ) U + L 2 , M where # = A (= + 1)  ; +1  −; −1 .  = L

+ S

= L

1

ñ óò ò ó ò ó

U + L 3, M

+ 1) 

< exp(T U 2 ) U + L , XÈU < −2 2 M 2 2 4  < where # = T O˜T . 4 (2 − > )  2  P  = L



:

: exp(T"U 2 ) U + L 2 ® M

J

−1 2

M

2 1+

: ,

=
ï ^ (ñó ò )` (ñó ò ò ó ) ª . Solution in parametric form:

ñóò ò ó ò ó

m (U ) U + L 3. U M M Here, m = m (U ), n = n (U ) is the general solution of the generalized Emden–Fowler equation: n ol

o = $Im 9 n ; ( n o
) < , where # = $C ; +2 < − 9 −3  1−; − < .  = L 1; + <

16.

ñóò ò ó ò ó

=


(ñ óò

Solution:

−1

)–1 (ñó ò ò ó )2 .


1 − #


(2 − # ) L

=



L






2




 L L

1

1

1

1

exp( L ln( L

(L

1 1

= L 1; +2 < − 9

m (U ),

2−

1

 + L 2 ) 1− O + L

 )+L

O

Xvn (U ) M

if # ≠ 1, # ≠ 2; if # = 1;

3

 + L 2) + L

3

−3

3

if # = 2.

Page 570

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

17.

ñóò ò ó ò ó

=

ñóò ñóò ò ó .   X (L 1 O =


ñ

X (L

1

+1

+L

2)

ln + L

2)

J

−1 2

J

−1 2

+L

M

+L

M

1

1 2

X exp(T

U 2) U + L 2, M

= L

2 1

X exp(T"U 2 ) U + L 3 , M

1

J exp(T"U 2 ) U + L , 2 M

−1 2

XÈU

= L

1

X exp(T U 2 ) U + L 3 , M

# = A 4 4 

where

−4

.

 = 2U «ZA exp(T"U 2 ).

=
ñ –1 ñó ò (ñó ò ò ó )3 . Solution in parametric form:

X U exp(T U 2 ) «

1

J

−1 2

1

J

−1 2

XZ«

U + L 3, M

ñóò ò ó ò ó =
ññó¿ò ñóò ò ó . Solution in parametric form:

M

2 1

exp(T"U 2 ),

where

= 'U ,

where

# = T 2

−2

where

# = T 

= L

U + L 3,

# =A

1 4 −2 . 2

 

.

ñ óò ò ó ò ó

=
ñ –2 (ñ ó ò )–1 . Solution in parametric form: 1

Xv«

J exp  T

−3 2

1 2

U 2  U + L 3, M

= L

1

«

−1

,

−4 4

 .

ñ óò ò ó ò ó

=
ñ –1 (ñ ó ò )–3 (ñ ó ò ò ó )3 . Solution in parametric form: 1

J  exp(T"U 2 ) U + L 3 , M

−3 2

X «

= ˜«

−1

exp(T"U 2 ),

where

# = T 8 2.

ñóò ò ó ò ó

=
ñ (ñó ò )–3 ñó ò ò ó . Solution in parametric form:

 = L 25.

 .

ñóò ò ó ò ó

 =L 24.

−4 2

In the solutions of equations 20–25, the following notation is used:

 = L 23.

# = T/

where

=
( ñ ó ò )3 ( ñ ó ò ò ó )2 . Solution in parametric form:

 =L 22.

if # = −1.

3

ñ óò ò ó ò ó

 = L 21.

if # ≠ −1;

3

« = X exp(T"U 2 ) U + L 2 , M 20.

571

=
(ñó ò )–1 . Solution in parametric form:

 = L ]

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

ñóò ò ó ò ó

 = L 19.

 Ë Ÿ Ï¢¡

–1

Solution:  18.

=

1

X «

J

1 2

M

U + L 3,

= L

2 1

 ,

where

# = T 8

−4 2

 .

ñóò ò ó ò ó

=
ñ –2 (ñó ò )3 (ñó ò ò ó )2 . Solution in parametric form:

 = L

1

X 

J exp(T"U 2 ) U + L 3 , M

−1 2

= L

2 1

« ,

where

# = A  4 

−2

.

Page 571

572

]

THIRD-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 26–33, the following notation is used:

« = £ U (U + 1) − ln  £ U + £ U + 1  + L 2 , 26.

=
(ñó ò )–3 . Solution in parametric form:

5 1

−2

X U

7 1

−1

²

J

−1 2

«

J

−3 2

X ²

=
ñ –1 (ñ ó ò )–3 . Solution in parametric form:

M

ñ óò ò ó ò ó

5 1

XÈU

= L

4 1

U + L 3,

= L

M

J ²

−1 2

−1

U + L 3,

J

−3 2

«

ñóò ò ó ò ó

=
ñ –2 (ñó ò )–1 (ñó ò ò ó )3 . Solution in parametric form: −1 1

 .

U ,

# = 2 4 

where

−1

.

2 1

² ,

where

# = −8 (−  )−5 J 2.

XZ²

−1

«

J 

−3 2

M

M

=
ñ –1 ) 2 (ñó ò )–3 ñó ò ò ó . Solution in parametric form:

= L

6 1

« ,

where

U + L 3,

= L

6 1

«

−1

U + L 3,

= L

2 1

U «

,

−1

# = 4 3 (−  )−7 J 2.

where

,

where

−6 5

# = − 41 

 .

# = 2 2  .

ñóò ò ó ò ó

7 1

J ²

−3 2

XZU

−1

«

J 

1 2

ñóò ò ó ò ó

=
ñ –2 (ñó ò ò ó )2 . Solution in parametric form:

 = L 34.

M

=
( ñ ó ò )3 ( ñ ó ò ò ó )3 ) 2 . Solution in parametric form:

 = L 33.

U + L 3,

−6 4

# = − 41 

where

ñ óò ò ó ò ó

 = L 32.

J

−1 2

X «

1

=
ññó ò (ñó ò ò ó )3 ) 2 . Solution in parametric form:

 = L 31.

« ,

ñóò ò ó ò ó

 = L 30.

3 1

= L

=
ñ –2 ñó ò (ñó ò ò ó )3 . Solution in parametric form:

 = L 29.

£ U + 1 + L 3,

2 1

ñóò ò ó ò ó

 = L 28.

 = ²u« − U .

ñóò ò ó ò ó

 = 2 L 27.

U +1 , U

² =

−1 1

X U

−2

²

=
ñ –1 ) 2 ñó ò (ñó ò ò ó )3 . Solution in parametric form:

−1

M

M

U + L 3,

U + L 3,

= L

= L

1

U

8 1

−1

2



,

« ,

where

where

# =

 J .

−4 7 2

# = 2 .

ñóò ò ó ò ó

 = A L where # = T

5 1

X U (U J

2

− 1)(U

3

− 3U + L

J

−1 2 2)

M

U + L 3,

= L

8 2 1(

U

2

− 1) ,

1 4 −5 2 . 144

 

Page 572

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

35.

−1 1

2 5

− 3U + L

U J

5 1 2 2 ( 1

L

2)

J

−1 2

U + L 3, M

2 1

= L

−2 −1

U ,

where

# = 3



2 ),

where

8 6 # = − 243  

.

U

− 5) + L

3,

= L

3

− 3U + L

6 3 1(

U

− 3U + L

−5

.

=
ñ –1 ) 2 (ñó ò )–4 (ñó ò ò ó )3 . Solution in parametric form:

ñóò ò ó ò ó

5 1

2

X (U

− 1)(U 2

2 2 1(

U

3

− 1) (U

− 3U + L

2)

J

−3 2

4

(U

2

− 6U

+ 4L

2

16 4 1 (

− 6U

−1 2) ,

U − 3) U + L 3 , M

 J

16 −1 5 2 . 9



=
ñ (ñ ó ò )–7 ) 3 . Solution in parametric form:

ñ óò ò ó ò ó

7 1

 = L

3

X (U

− 3U + L

2)

J

1 4

U + L 3, M

 (4   )1 J 3 .

= A/L

U

2

+ 4L

2

U − 3),

−5 2

where # = A 72

=
ñ –5 ) 3 (ñ ó ò )3 (ñ ó ò ò ó )3 . Solution in parametric form:

ñ óò ò ó ò ó

5 1

 = A L

2

X (U

9 3 1(

= L

U

3

− 1)(U

− 3U + L

2)

J

1 2 [ 2)

− 3U + L

A (U

4

2

− 6U

+ 4L

2

J

−1 2

U − 3)]

J ,

3 2

M

U + L 3,

J .

where # = T 8 × 9−5  6 

−10 3

=
ñ (ñ ó ò ò ó )7 ) 5 . Solution in parametric form:

ñ óò ò ó ò ó

 = L where # = A 41.

573

=
( ñ ó ò )3 ( ñ ó ò ò ó )3 . Solution in parametric form:

where # = T

40.

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

ñ óò ò ó ò ó

= L

39.

3

X (U

 = L

38.

(Ï

=
ññ ó ò . Solution in parametric form:

 =A 37.

 Ë Ÿ Ï¢¡

ñ óò ò ó ò ó

 = L 36.

=

−7 1

3

X (U

15  2

−1 −1



J

−3 2 2)

− 3U + L

O



J

2

2 5

2 "P

U + L 3, M

= A/L

U

2

− 1)(U

− 6U

2

+ 4L

1(

3

− 3U + L

J

−1 2 , 2)

.

=
ñ –1 ) 2 (ñó ò )3 (ñó ò ò ó )8 ) 5 . Solution in parametric form:

ñóò ò ó ò ó

31 1

 = A L = L

X [A ( U

32 4 1 (

U

− 6U

where # = −15 × 2−10  2 

J

−1 2

2

− 1)]

2

+ 4L

2

J O 

−5 2

(U

3 2

− 3U + L

2)

J

5 4

(U

4

2

U − 3) U + L 3 , M

U − 3) ,

2

2 "P

J

3 5

.

Page 573

574 42.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

=


ñ

) (ñ óò

)–7 ) 3 .

–7 3

Solution in parametric form: 17 1

 = L

16 4 1 (

= A/L

ñóò ò ó ò ó

=


ñ

U

− 3U + L 2

− 6U

2)

+ 4L

2

J

1 4

4

[A ( U

J

−3 2

− 6U

2

+ 4L

2

− 6U

2

+ 4L

2

U − 3)]

M

−1

U + L 3,

U − 3) ,

 J (  4)−1 J 3.

−5 17 3

where # = A 72 43.

3

X (U

) (ñ óò ò ó

)10 ) 7 .

–5 2

Solution in parametric form: 29 1

 = A L

2 3 1(

= L where # = −

]

U

− 3U + L −1

− 3U + L

2  J O

−1 5 2

3 7

3 \P

J

−3 2

4

(U

2)

J

2

2)

4

(U 2

− 6U

+ 4L

2

U − 3)

J

−1 3

U − 3)

J

2 3

M

U + L 3,

,

.

In the solutions of equations 44–47, the following notation is used:

.

44.

28  3

3

X (U

ñóò ò ó ò ó

=


ñ

6(

6

U ) = A (U

) (ñ óò

4

− 15U

+ 20 L

2

3

U

− 45U

2

+ 12 L

2

+ 4L

2

2 2 ).

U + 27 − 8 L

)–11 ) 3 (ñó ò ò ó )3 .

–5 3

Solution in parametric form: 5 1

 = L = A/L where # = T 45.

ñ óò ò ó ò ó

=

3

X (U 1(

3

U

− 3U + L

J

 )−2 J 3 .

)–5 ) 2 (ñ

ó ò ò ó )7 ) 5 .




ñ (ñ óò

J

1 2

J

4

[A ( U

3 2 4 ( 2)

− 3U + L

9 10 3 (2 16

2)

U

− 6U

− 6U 2

+ 4L

2

2

J

−3 2

U − 3)]

.

6(

−1

U ) U + L 3, M

U − 3) ,

Solution in parametric form: 11 1

 = L

27 3 1 (

= L

46.

where # =

405  8

ñ óò ò ó ò ó

–7 4

=


ñ

U

−3

) (ñ ó ò

OA )3 ( ñ

3

X (U

− 3U + L

J



J

−3 2 4 ( 2)

− 3U + L

1 2

2 ·P

2)



O

J

−1 2

.

J

2

2 5

12 "P

U

6(

− 6U

2

+ 4L

2

U − 3)

J

4 3

M

U + L 3,

U ),

.

ó ò ò ó )8 ) 5 .

Solution in parametric form: 37 1

 = A L = L

X (U

64 4 1 (

U

3

− 6U

where # = −45 × 2−13  2 

− 3U + L 2

+ 4L

2

J O 

−5 4

2)

J

5 4

(U

J

4

4 3

U − 3) 2

12 "P

J

3 5

− 6U

2

+ 4L

2

J

1 3

U − 3)

[

.

6(

U )]−1 J

2

M

U + L 3,

,

.

Page 574

3.2. EQUATIONS OF THE FORM

47.

55 1

U

5 1

X (U

2)

J

−3 2

.

−1 2) [ 6(

− 3U + L 2  J O

−5 9 2

J

1 4

2

A 1)

M

J

2

4 7

3 \P

4

(U

2

− 6U

+ 4L

2

J

5 3

U − 3)

.

6(

U )]2 ,

U ) U + L 3, M

.

U + L 3,

8 3 1(

= L

U

A 3U + L 2 ),

where # = A

81 −5 3 (3 2





  )1 J 3 .

=
ñ –5 ) 3 ñ ó ò (ñ ó ò ò ó )3 . Solution in parametric form:

ñ óò ò ó ò ó

where # = T

XÈU (U

1

2

J

1 2

A 1)

3

(U

A 3U + L 2 )

J

−1 2

U + L 3, M

J

= L

A 1)

2

A 1)

U

J

3 2

3 2 1(

,

4 4 −4 3 . 243

 

=
ññ ó ò (ñ ó ò ò ó )7 ) 5 . Solution in parametric form:

ñ óò ò ó ò ó

3 1

 = L

2

X (U

−2

where # = A 5 

J

−3 2

A 1)

J

−1 2

(U

3

A 3U + L 2 )

J (U

2

A 1)

M

2 2 2J 5 . 3 eP

O

U + L 3,

= L

1

U (U

J

−1 2

,

=
(ñó ò )3 (ñó ò ò ó )8 ) 5 . Solution in parametric form:

ñóò ò ó ò ó

9 1

 = L where # = T

4 2   27

−3

O

−1 2

X U

J

5 4

2 2 3J 5 . 3 \P

U + L 3, M

= L

8 3 1(

U

A 3U + L 2 ),

=
ñ –4 ) 3 (ñó ò )–7 ) 3 . Solution in parametric form:

ñóò ò ó ò ó

 = L where # = A 53.

− 3U + L

=
(ñó ò )–7 ) 3 . Solution in parametric form:

 = L

52.

575

ñóò ò ó ò ó

 = L

51.

3

X (U

54 3 1 (

where # = T 28 × 37 

50.

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

=
ñ –1 ) 2 (ñó ò )–4 (ñó ò ò ó )11 ) 7 . Solution in parametric form:

= L

49.

 Ë Ÿ Ï¢¡

=

ñóò ò ó ò ó

 = A L

48.

Ï Ò˜Í Í Ò˜Í Ò

7 1

X (U

A 1) J

81 −5 13 3 (3 2





J

1 4

2

(U

3

J

−3 2

A 3U + L 2 )

  )1 J 3 .

U + L 3, M

= L

8 3 1(

U

−1

A 3U + L 2 ) ,

=
ñ –5 ) 3 (ñó ò )–5 ) 3 (ñó ò ò ó )3 . Solution in parametric form:

ñóò ò ó ò ó

 = L

−1 1

X (A"U

2

+L

A 1)3 J 2 (U 4 2 2J 3 where # = T 27   (3   )2 J 3. = L

1(

U

2

2 3

U − 1)(U

2

J

1 2

A 1) −1

(U

3

A 3U + L 2 )

J

−3 2

M

U + L 3,

A 3U + L 2 ) ,

Page 575

576 54.

THIRD-ORDER DIFFERENTIAL EQUATIONS =
ñ (ñó ò )–7 (ñó ò ò ó )7 ) 5 . Solution in parametric form:

ñóò ò ó ò ó

 = L

X (U 2 A 1)

(U 3 A 3U + L

2)

J

5 6

U + L 3, M

 (2  2  )2 J 5 .

9 1(

= L

J

−1 2

A"U 2 + L 2 U −1)(U 2 A 1)

,

−8 6

where # = A 5  55.

J

−3 2

7 1

=
ñ –4 (ñó ò )3 (ñó ò ò ó )8 ) 5 . Solution in parametric form:

ñóò ò ó ò ó

−1 1

 = L

8 3 1(

= L

U

J

5 4

2

X (U

A 1)

A 3U + L 2 )

2

(AU

J

1 3

+L

U − 1)

2

J

−1 2

3

(U

A 3U + L 2 )

J

−2 3

M

U + L 3,

,

where # = T 5  2  (2  2  )3 J 5 . 56.

=
ñ –5 ) 2 (ñó ò ò ó )7 ) 4 . Solution in parametric form:

ñóò ò ó ò ó

 = L

−7 1

where # = 4  57.

  J O˜T

−1 5 2

(U 3 A 3U + L

J

2

3 4

2  P

2)

J

−1 3

M

U + L 3,

= L

−1 2J 3 U A 1) (U 3 A 3U + L 2) ,

2 2 1(

.

=
ñ –1 ) 2 (ñó ò )–4 (ñó ò ò ó )5 ) 4 . Solution in parametric form:

ñóò ò ó ò ó

17 1

 = L = L

2

X (AU

18 1 (

A"U

2

  J OT

−5 9 2

where # = −64 

]

J

−3 2

X (U 2 A 1)

+L

2

2

+L

2

2

1 4

J

2 P

U − 1)(U

U − 1) (U

2

2

J

−3 2

A 1)

(U

3

A 3 U + L 2 )2 J

3

−1

A 1) ,

M

U + L 3,

.

In the solutions of equations 58–69, the following notation is used:

8

 2  * + L 2  −  * sin( ÑU ),   = , £ 3, 2 * + , L 2  −  * V £ 3 cos( ÑU ) − sin( ÑU )W , 2 = 2, L 1  8 2 2 * − 2 , 2 L 2  −  * V¤£ 3 cos( ÑU ) + sin( ÑU )W , 3 = 4, L 1  8 8 2 8 8 8 8 8 38 3 4 = 2 − 2 1 3, 5 = 5 2 4 + 32 , 1. 1

8

58.

1

=
ñ –1 ) 2 (ñó ò )3 (ñó ò ò ó )3 . Solution in parametric form:

ñóò ò ó ò ó

3 1

 = L 59.

=L

8 X

J 8 8

−1 2 2 3 1

=
ñ –1 ) 2 (ñó ò )–6 (ñó ò ò ó )3 . Solution in parametric form:

M

U + L 3,

8

4 2 1 2,

= L

where

# = − 6 

J , .

−9 2 3

ñóò ò ó ò ó

 = L

3 1

X

8

J 8 8

−3 2 2 4 1

M

U + L 3,

= L

8

8

2 −1 2 1 1 2,

where

# = 16

 J , .

−3 9 2 3

Page 576

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

60.

ñóò ò ó ò ó

 Ë Ÿ Ï¢¡

(Ï

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

577

)–9 ) 5 .


ñ (ñ óò

=

=

Solution in parametric form: 3 1

 = L 61.

ñóò ò ó ò ó

+L

M

) (ñ óò


ñ

=

8 3J 4 U 1 X

8

8 1 4,

= L

3,

# = −160

where

−4

, 6 (16 , 3  )4 J 5 .

)3 (ñó ò ò ó )3 .

–7 5

Solution in parametric form: 3 1

 = L 62.

ñ óò ò ó ò ó

8 3J 2 8 8 X


ñ (ñ óò ò ó )9 )

=

J

−1 2 2 4

1

7

U + L 3, M

8 J

5 5 2 1 1 ,

= L

× 5−5  6 

1 4

# =

where

J ,

−18 5 −6

.

.

Solution in parametric form: −3 1

 = L 63.

ñóò ò ó ò ó


ñ

=

8 X

J

−3 2 1

) (ñ óò

U + L 3, M

8

J 8

−1 2 1 1 2,

= L

7  2

# =

where

−1 −1



O



J

2

8 ,

2 7

P

3

.

)3 (ñó ò ò ó )12 ) 7 .

–1 2

Solution in parametric form:

 = L 64.

ñóò ò ó ò ó

15 1

8 9J 4 8 X

1


ñ

=

J 8

−1 2 4 2

) (ñ óò

M

U + L 3,

8

16 2 1 4,

= L

J ,

where # = 7 ⋅ 2−16  2 

−5 2 −9

O



J

2

8 ,

5 7 3

P

.

)–9 ) 5 .

–13 5

Solution in parametric form:

 = L 65.

ñóò ò ó ò ó

9 1

8 3J 4 8 X


ñ

=

J

−3 2 4

1

) (ñ óò

M

U + L 3,

= L

8

8 −1 1 4 ,

where

# = −160

J , (16 , 3  )4 J 5 .

−4 23 5 6



)–21 ) 5 (ñó ò ò ó )3 .

–7 5

Solution in parametric form:

 = L 66.

ñ óò ò ó ò ó

3 1

8 3J 2 8 X

1


ñ (ñ óò

=

J 8

−3 2 5 4

)–9 ) 4 (ñ

M

U + L 3,

8 J 8

5 2 −1 1 1 4 ,

= L

where # =



J

5  512

−1 17 5 −6

−3

5 1J 4  2 2J 7 O . 2 rP 32 , 3 P



J ,

O

2 ·P

1 5

.

ó ò ò ó )9 ) 7 .

Solution in parametric form:

67.

 = L

9 1

ñóò ò ó ò ó

=

X

8
ñ

J 8 J

−3 2 6 5 1 4

) (ñ óò

–13 8

M

U + L 3,

= L

8

J 8

25 −1 2 5, 1 1

where # =

35  8

O −

)3 (ñó ò ò ó )12 ) 7 .

Solution in parametric form:

 = L where # = 175 × 2−22  2 

39 1

X

J ,

8 9J 4 8 3J 5 8

−11 8 −9

1

O



J

4

−1 2 5

2

5 7

32 ,

J

3

P

M

U + L 3,

= L

8 J

64 8 5 1 4 ,

.

Page 577

578 68.

THIRD-ORDER DIFFERENTIAL EQUATIONS =
ñ –7 ) 2 (ñó ò ò ó )18 ) 13 . Solution in parametric form:

ñóò ò ó ò ó

 = L 69.

27 1

8 X

J 8

J

−3 2 −3 5 1 4

U + L 3, M

=
ñ –1 ) 2 (ñ ó ò )–6 (ñ ó ò ò ó )21 ) Solution in parametric form:

ñ óò ò ó ò ó

51 1

 = L

13

8 X

8

8 J

2 −1 2 5 1 1 4 ,

= L

where # = − 208 5 

 J , (2  2  )5 J 13.

−1 7 2 3

.

J 8 J 8

−3 2 9 5 5 1 4

J , (2  2  )8 J 13 .

U + L 3, M

8

8

50 −1 2 1 1 5,

= L

5 −7 13 2 3

where # = 208 × 5 

]

{



In the solutions of equations 70–81, the following notation is used:

{

{

{

{ {

U + L 2) cos U , 2 = tanh( U + L 2) + tan U , 3 = tanh( U + L 2) − tan U , 4 = 3 2 3 − 4, 9 9 9 9 9 9 92 1 = cosh U − sin( U + L 2), 2 = sinh U + cos(U + L 2), 3 = sinh U − cos(U + L 2), 4 = 3 2 3 − 2 1. 1 = cosh(

70.

=
ñ 2 (ñó ò )–7 ) 3 . 1 b . Solution in parametric form:

ñóò ò ó ò ó

 = L

{

X

1

J

1 4 1

U + L 3, M

{ {

4 1 1 2,

= L

2 b . Solution in parametric form:

 = L 71.

1

9 1J 4 U 1

X

+L

M

9

4 1 2,

= L

3,

=
ñ –5 ) 3 (ñó ò )5 (ñó ò ò ó )3 . 1 b . Solution in parametric form:

# = −3

where

# = 38 

−5

−5

 (2   )1 J 3.

 (   )1 J 3 .

ñóò ò ó ò ó

2 1

 = L

{ {

{

J

−1 2 1 2 3

X

U + L 3, M

{

2 1

 = L

9 1 J 2 9 −1 J 2 9

X

1

2

3

M

U + L 3,

J

3 3 2 1 1 ,

= L

2 b . Solution in parametric form:

72.

where

9 J

3 3 2 1 1 ,

= L

# = 64 × 3−7  8 

where

where

J .

−16 3

# = −256 × 3−7  8 

J .

−16 3

=
ñ (ñ ó ò )–1 ) 3 (ñ ó ò ò ó )7 ) 5 . 1 b . Solution in parametric form:

ñ óò ò ó ò ó

 = L

−2 1

{ X

{

J

−1 1 2 1 2

M

U + L 3,

{

J

{

1 2 1 1 3,

= L

where # = −

5  1J 3 2 2 2J 5 O O . 2  2 ·P 3 eP

where # =

5  1 J 3 4 2 2J 5 O O . 2  2 ·P 3 \P

2 b . Solution in parametric form:

 = L 73.

−2 1

9 −3 J 2 9 1 J 2 U 1 2 X

M

+L

3,

= L

9 J 9

−1 2 1 1 3,

=
ñ –2 ) 3 (ñó ò )3 (ñó ò ò ó )8 ) 5 . 1 b . Solution in parametric form:

ñóò ò ó ò ó

 = L

11 1

X

{

J

{ {

J

11 4 2 −1 2 2 3 1

M

U + L 3,

= L

{ {

12 3 3 1 1 2,

where

# =

5 2   432

J O 2

−7 3

2

3 \P

J

3 5

.

Page 578

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

2 b . Solution in parametric form:

9 5 J 4 9 2 9 −1 J 2 U 2 3 1 M ñóò ò ó ò ó =
ñ –5 ) 2 (ñóò ò ó )1 ) 2 . 11 1

 = L

74.

+L

X

9

12 3 1 2,

= L

3,

1 b . Solution in parametric form: 3 1

 = L

{

X

J

−3 2 1

U + L 3, M

{

2 −1 1 1 ,

= L

2 b . Solution in parametric form: 3 1

 = L

75.

9 −3 J 2 U 1

X

M

3 1

 = L

{

X

{ {

J

3 2 2 2 3 1

2 −1 1 1 ,

= L

2 2 1 1 3,

U + L 3,

= L

2 −1 2 1 1 3,

U + L 3,

= L

4 −1 −1 1 1 2 ,

3,

U + L 3, M

9

= L

+L

=
ñ –1 ) 2 (ñ ó ò )–4 (ñ ó ò ò ó )5 ) 2 . 1 b . Solution in parametric form:

ñ óò ò ó ò ó

3 1

76.

9 −3 J 2 9 2 9

X

2 3

1

M

=
ñ –10 ) 3 (ñó ò )–7 ) 3 . 1 b . Solution in parametric form:

ñóò ò ó ò ó

5 1

 = L

{

X

{

J

J

−5 4 −3 2 1 2

M

9 1 J 4 9 −3 J 2 U +L 1 2 M ñóò ò ó ò ó =
ñ –5 ) 3 (ñ óò )–17 ) 3 (ñóò ò ó )3 . 5 1

77.

X

1 b . Solution in parametric form: 2 1

 = L

{ {

{

J

−3 2 1 2 4

X

U + L 3, M

2 1

78.

9 1 J 2 9 −3 J 2 9

X

1

4

2

U + L 3, M

{

= L

=
ñ (ñ ó ò )–13 ) 7 (ñ ó ò ò ó )7 ) 5 . 1 b . Solution in parametric form:

ñ óò ò ó ò ó

 = L

2 1

{

X

J

{

J

1 3 11 6 1 2

M

U + L 3,

9 −3 J 2 9 11 J 6 U + L 3, 1 2 M ñóò ò ó ò ó =
ñ –10 ) 7 (ñóò )3 (ñóò ò ó )8 ) 5 .

79.

2 1

1 b . Solution in parametric form:

 = L

19 1

{

X

{

J

{

J

where

# = 16

J

 J (−  2)−1 J 2.

−2 7 2

−5 20 3

where

# = 38 

−16 3 20 3

5 where # = −  4

U + L 3,

= L



J 

# =− 

M

J (2  )1 J 3.

# = −3

where

19 12 4 3 −1 2 1 2 4

45 2 2 3J 5 × 7−3  2  −11 J 7 O . 16 7 eP 2 b . Solution in parametric form:

 J (  2)−1 J 2.

where

where # =

.

−2 7 2

9 J 9

9 J 9

3 eP

 J (−2  )1 J 2.

# =

{

J

3 5

−2 7 2

# = −

where

J

2

 J (2  )1 J 2.

{

J

J O 4

−7 3

−2 7 2

# = −32

9

9 −1 2 4, 1 1

= L

X

5 2   54

where

3 2 −1 1 1 2 ,

{

2 b . Solution in parametric form:

 = L

# = 2

4 −1 1 2 ,

9 3 2 4, 1 1

= L

where

1 2 −1 1 1 2 ,

= L

2 b . Solution in parametric form:

 = L

# =−

{

= L

3,

579

where

9 9

{

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

where

2 b . Solution in parametric form:

 = L

(Ï

{ {

2 b . Solution in parametric form:

 = L

 Ë Ÿ Ï¢¡

=

J .

2 3

J O 

2 3

−2 14 3

3 4

−2 14 3

−2

5  4

−2

{

J

O

{

O

J

  J O

3  16



2 ·P

2 ·P

J

.

.

3  6J 7 2 2 2 J 5 O . 2 ·P 7 eP 3  6J 7 4 2 2J 5 O . 2 ·P 7 eP

J

28 7 3 7 3 1 1 2 ,

where # =

19 1

 = L where # = −

45 × 7−3  2  2

X

9 5 J 4 9 4 J 3 9 −1 J 2 U 1 2 4

J O 4

−11 7

2

7 \P

J

3 5

M

+L

3,

= L

9 J

28 7 3 1 2 ,

.

Page 579

580

THIRD-ORDER DIFFERENTIAL EQUATIONS =
ñ –5 ) 2 (ñó ò ò ó )13 ) 10 . 1 b . Solution in parametric form:

ñóò ò ó ò ó

80.

 = L

{

−11 1

X

{

J

J

−11 6 −1 3 1 2

U + L 3, M

{

−11 1

9 −3 J 2 9 −1 J 3 U 1 2

X

M

+L

3,

= L

4 −1 2 3 1 1 2 ,

9 9 J

=
ñ –1 ) 2 (ñó ò )–4 (ñó ò ò ó )17 ) 10 . 1 b . Solution in parametric form:

ñóò ò ó ò ó

81.

 = L

19 1

{

X

{

J

{

J

19 6 8 3 4 1 2

U + L 3, M

{ {

]

19 1

9 −3 J 2 9 8 J 3 9

X

1

2

4

M

U + L 3,

= L



9 9

  )3 J 10 .

 J

10 −1 5 2 (−2 2 3



 J (2  2  )7 J 10 .

−5 9 2

where # = −540 

18 −1 2 1 1 5,

  )3 J 10.

 J

20 −1 5 2 (2 2 3

where # = where # =

18 3 2 1 1 4,

= L

2 b . Solution in parametric form:

 = L

J

4 −1 3 2 3 1 1 2 ,

2 b . Solution in parametric form:

 = L

{

J

= L

 J (−2  2  )7 J 10 .

−5 9 2

where # = −270 

In the solutions of equations 82–84, the following notation is used:

0 0

1

0 82.

2 3

U  + L 2 U − , 1 ln U + L 2 , 1 sin( , ln U ) + L

=L =L =L

ñóò ò ó ò ó

2

2

cos( , ln U ),

Ä

3

2

, ln U ).

=U ,

where

, =

|1 + 8 # |,

ü

=

¡

1 if # > −1 8, 2 if # = −1 8, 3 if # < −1 8.



=
ñ (ñó ò )–3 (ñó ò ò ó )3 . Solution in parametric form: −1

Ä +

a = U0 + ,

U + L 3, M

where

, = ü

| # − 1|,

¡

=

=
ñ –1 ) 2 (ñ ó ò ò ó )3 ) 2 . Solution in parametric form:

ñ óò ò ó ò ó

 = T 4 X U 20

1

M

U + L 3,

=U

2

0

2 1,

, = £ 1 + 8#

where

1 if # < 1, 2 if # = 1, 3 if # > 1.

−2 .

In the solutions of equations 85–100, the following notation is used:

:

=

L 1 ø Á (U ) + L 2 n Á (U ) for the upper sign, L 1 ®Á (U ) + L 2 ; Á (U ) for the lower sign,

™

: : : = U
+ @ , ’ 2 = ’ 12 ACU 2 2 , ’ 3 = A 23 U * where ø Á (U ) and n Á (U ) are the Bessel functions, and ®Á (U ) and functions. ’

85.

1 ) cos(

ñóò ò ó ò ó

 =X U

]

Ä

= (1 + , ) L 1 U  + (1 − , ) L 2 U −  , = L 1 ln U + L 1 + L 2 , = ( L 1 − , L 2 ) sin( , ln U ) + ( L 2 + , L

3

 = XÈU J 0 −1 + J 2 M U + L 3,

84.

1

=
ñ (ñó ò ) . Solution in parametric form: –2

1 2

83.

Ä

1

1

2

:

3

− 2 ’ 1 ’ 2, Á ( U ) are the modified Bessel ;

=
ñ _ (ñ ó ò )3 ,  ≠ –2. Solution in parametric form:

ñ óò ò ó ò ó

 =L

1

XÈU

Á

3 −2 2

:

J

−1 2

M

U + L 3,

= 'U

2

Á ,

where

@ =

1 −: 1 , # =T  ý +2 8@ 2

−2

.

Page 580

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

86.

ñóò ò ó ò ó

=
ñ (ñó ò )` (ñó ò ò ó )3 , Solution in parametric form:

 = L 87.

X U 3Á

1

:

X

1

U +L 3, M

1

:

X U

1

XÈU

:

−2

−2

’ 1’

2

J OU : 2 − ’

2 −1 3 1

U

= L

U + L 3, M

ñ óò ò ó ò ó

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

@ =

@ =

where

1

X

:

U −L 3 ’ M

: U J 2,

2 4 3 1

=
ñ –1 ) 2 (ñó ò )–3 (ñó ò ò ó )3 . Solution in parametric form:

581

2

= +3

, # =A

1

@

2

 ;

 ;

+3 − −3

.

J :

−4 3

= 'U

1−> 4  −3  3 J , # =T 3 − 2> 3 − 2>

1

P

’

2 1,

where



O˜T

2

2  P

@ = 13 , # = 94  6 

@ = 23 , # = − 61  3 

where

−2

, where

2

<.

−3

.

J .

−3 2

 J

4 3 2 . 3

@ = 13 , # = A

=
ñ (ñ ó ò )–3 . Solution in parametric form:

:

X

1

U + L 3, M

ñ óò ò ó ò ó

= L

J ’ , 2

2 −2 3 1

U

=
ñ –2 (ñ ó ò )3 (ñ ó ò ò ó )3 . Solution in parametric form:

:

X

1

J

−1 2 2

’ 1’

M

U + L 3,

1

X U

−1

:

−2

M

U + L 3,

= 'U

J :

−2 3

where

: U J 2,

2 2 3 1

= L

=
ñ (ñó ò ò ó )3 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

−1

’ 1,

16 @ = 13 , # = − 81 

@ = 13 , # =

where

where

@ = 13 , # = 2 

−1

−6 3

 .

9 6 −3 . 32

 

(T 6  ) 1 J 2 .

=
ñ –1 ) 2 (ñó ò )3 (ñó ò ò ó )3 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

 = L where @ = 13 , # = T 95.

where

: U Á 2,

U + L 3, M

ñóò ò ó ò ó

 =L 94.

: U Á ,

=
ñ –1 ) 2 (ñó ò ò ó )3 . Solution in parametric form:

ñóò ò ó ò ó

 = L 93.

1

2 2 1

= L

= L

(Ï

≠ 3  2.

=
ï –3 ñ (ñó ò ò ó )3 . Solution in parametric form:

 = L 92.

= L

S

 Ë Ÿ Ï¢¡

≠ –3.

U + L 3, M

U + L 3, M

ñóò ò ó ò ó

 =L 91.

1

:

−1

 = L 90.

’

=
ñ –1 ) 2 (ñ ó ò ò ó ) ª , Solution in parametric form:

 = L 89.

−1

XÈU

ñ óò ò ó ò ó

 = L 88.

1



=

1

X

: 5J 2 ’

J ’

−1 2 1

2

J T 6  )1 J 2 .

M

U + L 3,

= L

1

U

J ’

−4 3

2 2,

27 3 −5 2 ( 32

 

ñóò ò ó ò ó

=
ñ –2 (ñó ò )–3 . Solution in parametric form:

 = L

1

X U

:

’

J

−3 2 2

M

U + L 3,

= L

1

U 2J 3 ’

−1 2 ,

where

@ = 13 , # = − 16 81 

−6 6

 .

Page 581

582

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

=
ñ –2 (ñó ò )–3 (ñó ò ò ó )3 . Solution in parametric form:

96.

 =L

:

X

1

J ’

−3 2 2

’

3

M

=
ñ (ñó ò )–3 (ñó ò ò ó )3 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

97.

 = L

1

XÈU

−2

:

−2

’

J

3 2 2

U + L 3, M

= L

3

J :

2 −4 3 1

U

:

−1

2

−1 2 ,

’

’ 3,

.

 (A 6  )1 J 2.

−3 2

@ = 13 , # = −8 

where

−3

@ = 13 , # = 18 

where

=
ñ –2 (ñó ò )3 (ñó ò ò ó )3 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

98.

 = L

1

XZU

: 5J 2 ’

J

−1 2 3

M

U + L 3,

 =L 100.

1

XZU

−1

:

−2

U + L 3, M

= 'U

U

J ’ 2,

J :

−2

2 −2 3 1

= L

=
ñ –2 (ñó ò ò ó )3 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

99.

−4 3

where

’ 2,

@ = 13 , # = A

where @ = 13 , # = T

:

−3

XZU

1

 J

−2

256 −3 7 2 (2 3



’ 23 J 2 ’

3

 )−1 J 2.

U + L 3, M

= L

1

U

J :

−8 3

−2

’

 

A 6  )1 J 2 .

4 2 3 (2

 )−1 J 2 .

27 3 −1 ( 8

@ = 13 , # = A

where

=
ñ –1 ) 2 (ñó ò )–3 (ñó ò ò ó )3 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

 = L ]

= 'U 4 J

U + L 3,



2 3,

In the solutions of equations 101–138, the following notation is used:

U =X

M A (4 < ü

< 3

− 1)

−L

 =

2,

ü

A (4 <

3

− 1).

The function < = < (U ) is defined implicitly by the above elliptic integral of the first kind. For the upper sign, the function < coincides with the classical elliptic Weierstrass function < = < (U + L 2 , 0, 1). In the solution given below, we can take < to be the parameter instead of U and use the explicit dependence U = U ( < ). 101.

ñóò ò ó ò ó

=
(ñó ò )5 . Solution in parametric form: 2 1

 = L 102.

J

−1 2

=
ñ 2 ñó ò (ñó ò ò ó )3 . Solution in parametric form:

M

ñóò ò ó ò ó

 = L 103.

X«<

5 1

XÈU

J 

−1 2

=
ññó ò (ñó ò ò ó )5 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

 = L

7 1

XZU

J <

−1 2 2

M

M

U + L 3,

= L

1

U ,

where

# = A 3 2 

U + L 3,

= L

4 1

<

where

# = T 24 4 

U + L 3,

= L

6 1

 ,

,

where

# = − 91  3 

−3

−4

.

−5

.

(A 3  )−1 J 2 .

Page 582

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

104.

5 1

−1 1

J <

J

−3 2 −1 2

XZU

=
ñ 2 (ñó ò )–9 (ñó ò ò ó )3 . Solution in parametric form: 5 1

= L

2 1

M

U ,

U + L 3,

where

2 −1 , 1

= L

J (U − < ) U + L 3 , M

−3 2

XZU

2 1

X U

<

−1 2

= L

U

6 −1 1

U

<

,

M

U + L 3,

= L

1(

U − < ),

5 1

X U 3 (U − < )−1 J

2

=
ñ –2 ) 3 (ñó ò ò ó )6 ) 5 . Solution in parametric form:

ñóò ò ó ò ó

7 1

XÈU

<

−4 2

# = A 3 2  .

where

# = T 24

# =T

where

# =A

1 2

−6 5

 .

 (A 2  )1 J 2.

M

M

U + L 3,

U + L 3,

4 4 , 1

= L

= L

U

<

9 −3 3 , 1

U

where

# = A 5

3 16



J (A 3  )−1 J 2 .

−3 4

J O 

−1 −2 3



=
ñ –1 ) 2 (ñó ò )–1 ) 3 (ñó ò ò ó )9 ) 5 . Solution in parametric form:

ñóò ò ó ò ó

−1 1

= L

8 1(

J

2

1 5

18 "P

.

U − < )2 ,

=
ñ –15 ) 7 (ñó ò )5 . Solution in parametric form:

ñóò ò ó ò ó

 = L

13 1

X U

J (U 2 p < T 1)−1 J

11 2

=
ñ 2 (ñó ò )–23 ) 7 (ñó ò ò ó )3 . Solution in parametric form:

ñóò ò ó ò ó

 = L 113.

where

where

X U 5 J 2 < 1 J 2 (U − < ) U + L 3 , M 2 1J 3 4J 5  12  O . where # = T 5  −1  1 J 2 O åP 18 P

112.

(A 3  )−1 J 2 .

=
ñ –5 ) 4 (ñó ò )3 (ñó ò ò ó )1 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

 = L

111.

−2

# = A 6

=
ñ (ñó ò )–3 ) 2 (ñó ò ò ó )5 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

 = L 110.

U + L 3,

ñóò ò ó ò ó

 = L 109.

M

=
ñ –5 (ñó ò )5 . Solution in parametric form:

 = L 108.

J

−1 2

X 

ñóò ò ó ò ó

 = L 107.

583

=
(ñó ò )3 (ñó ò ò ó )1 ) 2 . Solution in parametric form:

 = L 106.

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

ñóò ò ó ò ó

 = L 105.

 Ë Ÿ Ï¢¡

=

−5 1

X U

2

M

U + L 3,

J (U 3  +3U 2 s < T 1) U + L 3 , M

−7 2

= L

= L

14 7 , 1

U

2 1

where

U (U 2 s < T 1),

# = A 3 × 7−4  2 

where # = T

J .

−13 7

J 

J

24 −2 7 −5 7 . 49



=
ñ (ñó ò )–11 ) 4 (ñó ò ò ó )5 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

 = L where # = A

3 2(

1

XÈU

−3

2

(U 2
A 6   )3 J 4(T 6  )−1 J 2 .

M

U + L 3,

= L

U  + 3U 2 p < T 1),

3 −6 3 ( 1

U

Page 583

584 114.

THIRD-ORDER DIFFERENTIAL EQUATIONS =
ñ –15 ) 8 (ñó ò )3 (ñó ò ò ó )1 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

5 1

 = L 115.

(U 3  +3U 2 <…T 1)

3 1

2

XÈU 8(U 2 … < T 1) U + L 3, M

=
ñ –1 ) 2 (ñó ò )–1 ) 3 (ñó ò ò ó )23 ) Solution in parametric form: 9 1

8 −8 , 1

= L

U

X U

15

1

3

U 3( U 2 … < T 1) ,



J (T 6  )−1 J 2.

−1 8

# = T 15

where

J  J (18  )−7 J 15.

−1 15 2 3

.

J (U 3  + 3U 2 I < T 1)(U 2 I < T 1)1 J

−29 2

2

M

U + L 3,

= L

2

U  + 3U 2 p < T 1) ,

8 −12 3 ( 1

U

12  1 J 3  2 8 J 15  J O O . åP 18 "P

−1 1 2

4 1

X U

−5

J

−1 2

(U 2
M

=
ñ 2 (ñó ò )–33 ) 7 (ñó ò ò ó )3 . Solution in parametric form:

U + L 3,

= L

7 −7 , 1

U

where

# = A 3 × 7−4  2 

J .

−8 7

ñóò ò ó ò ó

5 1

J (U 3  −4U 2 ± < A 6) U + L 3 , M

−7 2

XZU

= L

U
12 −6 2 ( 1

U

where # = T

J  J

24 −12 7 5 7 . 49



=
ñ (ñó ò )–27 ) 13 (ñó ò ò ó )5 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

J (U 2 p < T 1)2 U + L 3 , M 24 (6   )1 J 13(A 39  )−1 J 2. where # = − 13 −11 1

 = L

−13 2

X U

= L

2 1

U (U 3  − 4U 2 I < A 6),

=
ñ –20 ) 13 (ñ ó ò )3 (ñ ó ò ò ó )1 ) 2 . Solution in parametric form:

ñ óò ò ó ò ó

25 1

 = L where # = A 121.

3 64

# =A

=
ñ –20 ) 7 (ñ ó ò )5 . Solution in parametric form:

 = L

120.

where

ñ óò ò ó ò ó

 = L

119.

= L

ñóò ò ó ò ó

where # = A 15

118.

U + L 3, M

=
ñ –2 ) 3 (ñó ò ò ó )22 ) 15 . Solution in parametric form:

 = L

117.

J

−1 2

ñóò ò ó ò ó

 = L 116.

−6

XÈU

6 169



J (U 3  − 4U 2 I < A 6)−1 J

23 2

X U

J (A 39  )−1 J 2.

2

M

U + L 3,

= L

26 13 , 1

U

−6 13

=
ñ –2 ) 3 (ñ ó ò ò ó )27 ) 20 . Solution in parametric form:

ñ óò ò ó ò ó

 = L where # = 20

19 1

  J O˜A

−1 2 3

−20

XZU 2

18 "P

2

(U 2
J

7 20

M

U + L 3,

= L

U

3

U
18 −18 2 ( 1

.

Page 584

3.2. EQUATIONS OF THE FORM

122.

=
ñ –1 ) 2 (ñó ò )–1 ) 3 (ñó ò ò ó )33 ) Solution in parametric form:

ñóò ò ó ò ó

11 1

 = L

10

XZU

12  1 J 3  2  J O ˜O A åP 18 "  P

5 1

J

1 2

U + L 3, M

J

13 20

= L

2

U  − 4U 2 C < A 6) ,

2 2 3 ( 1

U

.

U + L 3, M

<

7 −2 ( 1

= L

pA 2U'< 2 ),

−7 5

# = −192 

where

 .

1

X /<

−2

( uA 2U'< 2 )

J

−1 2

M

13 1

X < 3 ( uA 2U'< 2 )

J

−1 2

M

=
(ñó ò )3 (ñó ò ò ó )7 ) 5 . Solution in parametric form:

U + L 3,

= L

U + L 3,

= L

6 1

<

8 −2 , 1

 ,

where

where

5  6

# =T

J

3 4 −1 2 . 4

# =A

 

−2



O

J

2

3 5

6 "P

.

ñóò ò ó ò ó

17 1

X<

J

−3 −1 2



U + L 3, M

= L

<

14 −2 ( 1

uA 2U'< 2 ),

where # = T

=
ñ –1 ) 2 (ñ ó ò )–4 . Solution in parametric form:

ñ óò ò ó ò ó

11 1

< U M ( pA 2U'<

X

+L

J

2 )3 2

3,

= L

<

14 1

2

=
ñ –5 ) 2 (ñó ò ò ó )3 . Solution in parametric form:

pA 2U'<

2

,

where

5 2   8

−3

# = 192

2



O

6 "P

J

2 5

.

J .

−7 11 2



ñóò ò ó ò ó

−1 1

J

−3 2

X ( pA 2U'< 2 )

(U +2 < ) U + L

M

=
ñ (ñ ó ò )3 (ñ ó ò ò ó )8 ) 5 . Solution in parametric form:

ñ óò ò ó ò ó

 = L 130.

−2

=
ññó ò (ñó ò ò ó )8 ) 5 . Solution in parametric form:

 = L 129.

X«<

ñóò ò ó ò ó

 = L 128.

585

=
ñ –5 ) 2 ñ ó ò (ñ ó ò ò ó )3 . Solution in parametric form:

 = L 127.

(U 3  − 4U 2
ñ óò ò ó ò ó

 = L 126.

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

=
(ñ ó ò )–4 . Solution in parametric form:

 = L 125.

(Ï

ñ óò ò ó ò ó

 = L 124.

 Ë Ÿ Ï¢¡

=

.

−1 1 2

where # = T 20  123.

20

Ï Ò˜Í Í Ò˜Í Ò

23 1

X <

J ( …A 2U'< 2)5 J

−1 2

4

M

ñóò ò ó ò ó =
ñ (ñóò ) (ñóò ò ó ) ) 3

U + L 3,

= L

3,

= L

6 1(

3 3 1J 2 where # = − 16   .

−1

pA 2U'< 2 ) ,

16 1 (

U +2 < ),

2

U + L 3,

where # =

10 2   27

−4

2 2 3J 5 . 3 CP

O

7 5

. Solution in parametric form:

 = L

11 1

where # = −10  2 

J

−3 2

X ( pA 2U'< 2 ) −4

O

2 2 2J 5 . 3 eP

(U + 2 < )−1 J

M

= L

7 1

< (u  A 2' U <

J

2 −1 2

)

,

Page 585

586 131.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

) (ñ óò


ñ

=

)–7 ) 3 .

–1 2

Solution in parametric form: 23 1

 = L

ñóò ò ó ò ó

(U + 2 < ) U + L

32 1 (

= L

3,

M

U + 2 < )2 ,

 J (6   )1 J 3.

) (ñ óò ò ó


ñ

=

J

1 4

−5 7 2

where # = −648  132.

X ( pA 2U'< 2 )

–5 3

)3 .

Solution in parametric form:

 = L 133.

ñ óò ò ó ò ó

X < ( pA 2U'< 2 )

1

=

) (ñ ó ò


ñ

J

1 2

U + L 3, M

9 1(

= L

pA 2U'< 2 )3 J 2 ,

where

# =A

J

1 3 −1 3 . 324

 

)–7 ) 3 .

–5 6

Solution in parametric form: 25 1

 = L

ñóò ò ó ò ó

=

32 1 (

= L

3,

M

U + 2 < )−2 ,

J (6   )1 J 3.



) (ñ óò


ñ

(U + 2 < )−2 U + L

−5 23 6

where # = −648  134.

J

1 4

X ( pA 2U'< 2 )

)–2 ) 3 (ñó ò ò ó )3 .

–5 3

Solution in parametric form:

 = L

−1 1

X (U 2 2 < T 1)( 3A 2U'< 2 )

1 3   where # = − 324

135.

ñóò ò ó ò ó

=


ñ (ñ óò

J

1 2

(U + 2 < )−2 U + L

M

= L

3,

7 1(

J

3 2

pA 2U'< 2 )

(U + 2 < )−2 ,

J (6   )2 J 3.

−1 3

)11 (ñó ò ò ó )7 ) 5 .

Solution in parametric form: 31 1

 = L

X ( uA 2U'< 2 ) 10 −12

where # = −20  136.

ñóò ò ó ò ó

=


ñ

5



J

−3 2

(U + 2 < )13 J

6

M

U + L 3,

= L

J

−1 2

U
27 2 1 (

,

(2  2  )2 J 5 .

(ñó ò )3 (ñó ò ò ó )8 ) 5 .

Solution in parametric form: 43 1

 = L

where # = 20 2  137.

ñóò ò ó ò ó

=


ñ

J

−1 2

< T 1) X (U 2 p −8

) (ñ óò

–1 2

( uA 2U'< 2 )

J

5 4

(U + 2 < )−4 J

3

M

U + L 3,

= L

16 1 (

U + 2 < )−1 J 3 ,

(2  2  )3 J 5 . )–4 (ñó ò ò ó )4 ) 5 .

Solution in parametric form:

 = L

47 1

X (U 2 l < T 1)( 3A 2U'< 2 )

where # = −320 

J O 

−7 11 2



2

4  P

J

−3 2

J

4 5

(U + 2 < )4 J

3

M

U + L 3,

= L

2

−1

U
54 2 1 (

.

Page 586

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

138.

 Ë Ÿ Ï¢¡

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

587

=
ñ –5 ) 2 (ñó ò ò ó )11 ) 5 . Solution in parametric form:

ñóò ò ó ò ó

X ( uA 2U'< 2 )

−13 1

 = L

where # =

]

=

J

−3 2

(U + 2 < )1 J

3

U + L 3, M

= L

14 1 (

pA 2U'< 2 ) (U + 2 < )4 J 3 , −1

1J 5 5 3J 2   O . 4 4 "P 2

In the solutions of equations 139 and 140, the following notation is used:

U 

’ =X

139.

H =

1−

: 1X U

:



M

=
ñ –1 (ñó ò )` ñó ò ò ó , Solution in parametric form:

;

2− −1

H

−1

 É



M

if , ≠ 0, , ≠ −1; if , = 0; if , = −1.

≠ 0.

exp(− 21 ’ ) U + L

X U ;

2

U

ñóò ò ó ò ó

1

1 1 U  +1 + U  + L , +1 , U + ln |U | + L 2 1 ln |U | − + L 2





=
ñ _ (ñó ò )–1 (ñó ò ò ó )2 , Solution in parametric form:

 = L ]

M , H (U )



ñóò ò ó ò ó

 =L 140.

U

−1






= 'U 1 J : ,

3,

where

, = 1 ý , # = −2  −: .

≠ 1.

U + L 3,

=

É ,

where

, =

2

= −1

, # = ;

−1 1−



; .

In the solutions of equations 141–170, the following notation is used:

² =

® where ® = X 141.

U ²

M

U

+L

A (4U 2 3 = 4 UÈ® 1 T 2

® 1 = 2UÈ®lTç² , ® 2 = U −1 (²$® 1 − 1), ® 4 = ® 2 ® 3 − 8® 12 , ® 5 = 2²$® − U 2 ,

− 1),

® 22 ,

is the incomplete elliptic integral of the second kind in the form of Weierstrass.

=
(ñó ò )–7 . Solution in parametric form: 4 1

J ²

−3 2

X U

−1

U + L 3, M

= L

®

5 −1 1, 1

U

where

# = T 3

−10 8

 .

ñóò ò ó ò ó

=
ñ –4 ñó ò (ñó ò ò ó )3 . Solution in parametric form:

 = L 143.

3

ñóò ò ó ò ó

 = L 142.

ü

−1 1

XZU

J

J

®

−3 2 −1 2 1

=
ññó ò (ñó ò ò ó )7 ) 4 . Solution in parametric form:

M

U + L 3,

= L

4 −1 , 1

U

where

# = A 24 4  .

ñóò ò ó ò ó

 = L

11 1

X U 5J

®

J ²

2 −1 2 1

−1

M

U + L 3,

= L

6 1

² ,

where

# =T

2  3

−2

O˜T



2

3  P

J

3 4

.

Page 587

588 144.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

)3 (ñó ò ò ó )5 ) 4 .


(ñ óò

=

Solution in parametric form: 13 1

 = L 145.

ñóò ò ó ò ó


ñ (ñ óò

=

−2

XZU

J

−3 2

²

U + L 3, M

®

10 −1 1, 1

= L

U

# = −4 2 

where

−3



OT

J

2

1 4

3 "P

.

)–7 .

Solution in parametric form:

146.

ñ óò ò ó ò ó


ñ

=

–4

óò

(ñ

J ²

X®

7 1

 = L

−3 2 1

óò ò ó

)3 ( ñ

−1

U + L 3, M

10 1

= L

UÈ®

−1 1 ,

−10 7

# = T 3

where

 .

)3 .

Solution in parametric form:

 = L 147.

ñóò ò ó ò ó

XÈU

1

J


ñ (ñ óò ò ó )7 )

=

J

®

®

−1 2 −3 2 2 1 4

²

−1

M

U + L 3,

®

6 −1 1 1 ,

= L

# = A 24 6 

where

−1

.

.

Solution in parametric form:

148.

ñ óò ò ó ò ó

) (ñ ó ò


ñ

=

X® 12 ²

7 1

 = L

−1

M

= L

®

2 1 2,

where

# = −4

−1 −1





OA

J

2

3 4

6 "P

.

ó ò ò ó )5 ) 4 .

)3 ( ñ

–1 2

U + L 3,

Solution in parametric form: 11 1

 = L 149.

ñóò ò ó ò ó

=

X UÈ®

J ²

®

−3 −1 2 1 2

) (ñ óò


ñ

−1

U + L 3, M

®

10 2 −2 1 1 ,

= L

U

where

# =

J O˜A 

1 2   2

−5 2

J

2

5 4

6  P

.

)–5 ) 3 (ñó ò ò ó )3 .

–1 2

Solution in parametric form: −1 1

 = L 150.

ñ óò ò ó ò ó

=


ñ

J

XÈUÈ®

®

−1 2 2 1

) (ñ ó ò

−1

²

)–4 ) 3 (ñ

–1 2

U + L 3, M

óò ò ó

®

8 2 1 2,

= L

where # = T

J

1 2 −1 2 (12 27

 

  )2 J 3 .

)3 .

Solution in parametric form:

 = L 151.

ñóò ò ó ò ó

1

=

J

XÏ®


ñ

® ®

−5 2 2 3 1

) (ñ óò

−1

²

U + L 3, M

® ®

10 −3 2 1 1 2,

= L

where # = T

J A 3   )1 J 3.

16 2 −1 2 ( 27

 

)3 (ñó ò ò ó )8 ) 7 .

–1 2

Solution in parametric form:

 = L 152.

ñ óò ò ó ò ó

37 1

=

X® 17 J

®

J

®

4 −1 2 3 2


ñ (ñ óò ò ó

²

−1

M

U + L 3,

®

32 2 1 3,

= L

where # = T 7 × 2−10  2 

J O 

−5 2

J

2

1 7

6  P

.

)13 ) 7 .

Solution in parametric form:

 = L

13 1

X

® 1−5 J 2 ²

−1

M

U + L 3,

= L

®

J

®

5 −3 2 2, 1 1

where

# =

7  2

−1 −1



O



2

6 "P

J

6 7

.

Page 588

153.

−1 1

11 1

1

J ²

® ®

® 13 J 4 ® 3−3 J 2 ² X

−1

−1

® 1−1 J 2 ® 3−3 J 2 ® 4 ² X

−1

=
ñ –7 ) 6 (ñó ò ò ó )2 ) 3 . Solution in parametric form: 9 1

® 1−5 J 2 ® 35 ² X

−1

M

U + L 3,

U + L 3, M

ñóò ò ó ò ó

# = A 3 × 225 

−16 13



.

M

U + L 3, M

®

16 −1 1 3 ,

= L

U + L 3,

J

®

® ®

10 −3 6 1 1 3,

# = T 12 6  2 .

where

# = A 3 × 225 

where

®

where

where

# = A 54

5 1 2 −1 1 1 3 ,

= L

= L

J

®

3 1 2 1 1 ,

= L

−16 11



10 −2

# = T 192

J O 2

−3 13 6



.



.

2

2 3

J

9 eP

=
ñ –1 ) 2 (ñó ò )–4 ) 3 (ñó ò ò ó )7 ) 3 . Solution in parametric form:

.

ñóò ò ó ò ó

3 1

X®

J

® ®

−5 2 −1 3 4 1

−1

²

U + L 3, M

® ®

2 −3 2 1 1 4,

= L

# = 8  1 J 2 (2  3)1 J 3.

where

=
ñ –3 ) 4 (ñó ò )3 (ñó ò ò ó )8 ) 7 . Solution in parametric form:

ñóò ò ó ò ó

67 1

XÏ® 17 J

J ²

® ®

4 −5 −1 2 3 4

−1

M

U +L 3,

= L

®

64 −4 1 3 ,

where

J O 

# = T 7⋅2−13  2 

−9 4

2

J

1 7

12 "P

=
ñ (ñó ò )–1 ) 2 (ñó ò ò ó )13 ) 7 . Solution in parametric form:

.

ñóò ò ó ò ó

19 1

X®

J

®

−5 2 4 3 1

²

−1

M

ñóò ò ó ò ó

U + L 3,

= L

J

®

®

3 −3 2 4, 1 1

where # =

7  2

−1 −1



=
(ñó ò )2 . Solution in parametric form:

 = L 162.

J

−1 2 −1 2 2 3 1

=
ñ –7 (ñó ò )7 (ñó ò ò ó )3 . Solution in parametric form:

 = L 161.

X®

ñóò ò ó ò ó

 = L 160.

U + L 3, M

=
ñ 3 (ñó ò )–13 . Solution in parametric form:

 = L 159.

−1

ñóò ò ó ò ó

 = L 158.

where

16 1 3,

589

=
ñ –7 (ñó ò )3 (ñó ò ò ó )3 . Solution in parametric form:

 = L 157.

XÏ® 13 J 4 ²

13 1

ñóò ò ó ò ó

 = L 156.

®

= L

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

=
ñ (ñó ò )–13 . Solution in parametric form:

 = L 155.

(Ï

=

ñóò ò ó ò ó

 = L 154.

 Ë Ÿ Ï¢¡

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

−1 1

XZ²

−1

M

U + L 3,

= L

1

XÈU ²

−1

= L

8 2 , 1

=
ñ 1 ) 2 ñó ò (ñó ò ò ó )3 . Solution in parametric form:

ñóò ò ó ò ó

 = L

7 1

X UÈ®

J

−1 2

M

U + L 3,

U

M

U + L 2,

where

where

OA

3

J

1 2

 P

# = A 6

# = T 24 4 



O

2

12 "P

−1 −1



J

6 7

.

.

J .

−7 2

Page 589

590 163.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

=
ññó ò (ñó ò ò ó )4 . Solution in parametric form: 5 1

 = L 164.

−1 1

5 1

J ²

X®

−3 2

−1

7 1

XZUÈ®

J

® 5²

−3 2

−1

=
ñ (ñó ò )–9 ) 5 (ñó ò ò ó )4 . Solution in parametric form:

ñóò ò ó ò ó

J ²

® ñóò ò ó ò ó = 
ñ –7 ) 5 (ñóò 11 1

2 −1 4

XÈU

−1

2 1

= L

² ,

−1

X U ²

M

U + L 2, M

®

2 −1 , 1

U + L 3,

= L

6 2 −1 , 1

®

16 1 5,

= L

where

.

−2 −1



.

 J .

−1 5 2

# = T 6

# = T 48

where

−5

# = 9

where

where

®

U

1 6 # = − 162  

where

= L

U + L 3, M

2 # = − 1125  4

−3

 J .

−3 7 2

(12   )4 J 5.

)3 (ñó ò ò ó )–1 . Solution in parametric form:

XZUÈ® 3 J

1

J ²

®

2 −1 2 5

−1

=
ñ –1 ) 3 (ñó ò ò ó )9 ) 7 . Solution in parametric form:

ñóò ò ó ò ó

1

®

J ²

2 −5 2

XÈU

−1

U + L 3, M

U + L 3, M

= L

1

®

J ,

5 2

J

®

9 3 −3 2 , 1

= L

U

where

where

# =

=
ñ –1 ) 2 (ñó ò )1 ) 3 (ñó ò ò ó )12 ) 7 . Solution in parametric form:

ñóò ò ó ò ó

7 where # = −  4

J

  J O

−1 1 2

XÈU 1 J

23 1

 = L

]

6 1

= L

U + L 3, M

=
ñ 1 ) 2 (ñó ò )–6 (ñó ò ò ó )3 . Solution in parametric form:

 = L 170.

U + L 3, M

ñóò ò ó ò ó

 = L 169.

U + L 3, M

=
ñ –7 ) 2 (ñó ò )2 . Solution in parametric form:

 = L

168.

J

−3 2

X U ²

ñóò ò ó ò ó

 = L 167.

−1

=
(ñó ò )3 (ñó ò ò ó )–1 . Solution in parametric form:

 = L 166.

J ²

®

2 −1 2

ñóò ò ó ò ó

 = L 165.

X U

1 3

3  P

O



®

J

2 7 4 2

J

® 5²

5 7

18 "P

−1

M

U + L 3,

= L

 J

# =

36 −2 2 5 . 5

7  2

  J O



−1 1 3

J

2

2 7

18 "P

.

®

32 2 1 5,

.

In the solutions of equations 171 and 172, the following notation is used:
1 2$  L 2+ U 2+ U  +1 if , ≠ −1, U 4 , + 1
  = exp ” X £ M , H = 1 Hѕ L 2 + U 2 + 2 $ ln |U | if , = −1. 4

171.

`

ñóò ò ó ò ó

+5

=
ñ – 4 (ñó ò )` . Solution in parametric form:

 = L

3 1

X U

J  J H

J

−1 2 3 4 −1 2

M

U +L 3,

= L

4 1

 ,

where

1 , = 12 (= −1), # = $C 2(  2

−1)



3 2

 .

(1− )

Page 590

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

172.

J + 2) U + L 3 , M

−1 2

`

=L

=
ñ 2 (ñó ò )` (ñó ò ò ó )3 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

U 1 J 2 ,

1

3 1

˜

’ = exp ”X

U £

X

˜

˜

−£

(U"’

)3 4

M2

U

˜ ˜

J £

,

+4•

` +4_

+5 +3

=
ñ _ (ñó ò )` (ñó ò ò ó ) ` +2_ Solution in parametric form:

ñóò ò ó ò ó

:  = L 1; + 2  : ; O

where # =  ;  transcendental equation −1 − −

+2

2

; P

:



( H + , − 1)( H + , ) 1−  = OL

ñ óò ò ó ò ó

=
ñ –1 (ñ ó ò )–1 (ñ ó ò ò ó ) ª , Solution in parametric form:

 = L 1< −3 X U 

S

−1

> −1 1−, where , = , # =  > −2 2 the transcendental equation

−1

;

;

S

≠ 1, −1

−4 3

2

 O−

M

177.

ñóò ò ó ò ó

X  *H

J ’

−1 2

J

−1 2

M

U + L 3,

=A



 P

1 2

U J ’ 1J 2,

U ;

+1 2

if = ≠ −1, if = = −1,

:

H

≠ –1.

−1

M

= L 1; +1 ’ ,

U + L 3, M

U

U H·P +1 2

P

; H = H (U ) is the solution of the 1 −1

U 

,

, =−

2(ý + 1) . = +1

≠ 2.

U + L 3, 2

=
ñ –1 (ñó ò )–1 (ñó ò ò ó )2 . Solution in parametric form:

$

+

2

2$ U ; = + 2ý + 3

+

2

  H

J OL

−1 2

$ , ’ = exp OX

ñóò ò ó ò ó

1

 $ .

.

ln | ,H − U | −

 = A±L

1−

2 1 2 1

= L

= +1 P U −1 J 2 ( L 2 + 12 $ ln |U |)

+4 +5 − 2( +1)

2− 2

’

U

U + L 3,

≠ –1,

J ’

:



=

−3 2

2( +1) +2 +3

+4 M

2



,

X U

+4

2

˜

# = 23 J 2   − ;2 − 1 $ (−2   );

176.

, = −= − 2, # = −23−  

where

–1

where

175.

591

=
ñ –` –2 (ñó ò )` (ñó ò ò ó )3 . Solution in parametric form:

 = L

174.

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

ñóò ò ó ò ó

 =  X (U H 173.

 Ë Ÿ Ï¢¡

=

= L

< $ , ’ = exp OX U

,H − U

=

1  U +L

,

<

2 −4 1

M

, ( ,H − U )−1 ’ 1 'J  ,

U ; H = H (U ) is the solution of H¥P 2.

 * , where H = T #*U +  * + L 2, ’ = exp OA #X M

U . H P ¥

H = A #*U +  * + L 2 , ’ = exp O'T X M

U . H¥P

=
ñ (ñó ò ) ñó ò ò ó . Solution in parametric form:

 =L

–1

1

XZ * J 2 ’ M

–1

U + L 3,

= A/L

1

H’ ,

where

Page 591

592

THIRD-ORDER DIFFERENTIAL EQUATIONS

3.2.5. Some Transformations Let us consider some transformations of the equation




 = #* 9 : ( 
); ( 

 ) < . 1 b . In the special case = = > = 0, the transformation  = 1 a , = ¨u a 2 reduces the equation 


 = #* 9 : to an equation of similar form (with other parameters):

¨ g

g
g = − #*a − 9

:

−2 −4

¨ : .

2 b . In the special case ƒ = > = 0, the transformation  = − X equation




 = #

M

U

[ H (U )]3 J

2

,

=

1

H (U )

reduces the

: ( 
);

to an equation of similar form (with other parameters):

H




*+*+*

= #2H

−

: ;

2 + +5 2 (

¨
); .

*

3 b . In the special case ý = 0, the substitution ‡ ( ) = "
 brings the equation 


 = #* 9 ( 
); ( 

 ) < to the generalized Emden–Fowler equation:

‡

 = #* 9 ‡ ; (‡
 )< , which is discussed in Sections 2.3 and 2.5.

4 b . In the special case ƒ = 0, the substitution  ( ) = ("
 )2 reduces the equation




 = #

: ( 
); ( 

 ) <

to the generalized Emden–Fowler equation:



= # × 21− < :  ; 

−1 2 (


)< , 

which is discussed in Sections 2.3 and 2.5.

3.3. Equations of the Form í ïwî²î²ïwî ï = ð ( í )ë ( í­ï î ) ’ ( í­ïwî²î ï ) 3.3.1. Equations Containing Power Functions 1. 2.

ñ óò ò ó ò ó

= ( ôñ

ñóò ò ó ò ó

= (
ñ

÷

ñóò ò ó ò ó

= (
ñ

÷

2

+

õÿñ

+ ö )–5 ) 4 .

This is a special case of equation 3.5.2.29 with  (¨ ) = 1. +

(­ñ ú )ñ óò

.

This is a special case of equation 3.5.2.1 with  ( ) = # 3.

+

(­ñ ú

) V ô (ñó ò )3 +

õÿñóò

W.

This is a special case of equation 3.5.2.3 with  ( ) =  ( #

 + $ + .  + $ + ), ( ) =  (#

 + $ + ).

Page 592

3.3. EQUATIONS OF THE FORM

4.

5.

Ï Ò˜Í Í Ò˜Í Ò

= â (Ï )ã (Ï

ÒÍ

›

) (Ï

Ò˜Í Í Ò

)

593

(þ + 1) (ñó ò )3 +
(ñó ò )2ú +1 ® , þ ≠ –3, þ ≠ –1. (þ + 3)2 The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.4: ¨

= −2 − 2(¡ + 1) ¨ + 2 # ¨ + S .  Q ( + 3)2

ñóò ò ó ò ó

=

ñ

–2

ñ óò ò ó ò ó

=

ñ

–2 15 ( 8

¬–

¡

V

ñ óò

)3 +


(ñ ó ò

)–13 W .

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.35: ¨2

= −2 ( 15 ¨ + 2 # ¨ −7 ). 4



6.

= ñ –2 V 3(ñ ó ò )3 +
(ñ ó ò )–7 W . The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.31: ¨2

= −2 (6¨ + 2 # ¨ −4 ).

ñ óò ò ó ò ó 

7.

= ñ –2 V 6(ñó ò )3 +
(ñó ò )–4 W . The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.64: ¨2

= −2 (12¨ + 2 # ¨ −5 J 2 ).

ñóò ò ó ò ó 

8.

= ñ –2 V (ñó ò )3 +
(ñó ò )–3 W . The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.6: ¨2

= −2 (2¨ + 2 # ¨ −2 ).

ñóò ò ó ò ó 

9.

ñ óò ò ó ò ó

13.

14.

15.

ñóò

=

ñ

–2

=

ñ

–2 3 ( 8

)3 +


(ñ ó ò

)–7 ) 3 W .

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.26: ¨

= −2 (− 3 ¨ + 2 # ¨ −5 J 3 ). 16

ñóò ò ó ò ó

9 ( 200

)3 +


(ñ óò

)–7 ) 3 W .

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.10: ¨2

= −2 (− 9 ¨ + 2 # ¨ −5 J 3 ). 100

ñóò ò ó ò ó

V

ñóò

)3 +


(ñ óò

)–7 ) 3 W .

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.12: ¨2

= −2 ( 3 ¨ + 2 # ¨ −5 J 3 ). 4



12.

V–

–2



11.

óò

ñ



10.

V – 3 (ñ 32

=

)3 +
(ñó ò )–7 ) 3 W . ¨ The substitution ( ) = ( 
)2 leads to a second-order equation of the form 2.4.2.66: ¨2

= −2 ( 63 ¨ + 2 # ¨ −5 J 3 ). 4  ñ óò ò ó ò ó = ñ –2 V – 725 (ñ óò )3 +
(ñ óò )–9 ) 5 W . The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.29: ¨2

= −2 (− 5 ¨ + 2 # ¨ −7 J 5 ). 36  ñ óò ò ó ò ó = ñ –2 V – 19 (ñ óò )3 +
W . The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.14: ¨2

= −2 (− 2 ¨ + 2 # ¨ −1 J 2 ). 9  –2 V ñ óò ò ó ò ó = ñ – 252 (ñ óò )3 +
W . The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.8: ¨2

= −2 (− 4 ¨ + 2 # ¨ −1 J 2 ). 25 

ñóò ò ó ò ó

=

ñ

V

–2 63 ( 8

ñóò

Page 593

594 16.

17.

18.

19.

20.

21.

THIRD-ORDER DIFFERENTIAL EQUATIONS

V 10(ñó ò )3 +
eW . The substitution ¨ ( ) = (
 )2 leads ¨2

= −2 (20¨ + 2 # ¨ −1 J 2 ).  ñóò ò ó ò ó = ñ –2 V – 496 (ñóò )3 +
(ñóò )2 W . The substitution ¨ ( ) = (
 )2 leads ¨

= −2 (− 12 ¨ + 2 # ¨ 1 J 2 ). 49  ñóò ò ó ò ó = ñ –2 V
(ñóò )5 – 253 (ñóò )3 W . The substitution ¨ ( ) = (
 )2 leads ¨2

= −2 (2 # ¨ 2 − 6 ¨ ). 25  ñóò ò ó ò ó = ñ –2 V
(ñóò )5 + 253 (ñóò )3 W . The substitution ¨ ( ) = ( 
)2 leads ¨2

= −2 (2 # ¨ 2 + 6 ¨ ). 25 

ñóò ò ó ò ó

23.

24.

25.

26.

27.

ñ

–2

ñ –4 ) 3 ( 
ñóò

ñóò ò ó ò ó

=

ñóò ò ó ò ó

= (
ñ

4

ñóò ò ó ò ó

= (
ñ

2

ñóò ò ó ò ó

= (
ñ

–1

ñ óò ò ó ò ó

= (
ñ

–7 3

ñóò ò ó ò ó

= (
ñ

–4 3

ñóò ò ó ò ó

= (
ñ

–4 3

ñóò ò ó ò ó

= (
ñ

–2 3

+

(

to a second-order equation of the form 2.4.2.33:

to a second-order equation of the form 2.4.2.37:

to a second-order equation of the form 2.4.2.60:

to a second-order equation of the form 2.4.2.62:

).

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.40: ¨2

= −4 J 3 (2 # + 2 $ ¨ −1 J 2 ).  +

(­ñ

+

(

3

)(ñó ò )–13 .

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.39: ¨2

= (2 # 4 + 2 $ 3 )¨ −7 .



22.

=

)(ñó ò )–9 .

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.16: ¨2

= (2 # 2 + 2 $ )¨ −5 .  +

(­ñ

–2

)(ñó ò )–3 .

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.28: ¨2

= (2 # −1 + 2 $ −2 )¨ −2 . 

)

+

(­ñ

–10 3

)

+

(­ñ

–10 3

)

+

(­ñ

–7 3

)

+

(­ñ

–4 3

The substitution ¨ ( ) = ¨2

= (2 # −7 J 3 + 2 $ −10 J  The substitution ¨ ( ) = ¨2

= (2 # −4 J 3 + 2 $ −10 J 

)

)(ñ

óò

)–7 ) 3 .

)

(
 )2 leads to a second-order equation of the form 2.4.2.48: 3 ¨ −5 J 3 ) . )(ñó ò )–7 ) 3 .

(
 )2 leads to a second-order equation of the form 2.4.2.49: 3 ¨ −5 J 3 ) .

)

)(ñó ò )–7 ) 3 .

)

)(ñó ò )–7 ) 3 .

The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.24: ¨2

= (2 # −4 J 3 + 2 $ −7 J 3 )¨ −5 J 3 .  The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.90: ¨2

= (2 # −2 J 3 + 2 $ −4 J 3 )¨ −5 J 3 . 

Page 594

3.3. EQUATIONS OF THE FORM

28.

Ï Ò˜Í Í Ò˜Í Ò

= â (Ï )ã (Ï

ÒÍ

›

) (Ï

Ò˜Í Í Ò

595

)

= (
+ (­ñ –2 ) 3 )(ñó ò )–7 ) 3 . The substitution ¨ ( ) = ( 
)2 leads to a second-order equation of the form 2.4.2.89: ¨2

= (2 # + 2 $ −2 J 3 )¨ −5 J 3 .

ñóò ò ó ò ó 

29.

= (
ñ 2 + ( )(ñ ó ò )–7 ) 3 . The substitution ¨ ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.47: ¨2

= (2 # 2 + 2 $ )¨ −5 J 3 .

ñ óò ò ó ò ó 

30.

ñóò ò ó ò ó

= (
ñ 2 + (­ñ The substitution ¨ ¨2

= (2 # 2 + 2 $



31.

)(ñó ò )–7 ) 3 . ( ) = (
 )2 leads to a second-order equation of the form 2.4.2.46: )¨ −5 J 3 .

= (
ñ ÷ + (­ñ  )ñ ó ò (ñ ó ò ò ó )ú . This is a special case of equation 3.5.4.12 with  ( ) = #

ñ óò ò ó ò ó

 +$

.

3.3.2. Equations Containing Exponential Functions Tables 29–31 present the equations whose solutions are given in this subsection. TABLE 30 Solvable equations of the form 


 = # : 
exp[( 
)2 ]( 

 ) <

TABLE 29 Solvable equations of the form 


 = #   ( 
); ( 

 ) <

= >

Equation

>

ý

Equation

arbitrary ( > ≠ 1)

1

3.3.2.1

arbitrary ( > ≠ 2)

0

3.3.2.4

0

3

3.3.2.9

1

3.3.2.12

1

1

3.3.2.2

arbitrary (ý ≠ −1)

3 2

0

3.3.2.3

1

−1

3.3.2.14

3 2

3

3.3.2.7

3 2

− 12

3.3.2.5 3.3.2.8

3.3.2.11

3 2

1

2

arbitrary (= ≠ −1)

2

0

3.3.2.6

2

−1

3.3.2.13

3

1

3.3.2.10

TABLE 31 Other solvable equations of the type considered




 


 


 


 


 




Form of equation

Equation

= #   
exp[( 
)2 ]( 

 ) <

3.3.2.20

=#

3.3.2.15

=# =# =# =#

( 
); exp( 

 ), ( 
)−1 exp( 

 ) : 
exp( 

 ), −1 
exp( 

 )   
exp[( 
)2 +

= ≠ −1

3.3.2.16

ý ≠ −1 

 ]

3.3.2.17 3.3.2.18 3.3.2.19

Page 595

596

]

THIRD-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 1–6, the following notation is used:

U « = X exp(U 2 ) U + L 2 , 4 = X (1 ACU )  M + L 2 ,  = 2U « − exp(U 2 ), 5 M U 1.

ñóò ò ó ò ó =
B]Èñóò (ñóò ò ó )ª , Solution in parametric form:  = L

where , = 2.

1

1

−1

X «

=
ñ ó ò exp V (ñ ó ò )2 W (ñ ó ò ò ó ) ª , Solution in parametric form:

1

> −2

2.

= ln( L

<

2 −2 1

U ),

(2  2 ) < .

J

−1 2

5

1

−1

XZU

S



+ ln  £ 2 #

2

=U

U . # P

−1

«

−1

.

≠ 2.

1 −2

−3 −1

= ln O T

U + L 3,

U + L 3, M

(1 AIU ) <

1  < 2−>

, # =A

M

[ln( L 1< −2 U )]

J

−1 2

U + L 3, M

= L

1

4 ,

.

=
ñ –1 ) 2 ñó ò exp V (ñó ò )2 W (ñó ò ò ó )3 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

1

2

X «I [U

−1

+ ln( !«

J

−1 2

)]

=
ñ ó ò exp V (ñ ó ò )2 W (ñ ó ò ò ó )2 . Solution in parametric form: 1

XÈU

−1

U + L 3, M

ñ óò ò ó ò ó

exp(T"U ) ln OA

Q

U S # P ­

=L

J

−1 2

M

1

2



,

# = −£ 2 

where

U + L 3,

=L

1

U +1 V ln L U

2

−1

.

5 .

In the solutions of equations 7 and 8, the following notation is used:

« = £ U (U + 1)− ln V L

£ U + £ U +1  W , ² = 2

U +1 ,  = 1− U

=
B ] (ñ ó ò )3 (ñ ó ò ò ó )3 ) 2 . Solution in parametric form:

ñ óò ò ó ò ó

 = L 8.



−1

XÈU

1

ñ óò ò ó ò ó

 =L

7.

−2 −1

U + L 3, M

=
B] (ñó ò ò ó )3 ) 2 . Solution in parametric form:

 = 4L

]

J

−1 2

4

ñóò ò ó ò ó

where , =

6.

−1

XZU

=
B]Èñó ò ñóò ò ó . Solution in parametric form:

 = L

5.

+L

ñóò ò ó ò ó

 =L 4.

U

U

≠ 1.

1 1 , # =A  1−> 2(1 − > )

 =L 3.

S

= X exp(T"U ) M

1

X ²

−1

«

−1



=
ññ ó ò exp V (ñ ó ò )2 W (ñ Solution in parametric form:

ñ óò ò ó ò ó

 = − 12 L 1e X U

−2

²

−1

« [ln( L

J

−1 2

M

U + L 3,

= − ln( L

−3 1

« ),

where

 £ U + £ U +1  W .

# = 23 J 2  3  .

ó ò ò ó )3 ) 2 . J «

−3 2 1

−1

)]−1 J

2

M

U + L 3,

= L

1

 ,

where

# = −4

J .

−1 −3 2



Page 596

Ï Ò˜Í Í Ò˜Í Ò

3.3. EQUATIONS OF THE FORM

]

=

L

™

®

ø

1 0(

U ) + L 2 n 0 (U ) for the upper sign, U ) + L 2 ; 0 (U ) for the lower sign,

® 0 (U

−1

XZU

1

U ) are the modified Bessel

:

J

−1 2

M

U + L 3,

−1 2

1 8

= ln(T

#

U ).

=
ññó ò exp V (ñó ò )2 W (ñó ò ò ó )3 . Solution in parametric form:

:
U 2 −1 J 2 ln O˜A S U * Q #¯P M ≠ –1.

X

1

ñóò ò ó ò ó

=
B] (ñó ò )` (ñó ò ò ó )2 , Solution in parametric form:

where # = 12   ;

+1

, , ’ =X M

U 

J 

−1 2 −1

+L

ln( Œ" − U ) −

U

where # =  −:

U ’ ý +1P

O −

, , ’ =X

−1

M 

U

=

Œ" − U

=
ñ _ ñó ò exp V (ñó ò )2 Wñó ò ò ó , Solution in parametric form: −1

exp O

+L

’

=L

3,

U + L 3,

= + 1 P3M



Œ

,

U  + L 2,

= +1

Π=

2

ln( Œ" − U ) −

.

.

≠ –1.

J exp O − ’ U + L 3, ý + 1 P3M

1;

:

=’ ,

−1 2

+L

1

 =  (U ) is the solution of the transcendental equation

1;

ñóò ò ó ò ó

 = lXv

= U exp O −

’ , ý +1P

 =  (U ) is the solution of the transcendental equation

U , = − U  + L 2, Œ" − U Œ

Œ = ý + 1.

ñóò ò ó ò ó

=
B] (ñó ò )–1 (ñó ò ò ó )2 . Solution in parametric form:

 =L 14.

0(

ñóò ò ó ò ó

 = lXZU

13.

;

) and

=
B] (ñó ò )3 . Solution in parametric form:

 =L

12.

597

)

ñóò ò ó ò ó

 = 2L

11.

Ò˜Í Í Ò

1 0(

L

where ø 0 (U ) and n 0 (U ) are the Bessel functions, and functions.

10.

›

) (Ï

In the solutions of equations 9 and 10, the following notation is used:

:

9.

ÒÍ

= â (Ï )ã (Ï

1

X

ú

J

−1 2

M

U + L 3,

=U ,

ú

where

= exp O X

M

U

U − 2#  * + L

2

P

.

=
ñ –1 ñó ò exp V (ñó ò )2 WUñó ò ò ó . Solution in parametric form:

ñóò ò ó ò ó

 = L 1 XeU

J (U + #  * + L 2 )−1 ú

−1 2

M

U + L 3,

=L

1

ú

,

where

ú

= exp OX

U M U +#  * +L

2

P

.

Page 597

598

]

THIRD-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 15–19, the following notation is used:

˜

=L

Ä

ñóò ò ó ò ó

=

−

+ 1)(U + 1) 

(

Œ ¡

*

8

,

=L

−

1

1

Œ

(U + 2) 

−

* J 2,

Œ U  +1 U if B ≠ −1, B + 1 PuM = L 2 − X ln( L 1 − Œ ln |U |) U if B = −1, M 1 1 = ln 8 − X  − * J 2 8 −1 U − + L 2 . 2Œ M 2



ú

15.




1

−

1


(ñ óò )`

L




− X ln OL

2

1

exp(ñó ò ò ó ),



−

≠ –1.

Solution in parametric form:

 = where 16.

ñóò ò ó ò ó

¡ =

1

˜

XZ − *

Œ

2 −2

+

+1 +2

+

U + L 3, M

2

=

Xv − *

Œ

˜

−

+

+

+1

M

U + L 2,

= 12 (= − 1), # = 2−;!Œ .


(ñ óò

)–1 exp(ñó ò ò ó ).

Solution in parametric form:

 = 17.

˜

1

X exp  −U + 12  U + L 3 , Œ M

ñ óò ò ó ò ó

=


ñ _ ñ óò

óò ò ó

exp(ñ

=



),

˜

2

X exp  −U +  U + L 2 , Œ M

where

= 0, # = 2 Π.

¡

≠ –1.

Solution in parametric form:

ú

−1 2

exp(ñ

óò ò ó

 =X 18.

ñ óò ò ó ò ó

=


ñ

–1

ñ óò

J

U + L 3, M

= 2U ,

B = ý , # = 2−:

where

−1

Π.

).

Solution in parametric form:

ú

 =X 19.

ñóò ò ó ò ó

=

J

−1 2

U + L 3, M

= 2U ,

B = −1, # = Œ .

where

exp V (ñó ò )2 + ñó ò ò óW .


B]Èñóò

Solution in parametric form:

 = 20.

ñ óò ò ó ò ó

1 Xv 2Π=

−

* J 28


B ] ñ óò

−1

J

−1 2

Ä

exp[(ñ

óò

)2 ](ñ

M

U + L 3,

=

1 Xv 2Œ

−

* J 28

−1

M

U + L 2,

where

# =Π.

óò ò ó )ª .

Solution in parametric form:

 = XÈU − < H

−1

O ln






where

’ =X

U

M , H´U <

H =






#

H

−’ −L

J

−1 2 1

P

1 2− 1 U < + U 2− > 1−> U + ln |U | + L 2 1 ln |U | − + L 2

U

M

U + L 3,

1−

< +L

2

=’ , if > ≠ 2, > ≠ 1; if > = 1; if > = 2.

Page 598

3.3. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò˜Í Ò

= â (Ï )ã (Ï

ÒÍ

›

) (Ï

Ò˜Í Í Ò

599

)

3.3.3. Other Equations 1.

ñóò ò ó ò ó


ññóò

=

{cosh[ (ñó ò )2 ]}–2 ñó ò ò ó .

The substitution 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.4.1: ¨

= # [cosh( Œ ¨ )]−2 ¨
.



2.



ñóò ò ó ò ó

=
ññó ò {sinh[ (ñ„ó ò )2 ]}–2 ñó ò ò ó . The substitution 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.4.2: ¨2

= # [sinh( Œ ¨ )]−2 ¨2
.



3.

ñóò ò ó ò ó




ññóò

=


= £ ¨ ( ) leads to a second-order equation of the form 2.7.4.3: The substitution 2 ¨

= 2 # cosh( Œ ¨ )(¨
)3 J 2 . 2 

4.

cosh[ (ñó ò )2 ] (ñó ò ò ó )3 ) 2 .

ñóò ò ó ò ó




ññóò

=

sinh[ (ñ„ó ò )2 ] (ñó ò ò ó )3 ) 2 .


= £ ¨ ( ) leads to a second-order equation of the form 2.7.4.4: The substitution 2 ¨

= 2 # sinh( Œ ¨ )(¨
)3 J 2 . 2 

5.

ñ óò ò ó ò ó

=
cosh( ƒñ The substitution 2 ¨

= 2 # cosh( Π2



6.

ñóò ò ó ò ó ñóò ò ó ò ó

10. 11. 12. 13.

ó ò ò ó )3 ) 2 .


= £ ¨ ( ) leads to a second-order equation of the form 2.7.4.5: ) ¨ ( ¨
)3 J 2 . 



=




[cosh( ƒñ )]–2 (ñó ò )3 (ñó ò ò ó )2 .

=




[sinh( ƒñ )] (ñó ò )3 (ñó ò ò ó )2 .

The substitution 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.4.7: ¨2

= 1 # [cosh( Œ )]−2 ¨ (¨2
)2 . 2

ñóò ò ó ò ó



–2

The substitution 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.4.8: ¨

= 1 # [sinh( Œ )]−2 ¨ (¨
)2 . 2



9.

)3 ( ñ


= £ ¨ ( ) leads to a second-order equation of the form 2.7.4.6: The substitution 2 ¨

= 2 # sinh( Œ )¨ (¨
)3 J 2 . 2



8.

óò

sinh( ƒñ ) (ñ„ó ò )3 (ñó ò ò ó )3 ) 2 .




=



7.

) (ñ



ñ óò ò ó ò ó



=
cosh÷ ( ƒñ ) ñ ó ò (ñ ó ò ò ó )ú

. This is a special case of equation 3.5.4.12 with  ( ) = # cosh  ( Œ

ñóò ò ó ò ó

).

sinh÷ ( ƒñ ) ñó ò (ñó ò ò ó )ú .




=

This is a special case of equation 3.5.4.12 with  ( ) = # sinh  ( Œ

).

=
tanh÷ ( ƒñ ) ñ ó ò (ñ ó ò ò ó )ú . This is a special case of equation 3.5.4.12 with  ( ) = # tanh  ( Œ

ñ óò ò ó ò ó ñóò ò ó ò ó

=




).

coth÷ ( ƒñ ) ñó ò (ñó ò ò ó )ú .

This is a special case of equation 3.5.4.12 with  ( ) = # coth  ( Π=
ln÷ ( ƒñ ) ñ ó ò (ñ ó ò ò ó )ú . This is a special case of equation 3.5.4.12 with  ( ) = # ln  ( Œ

).

ñ óò ò ó ò ó

).

Page 599

600 14.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

=
ññó ò {cos[ (ñó ò )2 ]}–2 ñó ò ò ó . The substitution 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.5.1: ¨2

= # [cos( Œ ¨ )]−2 ¨2
.



15.



ñóò ò ó ò ó

=
ññó ò {sin[ (ñó ò )2 ]}–2 ñó ò ò ó . The substitution 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.5.2: ¨2

= # [sin( Œ ¨ )]−2 ¨2
.



16.

ñóò ò ó ò ó



=
[cos( ƒñ )] (ñó ò )3 (ñó ò ò ó )2 . The substitution 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.5.3: ¨2

= 1 # [cos( Œ )]−2 ¨ (¨2
)2 . 2 –2



17.

ñóò ò ó ò ó



=
[sin( ƒñ )] (ñó ò )3 (ñó ò ò ó )2 . The substitution 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.5.4: ¨2

= 1 # [sin( Œ )]−2 ¨ (¨2
)2 . 2 –2



18.



=
ññó ò cos[ (ñó ò )2 ] (ñó ò ò ó )3 ) 2 . 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.5.5: The substitution 2 ¨

= 2 # cos( Œ ¨ )(¨
)3 J 2 . 2

ñóò ò ó ò ó 

19.



=
ññ ó ò sin[ (ñ ó ò )2 ] (ñ ó ò ò ó )3 ) 2 . 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.5.6: The substitution 2 ¨

= 2 # sin( Œ ¨ )(¨
)3 J 2 . 2

ñ óò ò ó ò ó 

20.



=
cos( ƒñ ) (ñ„ó ò )3 (ñó ò ò ó )3 ) 2 . 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.5.7: The substitution 2 ¨

= 2 # cos( Œ )¨ (¨
)3 J 2 . 2

ñóò ò ó ò ó 

21.

22. 23. 24. 25. 26. 27.



=
sin( ƒñ ) (ñ„ó ò ) (ñó ò ò ó )3 ) 2 . 
= £ ¨ ( ) leads to a second-order equation of the form 2.7.5.8: The substitution 2 ¨

= 2 # sin( Œ )¨ (¨
)3 J 2 . 2

ñóò ò ó ò ó

3





ñóò ò ó ò ó =
cos÷ ( ƒñ ) ñ óò (ñ óò ò ó )ú

. This is a special case of equation 3.5.4.12 with  ( ) = # cos  ( Π=
sin÷ ( ƒñ ) ñ ó ò (ñ ó ò ò ó )ú . This is a special case of equation 3.5.4.12 with  ( ) = # sin  ( Œ

).

ñ óò ò ó ò ó

=
tan÷ ( ƒñ ) ñ ó ò (ñ ó ò ò ó )ú . This is a special case of equation 3.5.4.12 with  ( ) = # tan  ( Œ

).

ñ óò ò ó ò ó

=
cot÷ ( ƒñ ) ñó ò (ñó ò ò ó )ú . This is a special case of equation 3.5.4.12 with  ( ) = # cot  ( Œ

).

ñóò ò ó ò ó

).

=
(arcsin ñ ) ÷ ñó ò (ñó ò ò ó )ú . This is a special case of equation 3.5.4.12 with  ( ) = # (arcsin )  .

ñóò ò ó ò ó

=
(arctan ñ ) ÷ ñó ò (ñó ò ò ó )ú . This is a special case of equation 3.5.4.12 with  ( ) = # (arctan )  .

ñóò ò ó ò ó

Page 600

601

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

3.4. Nonlinear Equations with Arbitrary Parameters 3.4.1. Equations Containing Power Functions 3.4.1-1. Equations of the form  ( , )


 = ( , ). 1.

= ôñ ÷ . See Subsection 3.2.2. The substitution ¨ ( ) = ("
 )2 leads to the Emden–Fowler equation ¨2

= A 2   ¨ −1 J 2 , which is discussed in Section 2.3.

ñóò ò ó ò ó 

2.

3.

4.

ñóò ò ó ò ó

=

ô ƒï ÷ ñ

ñóò ò ó ò ó

=

ô ƒï ÷ ñ ú

ñóò ò ó ò ó

=

–1

.

This is a special case of equation 3.5.1.2 with  ( ) =   . On integrating the equation, we 1  have ! 

 − ( 
)2 =   +1 + L . 2 B +1 . See Subsections 3.2.3 and 3.2.5 (Item 1 b ). The transformation H =   leads to a second-order equation.

+3

+

−1

, ¨

=  
 

ôñ –5 ) 2 + õÿñ –7 ) 2 .

Using the transformation given in 3.5.2.15 (Item 2 b ), we reduce this equation to a constant coefficient nonhomogeneous linear equation. Solution in parametric form (  ≠ 0):

 =X where Y (U ) = − 5.

6.



+L



1

M

[ Y (U

U

J

)]3 2

+L

3,

=

,!U £ 3 , , =  1J 3.  −  * + L 2   * J 2 sin

= ô ï ñ –5 ) 2 + õ ï 3 ñ –7 ) 2 . The transformation  = 1 a , = ¨u a ¨ g

g
g = −  ¨ −5 J 2 −  ¨ −7 J 2 .

ñ óò ò ó ò ó

2

2

leads to an autonomous equation of the form 3.4.1.4:

= ü ( ôñ 2 + õÿñ + ö )–5 ) 4 . This is a special case of equation 3.5.2.29 with  (‡ ) = , .

ñ óò ò ó ò ó

7.

= ï ( ôñ 2 + õ ï 2 ñ + ö ï 4 )–5 ) 4 . This is a special case of equation 3.5.2.30 with  (ˆ ) ≡ 1.

8.

ñóò ò ó ò ó

= ü (ñ +

ñóò ò ó ò ó

=

9. 10.

1 , Y (U )

ñóò ò ó ò ó

ô ï

2

+

õï

+ ö )÷ .

The substitution H = +  2 + ' + ( leads to an equation H 


 = ,H  , whose solvable cases are outlined in Subsection 3.2.2.

üIñ ÷ (ô ï

2

+

õï

+ ö )– ÷

–2

.

This is a special case of equation 3.5.1.13 with  (¨ ) = , ¨  .

+ ý )–÷ –2ú –4 ñ ú .  +  ¨ ¨ ªŠ

Šª
Š = ¢ The transformation ˆ = , = leads to a simpler equation: u (˜ + ( (˜ + )2 M 3.2.2M and 3.2.3). where ¢ =  − ˜( (see Subsections

ñ óò ò ó ò ó

= (ô

ï

+ õ )÷ ( ö

ï

−3

ˆ  ¨2+ ,

M

Page 601

602 11. 12.

THIRD-ORDER DIFFERENTIAL EQUATIONS = õ ï 2÷ –1 ( ï – ô )–3 ñ –÷ . This is a special case of equation 3.5.1.11 with  (ˆ ) = 'ˆ −  .

ñóò ò ó ò ó

(ñ + ô ï 2 + õ ï + ö )ñ ó ò ò ó ò ó = ü ï ÷ . The substitution ¨ = +  2 + ' + ( leads to an equation of the form 3.4.1.2: ¨*¨p


 = ,!  .

3.4.1-2. Equations of the form 


 =  ( , , 
 ). 13.

ñóò ò ó ò ó

=

ôñ ÷ ñóò

+

õ ïƒú

.

Integrating yields a second-order equation: 14.



 =



B +1

= ô ï –÷ –2 ñ ÷ ñó ò – ô ï –÷ –3 ñ ÷ +1 . This is a special case of equation 3.5.2.5 with  (ˆ ) = ˆ  .

ñóò ò ó ò ó

15.

= ô ï –2÷ –4 ñ ÷ ñó ò – 2 ô ï –2÷ –5 ñ ÷ +1 . This is a special case of equation 3.5.2.8 with  (ˆ ) = ˆ  .

16.

ñ óò ò ó ò ó

17. 18. 19.

=

ƒñ –3 ñ óò

)

+ ôñ

–5 2

+

The transformation  = X [ Y (U )]−3 J 2 U , M equation: Y


− ŒY
+ ˜Y +  = 0.

+1

 +

+1

+L .

*+*+*

*

= – ï –2 ñó ò + ï –3 ñ + ô ï 1 ) 2 ñ –5 ) 2 . This is a special case of equation 3.5.2.31 with  (ˆ ) =  .

ñóò ò ó ò ó

= – ï –2 ñ ó ò + ï –3 ñ + ô ï –3 ) 4 ñ –5 ) 4 . This is a special case of equation 3.5.2.32 with  (ˆ ) =  .

ñ óò ò ó ò ó

= ôñ ÷ ñó ò + õÿñ ú (ñó ò )3 . This is a special case of equation 3.5.2.3 with  ( ) = 

ñóò ò ó ò ó

 and ( ) =  + .

= ( ôñ 2 + õÿñ + ö ) 4 (ñó ò )ú . This is a special case of equation 3.5.2.29 with  (ˆ ) = ˆ + .

25.

¡



= [ Y (U )]−1 leads to a constant coefficient linear

21.

24.

+

õÿñ –7 ) 2 .

= õÿñ ÷ (ñó ò )3 + ô (ñó ò )–5 . This is a special case of equation 3.5.2.4 with  ( ) =   .

23.

+1

ñóò ò ó ò ó

20.

22.



ñóò ò ó ò ó ñóò ò ó ò ó

–

ú

+5

= – ô 3 ñ + õ (ñó ò + ôñ )÷ . The substitution ¨ = 
+  leads to a second-order autonomous equation: ¨ 

 −  ¨ 
+  2¨ =  ¨  .

ñóò ò ó ò ó

= ô ï ( ï ñó ò – ñ )÷ . This is a special case of equation 3.5.2.18 with  (ˆ ) = ˆ  .

ñóò ò ó ò ó

= ô ï –÷ –2 ( ï ñó ò – ñ )÷ . This is a special case of equation 3.5.2.23 with  (ˆ ) = ˆ  .

ñóò ò ó ò ó

= ô ïƒ÷ ( ï ñó ò – ñ )ú . The substitution ¨ ( ) =  
− leads to a second-order generalized homogeneous equation: (¨2
 )
 =   ¨2+ .

ñóò ò ó ò ó

Page 602

603

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

26. 27. 28. 29.

= ô ï ÷ ( ï ñ ó ò – ñ )ú + õ ï  . The substitution ¨ ( ) =  
− leads to a second-order equation: (¨ 
 )
 =   ¨ + + '  .

ñ óò ò ó ò ó

= ô ïƒ÷ –5 ñ –÷ ( ï ñó ò – ñ )3 . This is a special case of equation 3.5.2.6 with  (ˆ ) = ˆ −  .

ñóò ò ó ò ó

= ô ï –÷ –4 ( ï ñó ò – 2ñ )÷ . This is a special case of equation 3.5.2.24 with  (ˆ ) = ˆ  .

ñóò ò ó ò ó

= ô ï ÷ ( ï ñ ó ò – 2ñ )ú + õ ï  . The substitution ¨ ( ) =  
− 2 leads to a second-order equation: ¨ 

 =  

ñ óò ò ó ò ó

30.

= ô ï 2÷ –7 ñ –÷ ( ï ñó ò – 2ñ )3 . This is a special case of equation 3.5.2.9 with  (ˆ ) = ˆ −  .

31.

ñóò ò ó ò ó

]

+1

¨ + + ' 

+1

.

ñóò ò ó ò ó

=

ô ƒï ÷ ñ ú (ï ñ óò

– 2 ñ )A .

The transformation a = 1  , H =   which is discussed in Section 3.2.

2

leads to the equation H g

g
g = −  (−1) - a − 

−2

+ −- −4 H + ( H g
)- ,

See also equations 3.3.1.2–3.3.1.30. 3.4.1-3. Equations of the form  ( , , 
 )


 + ( , , 
 )

 + % ( , , 
 ) = 0.

32.

ñ óò ò ó ò ó

+ ôññ ó ò ò ó – ô (ñ ó ò )2 = 0. 1 b . The substitution ¨ ( ) = ( 
)2 leads to a second-order generalized homogeneous equation: A £ ¨ ¨2

+  !¨2
− 2  ¨ = 0.





2 b . Particular solutions:

=L

1

exp( L

2

= 6(  + L 33.

ñóò ò ó ò ó

+

ôññóò ò ó

+

õÿñóò ò ó

Particular solution: 34.

ñ óò ò ó ò ó

+

ô 2 ñ óò ò ó

+ô

1

– ô (ñó ò )2 + =L

1

+

ô 0ñ

ñ óò

36.



=

2

 )−

õÿññ óò ò ó

1)

= 0. L 22 + L

L

– õ (ñ

óò

−1

2 2

−1

L 2,

. +(

.

)2 + ü .

, , where L is an arbitrary constant and Œ = Œ  are roots  0 , of the cubic equation Œ 3 + O  2 − Œ 2 +  1 Œ +  0 = 0.  0P =L  

Particular solutions:

35.

exp( L

ö^ñóò

 )−

+

= ô ïƒ÷ À ñó ò ò ó + õ ïƒú ( ï ñó ò – ñ )  + ö ï s . The substitution ¨ ( ) =  
− leads to a second-order equation: (¨u
 )
 =   ' +2¨  + (˜ .

ñóò ò ó ò ó ñóò ò ó ò ó

= ôññó ò ñóò ò ó – ô (ñóò )3 + 1 b . Particular solution:

õÿñóò

=L where the constants L 1 , L 2 , and L 2 b . Particular solution: where the constants L

1, L

−1

¨2
+

. exp( L 3  ) + L 2 exp(− L 3  ), are related by the constraint L

2 3

− 4 L

= L 1 cos( L 3  ) + L 2 sin( L 3  ), , and L 2 are related by the constraint L 2

2 3

−  (L

1 2

1

2 1

L 2L +L

−  = 0.

2 3

2 2)

L

2 3

+  = 0.

Page 603

604 37.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï ñ óò ò ó ò ó

óò ò ó

+ 3ñ

=

ô ï ÷ ñ ÷

.

The substitution ¨ ( ) = 

¨ 


 =  ¨  , which is discussed in leads to the equation p

ï ñóò ò ó ò ó

– ñó ò ò ó – 2 ô

2

ï ñóò ò ó ò ó

+ ( ô + 2)ñó ò ò ó = õ ( ï

Subsection 3.2.2. 38.

39.

40.

41.

42.

43.

44.

ï ññóò

+ ôñ

= 0.

Integrating yields a second-order equation: "

 =  2 + L3 +  . The transformation = Lp, 3 ¨ ,  6 1J 3  = ,H − , where , = O , leads to the first Painlev´e transcendent: ¨3‘'

‘ = ¨ 2 + 6 H L LÊP (see Paragraph 2.8.2-2). The substitution ¨ =  2.9.1.1: ¨3

 =  ¨  .

ñóò


+ 

+ ôñ )÷ . leads to a second-order autonomous equation of the form

ï ñóò ò ó ò ó

= – 32 ñó ò ò ó +

ô ï –÷ –2 ñ 2÷ (2ï ñ óò

– ñ ).

ï ñóò ò ó ò ó

= – 32 ñó ò ò ó +

ô ï –÷ –3 ñ 2÷ (2ï ñ óò

–ñ ) .

ï ñóò ò ó ò ó

+ ñó ò ò ó =

ô ï –÷

ï ñ óò ò ó ò ó

+ (1 – ô )ñ

óò ò ó

This is a special case of equation 3.5.3.37 with  (ˆ ) = ˆ 2  . 3

This is a special case of equation 3.5.3.39 with  (ˆ ) = ˆ 2  . –3

(ï

÷

–ñ ) .

ñóò

This is a special case of equation 3.5.3.47 with  (ˆ ) = ˆ  . =

÷

õ ï 2D (ï ñ óò

–ñ ) .

This is a special case of equation 3.5.3.42 with  (ˆ ) = 'ˆ  .

ï 2 ñ óò ò ó ò ó

+ 6ï

ñóò ò ó

ô ï 2÷ ñ ÷

+ 6ñó ò =

The substitution ¨ ( ) = 

ññ óò ò ó ò ó

+

1 2

ñ óò ñ óò ò ó

=

ô ï

.

leads to the equation ¨3


 =  ¨  , which is discussed in

2

Subsection 3.2.2. 45.

õ

+

+õ .

The transformation  =  (a ), = ( g
) leads to a fourth-order constant coefficient nonhomogeneous linear equation of the form 4.1.2.2: 2 g

g
g
g =  +  . 2

46.

ññóò ò ó ò ó

–

1 3

ñóò ñóò ò ó

=

ô ï

+õ .

1 b . On integrating the equation, we obtain !

 − 23 (
 )2 = 12    2 + ' + L . The substitution = ¨ 3 leads to a solvable equation of the form 2.8.1.5: ¨p

 = 13  12  2 + ' + L  ¨ −5 . 2 b . Particular solution: where the constants L

1,

=L

L 2 , L 3 , and L

1

3



47.

ññóò ò ó ò ó

+

1 2

ñóò ñóò ò ó

=


ï

)

–5 3

.

1

2



2

+L

3

 + L 4,

are related by two constraints

4

4L 6L

+L

L

1 4

L

3

− 43 L

2 3

− L

2

L

2 2

= ,

3

=.

The transformation  =  (a ), = ( g
)2 leads to an equation of the form 4.2.1.1: 2 g

g
g
g = #*

J .

−5 3

Page 604

605

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

48.

ññ óò ò ó ò ó

= ñ ó ò ñ ó ò ò ó + ôñ ó ò . Integrating yields a second-order constant coefficient linear equation:

 = L Solutions: = L 1 sinh( L 3  ) + L 2 cosh( L 3  ) + L 3−2 , =L

 ) + L 2 cos( L 3  ) − L 2 1 = −2   + L 1  + L 2.

49.

1

sin( L

3

– 2ñ ó ò ñ ó ò ò ó + 2 ô ï ñ ó ò – ôñ = 0. Integrating yields a second-order equation: "

 = L

ññ óò ò ó ò ó

=

1 ¨ , 2

Lp,

 = ,H ,

where

‘'‘

leads to the first Painlev´e transcendent: ¨3

= ¨ There is also the trivial solution = 0. 50.

51.

2

2

− .

−2 3 ,

+  . The transformation

, =O

6

LRP

J

1 5

,

+ 6 H (see Paragraph 2.8.2-2).

+ 3ñóÈ ò ñóò ò ó + 4ô ï ññóò + ôñ 2 = 0. Multiplying by 2 , we arrive at an exact differential equation. Integrating it yields Yermakov’s equation 2.9.1.2: 

 +  = L −3 . There is also the trivial solution = 0.

ññóò ò ó ò ó

ññ óò ò ó ò ó

+

1 2

ñ óò ñ óò ò ó

=

ü 7 ñ ñ óò ò ó

The transformation  =  (a ), geneous linear equation:

+ þxñ

óò

+

ô 7 ñ

+

õï

+ö .

2 = ( g
) leads to a fourth-order constant coefficient nonhomo-

2 g

g
g
g = A 2 ,! g

g
g + 2  g

g AC g
+ ' + ( .

¡

Here, the plus sign corresponds to  g
> 0 and the minus sign to  g
< 0. 52. 53. 54.

55. 56. 57. 58.

+ 3ñóÈ ò ñóò ò ó + ô ïƒ÷ ññóò = õ ïƒú . This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) = ' + .

ññóò ò ó ò ó

+ 3ñ ó ò ñ ó ò ò ó + ô [ññ ó ò ò ó + (ñ ó ò )2 ] = õ ï ÷ . This is a special case of equation 3.5.3.13 with  ( ) = '  .

ññ óò ò ó ò ó

– ñó ò ñóò ò ó = ô [ññóò ò ó – (ñóò )2 ] + õ ï + ö . This is a special case of equation 3.5.3.16 with  ( ) =  and ( ) = ' + ( . The substitution ¨ = !

 − (
 )2 leads to a first-order linear equation: u ¨ 
=  ¨ + ' + ( .

ññóò ò ó ò ó

+ (3ñó ò + 2 ôñ )ñó ò ò ó + 2 ô (ñó ò )2 + ô 2 ññó ò = õ ïƒ÷ .   This is a special case of equation 3.5.3.29 with  ( ) =   and ( ) = '    .

ññóò ò ó ò ó

+ (3ñó ò + ô ïƒ÷ ñ )ñó ò ò ó + ô ïƒ÷ (ñó ò )2 = 0. This is a special case of equation 3.5.3.17 with  ( ) =   .

ññóò ò ó ò ó

(ñ + ô )ñ„ó ò ò ó ò ó + õÿñó¿ ò ñóò ò ó + ö^ñ ÷ ñóò = 0. This is a special case of equation 3.5.3.21 with  ( ) = (  .

(ñ + ô ï )ñ„ó ò ò ó ò ó – ñó ò ñóò ò ó + õ ï ñóò ò ó = 0. A solution of this equation is any function that solves the first-order linear equation 
= L 1 + ( L 1 +  ) . L 1 +  Particular solution: = L 2 exp( L 1  ) − ( L 1  + 1). 2

L

1

Page 605

606 59.

THIRD-ORDER DIFFERENTIAL EQUATIONS (ñ +

ô ï )ñ ó ò ò ó ò ó

óò

= (ñ

+ ô )ñ

óò ò ó

õ ï ñ óò

+

õÿñ

–

.

1 b . Integrating yields a second-order constant coefficient equation:

 + L

= −( L +  ) .

2 b . Solutions:

 ) + L 2 cos( L 3  ) − (  + L 3−2 ) = L 1 sinh( L 3  ) + L 2 cosh( L 3  ) − (  − L 3−2 ) = − 16 ' 3 + L 1  + L 2 =L

60.

(ñ +

ô ï

+ õ )ñ

1

óò ò ó ò ó

sin( L

–

1 3

ñ óò ñ óò ò ó

óò ò ó

+ ö^ñ

Particular solution:

6L

ô ï

(ñ +

62.

ï ññ óò ò ó ò ó

=

ï ññóò ò ó ò ó

= ( ôñ +

63.

64.

65.

66.

67.

68.

+ õ )ñ„ó ò ò ó ò

ó

1

L 2 , L 3 , and L

1,

4L

61.

=

=L

where the constants L

if L

3

1

L

L

1

3



+L

2



2

+L

3

= −L

if L

= 0.

3

− 43 L 2

L

ö ïƒ÷

2 2

3

+ 6(  + ( ) L

+ 6 L

1

> 0, 2 3

< 0,

 + L 4,

are related by two constraints

− L

+ 3(ñó ò + ô )ñó ò ò ó =

+ s.

if L

4

2 3

4

ü ï

2 3

=L

+ 2 (L

1

=, ,

2

=°.

.

This is a special case of equation 3.5.3.23 with  ( ) = (˜  .

ï ñ óò ñ óò ò ó

+

ôññ óò ò ó

.

Integrating yields a second-order linear equation of the form 2.1.2.7:

 = L3  .

õ ï ñóò )ñ óò ò ó

.

Integrating yields the Emden–Fowler equation: "

 = L3 

ï (ñ ñóò ò ó ò ó

+ 3ñó ò ñóò ò ó ) + ô V ññóò ò ó + (ñóò )2 W =

ï 2 ñ ñóò ò ó ò ó

+ ï (3 ï



õ ïƒ÷

(see Section 2.3).

. This is a special case of equation 3.5.3.25 with  ( ) = '  .

ñóò

ï (ñ óò

+ 2 ôñ )ñó ò ò ó + 2 ô

)2 + ô ( ô – 1)ññó ò =

õ ïƒ÷

.

This is a special case of equation 3.5.3.29 with  ( ) =   and ( ) = '  + 

ñ 2 ñ óò ò ó ò ó

– 3ññó ò ñóò ò ó + 2(ñóò )3 =

ô ƒï ÷ ñ

ñ 2 ñ óò ò ó ò ó

+ 3þxññ

óò ñ óò ò ó

– 1)(ñ

3

−2

.

.

This is a special case of equation 3.5.3.26 with  ( ) =   . +þ

(þ

óò

)3 =

ô ï  ñ 2–ú

.

This is a special case of equation 3.5.3.27 with  ( ) =   and B =

ñóò ñóò ò ó ò ó

– (ñó ò ò ó )2 =

ôññóò ò ó

1 b . Particular solution: where the constants L

– ô (ñó ò )2 + õ . =L

L 2 , and L

1,

2 b . Particular solution: where the constants L

1,

1 3

=L

L 2 , and L

3 b . Particular solutions:

= A"

ü

3

exp( L

3

 )+L

2

exp(− L

3

 ),

are related by the constraint 4 L 1

cos( L

3

 )+L

2

sin( L

3

1

L 2(L

4 3

+ L

L

4 3−

2 3)

+  = 0.

 ),

are related by the constraint ( L

  +L .

+ 1.

¡

2 1+

L

2 2 )(

L 32 ) +  = 0.

Page 606

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

607

3.4.1-4. Other equations. 69.

2ñó ò ñóò ò ó ò

ó

– (ñó ò ò ó )2 = (ñó ò )2 + ôñ

70.

2ñó ò ñóò ò ó ò

ó

– 3(ñó ò ò ó )2 =

71.

2ñ

72.

2ñó ò ñóò ò ó ò

ó

– 3(ñó ò ò ó )2 =

ô ïƒ÷ –4 ñ

4–2

73.

2ñó ò ñóò ò ó ò

ó

– 3(ñó ò ò ó )2 =

ô ïƒ÷ (ñ óò

)2 +

õÿñ ú (ñ óò

74.

2ñ

óò ò ó

ôñ ÷ (ñ óò

)4 +

õ ï –1 ñ ú (ñ óò )7 ) 2 .

)2 +

2

+

õÿñ

+ö .

Differentiating with respect to  and dividing by "
 , we arrive at a fourth-order constant coefficient linear equation: 2



 = 2 Π

 + 2  +  .

ôñ

4–2

÷ (ñ óò )÷

.

This is a special case of equation 3.5.4.14 with  (ˆ ) = ˆ  .

óò ñ óò ò ó ò ó

– 3(ñ

óò ò ó

)2 =

ô ï 2÷ –8 ñ

4–2

÷ (ñ ó ò )÷

.

This is a special case of equation 3.5.4.16 with  (ˆ ) = ˆ  .

÷ (ñ óò )÷

. This is a special case of equation 3.5.4.15 with  (ˆ ) = ˆ  . )4 .

This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  + .

óò ñ óò ò ó ò ó

– 3(ñ

)2 =

This is a special case of equation 3.5.4.9 with  ( ) = 

75.

2ñó ò ñóò ò ó ò

ó

– 3(ñó ò ò ó )2 =

ô ïƒ÷ (ñ óò

76.

2ñó ò ñóò ò ó ò

ó

– (ñó ò ò ó )2 =

ïƒ÷ (ñ óò

77.

ï ñóò ñóò ò ó ò ó

õ ïƒú ñ

–1

(ñó ò )5 ) 2 .

This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) = ' + . )2 +

ôñ

2

+ 2 õÿñ + ö .

This is a special case of equation 3.5.4.5 with  ( ) = − Œ   . – 3 ï (ñó ò ò ó )2 + 3ñó ò ñóò ò ó =

ô ï ñ ÷ (ñ óò

)4 +

õÿñ ú (ñ óò

This is a special case of equation 3.5.4.11 with  ( ) =  78.

79.

80.

81.

82.

83.

 and ( ) =  + .

ñóò ò ó ò ó

=

ô ï –2÷

–5

(ï

ñóò

– ñ )÷ (ñó ò ò ó )3 .

ñóò ò ó ò ó

=

ô ï –4÷

–5

(ï

ñóò

– ñ )÷ (ñó ò ò ó )3 .

ñóò ò ó ò ó

= Vô

ï

ñ óò ò ó ò ó

= Vô

ï (ñ ó ò )÷

ñ óò ò ó ò ó

=

)5 .

 and ( ) =  + .

This is a special case of equation 3.5.4.18 with  (ˆ ) = ˆ  . This is a special case of equation 3.5.4.19 with  (ˆ ) = ˆ  . –5

õ ï 3 (ï ñ óò

+

– ñ )÷ W (ñó ò ò ó )3 .

This is a special case of equation 3.5.4.17 with  (ˆ ) = 'ˆ  . +

õÿñ (ñ óò )ú

+ ö (ñ

ó ò )  W (ñ ó ò ò ó

)3 + s(ñ

ó ò )A ( ñ ó ò ò ó

)2 .

This is a special case of equation 3.5.4.20 with  (ˆ ) = ˆ  , (ˆ ) = 'ˆ + , % (ˆ ) = (˜ˆ  , and Y (ˆ ) = °˜ˆ - .

ô ï ÷ (ñ ó ò ò ó )ú

+

õ ï  (ï ñ óò

The substitution ¨ ( ) =  
−

ï ñóò ò ó ò ó

+ ñó ò ò ó =

ô ïƒ÷ (ï ñ óò

– ñ )A + ö

ï

s

.

leads to a second-order equation.

– ñ )ú (ñó ò ò ó )÷ .

This is a special case of equation 3.5.5.5 with  ( H ) = !H + and (ˆ ) = ˆ  .

Page 607

608 84.

85. 86. 87.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

=


ïƒ÷ (ñ óò )ú (ï ñ óò

ñ óò ò ó ò ó

=

ô ï (ï ñ óò

A

– ñ ) (ñó ò ò ó )  .

The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ (
 = a ) leads to the equation ¨2g

g
g = − #*a +2¨ - (¨2g
)  (¨2g

g )3−  , which is discussed in Section 3.2. – ñ )÷ (ñ

óò ò ó

– ñ )ú (ñ

õ ï (ï ñ óò

)2 +

óò ò ó )

.

This is a special case of equation 3.5.4.21 with  (ˆ ) = ˆ  and (ˆ ) = 'ˆ + .

ññóò ò ó ò ó

=

ñóò ñóò ò ó

+ ôñ

÷ ú (ñ óò )÷ (ñ óò ò ó )ú V ñ ñóò ò ó

– (ñó ò )2 W .

– –

This is a special case of equation 3.5.5.7 with  (ˆ ) = ˆ  and (ˆ ) = ˆ + .

(ñó ò ò ó ò ó )2 = ô ( ï

2

– 2ï

ñóò ò ó

ñóò

+ 2ñ ) +

õÿñóò ò ó

+ö .

Differentiating with respect to  , we obtain 


 (2 



 −  factor to zero and integrating, we find the solution: =

1 720

6



+

1 48

'

4

+L

3

3



+L

2



2

+L

2

−  ) = 0. Equating the second

1

 + L 0.

The integration constants L‹© and the parameters  ,  , and ( are related by: 36 L

2 3

= 2 L

0

+ 2 L

2

+( .

This constraint is obtained by substituting the above solution into ¶ the original ¶ ¶ equation. In addition, to¶ the first factor there corresponds¶ the solution = L 2  2 + L 1  + L 0 , where the ¶ constants L·© are related by the constraint 2  L 0 + 2  L 2 + ( = 0.

3.4.2. Equations Containing Exponential Functions 3.4.2-1. Equations of the form 


 =  ( , , 
 ). 1.

ñ óò ò ó ò ó

=

ô ]

.

Autonomous equation. This is a special case of equation 3.5.1.1 with  ( ) =  ! . The substitution ‡ ( ) = ( 
)2 leads to a second-order equation: ‡

= A 2   ‡ −1 J 2 . The transformation  H =  ‡ −3 J 2 , ¨ = ‡
‡ leads to a first-order equation: H  Œ − 32 ¨  ¨2‘
= A 2 !H − ¨ 2 . 2.

3.



ñóò ò ó ò ó = ô] +_ ó

. ¨ = The substitution ? ¨2


 =   .

ô ó ñ ÷

.

ô ï ÷  ]

.

ñóò ò ó ò ó

=

ñ óò ò ó ò ó

=

ñóò ò ó ò ó

= ô (ñ +

+ ( ý Œ )

 

The transformation H = ž 4. 5.

The transformation H =  

õ ó

÷

+3

+ö ) –

−1

leads to an autonomous equation of the form 3.4.2.1:

, ¨ = 
  leads to a second-order equation.

 , ¨ =  
 leads to a second-order equation.

õ ó

.

 ¨ 


 =  ¨  , whose solvable cases are The substitution ¨ = +   + ( leads to the equation 3

outlined in Subsection 3.2.2. 6.

ñ óò ò ó ò ó

=

ô ] ñ óò

Solution: L

3

.

AC = X¥ L

2

+L

1

+ 2 Œ

−2

  

J

−1 2

M

.

Page 608

609

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

7.

ñ óò ò ó ò ó

=

ô ] ñ óò

+

õ ó

.

This is a special case of equation 3.5.2.2 with  ( ) =    and ( ) =  

   

 =   +   + L . Œ “

a second-order equation: 8. 9. 10.

ñóò ò ó ò ó

=

ô]Èñóò

+

õ  ] (ñ óò

ñ óò ò ó ò ó

=

ô ] ñ óò

+

õÿñ ÷ (ñ óò

)3 .

ñ óò ò ó ò ó

=

ôñ ÷ ñ óò

õ ] (ñ óò

)3 .

ñóò ò ó ò ó

=

ñóò ò ó ò ó

=2

. Integrating yields

)3 .

This is a special case of equation 3.5.2.3 with  ( ) =   and ( ) =     . This is a special case of equation 3.5.2.3 with  ( ) =   and ( ) =   . +

This is a special case of equation 3.5.2.3 with  ( ) =  11.



õ] (ñ óò

 and ( ) =   .

)3 + ô (ñó ò )–5 .

This is a special case of equation 3.5.2.4 with  ( ) =   . 12. 13. 14. 15.

2

ú ] (ñ óò )ú

(ñó ò )3 + ô

–5

.

This is a special case of equation 3.5.2.34 with  (ˆ ) = ˆ +

õ ó (ñ óò

.

– ôñ )÷ .

ñóò ò ó ò ó

=

ô 3ñ

ñóò ò ó ò ó

=

ô ó (ï ñ óò

– ñ )÷ .

ñóò ò ó ò ó

=

ô ó (ï ñ óò

– 2ñ )÷ .

+

−6

¨ 

 +  2 ¨ 
+  2 ¨ =    ¨  . leads to a second-order equation: u

The substitution ¨ = 
−  The substitution ¨ = 


−

¨ 
 )
 =    ¨  . leads to a second-order equation: (u

¨ 

 =    ¨  . 
− 2 leads to a second-order equation: u

The substitution ¨ = 

3.4.2-2. Other equations. 16. 17. 18. 19. 20.

ô ú ó ñ ú

= –3ñó ò ò ó + ô

ñ óò ò ó ò ó

+ 3 Ė

óò ñ óò ò ó

+

2

(ñ

óò

)3 =

ô _ ó – ]

.

ñ óò ò ó ò ó

+ 3 Ė

óò ñ óò ò ó

+

2

(ñ

óò

)3 =

ô ï ÷  – ]

.

=

õ 2D ó (ï ñ óò

+

+1

+ 2ñ .

This is a special case of equation 3.5.3.33 with  (ˆ ) = ˆ + .

 This is a special case of equation 3.5.3.1 with  ( ) =   : . This is a special case of equation 3.5.3.1 with  ( ) =   .

ï ñóò ò ó ò ó

+ (1 – ô

ï )ñ óò ò ó

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó =

÷

–ñ ) .

This is a special case of equation 3.5.3.44 with  (ˆ ) = 'ˆ  .

Solution: 21.

ú ó ñ ú ñóò

ñóò ò ó ò ó

ññóò ò ó ò ó

2

=L

2



+ 3ñóÈ ò ñóò ò ó =

ô ó

2

+L

ô]

. 1

 +L

0

+ 2 Œ

−3

   .

+õ .

This is a special case of equation 3.5.3.8 with  ( ) =   +  .

Page 609

610 22.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ññ óò ò ó ò ó

+ 3ñ

óò ñ óò ò ó

ó

 ññ óò

+ ô

õ ó

=

.



This is a special case of equation 3.5.3.10 with  ( ) =   23.

ó ññóò

õ ïƒ÷

.

õ ó

.

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô V ññóò ò ó + (ñóò )2 W =

ññóò ò ó ò ó

– ñó ò ñóò ò ó =

ññóò ò ó ò ó

+ (3ñó ò + 2 ôñ )ñó ò ò ó + 2 ô (ñó ò )2 +

ññóò ò ó ò ó

+ (3ñó ò +

=



25.

ïƒ÷ ññóò

=

.

and ( ) = '  .

This is a special case of equation 3.5.3.10 with  ( ) =   24.



and ( ) =  

 This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =   .

õ ó

.



This is a special case of equation 3.5.3.13 with  ( ) =   . 26.

ô V ñ ñóò ò ó

– (ñó ò )2 W +

õ ó

+ö .



This is a special case of equation 3.4.3.16 with  ( ) =  and ( ) =   + ( . The substitution ¨ = !

 − (
 )2 leads to a first-order linear equation: ¨u
=  ¨ +    + ( . 27. 28.

ô 2 ñ ñóò

=

õ ó

.

  This is a special case of equation 3.5.3.29 with  ( ) =   and ( ) =   (  +  ) .

ô ó ñ )ñ óò ò ó

+

ô ó (ñ óò

)2 = 0.



This is a special case of equation 3.5.3.17 with  ( ) =   . 29.

óò ò ó ò ó

( ñ + ô )ñ

+

õÿñ óò ñ óò ò ó

] ñ óò

+ ö

= 0.

This is a special case of equation 3.5.3.21 with  ( ) = (  .

30.

(ñ +

ô ï

+ õ )ñ

óò ò ó ò ó

óò

+ 3(ñ

2

+ ô )ñ

óò ò ó



+L

Solution: ( +  +  ) = L 31.

32.

ñ 2 ñ óò ò ó ò ó

– (ñó ò )3 +

ôñ 2 ñ óò

2

2

õ ó

=

ü  ó

= 1

 +L

.

0

+ 2, Œ

−3

   .

.




Integrating yields a second-order equation: 2  − (  )2 + 13  3 = Œ −1   we have an equation of the form 2.8.3.57 with , = −1: ! 

 − ( 
)2 + 13 

ñ 2 ñ óò ò ó ò ó

– (ñó ò )3 +

ñ 2 ñ óò ò ó ò ó

– 3ññ

ôñ 2 ñ óò

õï

=

exp(

ï

2

).





 2

+ L . For L = 0,  = Œ −1   −1 .




Integrating yields a second-order equation: 2  − (  )2 + 13  3 = 12 Œ −1 exp( Œ  2 ) + L .


For L = 0, we have an equation of the form 2.8.3.5.7 with , = −1: !  − (  )2 + 13  2 = −1 2 −1 1 exp( Œ  ) . 2 Œ 33.

óò ñ óò ò ó

Solution: ln | | = L 34.

ñ 2 ñ óò ò ó ò ó

+ 3þxññ

+ 2(ñ 2



óò ñ óò ò ó

2

óò

ô  ó ñ

)3 =

+L

1

 +L

+þ

(þ

0

3

+ Œ

– 1)(ñ

óò

.

−3

   .

)3 =

ô  ó ñ 2–ú

.

This is a special case of equation 3.5.3.27 with  ( ) =   35. 36.



and B =

ï 2 ññóò ò ó ò ó + ï (3ï ñóò + 2ô ñ )ñóò ò ó + 2ô ï (ñ óò )2 + ô (ô – 1)ññóò = õ ó

+ 1.

¡ .

This is a special case of equation 3.5.3.29 with  ( ) =   and ( ) = '  2ñó ò ñóò ò ó ò

ó

– (ñó ò ò ó )2 =

ü ó (ñ óò

)2 + ôñ

2

+ 2 õÿñ + ö .

−2

  .



This is a special case of equation 3.5.4.5 with  ( ) = − ,  .

Page 610

611

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

37. 38.

ó

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô  (ñ ó ò )2 + õ ] (ñ ó ò )4 .  This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =     .

ó

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô (ñóò )2 + õÿñ ú (ñóò )4 .  This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  + .

39.

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô ïƒ÷ (ñóò )2 + õ  ] (ñóò )4 . This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =     .

40.

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô (ñóò )2 + õ  ñ –1 (ñóò )5 ) 2 .   This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =    .

41. 42. 43. 44. 45.

ó

ó

ó

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô  (ñ ó ò )2 + õ ï ÷ ñ –1 (ñ ó ò )5 ) 2 .  This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) = '  .

ó

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô ïƒ÷ (ñóò )2 + õ ñ –1 (ñóò )5 ) 2 . 2ñó  This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =   .

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô] (ñóò )4 + õ ï –1   ] (ñóò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =   and ( ) =     .

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô] (ñóò )4 + õ ï –1 ñ ÷ (ñóò )7 ) 2 . 2ñó This is a special case of equation 3.5.4.9 with  ( ) =    and ( ) =   . 2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ôñ ÷ (ñ ó ò )4 + õ ï –1  ] (ñ ó ò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) = 

 and ( ) =   .

3.4.3. Equations Containing Hyperbolic Functions 3.4.3-1. Equations with hyperbolic sine. 1.

ñóò ò ó ò ó

=

ô

sinh( ƒñ )ñ„ó ò .

Solution: L 2.

3

AC = X V L

2

+L

1

+ 2 Œ

−2

sinh( Œ

4. 5.

M

.

) and ( ) =  sinh(“/ ).

 =  cosh( Œ ) +  cosh(“/ ) + L . Œ “

= ô sinh ÷ ( ƒñ )ñ ó ò + õ sinh( “ñ )(ñ ó ò )3 . This is a special case of equation 3.5.2.3 with  ( ) =  sinh  ( Œ

ñ óò ò ó ò ó ñóò ò ó ò ó

=

1 2

2

(ñó ò )3 + ô (sinh ƒñ )–ú

–3

(ñó ò )2ú

+1

) and ( ) =  sinh(“

).

.

This is a special case of equation 3.5.2.36 with  (ˆ ) = ˆ 2 + .

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó =

Solution:

6.

J

−1 2

= ô sinh( ƒñ )ñ„ó ò + õ sinh( ï ). This is a special case of equation 3.5.2.2 with  ( ) =  sinh( Œ

ñóò ò ó ò ó

Integrating yields a second-order equation: 3.

)W

2

=L

2

ô

2

ï

).

1 + L

0

sinh( +L

+ 2 Œ

−3

cosh( Π ).

+ 3ñ ó ò ñ ó ò ò ó = ô sinh ÷ ( ï ) + õ . This is a special case of equation 3.5.3.6 with  ( ) =  sinh  ( Œ  ) +  .

ññ óò ò ó ò ó

Page 611

612 7.

THIRD-ORDER DIFFERENTIAL EQUATIONS + 3ñ ó ò ñ ó ò ò ó = ô sinh ÷ ( ƒñ ) + õ . This is a special case of equation 3.5.3.8 with  ( ) =  sinh  ( Œ

ññ óò ò ó ò ó

)+ .

8.

+ 3ñóÈ ò ñóò ò ó + ô ïƒ÷ ññóò = õ sinhú ( ï ). + This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  sinh ( Œ  ).

9.

ññóò ò ó ò ó

10. 11. 12.

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô V ññóò ò ó + (ñóò )2 W =

sinh÷ (

õ

ï

).

This is a special case of equation 3.5.3.13 with  ( ) =  sinh  ( Œ  ). + (3ñ ó ò + ôñ sinh ÷ ï )ñ ó ò ò ó + ô sinh ÷ ï (ñ ó ò )2 = 0. This is a special case of equation 3.5.3.17 with  ( ) =  sinh   .

ññ óò ò ó ò ó

(ñ + ô )ñ„ó ò ò ó ò ó + õÿñó¿ ò ñóò ò ó + ö sinh÷ ( ƒñ )ñóò = 0. This is a special case of equation 3.5.3.21 with  ( ) = ( sinh  ( Œ

ñ 2 ñ óò ò ó ò ó

+ 3þxññóÈ ò ñóò ò ó + þ

(þ

– 1)(ñ„ó ò )3 =

ï )ñ 2–ú

sinh  (

ô

). .

This is a special case of equation 3.5.3.27 with  ( ) =  sinh  ( Π ) and B =

¡

+ 1.

3.4.3-2. Equations with hyperbolic cosine. 13.

ñóò ò ó ò ó

cosh( ƒñ )ñ„ó ò .

ô

=

Solution: L 14.

ñóò ò ó ò ó

AC = X V L

2

+L

1

+ 2 Œ

ï

cosh( ƒñ )ñ„ó ò + õ cosh(

=

ô

3

−2

cosh( Œ

)W

J

−1 2

M

.

).

This is a special case of equation 3.5.2.2 with  ( ) =  cosh( ΠIntegrating yields a second-order equation: 15.

ñóò ò ó ò ó

=

cosh÷ ( ƒñ )ñó ò + õ cosh( “ñ )(ñ„ó ò )3 .

ô

) and ( ) =  cosh(“/ ).

 =  sinh( Œ ) +  sinh(“/ ) + L . Œ “

This is a special case of equation 3.5.2.3 with  ( ) =  cosh  ( Œ

16.

= ôñ ÷ ñ ó ò + õ cosh( “ñ )(ñ ó ò )3 . This is a special case of equation 3.5.2.3 with  ( ) = 

17.

ñóò ò ó ò ó

18.

19.

ñ óò ò ó ò ó

=

1 2

21.

(ñó ò )3 + ô (cosh ƒñ )–ú

–3

(ñó ò )2ú

+1

).

.

This is a special case of equation 3.5.2.35 with  (ˆ ) = ˆ 2 + .

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó =

Solution: 20.

2

).

 and ( ) =  cosh(“ ).

= õ cosh( ƒñ )(ñ„ó ò )3 + ô (ñó ò )–5 . This is a special case of equation 3.5.2.4 with  ( ) =  cosh( Œ

ñóò ò ó ò ó

) and ( ) =  cosh(“

ññóò ò ó ò ó

2

=L

2



+ 3ñóÈ ò ñóò ò ó =

ô

ï

).

 +L

0

cosh(

2

ô

+L

1

cosh÷ (

ï

+ 2 Œ

−3

sinh( Π ).

).

This is a special case of equation 3.5.3.6 with  ( ) =  cosh  ( Π ).

+ 3ñ ó ò ñ ó ò ò ó = ô cosh÷ ( ƒñ ) + õ . This is a special case of equation 3.5.3.8 with  ( ) =  cosh  ( Œ

ññ óò ò ó ò ó

)+ .

Page 612

613

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

22. 23. 24. 25. 26.

27. 28.

+ 3ñ ó ò ñ ó ò ò ó + ô ï ÷ ññ ó ò = õ coshú ( ï ). + This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  cosh ( Œ  ).

ññ óò ò ó ò ó

+ 3ñ ó ò ñ ó ò ò ó + ô V ññ ó ò ò ó + (ñ ó ò )2 W = õ cosh÷ ( ï ). This is a special case of equation 3.5.3.13 with  ( ) =  cosh  ( Œ  ).

ññ óò ò ó ò ó

+ (3ñ ó ò + ôñ cosh÷ ï )ñ ó ò ò ó + ô cosh÷ ï (ñ ó ò )2 = 0. This is a special case of equation 3.5.3.17 with  ( ) =  cosh   .

ññ óò ò ó ò ó

(ñ + ô )ñ ó ò ò ó ò ó + õÿñ ó ò ñ ó ò ò ó + ö cosh÷ ( ƒñ )ñ ó ò = 0. This is a special case of equation 3.5.3.21 with  ( ) = ( cosh  ( Œ

ñ 2 ñ óò ò ó ò ó

+ 3þxññ

óò ñ óò ò ó

+þ

(þ

– 1)(ñ

óò

)3 =

ô

cosh (

ï )ñ 2–ú

). .

This is a special case of equation 3.5.3.27 with  ( ) =  cosh  ( Π ) and B =

¡

+ 1.

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô cosh÷ ( ï )(ñ ó ò )2 + õÿñ ú (ñ ó ò )4 . This is a special case of equation 3.5.4.7 with  ( ) =  cosh  ( Œ  ) and ( ) =  + . 2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô ï ÷ (ñ ó ò )2 + õ coshú ( ƒñ )(ñ ó ò )4 . + This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  cosh ( Œ

).

29.

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô cosh÷ ( ï )(ñ ó ò )2 + õ coshú ( ï )ñ –1 (ñ ó ò )5 ) 2 . + This is a special case of equation 3.5.4.8 with  ( ) =  cosh  ( Œ  ) and ( ) =  cosh (“/ ).

30.

2ñó ò ñóò ò ó ò

31.

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô ïƒ÷ (ñóò )2 + õ coshú ( ï )ñ –1(ñóò )5 ) 2 . 2ñó + This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =  cosh ( Œ  ).

32. 33. 34.

– 3(ñó ò ò ó )2 = ô cosh÷ ( ï )(ñó ò )2 + õ ïƒú ñ –1 (ñó ò )5 ) 2 . This is a special case of equation 3.5.4.8 with  ( ) =  cosh  ( Œ  ) and ( ) = ' + .

ó

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô cosh÷ ( ƒñ )(ñóò )4 + õ ï –1 coshú ( “ñ )(ñóò )7 ) 2 . + This is a special case of equation 3.5.4.9 with  ( ) =  cosh  ( Œ ) and ( ) =  cosh (“

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô cosh÷ ( ƒñ )(ñóò )4 + õ ï –1 ñ ú (ñóò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =  cosh  ( Œ

).

) and ( ) =  + .

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ôñ ÷ (ñóò )4 + õ ï –1 coshú ( ƒñ )(ñóò )7 ) 2 . + This is a special case of equation 3.5.4.9 with  ( ) =   and ( ) =  cosh ( Œ

).

3.4.3-3. Equations with hyperbolic tangent. 35. 36. 37.

ñ óò ò ó ò ó

= ô tanh( ƒñ )ñ ó ò . This is a special case of equation 3.5.2.1 with  ( ) =  tanh( Œ

= ô tanh( ƒñ )ñ ó ò + õ tanh( ï ). This is a special case of equation 3.5.2.2 with  ( ) =  tanh( Œ

ñ óò ò ó ò ó

). ) and ( ) =  tanh(“/ ).

+ 3ñ ó ò ñ ó ò ò ó = ô tanh÷ ( ï ) + õ . This is a special case of equation 3.5.3.6 with  ( ) =  tanh  ( Œ  ) +  .

ññ óò ò ó ò ó

Page 613

614 38.

39.

40.

41.

42.

43.

44.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ññ óò ò ó ò ó

óò ñ óò ò ó

+ 3ñ

=

tanh÷ ( ƒñ ) + õ .

ô

This is a special case of equation 3.5.3.8 with  ( ) =  tanh  ( Œ

ïƒ÷ ññóò

õ

tanhú (

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô V ññóò ò ó + (ñóò )2 W =

ññóò ò ó ò ó

+ (3ñó ò +

=

ï

)+  .

).

+ This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  tanh ( Π ).

õ

tanh÷ (

ï

).

This is a special case of equation 3.5.3.13 with  ( ) =  tanh  ( Π ).

ôñ

tanh÷

ï )ñ óò ò ó

tanh÷

ô

+

ï (ñ óò

)2 = 0.

This is a special case of equation 3.5.3.17 with  ( ) =  tanh   . (ñ + ô )ñ„ó ò ò ó ò

ó

+

+ ö tanh÷ ( ƒñ )ñó ò = 0.

õÿñó¿ò ñóò ò ó

This is a special case of equation 3.5.3.21 with  ( ) = ( tanh  ( Œ

ñ 2 ñ óò ò ó ò ó

+ 3þxññ

óò ñ óò ò ó

+þ

(þ

– 1)(ñ

óò

)3 =

ô

tanh (

ï )ñ 2–ú

). .

This is a special case of equation 3.5.3.27 with  ( ) =  tanh  ( Œ  ) and B = 2ñó ò ñóò ò ó ò

ó

ô

tanh÷ (

– 3(ñó ò ò ó )2 =

ï )(ñ óò

)2 + õ tanhú ( “ñ )(ñó ò )4 .

¡

+ 1.

+

This is a special case of equation 3.5.4.7 with  ( ) =  tanh  ( Œ  ) and ( ) =  tanh (“

óò ñ óò ò ó ò ó

45.

2ñ

46.

ò ñóò ò ó ò 2ñó

– 3(ñ

óò ò ó

tanh÷ (

ô

)2 =

ï )(ñ óò

)2 +

õÿñ ú (ñ óò

).

)4 .

This is a special case of equation 3.5.4.7 with  ( ) =  tanh  ( Π ) and ( ) =  + .

ó

ô ïƒ÷ (ñ óò

– 3(ñó ò ò ó )2 =

)2 + õ tanhú ( ƒñ )(ñó ò )4 .

+ This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  tanh ( Œ

47.

2ñó ò ñóò ò ó ò

48.

2ñ

49.

2ñó ò ñóò ò ó ò

50.

2ñ

ó

– 3(ñó ò ò ó )2 =

ô

tanh÷ (

ï )(ñ óò

)2 + õ tanhú (

óò ò ó

ô

tanh÷ (

ï )(ñ óò

)2 +

ï )ñ

–1

).

(ñó ò )5 ) 2 .

+

This is a special case of equation 3.5.4.8 with  ( ) =  tanh  ( Œ  ) and ( ) =  tanh (“/ ).

óò ñ óò ò ó ò ó

– 3(ñ

)2 =

õï ú ñ

–1

(ñ

ó ò )5 ) 2 .

This is a special case of equation 3.5.4.8 with  ( ) =  tanh  ( Π ) and ( ) = ' + .

ó

ô ïƒ÷ (ñ óò

– 3(ñó ò ò ó )2 =

)2 + õ tanhú (

ï )ñ

–1

(ñó ò )5 ) 2 .

+ This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =  tanh ( Π ).

óò ñ óò ò ó ò ó

– 3(ñ

óò ò ó

tanh÷ ( ƒñ )(ñ

ô

)2 =

óò

)4 +

õï

–1

tanhú ( “ñ )(ñ

This is a special case of equation 3.5.4.9 with  ( ) =  tanh  ( Œ tanh÷ ( ƒñ )(ñó ò )4 +

51.

2ñó ò ñóò ò ó ò

ó

– 3(ñó ò ò ó )2 =

ô

52.

ò ñóò ò ó ò 2ñó

ó

– 3(ñó ò ò ó )2 =

ôñ ÷ (ñ óò

õ ï –1 ñ ú (ñ óò )7 ) 2 .

This is a special case of equation 3.5.4.9 with  ( ) =  tanh  ( Π)4 +

õï

–1

ó ò )7 ) 2 . +

) and ( ) =  tanh (“

).

) and ( ) =  + .

tanhú ( ƒñ )(ñó ò )7 ) 2 .

This is a special case of equation 3.5.4.9 with  ( ) = 

 and ( ) =  tanh+ ( Π).

Page 614

615

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

3.4.3-4. Equations with hyperbolic cotangent. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.

ñóò ò ó ò ó = ô coth( ƒñ )ñ„óò . This is a special case of equation 3.5.2.1 with  ( ) =  coth( Œ

).

= ô coth( ƒñ )ñ„ó ò + õ coth( ï ). This is a special case of equation 3.5.2.2 with  ( ) =  coth( Œ

ñóò ò ó ò ó

ññ óò ò ó ò ó + 3ñ óò ñ óò ò ó = ô coth÷

) and ( ) =  coth(“/ ).

ï

( )+ õ . This is a special case of equation 3.5.3.6 with  ( ) =  coth  ( Œ  ) +  . + 3ñ ó ò ñ ó ò ò ó = ô coth÷ ( ƒñ ) + õ . This is a special case of equation 3.5.3.8 with  ( ) =  coth  ( Œ

ññ óò ò ó ò ó

)+  .

+ 3ñóÈ ò ñóò ò ó + ô ïƒ÷ ññóò = õ cothú ( ï ). + This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  coth ( Œ  ).

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô V ññóò ò ó + (ñóò )2 W = õ coth÷ ( ï ). This is a special case of equation 3.5.3.13 with  ( ) =  coth  ( Œ  ).

ññóò ò ó ò ó

+ (3ñó ò + ôñ coth÷ ï )ñó ò ò ó + ô coth÷ ï (ñó ò )2 = 0. This is a special case of equation 3.5.3.17 with  ( ) =  coth   .

ññóò ò ó ò ó

(ñ + ô )ñ„ó ò ò ó ò ó + õÿñó¿ ò ñóò ò ó + ö coth÷ ( ƒñ )ñóò = 0. This is a special case of equation 3.5.3.21 with  ( ) = ( coth  ( Œ

ñ 2 ñ óò ò ó ò ó

+ 3þxññ

óò ñ óò ò ó

+þ

(þ

– 1)(ñ

óò

)3 =

ô

coth (

ï )ñ 2–ú

). .

This is a special case of equation 3.5.3.27 with  ( ) =  coth  ( Π ) and B =

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô coth÷ 2ñó

ï )(ñ óò

)2 + õ cothú ( “

ñ )(ñ óò

¡

+ 1.

( )4 . +  This is a special case of equation 3.5.4.7 with  ( ) =  coth ( Œ  ) and ( ) =  coth (“

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô coth÷ ( ï )(ñóò )2 + õÿñ ú (ñóò )4 . This is a special case of equation 3.5.4.7 with  ( ) =  coth  ( Œ  ) and ( ) =  + . 2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô ïƒ÷ (ñóò )2 + õ cothú ( ƒñ )(ñóò )4 . + This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  coth ( Œ = ô coth÷

ï )(ñ óò

+ õ cothú ( ï )ñ

(ñó ò ) )

).

).

ò ñóò ò ó ò ó – 3(ñóò ò ó ) ( ) . 2ñó +  This is a special case of equation 3.5.4.8 with  ( ) =  coth ( Œ  ) and ( ) =  coth (“/ ). 2

2

–1

5 2

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô coth÷ ( ï )(ñ ó ò )2 + õ ï ú ñ –1 (ñ ó ò )5 ) 2 . This is a special case of equation 3.5.4.8 with  ( ) =  coth  ( Œ  ) and ( ) = ' + . 2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô ï ÷ (ñ ó ò )2 + õ cothú ( ï )ñ –1 (ñ ó ò )5 ) 2 . + This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =  coth ( Œ  ).

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô coth÷ ( ƒñ )(ñóò )4 + õ ï –1 cothú ( “ñ )(ñóò )7 ) 2 . + This is a special case of equation 3.5.4.9 with  ( ) =  coth  ( Œ ) and ( ) =  coth (“

69.

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô coth÷ ( ƒñ )(ñ óò

70.

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ôñ ÷ (ñóò )4 + õ ï –1 cothú ( ƒñ )(ñóò )7 ) 2 . + This is a special case of equation 3.5.4.9 with  ( ) =   and ( ) =  coth ( Œ

)4 + õ ï –1 ñ ú ( ñ ó ò )7 ) 2 .

This is a special case of equation 3.5.4.9 with  ( ) =  coth  ( Œ

).

) and ( ) =  + . ).

Page 615

616

THIRD-ORDER DIFFERENTIAL EQUATIONS

3.4.4. Equations Containing Logarithmic Functions 3.4.4-1. Equations of the form 


 =  ( , , 
 ). 1.

ñ óò ò ó ò ó

=

ñ óò ò ó ò ó

=

ñóò ò ó ò ó

=

ñóò ò ó ò ó

=

ï

ôñ

(

ô ï

–3

ô ï

–4

+þ

ln ñ ).

This is a special case of equation 3.5.1.16 with  ( H ) =  ln H . 2.

ln ï ).

( ƒñ + þ

This is a special case of equation 3.5.1.17 with  ( H ) =  ln H . 3.

(ln ñ – 2 ln ï ).

This is a special case of equation 3.5.1.7 with  ( H ) =  ln H . 4.

ô

ln( ƒñ )ñó ò .

This is a special case of equation 3.5.2.1 with  ( ) =  ln( Π5.

ñóò ò ó ò ó

=

ñ óò ò ó ò ó

=

ô

ln( ƒñ )ñó ò + õ ln(

ô

ln÷ ( ƒñ )ñ

ï

).

This is a special case of equation 3.5.2.2 with  ( ) =  ln( Π6.

7.

8.

+ õ lnú ( “ñ )(ñ

óò

óò

) and ( ) =  ln(“/ ).

)3 .

This is a special case of equation 3.5.2.3 with  ( ) =  ln  ( Œ ln÷ ( ƒñ )ñó ò +

õÿñ ú (ñ óò

ñóò ò ó ò ó

=

ô

ñóò ò ó ò ó

=

ôñ ÷ ñóò

ñóò ò ó ò ó

=

ñ óò ò ó ò ó

=

ñóò ò ó ò ó

=

ñóò ò ó ò ó

=

ñóò ò ó ò ó

=

ô 3ñ

ñ óò ò ó ò ó

=

ô

ln ï ( ï

ñ óò

– ñ )÷ .

ñóò ò ó ò ó

=

ô

ln ï ( ï

ñóò

– 2ñ )÷ .

)3 .

This is a special case of equation 3.5.2.3 with  ( ) =  ln  ( Œ + õ lnú ( ƒñ )(ñó ò )3 .

This is a special case of equation 3.5.2.3 with  ( ) =  9.

).

ô ï

–3

ô ï

–5

(ln ñ – ln ï )( ï

+

) and ( ) =  ln (“

).

) and ( ) =  + .

 and ( ) =  ln+ ( Π).

– ñ ).

ñ„óò

This is a special case of equation 3.5.2.5 with  (ˆ ) =  ln ˆ . 10.

(ln ñ – 2 ln ï )( ï

ñ óò

– 2ñ ).

This is a special case of equation 3.5.2.8 with  (ˆ ) =  ln ˆ . 11.

ôñ –5 ) 2 (2 ln ñóò

– ln ñ ).

ôñ –5 ) 4 (4 ln ñóò

– ln ñ ).

This is a special case of equation 3.5.2.27 with  (ˆ ) = 2  ln ˆ . 12.

This is a special case of equation 3.5.2.28 with  (ˆ ) = 4  ln ˆ . 13.

14.

15.

+ õ ln ï (ñó ò – ôñ )÷ .

This is a special case of equation 3.5.2.16 with  ( , ¨ ) =  ¨  ln  . This is a special case of equation 3.5.2.20 with  ( , ¨ ) =  ¨  ln  . This is a special case of equation 3.5.2.21 with  ( , ¨ ) =  ¨  ln  .

Page 616

617

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

3.4.4-2. Other equations. 16. 17.

ñ óò ò ó ò ó

= –3ñ

+ ô ( ï + ln ñ )÷ (ñ

óò ò ó

óò

+ ñ ) + 2ñ .

This is a special case of equation 3.5.3.33 with  (ˆ ) =  ln  ˆ .

ï ñóò ò ó ò ó

= õ (ï

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó =

ô

ln÷ ( õ

ññ óò ò ó ò ó

+ 3ñ

óò ñ óò ò ó

ô

ln÷ ( õÿñ ).

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô V ññóò ò ó + (ñóò )2 W =

ñóò

–ñ +

ln ï )ñó ò ò ó .

ô

This is a special case of equation 3.5.3.45 with  (ˆ ) = 'ˆ . 18. 19. 20. 21. 22. 23.

24.

ï

).

This is a special case of equation 3.5.3.6 with  ( ) =  ln  ( ' ). =

This is a special case of equation 3.5.3.8 with  ( ) =  ln  (  ).

ïƒ÷ ññóò

=

õ

lnú (

ï

).

+ This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  ln ( Œ  ). ln÷ (

õ

ï

).

This is a special case of equation 3.5.3.13 with  ( ) =  ln  ( Π ).

óò ò ó ò ó

( ñ + ô )ñ

+

+ ö ln÷ ( ƒñ )ñ

õÿñ óò ñ óò ò ó

óò

= 0.

This is a special case of equation 3.5.3.21 with  ( ) = ( ln  ( Œ

ñ 2 ñ óò ò ó ò ó

+ 3þxññóÈ ò ñóò ò ó + þ

(þ

– 1)(ñ„ó ò )3 =

ln  ( õ

ô

ï )ñ 2–ú

). .

This is a special case of equation 3.5.3.27 with  ( ) =  ln  ( ' ) and B =

ò ñóò ò ó ò 2ñó

ó

– 3(ñó ò ò ó )2 =

ôñ

2ñó ò ñóò ò ó ò

ó

– 3(ñó ò ò ó )2 =

ô

4

(ln ñó ò – 2 ln ñ ).

¡

+ 1.

This is a special case of equation 3.5.4.14 with  (ˆ ) =  ln ˆ .

25.

ln÷ (

ï )(ñ óò

)2 + õ lnú ( “ñ )(ñó ò )4 .

+

This is a special case of equation 3.5.4.7 with  ( ) =  ln  ( Œ  ) and ( ) =  ln (“

óò ñ óò ò ó ò ó

26.

2ñ

27.

2ñó ò ñóò ò ó ò

– 3(ñ

óò ò ó

ln÷ (

ô

)2 =

ï )(ñ óò

)2 +

õÿñ ú (ñ óò

).

)4 .

This is a special case of equation 3.5.4.7 with  ( ) =  ln  ( Π ) and ( ) =  + .

ó

ô ïƒ÷ (ñ óò

– 3(ñó ò ò ó )2 =

)2 + õ ln ú ( ƒñ )(ñó ò )4 .

+ This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  ln ( Œ

ó

– 3(ñó ò ò ó )2 =

ô

ln÷ (

óò ò ó

ô

ln÷

28.

2ñó ò ñóò ò ó ò

29.

2ñ

30.

ò ñóò ò ó ò 2ñó

ó

– 3(ñó ò ò ó )2 =

ô ïƒ÷ (ñ óò

31.

2ñó ò ñóò ò ó ò

ó

– 3(ñó ò ò ó )2 =

ô

ï )(ñ óò

)2 + õ lnú (

ï )ñ

–1

(ñó ò )5 ) 2 .

).

+

This is a special case of equation 3.5.4.8 with  ( ) =  ln  ( Œ  ) and ( ) =  ln (“/ ).

óò ñ óò ò ó ò ó

– 3(ñ

)2 =

ï (ñ ó ò

)2 +

õï ú ñ

–1

(ñ

ó ò )5 ) 2 .

This is a special case of equation 3.5.4.8 with  ( ) =  ln   and ( ) = ' + . )2 + õ ln ú (

ï )ñ

–1

(ñó ò )5 ) 2 .

+ This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =  ln ( Œ  ). ln÷ ( ƒñ )(ñó ò )4 +

õï

–1

lnú ( “ñ )(ñó ò )7 ) 2 .

This is a special case of equation 3.5.4.9 with  ( ) =  ln  ( Œ

+

) and ( ) =  ln (“

).

Page 617

618 32.

THIRD-ORDER DIFFERENTIAL EQUATIONS 2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ôñ ÷ (ñóò )4 + õ ï –1 lnú ( ƒñ )(ñóò )7 ) 2 . + This is a special case of equation 3.5.4.9 with  ( ) =   and ( ) =  ln ( Œ

33.

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô ln÷ ( ƒñ )(ñóò )4 + õ ï –1 ñ ú (ñóò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =  ln  ( Œ

34.

ññóò ò ó ò ó

).

) and ( ) =  + .

ln ñ – ñó ò ñóò ò ó ln ñóò ò ó + ôñóÈò ñóò ò ó = 0. k . Integrating yields a second-order autonomous equation of the form 2.9.1.1: /

 =  

3.4.5. Equations Containing Trigonometric Functions 3.4.5-1. Equations with sine. 1.

ñóò ò ó ò ó

=

ô

sin( ƒñ )ñ„ó ò .

Solution: L 2.

3

AC = X V L

+L

2

1

4. 5.

7. 8. 9. 10. 11. 12.

sin( Œ

)W

J

−1 2

M

.

) and ( ) =  sin(“/ ). Integrating

 = −  cos( Œ ) −  cos(“/ ) + L . Œ “

= ô sin÷ ( ƒñ )ñó ò + õ sin( “ñ )(ñ„ó ò )3 . This is a special case of equation 3.5.2.3 with  ( ) =  sin  ( Œ

ñóò ò ó ò ó ñóò ò ó ò ó

=

1 2

2

(ñó ò )3 + ô (sin ƒñ )–ú

–3

(ñó ò )2ú

+1

) and ( ) =  sin(“

).

.

This is a special case of equation 3.5.2.36 with  (ˆ ) = ˆ 2 + .

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó =

Solution:

6.

−2

= ô sin( ƒñ )ñ„ó ò + õ sin( ï ). This is a special case of equation 3.5.2.2 with  ( ) =  sin( Œ

ñóò ò ó ò ó

yields a second-order equation: 3.

− 2 Œ

2

=L

2



ô

2

sin( +L

1

ï

).

 +L

0

+ 2 Œ

−3

cos( Π ).

+ 3ñ ó ò ñ ó ò ò ó = ô sin ÷ ( ï ) + õ . This is a special case of equation 3.5.3.6 with  ( ) =  sin  ( Œ  ) +  .

ññ óò ò ó ò ó

+ 3ñ ó ò ñ ó ò ò ó = ô sin ÷ ( ƒñ ) + õ . This is a special case of equation 3.5.3.8 with  ( ) =  sin  ( Œ

ññ óò ò ó ò ó

)+  .

+ 3ñ ó ò ñ ó ò ò ó + ô ï ÷ ññ ó ò = õ sinú ( ï ). This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  sin + ( Œ  ).

ññ óò ò ó ò ó

+ 3ñ ó ò ñ ó ò ò ó + ô V ññ ó ò ò ó + (ñ ó ò )2 W = õ sin÷ ( ï ). This is a special case of equation 3.5.3.13 with  ( ) =  sin  ( Œ  ).

ññ óò ò ó ò ó

+ (3ñ ó ò + ôñ sin ÷ ï )ñ ó ò ò ó + ô sin ÷ ï (ñ ó ò )2 = 0. This is a special case of equation 3.5.3.17 with  ( ) =  sin   .

ññ óò ò ó ò ó

(ñ + ô )ñ ó ò ò ó ò ó + õÿñ ó ò ñ ó ò ò ó + ö sin÷ ( ƒñ )ñ ó ò = 0. This is a special case of equation 3.5.3.21 with  ( ) = ( sin  ( Œ

).

+ 3þxññ ó ò ñ ó ò ò ó + þ (þ – 1)(ñ ó ò )3 = ô sin  ( ï )ñ 2–ú . This is a special case of equation 3.5.3.27 with  ( ) =  sin  ( Œ  ) and B =

ñ 2 ñ óò ò ó ò ó

¡

+ 1.

Page 618

619

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

3.4.5-2. Equations with cosine. 13.

ñ óò ò ó ò ó

ô

Solution: L 14.

ñ óò ò ó ò ó

óò

cos( ƒñ )ñ

=

ô

AC = X V L

3

óò

cos( ƒñ )ñ

=

. +L

2

+ õ cos(

1

− 2 Œ

ï

).

−2

J

−1 2

)W

cos( Œ

M

.

This is a special case of equation 3.5.2.2 with  ( ) =  cos( ΠIntegrating, we obtain a second-order equation: 15. 16.

ñóò ò ó ò ó

=

cos÷ ( ƒñ )ñó ò + õ cos( “ñ )(ñ„ó ò )3 .

ô

) and ( ) =  cos(“/ ).

 =  sin( Œ ) +  sin(“/ ) + L . Œ “

This is a special case of equation 3.5.2.3 with  ( ) =  cos  ( Œ

ñóò ò ó ò ó

=

ñóò ò ó ò ó

=

ôñ ÷ ñóò

+ õ cos( “ñ )(ñ„ó ò )3 .

õ

cos( ƒñ )(ñ„ó ò )3 + ô (ñó ò )–5 .

This is a special case of equation 3.5.2.4 with  ( ) =  cos( Π18.

19.

20. 21. 22. 23. 24. 25. 26. 27.

ñóò ò ó ò ó

=

).

 and ( ) =  cos(“ ).

This is a special case of equation 3.5.2.3 with  ( ) =  17.

) and ( ) =  cos(“

1 2

2

(ñó ò )3 + ô (cos ƒñ )–ú

–3

(ñó ò )2ú

+1

).

.

This is a special case of equation 3.5.2.35 with  (ˆ ) = ˆ 2 + .

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó =

Solution:

2

ññ óò ò ó ò ó

+ 3ñ

ññ óò ò ó ò ó

ô

ï

cos(

).



2

óò ñ óò ò ó

=

ô

cos÷ (

+ 3ñ

óò ñ óò ò ó

=

ô

cos÷ ( ƒñ ) + õ .

ññ óò ò ó ò ó

+ 3ñ

óò ñ óò ò ó

+ô

ññ óò ò ó ò ó

+ 3ñ

óò ñ óò ò ó

+ ô V ññ

ññ óò ò ó ò ó

+ (3ñ

=L

2

+L

1

 +L

0

ï

−3

− 2 Œ

sin( Π ).

).

This is a special case of equation 3.5.3.6 with  ( ) =  cos  ( Œ  ). This is a special case of equation 3.5.3.8 with  ( ) =  cos  ( Œ

ï ÷ ññ óò

cosú (

õ

=

ï

)+ .

).

This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  cos + ( Π ).

óò ò ó

+ (ñ

óò

)2 W =

+

ô

cos÷ (

õ

ï

).

This is a special case of equation 3.5.3.13 with  ( ) =  cos  ( Π ).

óò

+

ôñ

cos÷

ï )ñ ó ò ò ó

cos÷

ï (ñ ó ò

)2 = 0.

This is a special case of equation 3.5.3.17 with  ( ) =  cos   . ( ñ + ô )ñ

óò ò ó ò ó

+

+ ö cos÷ ( ƒñ )ñ

õÿñ óò ñ óò ò ó

óò

= 0.

This is a special case of equation 3.5.3.21 with  ( ) = ( cos  ( Œ

ñ 2 ñ óò ò ó ò ó

+ 3þxññ

óò ñ óò ò ó

+þ

(þ

– 1)(ñ

óò

)3 =

ô

cos (

ï )ñ 2–ú

). .

This is a special case of equation 3.5.3.27 with  ( ) =  cos  ( Œ  ) and B = 2ñ

óò ñ óò ò ó ò ó

– 3(ñ

óò ò ó

)2 =

ô

cos÷ (

ï )(ñ óò

)2 +

õÿñ ú (ñ óò

¡

+ 1.

)4 .

This is a special case of equation 3.5.4.7 with  ( ) =  cos  ( Π ) and ( ) =  + .

Page 619

620 28. 29. 30.

THIRD-ORDER DIFFERENTIAL EQUATIONS 2ñó ò ñó ò ò ó ò ó – 3(ñó ò ò ó )2 = ô ïƒ÷ (ñó ò )2 + õ cosú ( ƒñ )(ñó ò )4 . This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  cos + ( Œ

).

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô cos÷ ( ï )(ñóò )2 + õ cosú ( ï )ñ –1(ñóò )5 ) 2 . 2ñó This is a special case of equation 3.5.4.8 with  ( ) =  cos  ( Œ  ) and ( ) =  cos + (“/ ). 2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô cos÷ ( ï )(ñóò )2 + õ ïƒú ñ –1 (ñóò )5 ) 2 . This is a special case of equation 3.5.4.8 with  ( ) =  cos  ( Œ  ) and ( ) = ' + .

31.

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô ïƒ÷ (ñóò )2 + õ cosú ( ï )ñ –1(ñóò )5 ) 2 . 2ñó This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =  cos + ( Œ  ).

32.

2ñó ò ñóò ò ó ò

33. 34.

– 3(ñó ò ò ó )2 = ô cos÷ ( ƒñ )(ñó ò )4 + õ ï –1 cosú ( “ñ )(ñó ò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =  cos  ( Œ ) and ( ) =  cos + (“

ó

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô cos÷ ( ƒñ )(ñ ó ò )4 + õ ï –1 ñ ú (ñ ó ò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =  cos  ( Œ

).

) and ( ) =  + .

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ôñ ÷ (ñ ó ò )4 + õ ï –1 cosú ( ƒñ )(ñ ó ò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =   and ( ) =  cos + ( Œ

).

3.4.5-3. Equations with tangent. 35. 36. 37. 38. 39. 40. 41. 42. 43.

ñóò ò ó ò ó

= ô tan( ƒñ )ñ„ó ò . This is a special case of equation 3.5.2.1 with  ( ) =  tan( Œ = ô tan( ƒñ )ñ„ó ò + õ tan( ï ). This is a special case of equation 3.5.2.2 with  ( ) =  tan( Œ

ñóò ò ó ò ó

). ) and ( ) =  tan(“/ ).

+ 3ñ ó ò ñ ó ò ò ó = ô tan÷ ( ï ) + õ . This is a special case of equation 3.5.3.6 with  ( ) =  tan  ( Œ  ) +  .

ññ óò ò ó ò ó

+ 3ñ ó ò ñ ó ò ò ó = ô tan÷ ( ƒñ ) + õ . This is a special case of equation 3.5.3.8 with  ( ) =  tan  ( Œ

ññ óò ò ó ò ó

)+ .

+ 3ñóÈ ò ñóò ò ó + ô ïƒ÷ ññóò = õ tanú ( ï ). This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  tan + ( Œ  ).

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô V ññóò ò ó + (ñóò )2 W = õ tan÷ ( ï ). This is a special case of equation 3.5.3.13 with  ( ) =  tan  ( Œ  ).

ññóò ò ó ò ó

+ (3ñó ò + ôñ tan÷ ï )ñó ò ò ó + ô tan÷ ï (ñó ò )2 = 0. This is a special case of equation 3.5.3.17 with  ( ) =  tan   .

ññóò ò ó ò ó

(ñ + ô )ñ„ó ò ò ó ò ó + õÿñó¿ ò ñóò ò ó + ö tan÷ ( ƒñ )ñóò = 0. This is a special case of equation 3.5.3.21 with  ( ) = ( tan  ( Œ

).

+ 3þxññóÈ ò ñóò ò ó + þ (þ – 1)(ñ„óò )3 = ô tan ( ï )ñ 2–ú . This is a special case of equation 3.5.3.27 with  ( ) =  tan  ( Œ  ) and B =

ñ 2 ñ óò ò ó ò ó

¡

+ 1.

Page 620

621

3.4. NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

44. 45. 46.

2ñó ò ñó ò ò ó ò ó – 3(ñó ò ò ó )2 = ô tan÷ ( ï )(ñó ò )2 + õ tanú ( “ñ )(ñó ò )4 . This is a special case of equation 3.5.4.7 with  ( ) =  tan  ( Œ  ) and ( ) =  tan + (“

).

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô tan÷ ( ï )(ñóò )2 + õÿñ ú (ñóò )4 . 2ñó This is a special case of equation 3.5.4.7 with  ( ) =  tan  ( Œ  ) and ( ) =  + . 2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô ïƒ÷ (ñóò )2 + õ tanú ( ƒñ )(ñóò )4 . This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  tan + ( Œ

).

47.

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô tan÷ ( ï )(ñóò )2 + õ tanú ( ï )ñ –1(ñóò )5 ) 2 . 2ñó This is a special case of equation 3.5.4.8 with  ( ) =  tan  ( Œ  ) and ( ) =  tan + (“/ ).

48.

2ñó ò ñóò ò ó ò

49.

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô ï ÷ (ñ ó ò )2 + õ tanú ( ï )ñ –1 (ñ ó ò )5 ) 2 . This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =  tan + ( Œ  ).

50.

– 3(ñó ò ò ó )2 = ô tan÷ ( ï )(ñó ò )2 + õ ïƒú ñ –1 (ñó ò )5 ) 2 . This is a special case of equation 3.5.4.8 with  ( ) =  tan  ( Œ  ) and ( ) = ' + .

ó

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô tan÷ ( ƒñ )(ñ ó ò )4 + õ ï –1 tanú ( “ñ )(ñ This is a special case of equation 3.5.4.9 with  ( ) =  tan  ( Œ

51.

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô tan÷ ( ƒñ )(ñóò )4 + õ ï –1 ñ ú (ñóò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =  tan  ( Œ

52.

2ñó ò ñóò ò ó ò

ó ò )7 ) 2 .

) and ( ) =  tan + (“

).

) and ( ) =  + .

– 3(ñó ò ò ó )2 = ôñ ÷ (ñó ò )4 + õ ï –1 tanú ( ƒñ )(ñó ò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =   and ( ) =  tan + ( Œ

ó

).

3.4.5-4. Equations with cotangent. 53. 54. 55. 56. 57. 58. 59.

ñ óò ò ó ò ó

= ô cot( ƒñ )ñ ó ò . This is a special case of equation 3.5.2.1 with  ( ) =  cot( Œ

= ô cot( ƒñ )ñ ó ò + õ cot( ï ). This is a special case of equation 3.5.2.2 with  ( ) =  cot( Œ

ñ óò ò ó ò ó

). ) and ( ) =  cot(“/ ).

+ 3ñóÈ ò ñóò ò ó = ô cot÷ ( ï ) + õ . This is a special case of equation 3.5.3.6 with  ( ) =  cot  ( Œ  ) +  .

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó = ô cot÷ ( ƒñ ) + õ . This is a special case of equation 3.5.3.8 with  ( ) =  cot  ( Œ

ññóò ò ó ò ó

)+ .

+ 3ñóÈ ò ñóò ò ó + ô ïƒ÷ ññóò = õ cotú ( ï ). This is a special case of equation 3.5.3.10 with  ( ) =   and ( ) =  cot + ( Œ  ).

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ô V ññóò ò ó + (ñóò )2 W = õ cot÷ ( ï ). This is a special case of equation 3.5.3.13 with  ( ) =  cot  ( Œ  ).

ññóò ò ó ò ó

+ (3ñó ò + ôñ cot÷ ï )ñó ò ò ó + ô cot÷ ï (ñó ò )2 = 0. This is a special case of equation 3.5.3.17 with  ( ) =  cot   .

ññóò ò ó ò ó

Page 621

622 60. 61. 62. 63. 64. 65.

THIRD-ORDER DIFFERENTIAL EQUATIONS (ñ + ô )ñ ó ò ò ó ò ó + õÿñ ó ò ñ ó ò ò ó + ö cot÷ ( ƒñ )ñ ó ò = 0. This is a special case of equation 3.5.3.21 with  ( ) = ( cot  ( Œ

).

+ 3þxññ ó ò ñ ó ò ò ó + þ (þ – 1)(ñ ó ò )3 = ô cot ( ï )ñ 2–ú . This is a special case of equation 3.5.3.27 with  ( ) =  cot  ( Œ  ) and B =

ñ 2 ñ óò ò ó ò ó

+ 1.

¡

2ñ ó ò ñ ó ò ò ó ò ó – 3(ñ ó ò ò ó )2 = ô cot÷ ( ï )(ñ ó ò )2 + õ cotú ( “ñ )(ñ ó ò )4 . This is a special case of equation 3.5.4.7 with  ( ) =  cot  ( Œ  ) and ( ) =  cot + (“ 2ñó ò ñó ò ò ó ò ó – 3(ñó ò ò ó )2 = ô cot÷ ( ï )(ñó ò )2 + õÿñ ú (ñó ò )4 . This is a special case of equation 3.5.4.7 with  ( ) =  cot  ( Œ  ) and ( ) =  + .

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô ïƒ÷ (ñóò )2 + õ cotú ( ƒñ )(ñóò )4 . 2ñó This is a special case of equation 3.5.4.7 with  ( ) =   and ( ) =  cot + ( Œ

).

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô cot÷ ( ï )(ñóò )2 + õ cotú ( ï )ñ –1(ñóò )5 ) 2 . 2ñó This is a special case of equation 3.5.4.8 with  ( ) =  cot  ( Œ  ) and ( ) =  cot + (“/ ).

66.

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô cot÷ ( ï )(ñóò )2 + õ ïƒú ñ –1 (ñóò )5 ) 2 . This is a special case of equation 3.5.4.8 with  ( ) =  cot  ( Œ  ) and ( ) = ' + .

67.

2ñó ò ñóò ò ó ò

68.

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô cot÷ ( ƒñ )(ñóò )4 + õ ï –1 cotú ( “ñ )(ñóò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =  cot  ( Œ ) and ( ) =  cot + (“

69. 70.

).

– 3(ñó ò ò ó )2 = ô ïƒ÷ (ñó ò )2 + õ cotú ( ï )ñ –1 (ñó ò )5 ) 2 . This is a special case of equation 3.5.4.8 with  ( ) =   and ( ) =  cot + ( Œ  ).

ó

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ô cot÷ ( ƒñ )(ñóò )4 + õ ï –1 ñ ú (ñóò )7 ) 2 . 2ñó This is a special case of equation 3.5.4.9 with  ( ) =  cot  ( Œ

).

) and ( ) =  + .

2ñó ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ôñ ÷ (ñóò )4 + õ ï –1 cotú ( ƒñ )(ñóò )7 ) 2 . This is a special case of equation 3.5.4.9 with  ( ) =   and ( ) =  cot + ( Œ

).

3.5. Nonlinear Equations Containing Arbitrary Functions 3.5.1. Equations of the Form — ( æ , â ) â$äãÑãÑäã ä + ˜ ( æ , â ) = 0 3.5.1-1. Arguments of the arbitrary functions are  or . 1.

ñóò ò ó ò ó

= ! (ñ ). The substitution ¨ ( ) = (
 )2 leads to a second-order equation: ¨u

= A 2  ( )¨ −1 J 2 . In par ticular, with  ( ) =   the obtained equation is an Emden–Fowler equation; see Section 2.3.

2.

ñ óò ò ó ò ó

= ! ( ï )ñ

–1

.

1 b . On integrating the equation, we have ! 

 − 12 ( 
)2 = Xv ( )  + L . The substitution =¨

2



reduces the latter equation to the form ¨   =

2 b . The transformation  ‡ g

g
g = −a −2  (1 a )‡ −1.

= 1 a ,

= ‡ a

2

1 2

M Xv ( )  + L2S ¨ Q M

−3

.

leads to an equation of the same form:

Page 622

623

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

3.

4. 5.

= ï –3 ! (ñ ). The substitution a = ln | | leads to an autonomous equation of the form 3.5.5.9: /
g
g
g − 3
g
g + 2
g =  ( ).

ñóò ò ó ò ó

(ñ + ô ï 2 + õ ï + ö )ñó ò ò ó ò ó = ! ( ï ). The substitution ¨ = +  2 + ' + ( leads to an equation of the form 3.5.1.2: ¨*¨ 


 =  ( ).

ó

ó

( ôñ + õ )ñó ò ò ó ò ó + õ ñ = ! ( ï ). Integrating yields a second-order equation:

   +   ) 

 − 12  ( 
)2 −   
+  

(

= Xv ( )  + L .

M

3.5.1-2. Arguments of the arbitrary functions depend on  and . 6.

7.

8.

= ï –2 ! (ñ ï –1 ). The transformation a = ln | |, ¨ =  ¨2g

g
g − ¨2g
=  (¨ ).

ñ óò ò ó ò ó

= ï –4 ! (ñ ï –2 ). The transformation a =  ¨2g

g
g = −  (¨ ).

ñ óò ò ó ò ó

−1

, ¨ = 

−1

leads to an autonomous equation of the form 3.5.5.9:

−2

leads to an autonomous equation of the form 3.5.1.1:

= ï –  –3 ! ( ï  ñ ). Generalized homogeneous equation.

ñ óò ò ó ò ó

1 b . The transformation a = ln  , H =  

leads to an autonomous equation.

2 b . The transformation H =   , ¨ =  
 leads to a second-order equation.

9.

10.

11.

= ñ ï –3 ! ( ïƒ÷ ñ ú ). Generalized homogeneous equation. The transformation H =   + , ¨ second-order equation.

ñóò ò ó ò ó

=  
  leads to a

= ! (ñ + ô ï 3 + õ ï 2 + ö ï + ü ). The substitution ¨ = +  3 + ' 2 + (˜ + , leads to an autonomous equation of the form 3.5.1.1: ¨3


 =  (¨ ) + 6  .

ñ óò ò ó ò ó ï (ï

óò ò ó ò ó

– ô )3 ñ

= ! (ñ

ï

–2

),

The transformation ˆ = ln û û 3.5.5.9: ¨3Šª

Šª
Š − 3¨2Šª

Š + 2¨2Š
û

ô

≠ 0.

 − , ¨ =  ûû ¨  =  −3  û ( ).

2

leads to an autonomous equation of the form

ô ï  ï

12.

+ õÿñ + ö . + ñ + P This is a special case of equation 5.2.6.19 with B = 3.

13.

(ô

ñóò ò ó ò ó ï

2

= (ô

+

õï

ï

+

õÿñ

+ ö )–2 !IO

+ ö )2 ñó ò ò ó ò

ó

=I ! O

ô ï

ñ

2

+

õï

+ö P

.

 , ¨ = 2 M + '   + ( ¨ Š
=  (¨  ). + ' + ( of the form 3.5.5.9: ¨3ªŠ

Šª
Š + (4 !( −  2 )2 The transformation ˆ = X



2

leads to an autonomous equation

Page 623

624 14.

15.

16.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

ñ

ñ –2 I ! O ï ô

. +õ ï +ö P Setting  (‡ ) = ‡ 2  1 (‡ ), we have equation 3.5.1.13 with the function  =

2

ñóò ò ó ò ó =   ó ! (ñ  – ó

1

(instead of  ).

).  This is a special case of equation 3.5.3.32 with  =  = ( = 0. The substitution ¨ ( ) =  −  leads to an autonomous equation.

ó

= ñ! (   ñ ú ).  This is a special case of equation 3.5.5.21. The transformation H = ! + , ¨ ( H ) = 
  leads to a second-order equation.

ñ óò ò ó ò ó

17.

= ï –3 ! ( ïƒú ] ). The transformation H =  +   , ¨ ( H ) =  
 leads to a second-order equation.

18.

ñ óò ò ó ò ó

19.

ñóò ò ó ò ó

= ! (ñ + ô  The substitution ¨ ¨2


 =  (¨ ).

ó

ó

) – ô 3   .  = +  ž  leads to an autonomous equation of the form 3.5.1.1:

= H ( ï , ñ ). The transformation  = 1 a , −a −4  (1 a , ¨u a 2 ).

ñóò ò ó ò ó

= ¨u a

2

¨ g

g
g = leads to an equation of the same form: u

3.5.2. Equations of the Form — ( æ , â , â$ä ã ) âäãÑãÑäã ä + ˜ ( æ , â , âä ã ) = 0 3.5.2-1. Arguments of the arbitrary functions depend on  and . 1.

ñ óò ò ó ò ó

= ! (ñ )ñ

Solution: L 2.

ñ óò ò ó ò ó

3

= ! (ñ )ñ

óò

.

AC = X

óò

Q

L

2

+L

1

+2X  ( )

+ " ( ï ).

S M

J

−1 2

M

, where  ( ) = X



 = X  ( )

Integrating yields a second-order equation:

ñóò ò ó ò ó

M

 ( ) M

.

+ X ( )  + L .

M

3.

= ! (ñ )ñó ò + " (ñ )(ñó ò )3 . The substitution H ( ) = (
 )2 leads to a second-order linear equation: H

= 2 ( ) H + 2  ( ).

4.

ñ óò ò ó ò ó = ! (ñ )(ñ óò ) + ô (ñ óò ) . The substitution H ( ) = A (
 )2 leads to Yermakov’s equation 2.9.1.2: H

= 2  ( ) H + 2 !H



5.

3

ñóò ò ó ò ó

=

ñ ï –3 I ! O ï

–5

(ï

P



ñóò

−3

.

– ñ ).

The transformation H =   , ¨ =  −2 ( 
− )2 leads to a second-order linear equation: ¨2‘'

‘ = 2  ( H ) + 2. Integrating the latter equation twice, we arrive at a first-order homogeneous equation for ( ):


= H…A 6.

ñóò ò ó ò ó

=

‘

Q

2

H + L 2H + L

ñ ï –5 I ! O ï

P

(ï

1 +2

ñóò

X ‘ ( H − a )  (a ) Ýa S M 0

–ñ ) .

J

1 2

,

where

H = 

, H

0

is an arbitrary number.

3

The transformation H =   , ¨ ¨2‘'

‘ = 2  ( H )¨ + 2.

= 

−2

( 
 − )2 leads to a second-order linear equation:

Page 624

625

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

7.

ñóò ò ó ò ó

=

ñ ï –3 I ! O ï P

(ï

ñóò

–ñ )+

ñ ï –5 3 " O ï

(ï

P

ñóò

–ñ ) . 3

This is a special case of equation 3.5.3.38 with , = −1. The transformation a = ln  , H =   , followed by the substitution ¨ ( H ) = ( H g
)2 , leads to a second-order linear equation: ¨2‘'

‘ = 2 ( H )¨ + 2  (H ) + 2. 8.

ñóò ò ó ò ó

=

ñ ï –5 I ! O ï 2 (ï ñ óò P

– 2ñ ).

The transformation a = 1  , H =   2 leads to an autonomous equation: H g

g
g =  ( H ) H g
. The substitution ¨ ( H ) = ( H g
)2 then yields the second-order linear equation ¨u‘'

‘ = 2  ( H ), whose solution is given by:

¨ =L H +L 2 9.

ñóò ò ó ò ó

=

‘

1

+ 2 X ‘ ( H − ˆ )  (ˆ ) ˆ ,

ñ ï –7 I ! O ï 2 (ï ñ óò P

– 2ñ ) .

ñ ï –5 I ! O ï 2 (ï ñ óò P

– 2ñ ) +

H 0 is an arbitrary number.

M

0

3

The transformation a = 1  , H =   2 , followed by the substitution ¨ ( H ) = ( H g
)2 , leads to a ‘'‘ second-order linear equation: ¨u

= 2  ( H )¨ . 10.

ñóò ò ó ò ó

=

ñ ï –7 3 " O ï 2 (ï ñ óò P

– 2ñ ) . 3

The transformation a = 1  , H =   2 , followed by the substitution ¨ ( H ) = ( H g
)2 , leads to a second-order linear equation: ¨u‘'

‘ = 2 ( H )¨ + 2  ( H ). 11.

ñóò ò ó ò ó + „! ñóò + I" ñ

= –3 %„ñ

–4

(ñó ò )2 – 3( %„òóÈñ

3

+ 2%

2

)ñ

–6

ñóò – % 3 ñ

–8

– 3 %%„ó òñ

–5

– ( !% + %„ò òó

ó )ñ

–2

.

Here,  =  ( ), = ( ), and % = % ( ) are arbitrary functions. % ( )  1 J 3 ¨ ¨ = ¨ L +3X Solution: ¨ 3M S , where = ( ) is the general solution of the

¨ 


 Q +  ( )2 ¨ 
+ ( )¨ = 0. linear equation: 3 12.

[ ôñ + ! ( ï )]ñ

óò ò ó ò ó

+ " (ñ )ñ

óò

+

! ó ò ò ó ò ó ( ï )ñ

+ % ( ï ) = 0.

The equation admits a first integral: [








+  ( )]   − 12  (  )2 −   ( )  +   ( ) + Xw ( )

+ X¯% ( )  = L .

M

M

3.5.2-2. Arguments of the arbitrary functions depend on  , , and
 . 13.

ñóò ò ó ò ó

= ! (ñó ò ).

Solution in parametric form:

U  =X k* M , Y ( U ) 2

14.

15.

ñóò ò ó ò ó

= ! ( ï , ñó ò ).

ñóò ò ó ò ó

= ! (ñ , ñó ò ).

=X

* U MU k 3

Y (U )

,

where

Y =A L Q

1

+ 2 Xv (U ) U S

J

1 2

M

.

¨ 

 =  ( , ¨ ). The substitution ¨ ( ) = 
leads to a second-order equation: u Autonomous equation (it is a special case of equation 3.5.5.9).

1 b . The substitution ‡ ( ) = (
 )2 leads to a second-order equation: ‡

= A 2‡



J  ( , A ‡ 1 J 2 ).

−1 2

Page 625

626

THIRD-ORDER DIFFERENTIAL EQUATIONS 2 b . The transformation

 = X [ Y (U )]−3 J

2

M

= [ Y (U )]−1

U ,

(1)

leads to an analogous equation with respect to Y = Y (U ):

Y




*+*+*

= −Y

J  (Y

−5 2

−1

J Y
).

−1 2

, −Y

*

Note two important cases of transforming equations of special form:




 =  ( ) 


 = #  16. 17.

18.

20. 21. 22.

‰l‰-transformation ‰-‰-‰-‰-‰-‰-‰-‰-‰-(1)‰D

*+*+*

+ ! ( ï , ñó ò – ôñ ).

ñóò ò ó ò ó

=

ô 3ñ

ñóò ò ó ò ó

= (3 ô

ñóò ò ó ò ó

=

ñóò ò ó ò ó

=

ñ óò ò ó ò ó

= ! (ï , ï

ñ óò

ñóò ò ó ò ó

= ! (ï ,

ï ñóò

The substitution ¨ = 
−  2

ï

–ô

3

ï 3 )ñ

leads to a second-order equation: ¨ 

 +  ¨ 
+  2 ¨ =  ( , ¨ ).

+ ! ( ï , ñó ò + ô

ï ñ

).

The substitution ¨ = 
+  leads to a second-order equation: ¨ 

 −  ¨ 
+ (  2  2 − 2  )¨ =  ( , ¨ ).

ï ! (ï ñ óò

– ñ ).

The substitution H =  "H 

 = H 
+  3  ( H ). 19.

Y


= − Y −5 J 2  ( Y −1 ), *+*+* Y


= − #2Y −  −5 J 2 .

‰l‰-transformation ‰-‰-‰-‰-‰-‰-‰-‰-‰-(1)‰D

ï –3 ! (ï ñ óò


− leads to a second-order equation of the form 2.9.2.20 with B = 1:

– ñ ).

The transformation a = ln | |, H =  
 − leads to the second-order autonomous equation H g

g − 2 H g
=  ( H ), which is reduced, with the aid of the substitution ¨ ( H ) = 12 H g
, to the Abel equation ¨*¨2‘
− ¨ = 14  ( H ) (for some functions  , solutions of this Abel equation are given in Subsection 1.3.1). – ñ ).

The substitution ¨ =  
−

¨ 
 )
 =  ( , ¨ ). leads to a second-order equation: (u

– 2ñ ).

¨ 

 =   ( , ¨ ). The substitution ¨ =  
− 2 leads to a second-order equation: u

ï 3 ñ óò ò ó ò ó

= ! (ï , ï

ñ óò + ôñ ) – ô (ô + 1)(ô + 2)ñ . The substitution ¨ =  
+  leads to a second-order equation:  2 ¨ 

 − (  + 2) ¨ 
+ (  + 1)( + 2)¨ =  ( , ¨ ).

23.

ñ óò ò ó ò ó

=

ï –2 H O ñ óò

–

ï

ñ

.

P

The substitution H =  
− leads to the second-order equation "H 

 = H 
+  ( H  ), which is a special case of the equation 2.9.4.22 with B = −1, = 1, , = −1,  (ˆ ) = ˆ´ (ˆ ). 24.

ñóò ò ó ò ó = ï –4 I ! Oiñóò

¡

ñ –2 ï

P

.

The transformation a = 1  , H =   2 yields H g

g
g = −  (− H g
). The substitution ¨ = − H g
leads to a second-order autonomous equation of the form 2.9.1.1: ¨Ig

g =  (¨ ).

Page 626

627

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

25.

ñóò ò ó ò ó

=

ñ

1

ï 4 !IO ï

2

ñ

ñóò

,

–2ï

.

P

The transformation a = 1  , H =   2 leads to an equation of the form 3.5.2.15: H g

g
g = −  ( H , − H g
), which admits, with the aid of the substitution ¨ ( H ) = ( H g
)2 , reduction ‘'‘ of its order: ¨3

= A 2¨ −1 J 2  ( H , A ¨ 1 J 2 ). 26.

ñ óò ò ó ò ó

=

ñ ï –3 ! (ï ñ óò ¦ñ

).


 , ¨ = 

The transformation H =  2¨ − H ¨ +  ( H ). 27.

ñóò ò ó ò ó

=

ñ óò ñ –5 ) 2 e ! ” 7 ñ

2



  leads to a first-order equation: (¨ + H − H 2)¨2‘
=

.

•

¨

= The substitution ¨ ( ) = (
 )2 leads to an equation of the form 2.9.1.8: u   where  (ˆ ) = A 2ˆ −1 J 2  (Aˆ 1 J 2 ). 28.

ñ óò ò ó ò ó

=

ñ –5 ) 4 !

ñ óò

”

ñóò ò ó ò ó

= ( ôñ

2

+

õÿñ

–5 4

ñ óò ò ó ò ó

= ï ( ô ñ

2

−3

¨

¥” ü

+ õ ï 2ñ

+ö ï



2

+( •

+

) !e”

4 –5 4

)

The transformation a = 1  , H =  

2

,

ñóò ò ó ò ó

= –ï

–2

ñóò

+

ï –3 ñ

+

 (ˆ ) = A 2ˆ 5 J 2  (Aˆ 1 J 2 ).

where

ï ñ ó ò – 2ñ ( ô ñ 2 + õ ï 2ñ + ö ï

) •

4 )1 4

.

leads to an equation of the form 3.5.2.29:

−5 J 4 H g

g
g = −( !H 2 + ˜H + ( )  ”

31.

J ,

−1 2

ñ óò ( ô ñ 2 + õÿñ

) !e”

+ö )

J   ¨*

−3 2

. + ö )1 ) 4 • The substitution ¨ ( ) = (
 )2 leads to an equation of the form 2.9.1.21:

¨

= ¨  30.

 (¨u  ),

.

ñ 1) 4 •

The substitution ¨ ( ) = ( 
)2 leads to an equation of the form 2.9.1.9: ¨

=   where  (ˆ ) = A"ˆ −1 J 2  (Aˆ 1 J 2 ). 29.

−3

ï ñ óò – ñ ï 1 ) 2 ñ –5 ) 2 e ! ” 7 ï ñ

( !H

•

2

−H g


+ ˜H + ( )1 J 4 •

.

.

The transformation a = ln  , H =   , followed by the substitution ¨ ( H ) = ( H g
)2 , leads to a second-order equation of the form 2.9.1.8: ¨p‘'

‘ = H −3  (¨u H ), where  (ˆ ) = A 2ˆ −1 J 2   A £ ˆ  . 32.

ñóò ò ó ò ó

= –ï

–2

ñóò

+

ï –3 ñ

+

ï ñ óò – ñ ï –3 ) 4 ñ –5 ) 4 e ! ” ï 3) 4 1) 4 ñ •

.

The transformation a = ln  , H =   , followed by the substitution ¨ ( H ) = ( H g
)2 , leads ¨ ‘'

‘ = H −3 J 2  (¨ H −1 J 2 ), where  (ˆ ) = to a second-order equation of the form 2.9.1.9: p A 2ˆ −1 J 2 l A £ ˆ  . 33.

ñóò ò ó ò ó

ñóò !  ñ óò 2 + ôñ

. ¨ The substitution ( ) = (
 )2 +  2.9.1.1: ¨3

= 2  (¨ ).  =

leads to a second-order autonomous equation of the form

Page 627

628 34.

35.

36.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

= 2 2 (ñ ó ò )3 +  6 ] ! (  ] ñ ó ò )ñ ó ò . This is a special case of equation 3.5.2.48 with j ( ) = 

ñóò ò ó ò ó

 .

ñ óò

ñóò . cosh ƒñ • This is a special case of equation 3.5.2.48 with j ( ) = cosh Œ ñóò ò ó ò ó

=

1 2

2

(ñó ò )3 + (cosh ƒñ )–3 !e”

−2

(ñó ò )3 + (sinh ƒñ )–3 !e”

7

.

ñ óò

ñóò . sinh ƒñ • This is a special case of equation 3.5.2.48 with j ( ) = sinh Œ =

1 2

2

7

.

37.

= ñ tanh ï + ! ( ï , ñ„ó ò – ñ tanh ï ). This is a special case of equation 3.5.2.47 with Y ( ) = cosh  .

38.

ñóò ò ó ò ó

ñóò ò ó ò ó

= (sinh ñ )–2 (ñó ò )3 + (tanh ñ )3 !  ñó ò tanh ñ  ñó ò . ü This is a special case of equation 3.5.2.48 with j ( ) = coth .

39.

= ñ coth ï + ! ( ï , ñ ó ò – ñ coth ï ). This is a special case of equation 3.5.2.47 with Y ( ) = sinh  .

40.

ñ óò ò ó ò ó

= –(cosh ñ )–2 (ñ ó ò )3 + (coth ñ )3 !u ñ ó ò coth ñ  ñ ó ò . ü This is a special case of equation 3.5.2.48 with j ( ) = tanh .

41.

ñóò ò ó ò ó

42.

ñ óò ò ó ò ó

ñóò ò ó ò ó

2

(ñó ò )3 + (cos ƒñ )–3 !e”

ñ óò

ñóò . cos ƒñ • This is a special case of equation 3.5.2.48 with j ( ) = cos Œ = – 12

2

.

ñ óò

ñóò . sin ƒñ • This is a special case of equation 3.5.2.48 with j ( ) = sin Œ = – 12

(ñó ò )3 + (sin ƒñ )–3 !e”

7

7

43.

= ñ tan ï + ! ( ï , ñ„ó ò + ñ tan ï ). This is a special case of equation 3.5.2.47 with Y ( ) = cos  .

44.

ñóò ò ó ò ó

.

ñóò ò ó ò ó

= (sin ñ )–2 (ñó ò )3 + (tan ñ )3 !  ñó ò tan ñ  ñó ò . ü This is a special case of equation 3.5.2.48 with j ( ) = cot .

45.

= –ñ cot ï + ! ( ï , ñ ó ò – ñ cot ï ). This is a special case of equation 3.5.2.47 with Y ( ) = sin  .

46.

ñóò ò ó ò ó

= (cos ñ )–2 (ñó ò )3 + (cot ñ )3 !  ñó ò cot ñ  ñó ò . ü This is a special case of equation 3.5.2.48 with j ( ) = tan .

47.

ñóò ò ó ò ó

ñ óò ò ó ò ó

=

r óò ò óò ó r ñ

+I ! O

ï , ñ óò

–

Y
 The substitution ¨ = 
− Y

r óò r ñ P

,

r

=

r (ï

).

leads to a second-order equation:





2 ¨ 

 + Y  ¨ 
+ 2 Y   − ( Y  ) S ¨ =  ( , ¨ ). Y Q Y Y 2

Page 628

629

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

48.

ñó 1 òò s ]f] (ñ óò )3 + –3 ! ” 7 ò ñ óò , = (ñ ). • 2 s s s s s
2 The substitution H = (  ) leads to a second-order equation of the form 2.9.1.46: ñ óò ò ó ò ó

=

j

H

=  H + 2j  j

49.

−3

u”A

H . j •

= H ( ï , ñ , ñó ò ). Let  ≠ Y ( )
 + j ( ) + ¶ ( ), i.e., the equation is nonlinear. Then its order can be reduced by one if the right-hand side of the equation has the following form:

ñóò ò ó ò ó

 ( , , 
) = 

−2

«E¦/` (‡ , ¨ ) + X V 2 / 


 ¨ + (  2  



 + 2 / 
 


 )‡ − (2  


 +  


 ) «

where

−1

˜ − (2 / 
 


 +  2  



 − 2 , / 


 ) W M

t§ ,



¨ =   −  + −, ˜ ;  «  « ` = ` (‡ , ¨ ),  =  ( ), and = ( ) are arbitrary functions; , is an arbitrary constant. ˜ In this case, the transformation a = X  −1  , ‡ =  −1 « −1 + , followed by the substitution M H (‡ ) = ‡ g
, leads to a second-order equation: H 2 H 

 + H ( H 
)2 −3 ,HH 
+3 , 2 H − , 3 ‡ = ` (‡ , H − ,!‡ ). « = exp O − , X

˜

 ,  P v

M



=X

 M , 2  «

‡ =

3.5.3. Equations of the Form — ( æ , â , âäã )âäãÑãÑäã ä + ˜ ( æ , â , âäã )âäãÑã ä + ™

+

˜

,

( æ , â , âä ã ) = 0

3.5.3-1. The arbitrary functions depend on  or . 1.

ñóò ò ó ò ó

+ 3 ƒñóÈ ò ñóò ò ó +

Solution:   = L 2.

2

2 2



(ñó ò )3 = ! ( ï ) 

+L

 +L

0

+

Œ

] 

.

X ( − a )2  (a ) a , where  2  0 M

ñóò ò ó ò ó

 ( ) M

S.

= V ! (ñ )ñó ò + " ( ï )WUñó ò ò ó .

Integrating yields a second-order equation:

5.

ï 3 ñ óò ò ó ò ó

+ ô ï 2 ñ óò ò ó

+õ ï  ñ óò



 = L exp v X  ( ) Q M

+ Xw ( )  S .

M

= ! (ñ ). The substitution a = ln | | leads to an autonomous equation of the form 3.5.5.9:







g g g + (  − 3) g g + (  −  + 2) g =  ( ).

+ 3 ôñ ó ò ò ó + 2 ô 2 ñ ó ò ) = ! ( ï ). Integrating yields a second-order equation:

ñ (ñ ó ò ò ó ò ó

2! 

 + 2  ! 
− ( 
)2 = 

6.

is an arbitrary number.

= ! (ñ )ñó ò ñó ò ò ó . Integrating yields a second-order autonomous equation of the form 2.9.1.1: /

 =  ( ),

Q

4.

0

ñóò ò ó ò ó

where  ( ) = L exp ù 3.

1

–

ññ óò ò ó ò ó

+ 3ñ

Solution:

óò ñ óò ò ó 2

=L

= ! ( ï ). 2

2

+L

1 + L

−2

  2 XZ 2    ( )  + L2S . M Q



2 0 + X  (  − a )  ( a ) a , where  M 0

0

is an arbitrary number.

Page 629

630 7.

8.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ññ óò ò ó ò ó

+

= ! ( ï ).

ôñ óò ñ óò ò ó

Integrating yields a second-order equation: ! 

 + 12 (  − 1)( 
)2 = X

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó = ! (ñ ).

ññ óò ò ó ò ó

–ñ

The substitution ¨ 2 l A £ ¨  .

9.

óò ñ óò ò ó

=

2

¨ 


 = leads to an autonomous equation of the form 3.5.1.1: p

= ! ( ï )ñ 2 .



 = X  ( )  + L2S . Q M

Integrating yields a second-order linear equation: 10.

 ( )  + L . M

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + ! (ï )ññóò = " (ï ).

ññóò ò ó ò ó

+ 3ñóÈ ò ñóò ò ó + 4 ! (ï )ññóò +

¨ 

 + The substitution ¨ = ! 
leads to a second-order linear nonhomogeneous equation: p ¨  ( ) = ( ).

11.

2

!„ó ò (ï )ñ

2

= 0.

Multiplying by , we arrive at an exact differential equation. Integrating it yields Yermakov’s equation 2.9.1.2: 

 +  ( ) = L −3 . There is also the trivial solution = 0. 12.

ññóò ò ó ò ó

+

= ! (ñ )ñó ò + " ( ï ).

ôñóÈò ñóò ò ó

Integrating yields a second-order equation: ! 

 + 12 (  − 1)( 
)2 = X\ ( ) + 3ñóÈ ò ñóò ò ó + ô V ññóò ò ó + (ñóò )2 W

ññóò ò ó ò ó

13.

Solution: 2

 14.

0

=L

3

  − + L 2  + L

= ! (ï

+ XC ( )  + L .

M

).



1

M

g + 2 X  ( − a )  −   (a ) a , M 0

where  (a ) = X

g    (a ) a , M

is an arbitrary number.

ññóò ò ó ò ó

+ (3ñó ò + 2 ôñ )ñó ò ò ó + 2 ô (ñó ò )2 +

ô 2 ñ ñóò

= ! ( ï ).

Integrating the equation twice, we arrive at a first-order separable equation:

   ! 
= L 2  + L 15.

ññóò ò ó ò ó

=

ñóò ñóò ò ó

+ ! ( ï )ññó ò ò ó .

Integrating yields a second-order linear equation: 16. 17.

= ! ( ï ) V ññó ò ò ó – (ñó ò )2 W + " ( ï ).

ññóò ò ó ò ó

– ñó ò

ññóò ò ó ò ó

+ [3ñó ò + ! ( ï )ñ ]ñó ò ò ó + ! ( ï )(ñó ò )2 = 0.



 = L exp ¯ X  ( )  S . Q M

The substitution ¨ = ! 

 − ( 
)2 leads to a first-order linear equation: ¨ 
=  ( )¨ + ( ).

Solution: 18.

ñóò ò ó



 g  (a ) a . 1 + X  ( − a )  M 0

2

=L

3

 +L

2

+L

1

 X  ( − a )  − Ž (g ) a , where  (a ) = X  (a ) a . M M 0

+ [3ñ ó ò + ! ( ï )ñ ]ñ ó ò ò ó + ! ( ï )(ñ ó ò )2 + " ( ï )ññ ó ò + % ( ï ) = 0. The substitution ¨ = ! 
leads to a second-order linear nonhomogeneous equation: ¨2

 +  ( )¨2
+ ( )¨ + % ( ) = 0.

ññ óò ò ó ò ó

Page 630

631

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

19.

20.

+ ( ! – 1)ñó ò ñó ò ò ó + !"Iññó ò + "Ió ò ñ 2 = 0, ! = ! ( ï ), " = " (ï ). A solution of this equation is any function that solves the second-order linear equation 

 + ( ) = 0.

ññóò ò ó ò ó

(ñ + ô )ñ„ó ò ò ó ò

ó

+

= ! ( ï ).

õÿñó¿ò ñóò ò ó

Having integrated the equation, we obtain ( +  ) 

 + 12 (  − 1)( 
)2 = X =¨

 ≠ −1 , the substitution



2 +1

−  leads to the equation:

 ( )  + L . For M

−3 ¨ 

 =  + 1 v X  ( )  + L2S ¨  +1 2 Q M

= 0 and  ( ) = Π  , see Section 2.3).

(with L 21.

(ñ + ô )ñ„ó ò ò ó ò ó + õÿñó¿ ò ñóò ò ó + ! (ñ )ñóò = 0. Having integrated the equation, we obtain a second-order autonomous equation: ( +  ) 

 + 12 (  − 1)( 
)2 + X

 ( )

=L ,

M

which is reduced with the aid of the substitution ¨ ( ) = ("
 )2 to a first-order linear equation: ( +  )¨
+ (  − 1)¨ + 2 X



22.

23.

(ñ + ô ï




( +  )   + 12 (  − 1)(  )2 + Xv ( )

+ õ )ñ„ó ò ò ó ò ó + 3( ñ óò + ô )ñóò ò ó = ! (ï 2



2

+L

1

 +L

number.

ï (ñ ñ óò ò ó ò ó

) + ô V ññ

óò ñ óò ò ó

+ 3ñ

óò ò ó

Solution: 2



0

=L

3



2−

 +L  +L 2

= XÈ ( )  + L .

M



0

+ X  ( − a )2  (a ) a , where  M 0

0

is an arbitrary

+ (ñ

óò

)2 W = ! ( ï ).



+ 2 X  ( − a )a −   (a ) a , M 0

1

where  (a ) = X a 

−1

 (a ) a ; M

is an arbitrary number.

ñ 2 ñ óò ò ó ò ó

– 3ññó ò ñóò ò ó + 2(ñóò )3 = ! (ï )ñ 3 .

Solution: ln | | = L 27.

).

M

[ñ + ! (ï )]ñ„ó ò ò ó ò ó = [ñó ò + !„ó ò ( ï )]ñó ò ò ó + ô! ( ï )ñó ò – ô!„ó ò ( ï )ñ . Integrating yields a second-order constant coefficient linear equation of the form 2.1.9.1: 

 + L = −(  + L )  ( ). There is also the trivial solution = 0.

25.

26.

= 2L .

M

(ñ + ô )ñ ó ò ò ó ò ó + õÿñ ó ò ñ ó ò ò ó + ! (ñ )ñ ó ò = " ( ï ). Having integrated the equation, we obtain a second-order equation:

Solution: ( +  +  )2 = L 24.

 ( )

2

2

+L

1 + L

0+

1 2

 X  ( − a )2  (a ) a , where  M 0

+ 3(ù – 1)ññ„ó ò ñóò ò ó + (ù – 1)(ù – 2)(ñ„óò )3 = ! (ï )ñ Solution for B ≠ 0:

ñ 2 ñ óò ò ó ò ó

 = L 2

2

+L

1

 +L

0

+

B

 X  (  − a )2  ( a ) a , 2 0 M

3–

where 

÷

0

is an arbitrary number.

.

0

is an arbitrary number.

For the case B = 0, see equation 3.5.3.26.

Page 631

632 28.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ3¬!„ñ óò ò ó ò ó

3 2

+

! ó ò ñ óò ò ó

1 2

+

= " ( ï ),

! ó ò ò ó ñ óò 

= ! ( ï ).

!

Having integrated the equation, we obtain a second-order equation: 2  ! 

 +  
! 
−  ( 
)2 = 2 X ( )  + L .

M

29.

30.

!„ññóò ò ó ò ó

+ (3 !„ñó ò + 2 !„óò ñ )ñó ò ò ó + 2 !„ó ò (ñó ò )2 +

!„ó ò ò óññóò

= " ( ï ),

!

= ! ( ï ).


Integrating the equation twice, we arrive at a first-order separable equation:  ( )!  = L 2 + L

1

ñóò ò ó ò ó

!„ñóò

+

+ X  (  − a ) ( a ) a . M 0

+ "Iñ = –(ù + 2) %„ñ ÷ – [(2ù + 1) %ƒòó + 3ù%

ñóò ò ó – (ù – 1)(ù + 1)%„ñ ÷ –2 (ñóò )2 ñ ÷ –1 ]ñ ÷ –1 ñóò – % 3 ñ 3÷ –2 – 3 %%„óò ñ 2÷

–1 2

–1

– ( !% +

%„ò òó ó )ñ ÷

.

Here,  =  ( ), = ( ), and % = % ( ) are arbitrary functions. 1

1−  , where ¨ = ¨ ( ) is the general solution Solution: = ¨ L + (1 − B ) X % ( )¨  −1  S Q ¨3


M ¨ 2
of the linear equation:   +  ( )  + ( ) = 0.

31.

+ ! ( ï )ñó ò

ñóò ñóò ò ó ò ó

+ " ( ï )ññó ò ò ó + % ( ï )(ñó ò )2 + V "Ió ò ( ï ) + ! ( ï )" ( ï )WNññó ò + " 2 ( ï )ñ

ñóò ò ó

2

= 0.

The solution satisfies the second-order linear equation 

 − H ( , L ) 
+ ( ) = 0, where H = H ( , L ) is the general solution of the Riccati equation H 
+ H 2 +  ( ) H − ( ) + % ( ) = 0.

3.5.3-2. Arguments of arbitrary functions depend on  and . 32.

ñ óò ò ó ò ó

+

ôñ óò ò ó

õÿñ óò

+

+

ö^ñ

=

ó

  ! (ñ  –

).

 The substitution ¨ ( ) =  −  leads to an autonomous equation of the form 3.5.5.9: ¨ 


 + (3 Œ +  )¨ 

 + (3 Œ

33.

ó

ñóò ò ó ò ó

= –3ñó ò ò ó + 2ñ + ! ( 



ó ñ )(ñ óò

2

+ 2 Œ +  )¨ 
+ ( Œ

3

+ Œ

2

+ Œ + ( )¨ =  (¨ ).

+ ñ ).



The transformation H =  , ¨ =  2 (
 + ) leads to a second-order linear equation: ¨ ‘'

‘ = 2  ( H ) + 6. Integrating the latter, we find the solution:

X ü

34.

35.

36.

ï ñóò ò ó ò ó

3H

2

+L

2

M

H

H +L

+ 2 ` (H )

1

+ 3ñó ò ò ó = ! ( ï

ñ

The substitution ¨ ( ) = 

ï 2 ñ óò ò ó ò ó

+ 6ï

ñóò ò ó

).

+

ô ï 2 ñ óò ò ó

+ 6ñó ò = ! ( ï

+

õ ï ñóò

3,

where

H =



, ` (H ) = X

Q

X  ( H ) žH S H . M M

¨ 


 =  (¨ ). leads to an autonomous equation of the form 3.5.1.1: C

The substitution ¨ ( ) =  ¨2


 =  (¨ ).

ï 3 ñ óò ò ó ò ó

= A" + L

2

2

2

ñ

). leads to an autonomous equation of the form 3.5.1.1:

= ! ( ïƒú

]

).

The transformation a = ln  , Œ ¨ = Œ + a leads ? to an autonomous equation of the form ¡





¨ ¨ ¨ 3.5.5.9: g g g + (  − 3) g g + (  −  + 2) g =  (   ) + ¡ (  −  + 2). Œ

Page 632

633

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

37.

ï 3 ñ óò ò ó ò ó

= – 32 ï

2

ñóò ò ó

+I ! O

ñ

(2 ï

7 ï P

ñóò

– ñ ).

1 The transformation a = £ , H =   
−   2 H g

g = 8  (a ) + 1, whose solution is given by:

H = 14 a 2 + L 2 a + L

1 2

 2 leads to the second-order linear equation

g

g 0 (a − ˆ )  (ˆ ) M ˆ , a 0 is an arbitrary number. Passing on to the variables  , a =  −1 J 2 , we obtain a separable equation. 38.

39.

1

+4X

ï 3 ñ óò ò ó ò ó

= –3( ü +1) ï 2 ñó ò ò ó + ü ( ü +1)(2 ü +1)ñ + ! ( ï  ñ )( ï ñó ò + üIñ )+ ï 2  " ( ï ñ )( ï ñó ò + üIñ ) . The transformation a = ln  , H =   , followed by the substitution ¨ ( H ) = ( H g
)2 , leads to a ‘'‘ second-order linear equation: ¨

= 2 ( H )¨ + 2  ( H ) + 6 , 2 + 6 , + 2.

ï 4 ñ óò ò ó ò ó

3

= – 32 ï

3

ñ óò ò ó

+

ñ

(2 ï

! O 7 ï P

ñ óò

1 The transformation a = £ , H = (   2 H g

g = 16  (a ) H + 12 . 40.

ó

ñ 2 ñ óò ò ó ò ó

–ñ ) . 3


−

1 2

)2 leads to a second-order linear equation:

ó

= –3ñ 2 ñó ò ò ó + 2ñ 3 + ñ 2 ! (  ñ )(ñó ò + ñ ) + " (  ñ )(ñó ò + ñ )3 .  The substitution H ( ) =  , followed by reduction of the equation order and the substitution ¨ ( H ) = ( H 
)2 , leads to a second-order linear equation: ¨u‘'

‘ = 2 H −2 ( H )¨ + 2  ( H ) + 6.

3.5.3-3. Arguments of arbitrary functions depend on  , , and
 . 41.

42.

43. 44.

45.

46.

47.

ï ñ óò ò ó ò ó

= ! ( ï ñ ó ò – ñ )ñ ó ò ò ó . The substitution H =  
− "H 

 = [  ( H ) + 1]H 
.

leads to a second-order equation of the form 2.9.2.21:

ï ñóò ò ó ò ó

+ (1 – ô )ñó ò ò ó = ï 2 Dl! ( ï The substitution H =  
− "H 

 = !H 
+  2  +1  ( H ).

ñóò

– ñ ). leads to a second-order equation of the form 2.9.2.20:

ï ñ óò ò ó ò ó

+ ( ô + 2)ñ ó ò ò ó = ! ( ï , ï ñ ó ò + ôñ ). The substitution ¨ =  
+  leads to a second-order equation: ¨ 

 =  ( , ¨ ).

ó

ï ñ óò ò ó ò ó

+ (1 – ô ï )ñ ó ò ò ó =  2 D ! ( ï ñ ó ò – ñ ). The substitution H =  
− leads to a second-order equation of the form 2.9.2.17: H 

 − !H 
=   2   ( H ).

ï ñóò ò ó ò ó

= ! ( ï ñó ò – ñ + ô The substitution ‘ H =  "H 

 = V l ln(   )  + 1W

ï 3 ñ óò ò ó ò ó

+ ï 2 ñó ò ò ó = ! The transformation a 2.9.6.2: H g

g − H g
=  ( H equation ¨*¨2‘
− ¨ =

ï 4 ñ óò ò ó ò ó

+

ï 3 ñ óò ò ó

ln ï )ñó ò ò ó . 
− leads to a second-order equation of the form 2.9.2.39: H 
.

( ï ñó ò – ñ ). = ln | |, H =  
 − leads to an autonomous equation of the form ), which is reduced, with the aid of the substitution ¨ = H g
, to the Abel  (¨ ) (see Subsection 1.3.1).

=I ! Oiñóò –

The substitution ¨ ( ) = 

ï

ñ

P

.


− leads to an equation of the form 2.9.1.8: ¨ 

 = 

−3

 (¨u  ).

Page 633

634

THIRD-ORDER DIFFERENTIAL EQUATIONS

3.5.4. Equations of the Form — ( æ , â , â$ä ã ) âäãÑãÑäã ä +

› ¦

˜ ¦ ( æ , â , âäã )(âäãÑã ä )¦ = 0

3.5.4-1. Arbitrary functions depend on  or . 1.

2.

3.

+ (ñ ó ò ò ó )2 – 14 (ñ ó ò )2 – 14 ññ ó ò – ! ( ï )ñ 2 = 0. This is a special case of equation 3.5.4.4. The solution satisfies the second-order linear equation 

 + 12 
 − H ( , L ) = 0, where H = H ( , L ) is the general solution of the Riccati equation H 
+ H 2 − 12 H =  ( ).

ññ óò ò ó ò ó

+ (ñ ó ò ò ó )2 – ñ ó ò ñ ó ò ò ó – ! ( ï )ñ 2 = 0. The solution satisfies the second-order linear equation 

 − H ( , L ) = 0, where H = H ( , L ) is the general solution of the Riccati equation H 
+ H 2 =  ( ).

ññ óò ò ó ò ó

+ (ñó ò ò ó )2 + [2 % ( ï ) – 1]ñ„óÈ ò ñóò ò ó + [ ! (ï ) + % (ï )]ññ„óò ò ó + % ( ï )[ % ( ï ) – 1](ñ„ó ò )2 + [ %„òó ( ï ) + ! ( ï ) % ( ï )]ññ„ó ò = 0. This is a special case of equation 3.5.4.4 with ( ) = 0. The solution satisfies the second-

ññóò ò ó ò ó

−1 order linear equation 

 + % ( )
 − H ( , L ) = 0, where H ( , L ) =  ( ) L + X  ( )  S ,

Q

 ( ) = exp − X  ( )  S . Q M

4.

5.

6.

7.

+ (ñó ò ò ó )2 + [2 % ( ï ) – 1]ñ„óÈ ò ñóò ò ó + [ ! (ï ) + % (ï )]ññ„óò ò ó + % ( ï )[ % ( ï ) – 1](ñ ó ò )2 + [ % òó ( ï ) + ! ( ï ) % (ï )]ññ ó ò = " ( ï )ñ 2 . The solution satisfies the second-order linear equation "

 + % ( )
 − H ( , L ) = 0, where H = H ( , L ) is the general solution of the Riccati equation H 
+ H 2 +  ( ) H = ( ).

ññóò ò ó ò ó

ò ñóò ò ó ò ó – (ñóò ò ó )2 + ! (ï )(ñóò )2 = ôñ 2 + 2 õÿñ + ö . 2ñó Differentiating both sides of the equation with respect to  and dividing by
 , we arrive at a fourth-order linear equation: 



 +  

 + 12  

 =  +  . ó

2ñ ó ò ñ ó ò ò ó ò ó – (ñ ó ò ò ó )2 – ƒñ ó ò ñ ó ò ò ó + H ( ï )(ñ ó ò )2 =   ( ôñ 2 + 2 õÿñ + ö ).   Multiplying both sides by  −  , we arrive at an equation of the form 3.5.4.13 with  ( ) =  −   and ( ) =  −   ( ).

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ! (ï )(ñóò )2 + " (ñ )(ñóò )4 . 2ñó Solution:  = X ¨ M2 +L , X M ‡ 2( ) ( ) where ‡ = ‡ ( ) and ¨ = ¨ ( ) are the general solutions of the second-order linear equations: 4‡

8.

2ñ

óò ñ óò ò ó ò ó

– 3(ñ

óò ò ó



− ( )‡ = 0  

)2 = ! ( ï )(ñ

The substitution ¨ ( ) =

óò




4¨2

 +  ( )¨ + ( ) = 0. ü

9.

M

)2 + " ( ï ) ñ

and –1

(ñ



4¨   +  ( )¨ = 0.

ó ò )5 ) 2 .

leads to a second-order nonhomogeneous linear equation:

ò ñóò ò ó ò ó – 3(ñóò ò ó )2 = ! (ñ )(ñóò )4 + ï –1 " (ñ )(ñóò )7 ) 2 . 2ñó Taking to be the independent variable, we obtain an equation of the form 3.5.4.8 for  =  ( ): 2



− 3(

)2 = −  ( )(
)2 − ( ) −1 (
)5 J 2 .  







Page 634

635

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

10.

11.

2ï

– ï (ñó ò ò ó )2 + ùñó ò

ñóò ñóò ò ó ò ó

ñóò ò ó

+

H (ï

óò ñ óò ò ó

=

ï ! (ñ )(ñ óò

Multiplying both sides by   and ( ) =   −1  ( ).

ï ñ óò ñ óò ò ó ò ó

– 3 ï (ñ

óò ò ó

)2 + 3 ñ

−1

ï 1–÷ (ô ñ

)(ñó ò )2 =

2

+ 2 õÿñ + ö ).

, we arrive at an equation of the form 3.5.4.13 with  ( ) =   )4 + " (ñ )(ñ

óò

)5 .

Taking to be the independent variable, we obtain an equation of the form 3.5.3.10 for  =  ( ): 


+ 3


= − ( ) −  ( )
.



12.

ñóò ò ó ò ó = ! (ñ )ñ óò (ñóò ò ó )ú 3

L 13.

¡

AI = X ¦ 2 X

Solution for

2 !„ñó ò ñóò ò ó ò

ó



.

≠ 1:

Solution for

L

 

(1 −

Q

) ( ) + L

¡

2

1 1−

+

S

+L

M

1

1 −2

§

M

,

where

 ( )=X  ( ) M

.

= 1:

¡ 3

AI = X Q

L

– ! (ñó ò ò ó )2 +

2

X  Ž (

)

!„óò ñóò ñóò ò ó

M

+L

1

S

1 −2

M

,

+ " ( ï )(ñó ò )2 =

where

ôñ

2

 ( ) = Xv ( )

+ 2 õÿñ + ö ,

!

M

.

= ! ( ï ).

Differentiating both sides of the equation with respect to  and dividing by
 , we arrive at a fourth-order linear equation:  



 + 32  



 + ( + 12  

 )

 + 12 

 =  +  .

3.5.4-2. Arguments of arbitrary functions depend on  , , and
 . 14.

2ñ

óò ñ óò ò ó ò ó

– 3(ñ

óò ò ó

)2 =

ñ 4I ! O

ñ óò ñ 2P

.

O

ï ñ óò ñ 2

The substitution ¨ ( ) = (
 )−1 J 2 leads to a second-order autonomous equation of the form 2.9.1.1: ¨3

 =  (¨ ), where  (¨ ) = − 14 ¨ 5  (¨ −2 ). 15.

2ñ

óò ñ óò ò ó ò ó

– 3(ñ

óò ò ó

)

2

= ï –4 ñ 4 !

.

P

The substitution ¨ ( ) = (
 )−1 J 2 leads to a second-order equation of the form 2.9.1.9: ¨2

 =  −3 J 2  (¨  −1 J 2 ), where  (ˆ ) = − 1 ˆ 5  (ˆ −2 ). 4 16.

2ñ

óò ñ óò ò ó ò ó

– 3(ñ

óò ò ó

)2 =

ï –8 ñ 4 ! O

ï 2 ñ óò ñ 2

P

.

The substitution ¨ ( ) = (
 )−1 J 2 leads to a second-order equation of the form 2.9.1.8: ¨2

 =  −3  (¨  −1 ), where  (ˆ ) = − 1 ˆ 5  (ˆ −2 ). 4 17.

18.

ñóò ò ó ò ó

= Vï

! (ï ñ óò

ñóò ò ó ò ó

=

ï ñ óò ï –5 I ! O ï

–ñ )+ô

ï

W (ñó ò ò ó )3 . ¨ g
− ¨ The Legendre transformation  = ¨ug
, = a 2 ¨2g

g
g = −  (¨ )(¨2g
)3 −  (¨2g
)−5 . 3

–ñ 2

P

–5

leads to an equation of the form 3.5.2.4:

(ñó ò ò ó )3 .

The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ leads to an equation of the form 3.5.2.27: ¨2g

g
g = ¨ −5 J 2  (¨2g
¨ −1 J 2 ), where  (ˆ ) = −ˆ −5  (ˆ −2 ).

Page 635

636 19.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

=

ï ñ óò ï –5 I ! O ï

–ñ

P

4

(ñó ò ò ó )3 .

¨ g
− ¨ leads to an equation of the form 3.5.2.28: The Legendre transformation  = ¨ug
, = a 2 ¨ g

g
g = ¨ −5 J 4  (¨ g
¨ −1 J 4 ), where  (ˆ ) = −ˆ −5  (ˆ −4 ).

20.

= [ ï ! (ñó ò ) + ñ" (ñó ò ) + % (ñó ò )](ñó ò ò ó )3 + r (ñó ò )(ñó ò ò ó )2 . ¨ g
, = a ¨2g
− ¨ leads to a linear equation: The Legendre transformation  = u

ñóò ò ó ò ó

¨ g

g
g = − Y (a )¨ g

g − [  (a ) + aÝ (a )]¨ g
+ (a )¨ − % (a ).

21.

= ï ! ( ï ñ ó ò – ñ )(ñ ó ò ò ó )2 + ï " ( ï ñ ó ò – ñ )(ñ ó ò ò ó )  . ¨ g

g
g = −  (¨ )2 ¨ g
¨2g

g − The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ leads to the equation 3 ?
¨ 2 ¨
2 ¨

¨ p ¨

2 ¨




2 ¨ ( ) g ( g g )3−  . Lowering its order with the substitution H ( ) = g g ( H = g g g g
), we have a Bernoulli equation: H ?
= −  (¨ ) H − (¨ ) H 3−  .

ñ óò ò ó ò ó

3.5.5. Other Equations 3.5.5-1. Equations of the form  ( , , 
 , 

 )


 + 4 ( , , 
 , 

 ) = 0. 1.

ñóò ò ó ò ó

= ! (ñó ò ò ó ).

Solution in parametric form:

 =X k 2.

ñóò ò ó ò ó

g 1

a1 ,  (a 1 ) M

=X

k

g 2

a1 g1 a2 a2 X M .  (a 1 ) k 3  (a 2 ) M

= ! (ñ )ñó ò " (ñóò ò ó ).

Integrating the equation and substituting ¨ ( ) = 12 (
 )2 , we arrive at a first-order equation:

ˆ =X  ( ) +L , where ˆ = ¨
. M  (ˆ ) M Solving this equation for ¨u
, we obtain a separable equation.  X

3.

ñóò ò ó ò ó = ! (ñ )" (ñóò ) % (ñóò ò ó ). The substitution ¨ ( ) = 12 (
 )2 leads to a second-order equation: ¨

=  ( ) Y (¨ ) % (¨
),  

where

 A £ 2¨  Y (¨ ) = A , £ 2¨

whose solvable cases for some functions  , , and % are outlined in Section 2.7. 4.

5. 6.

= ï ! ( ï ñ ó ò – ñ )" (ñ ó ò ò ó ). The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ , where ¨ = ¨ (a ), leads to an equation of







the form 3.5.5.2: ¨ g g g = −  (¨ )¨ g (1 ¨ g g )(¨ g g )3 .

ñ óò ò ó ò ó

ï ñóò ò ó ò ó

+ ñó ò ò ó = ! ( ï

ñóò

– ñ )" ( ï

ñóò ò ó

).

The substitution ¨ ( ) =  
− leads to an equation of the form 2.9.4.36: ¨ 

 =  (¨ ) (¨ 
).

= ! ( ï )" ( ï 2 ñó ò ò ó – 2 ï ñó ò + 2ñ ). ¨ 
= The substitution ¨ ( ) =  2 

 − 2 
 + 2 leads to a first-order separable equation: u ¨  2  ( ) ( ).

ñóò ò ó ò ó

Page 636

637

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

7.

ñóò ò ó ò ó

=

ñ óò ñ óò ò ó ñ

+ ¬9ñó ò ò ó –

(ñ

ñ

óò

)2

®l!e”

The transformation a = 
 , ¨  (a ) (¨ ). 8. 9.

11.

13.

.

¨ g
= = 

  leads to a first-order separable equation: u

= H ( ï , ñó ò , ñó ò ò ó ). The substitution ‡ ( ) = 
leads to a second-order equation: ‡

 =  ( , ‡ , ‡
 ).

ñóò ò ó ò ó = H (ñ , ñóò , ñóò ò ó ). Autonomous equation. The substitution ¨ ( ) = ("
 )2 leads to a second-order equation: ¨

= A 2   , A £ ¨ , 1¨
 . 2 £ ¨ 

ñóò ò ó ò ó

= ñH (ñó ò ¦ñ , ñóò ò ó¦ñ ). This is a special case of equation 3.5.5.9. The transformation a =
  , ¨ = 

  leads to a first-order equation: ( ¨ − a 2 )¨2g
= −a ¨ +  (a , ¨ ). = ï –  –3 H ( ï  ñ , ï  +1 ñ ó ò , ï  +2 ñ ó ò ò ó ). Generalized homogeneous equation. The transformation a = ln  , H =   , followed by the substitution ¨ ( H ) = ( H g
)2 , leads to a second-order equation:

ñ óò ò ó ò ó

¨ '‘

‘ = A 3( , + 1)¨

12.

•

ñóò ò ó ò ó



10.

ñ óò ñ óò ò ó "I” ñ • ñ

J ¨ ‘
− 6 , 2 − 12 , − 4 A 2 , ( , + 1)( , + 2) H ¨ −1 J 2 A 2¨ −1 J 2  H , A ¨ 1 J 2 − ,H , 12 ¨ ‘
T (2 , + 1)¨ 1 J 2 + , ( , + 1) H  .

−1 2

= ñ ï –3 H ( ï ñ ú , ï ñó ò ¦ñ , ï 2 ñóò ò ó¦ñ ). Generalized homogeneous equation. The transformation a =   + , H =  second-order equation.

ñóò ò ó ò ó ñóò ò ó ò ó

=

ñ ï –3 H (ï ñ óò ¦ñ , ï 2 ñ óò ò ó¦ñ

).

This is a special case of equation 3.5.5.12. The transformation H =  ‘ leads to a first-order equation: (¨ + H − H 2 )¨2
= 2¨ − H ¨ +  ( H , ¨ ). 14. 15. 16.


 leads to a


 , ¨ = 

2



 

= H ( ï , ï ñó ò – ñ , ñó ò ò ó ). The substitution H =  
− leads to a second-order equation: "H 

 = H 
+  2  ( , H , H 
 ).

ñóò ò ó ò ó

= ! ( ï , ñó ò ò ó + ñó ò + ñ ) + ñ . ¨ 
=  ( , ¨ ) + ¨ . The substitution ¨ = 

 + 
 + leads to a first-order equation: u

ñóò ò ó ò ó

= ! ( ï , ñó ò ò ó – ñó ò + ñ ) – ñ . ¨ 
=  ( , ¨ ) − ¨ . The substitution ¨ = 

 − 
 + leads to a first-order equation: u

ñóò ò ó ò ó

17.

= H ( ï , ñ , ñ ó ò , ñ ó ò ò ó ). The Legendre transformation  = ¨ug
,

18.

ñóò ò ó ò ó

ñ óò ò ó ò ó

¨ g
− ¨ leads to the equation =a2 ¨ g

g
g = −   ¨ g
, a ¨ g
− ¨ , a , 1 ¨ g

g  (¨ g

g )3 .

= ñó ò ! (ññóò ò ó – ñóò 2 ). 1 b . Particular solution:

=L where the constants L 1 , L 2 , and L 2 b . Particular solution: where the constants L

1, L

1 3

exp( L 3  ) + L 2 exp(− L 3  ), are related by the constraint L

= L 1 cos( L 3  ) + L 2 sin( L 3  ), , and L 3 are related by the constraint L 2

2 3

.

=   4L

1

L 2L

2 3

+   −( L

2 1

+L

2 2)

2 3

L

2 3

 = 0.

Page 637

638 19.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ñóò ò ó ò ó

– (ñó ò ò ó )2 = ! (ññó ò ò ó – ñó ò 2 ). 1 b . Particular solution: = L 1 exp( L where the constants L

1,

L 2 , and L

2 b . Particular solution: where L 20.

1,

ñ óò ò ó ò ó

L 2 , and L

ó

ó

3

 )+L

3

2

exp(− L

=L

ó

 ),

are related by the constraint 4 L

3

cos( L

1

3

 )+L

2

sin( L

are related by the constraint ( L

=  – H (  ñ ,   The substitution H = ž

3

ó

2 1

 ), + L 22 ) L

1

L 2L

4 3+

 (4 L 1 L 2 L

2 3 ) = 0.

3

4 3

+ l −( L

2 1

+L

2 2)

L

2 3

 = 0.

 ñ óò ,   ñ óò ò ó

). leads to an autonomous equation of the form 3.5.5.9:

H 


 − 3 ŒH 

 + 3 Π2 H 
− Œ 3 H =  ( H , H 
− ŒH , H 

 − 2 ŒH 
+ Π2 H ). 21.

ó

ñóò ò ó ò ó

= ñH (  ñ , ñó ò ¦ñ , ñóò ò ó¦ñ ). Equation invariant under “translation–dilatation” transformation.  H =  , ¨ = 
  leads to a second-order equation:

H 2 (¨ + Œ )2 ¨ '‘

‘ + H 2 (¨ + Œ )(¨ ‘
)2 + H (¨ + Œ )(4¨ + Œ )¨ ‘
+ ¨ 22.

3

The transformation

=   H , ¨ , H (¨ + Œ )¨ ‘
+ ¨

= ï –3 H ( ï ú  ] , ï ñ ó ò , ï 2 ñ ó ò ò ó ). Equation invariant under “dilatation–translation” transformation. H =  +  , ¨ =  
 + leads to a second-order equation:

2

.

ñ óò ò ó ò ó

H 2¨

2

¡ ¨ '‘

‘ + H 2 ¨ (¨ ‘
)2 + H ¨

2

¨ ‘
− 3 H ¨*¨ ‘
+ 2¨ − 2 ¡

=  (H , ¨ −

¡

The transformation , H ¨*¨ ‘
− ¨ +

¡

).

3.5.5-2. Equations of the form  ( , , 
 , 

 , 


 ) = 0. 23.

ññóò ò ó ò ó

– 13 ñó ò ñóò ò ó = Particular solution:

ï ! (ñ óò ò ó ò ó

) + " (ñó ò ò ó ò ó ).

L 1  3 + 12 L 2  2 + L 3  + L 4 , where the constants L 1 , L 2 , L 3 , and L 4 are related by two constraints =

1 6

2L 3L Here, L 24.

3

and L

4

ññ óò ò ó ò ó

1

L

1 4

L

3

−L

−L

2

L

2 2 3

= 3  ( L 1 ), = 3 ( L 1 ).

are defined in terms of two arbitrary constants L

– 13 ñ ó ò ñ ó ò ò ó = Particular solution:

ñ ó ò ò ó ! (ñ ó ò ò ó ò ó

)+

ï " (ñ ó ò ò ó ò ó

) + % (ñ

óò ò ó ò ó

1

and L

2.

).

L 1  3 + 12 L 2  2 + L 3  + L 4 , where the constants L 1 , L 2 , L 3 , and L 4 are related by two constraints =

2 3

1 6

L 1 L 3 − 13 L L 1 L 4 − 13 L 2 L Here, L 25.

3

and L

H ( ï , ñ ñóò ò ó

4

2 2

=L

 ( L 1 ) + ( L 1 ), 3 = L 2  ( L 1 ) + % ( L 1 ). 1

are defined in terms of two arbitrary constants L

1

and L

2.

– ñó ò 2 , ññó ò ò ó ò ó – ñó ò ñóò ò ó ) = 0. The substitution ¨ = ! 

 − (
 )2 leads to a first-order equation:  ( , ¨ , ¨u
) = 0.

Page 638

639

3.5. NONLINEAR EQUATIONS CONTAINING ARBITRARY FUNCTIONS

26.

H  ñóò ò ó ¦ñ , ñ ñóò ò ó

– (ñó ò )2 , 1 b . Particular solution: where the constants L

1,

ñóò ò ó ò ó ¦ñóò , ñ óò ñóò ò ó ò ó =L

1

exp( L

L 2 , and L

3

are related by the constraint

2 3,

 L 2 b . Particular solution: where the constants L

27.

HwO

ñ óò ò ó , ñ óò ñ óò

–ñ

L 2L

=L

1

cos( L

3

ñ óò ò ó ò ó ñ óò

= 0.

Particular solution:

P

3

2

2 3,

L

exp(− L

−4 L

 )+L

2

1

3

4 3

L 2L

sin( L

 ),

3

 = 0.

 ),

are related by the constraint

2 1

,

 )+L

2 3,

1

  − L 32 , −( L

ñ óò ò ó ñ óò

3

4L

L 2 , and L

1,

– (ñó ò ò ó )2  = 0.

2 2)

+L

L 32 , −( L

HwOiñ

ñ óò ò ó ò ó ñ óò

+ ôñó ò ò ó ,

ñD

+1

ñ óò ò ó ò ó ñ óò

1

exp( L

2 1

+L

2 2)

 ) + L 3, where L 1 is an arbitrary constant and the constants L 2 and L  ( L 2 , − L 2 L 3 , L 22 ) = 0. 28.

=L

2 3,−

L

 = 0.

L

4 3

2

are related by the constraint

2

= 0.

P

A solution of this equation is any function that solves the following second-order autonomous equation of the form 2.9.1.1: 

 = L 1 −  + L 2 , where the constants L 29.

HwO

ñ óò ò ó ò ó ñ óò

+ ñó ò ò ó ,

]

and L

1

ñ óò ò ó ò ó ñ óò

P

2

are related by the constraint  (L

2, −

L 1 ) = 0.

= 0.

A solution of this equation is any function that solves the following second-order autonomous equation of the form 2.9.1.1: 

 = L 1  −  + L 2 , where the constants L 30.

1

HwO r ò]

ñ óò ò ó ò ó , ñ óò ò ó ñ óò

1

–

r

and L

r ò ]

2

are related by the constraint  ( L

ñ óò ò ó ò ó ñ óò

P

r

= 0,

r (ñ

=

L 1 ) = 0.

2, −

).

A solution of this equation is any function that solves the following second-order autonomous equation of the form 2.9.1.1: 

 = L 1 Y ( ) + L 2 , where the constants L 31.

H (ñ óò ò ó , ï ñ óò ò ó

1

and L

– ñ ó ò , 2ññ Particular solution:

óò ò ó

2

are related by the constraint  ( L

–ñ

óò 2 , ñ óò ò ó ò ó

L 2 ) = 0.

) = 0.

H (ñ óò ò ó ò ó , ï ñ óò ò ó ò ó

Solution:

– ñó ò ò ó ,

ï 2 ñ óò ò ó ò ó

– 2ï

=L

1



2

 + L 3, where the constants L 1 , L 2 , and L 3 are related by  (2 L 1 , − L 2 , 4 L 1 L 32.

=L

1,

3

+L

2

+ 2ñó ò , 2



2

+L

ï 3 ñ óò ò ó ò ó

– 3ï

2

ñóò ò ó

3

−L

+ 6ï

2 2 , 0)

ñóò

= 0.

– 6ñ ) = 0.

 + L 4, where the constants L 1 , L 2 , L 3 , and L 4 are related by  (6 L 1 , −2 L 2 , 2 L 3 , −6 L 4 ) = 0. 1



ñóò ò ó

+L

3

Page 639

Chapter 4

Fourth-Order Differential Equations 4.1. Linear Equations 4.1.1. Preliminary Remarks 1 b . A nonhomogeneous linear equation of the fourth order has the form



4





 + 

3




 + 



 + 

2


+ 

1

= ( ),

0

  =   ( ).

(1)

Let 0 = 0 ( ) be a nontrivial particular solution of the corresponding homogeneous equation (with ≡ 0). Then the substitution = 0 ( ) XZH ( )  (2) leads to a third-order linear equation:

H
+ (4  4 0


+ 3  3 0

+ 2  2 0
+  1 0 ) H = , where the prime denotes differentiation with respect to  . 2 b . Let 1 = 1 ( ) and 2 = 2 ( ) be two nontrivial linearly independent particular solutions of equation (1) with ≡ 0. Then the substitution = 1 X 2¨  − 2 X 1¨  M M 

4 0

H


+ (4 


+

4 0

3 0)

H

+ (6 

M



+ 3

4 0

leads to a second-order linear equation:

where

¢

1

 4¢

1

= 1


2

¨

+ (3  ¢ 4


−

¢

1 2,

2

+  3¢

1)

2

= 1



2


+

3 0

2 0)

¨
+ [  (3 ¢ 4

−





¢

1 2,

3

3

+ 2 ) + 2  3 ¢

= 1




2

2

+  2¢

1]

¨ = ,




, =


−


. 1 2 1 2

−

1 2

See also Subsections 0.4.1 and 0.4.2.

4.1.2. Equations Containing Power Functions 4.1.2-1. Equations of the form  4 ( ) 



 +  0 ( ) = ( ). 1.

ñ óò ò ó ò ò ó ó

+ ôñ = 0. 1 b . Solution for  = 0: 2 b . Solution for  = 4 , =L

1

4

ñóò ò ó ò ò ó ó + ƒñ = ô ï 1

4

=L

3 +õ ï

1

+L

2

 + L 3

2

+L

3

sinh ,! cos ,! + L

4

 3.

> 0:

cosh ,! cos ,! + L

3 b . Solution for  = − , 2.

=L 2

cosh ,! sin ,! + L

4

sinh ,! sin ,! .

< 0: 1

cos ,! + L

2 +ö ï

+ s,

2

sin ,! + L

3

cosh ,! + L

4

sinh ,! .

≠ 0.

Solution: = (  3 + '  2 + (˜ + ° ) + ¨ ( ), where ¨ ( ) is the general solution of equation Œ 4.1.2.1: ¨3



 + Œ ¨ = 0.

Page 641

642 3. 4.

5. 6.

FOURTH-ORDER DIFFERENTIAL EQUATIONS =ô ï ñ +õ . This is a special case of equation 5.1.2.3 with B = 4.

ñ óò ò ó ò ò ó ó

=ô ï _ ñ . This is a special case of equation 5.1.2.4 with B = 4. For ý = −2, −4, −6, −8, and −9, see equations 4.1.2.5, 4.1.2.6, 4.1.2.7, 4.1.2.8, and 4.1.2.12, respectively. The transformation  = a −1, = ‡"a −3 leads to an equation of the same form: ‡ g

g
g
g = a −: −8 ‡ .

ñ óò ò ó ò ò ó ó

ï 2 ñ óò ò ó ò ò ó ó

= ôñ . This is a special case of equation 5.1.2.6 with B = 2.

ï 4 ñ óò ò ó ò ò ó ó

ôñ

=

.

Solution: =L

7. 8.

9.

10.

1

  1 + L 2  2 + L 3  3 + L 4  4 ,

ï 6 ñ óò ò ó ò ò ó ó

1,2

= 32 C A 

+£  + 1 

5 4

J

1 2

, ,

3,4

= 32 C A 

5 4

−£  + 1 

J

1 2

.

= ôñ . This is a special case of equation 5.1.2.7 with B = 2.

ï 8 ñ óò ò ó ò ò ó ó

= ôñ . The transformation  = a −1 , form 4.1.2.1: ¨3g

g
g
g =  ¨ .

ï

(ô

ï

+ õ )4 ( ö

+ ý )4 ñ

óò ò ó ò ò ó ó

=¨ a

−3

ï

2

+

õï

+ ö )4 ñó ò ò ó ò ò ó

ó

=

üIñ

leads to a constant coefficient linear equation of the

üIñ

.  +  The transformation ˆ = ln û , ¨ (û ˜ + ûû M û û equation.

(ô

=

=

leads to a constant coefficient linear

( (˜ + )3

M

.

 ,¨ = leads to a constant coefficient M   + ' + ( (  2 + ' + ( )3 J 2 3 ¨



2 ¨

¨ ŠªŠ +  169 c 2 − ,  = 0, where c =  2 − 4 !( . linear equation: ªŠ ŠªŠªŠ − 52 c The transformation ˆ = X

ï

11.

(ô

12.

ï 9 ñ óò ò ó ò ò ó ó

13.

,

+ õ )2 ( ö

ï

+ ý )6 ñ

2

óò ò ó ò ò ó ó

=

üIñ

,

=¨ a

.

 +  ¨ The transformation ˆ = , = leads to an equation of the form 4.1.2.5: (˜ + ( (˜ + )3 M M 2 ¨2Šª

Šª

ŠªŠ −4 ¨ ˆ =, ¢ , where ¢ =  − ˜( . M = ôñ + õ ï 4 . The transformation  = a

(ô

ï

+ õ )9 ñó ò ò ó ò ò ó

ó

= (ö

ï

−1

−3

leads to an equation of the form 4.1.2.3: 4¨ a = a ¨ +  .

+ ý )ñ . (˜ + The transformation ˆ = M , ¨ = leads to an equation of the form 4.1.2.3:  +  (  +  )3 ¨2Šª

Šª

ŠªŠ = ¢ −4 ˆ ¨ , where ¢ =  − ˜ ( .

M

4.1.2-2. Equations of the form  4 ( )



 +  1 ( )
 +  0 ( ) = ( ). 14.

ñ óò ò ó ò ò ó ó + ôñ óò + õÿñ = 0. This is a special case of equation 4.1.2.41 with 

2

=

3

= 0.

Page 642

643

4.1. LINEAR EQUATIONS

15. 16. 17. 18.

19.

20.

21.

22.

+ 2 ôñó ò – ô 2 ï 2 ñ = 0. This is a special case of equation 4.1.2.25 with B = 1.

ñóò ò ó ò ò ó ó

+ 4 ô ï ñó ò + (2 ô – ô 2 ï 4 )ñ = 0. This is a special case of equation 4.1.2.25 with B = 2.

ñóò ò ó ò ò ó ó

+ ( ô 1 ï + õ 1 )ñó ò + ( ô 2 ï + õ 2 )ñ = 0. This is a special case of equation 5.1.2.35 with B = 4.

ñóò ò ó ò ò ó ó

+ ô ï (2 õ – 3 ô The substitution ¨ = ¨2

 +  ¨2
+ (2  − 

ñóò ò ó ò ò ó ó

+ õ (2 ô – õ + ô 2 ï 2 )ñ = 0. 

 −  
 +  leads to a second-order equation of the form 2.1.2.31: +  2  2 )¨ = 0.

–ô

2

ï 2 )ñ óò

+ ô ï ñó ò – ô ï –1 ñ = õ ïƒ÷ . For  = 0, a particular solution is: 0 =  . The substitution H =  linear equation.

ñóò ò ó ò ò ó ó

+ ô ï  ñ ó ò – 2 ô ï  –1 ñ = õ For  = 0, a particular solution is: linear equation.

ñ óò ò ó ò ò ó ó

ï ÷

. 2 0 =  . The substitution H = 


− 2 leads to a third-order

+ ô ï  ñ ó ò – 3 ô ï  –1 ñ = õ ï ÷ . For  = 0, a particular solution is: 0 =  3 . The substitution H =  linear equation: H 


 +   H = '  +1 (for  = 0, see 3.1.2.7).


− 3 leads to a third-order

ñ óò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

+

ô ï  ñ óò

ï –1 ñ

+ ôƒü

=

õ ïƒ÷

.




 +  

Integrating yields a third-order linear equation: 23.

24. 25.

26. 27.

+1

+L .

¡

¡

+ 2 ôù ïƒ÷ –1 ñó ò + ô [ù (ù – 1)ïw÷ –2 – ô ï 2÷ ]ñ = 0. The substitution ¨ = 

 +   leads to a second-order equation of the form 2.1.2.7: ¨2

 −   ¨ = 0.

ñóò ò ó ò ò ó ó

+ (ô ï + õ )ï Particular solution:

 ñ óò

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó + (ô ï ñóò ò ó ò ò ó ó

+ (ô

ï

ñóò ò ó ò ò ó ó + (ô ï ï ñ óò ò ó ò ò ó ó

0

+ õ ) ï  ñ óò + õ ) ï +

0

ñóò 0

õ )ñ óò 3

Particular solution:

30.

   B +1

+ õ ï ñó ò – ô ( ô 3 + õ ï )ñ = 0. This is a special case of equation 4.1.6.4 with  = '  .

ñóò ò ó ò ò ó ó

Particular solution: 29.

=

+ ô ï  ñ ó ò + ô ( ü + 3) ï  –1 ñ = 0. The transformation  = a −1 , = ¨ a −3 leads to an equation of the form 4.1.2.22 with  = 0: ¨2g

g
g
g + (˜a +2¨2g
+ ( a + −1 ¨ = 0, where ( = − , = − , − 6.

ñ óò ò ó ò ò ó ó

Particular solution: 28.


− leads to a third-order

0

– ô ï  ñ = 0. =  +  . – 2ô

ï ñ

– 3ô

ï ñ

= 0.

ôƒõ ï  ñ

= 0.

= 0. 2

= (  +  ) . = (  +  )3 . + =

−





.

+ ô ï  +1 ñ ó ò – [ ô ( ï + 1)ï  Particular solution: 0 =   .



+

ï

+ 4]ñ = 0.

Page 643

644

FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.1.2-3. Equations of the form  4 ( )



 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ). 31.

ñóò ò ó ò ò ó ó

+ 2 ôñó ò ò ó + ô =

Solution: 32.

™

(L (L

2

ñ

= 0.

+L 1+L

 ) cos( ,! ) + ( L 2  ) exp( ,! ) + ( L

1

3

ñóò ò ó ò ò ó ó

+ ( ô + õ )ñó ò ò ó + ôƒõÿñ = 0. The case  =  is given in 4.1.2.31. Let  ≠  . 1 b . Solution for  = ƒ

> 0,  = ý

2

=L

2

> 0:

cos( ƒ… ) + L

1

sin( ƒ… ) + L

2

2

> 0,  = −ý

2

< 0:

=L

1

cos( ƒ… ) + L

2

sin( ƒ… ) + L

< 0,  = ý

2

> 0:

exp( ƒ… ) + L

2

exp(− ƒ… ) + L

=L

2 1

4 b . Solution for  = − ƒ =L

ñóò ò ó ò ò ó ó

+

ôñóò ò ó

+

1

< 0,  = −ý

2

exp( ƒ… ) + L

õ ƒï ÷ ñ óò

2

õÿù ïƒ÷ –1 ñ

+

2

3

cos(ý± ) + L

exp(ý± ) + L

3

4

4

sin(ý± ).

exp(−ý± ).

cos(ý± ) + L

4

sin(ý± ).

< 0:

exp(− ƒ… ) + L = s ïƒú .

3

exp(ý± ) + L

4




 +  
+ ' 

Integrating yields a third-order linear equation: 34.

3

2 b . Solution for  = ƒ

3 b . Solution for  = − ƒ

33.

+ L 4  ) sin( ,! ) if  = , 2 > 0, 2 < 0. 3 + L 4  ) exp(− ,! ) if  = − ,

2

exp(−ý± ).

=

° ¡

– 2 ô 2 ñó ò ò ó + ô 4 ñ – ( ô ï – õ )(ñó ò ò ó – ô 2 ñ ) = 0. This equation arises in the turbulence theory. Setting H ( ) = "

 −  second-order linear equation of the form 2.1.2.12:

ñóò ò ó ò ò ó ó

+1 2

H 

 −  2 H − Œ (  −  ) H = 0.

 +

+1

+L .

, one obtains a (1)

Let the following boundary conditions be given: (0) = 
(0) = 0,

(1) = 
(1) = 0,

(2)

The solution of the original equation satisfying the first two conditions in (2) can represented as:   −     2 =   X H  −  − X   H  . 

M

0

M

0

To meet the last two conditions in (2), one should take the solution of (1) that satisfies the integral relations X 35.

ñóò ò ó ò ò ó ó

+ (ô

ï

2

1

0

ñ óò ò ó ò ò ó ó

+

M

 =X

1

0

   H

M

 = 0.

+ õ )ñó ò ò ó – 2 ôñ = 0.

Particular solution: 36.

  − H

ô ï ÷ ñ óò ò ó

0

+ õ (ô

= 

ï ÷

2

+ .

– õ )ñ = 0.

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

The substitution ¨ = 

 + 

= cos   £   ,

2

= exp  − £ −   ,

= sin   £   . 2

= exp   £ −   .

¨ 

 + (   −  )¨ = 0. leads to a second-order linear equation: u

Page 644

645

4.1. LINEAR EQUATIONS

37.

ñóò ò ó ò ò ó ó

ô ïƒ÷ +1 ñ óò ò ó

+

– 4ô

ïƒ÷ ñóò

ïƒ÷ –1 ñ

+ 6ô

= 0.

Particular solutions: 1 =  , 2 =  . The substitution ¨ =  2 

 − 4 
 + 6 leads to a second-order linear equation of the form 2.1.2.7: ¨ 

 +   +1 ¨ = 0. 2

3

38.

+ 10 ô ïƒ÷ ñó ò ò ó + 10 ôù ïƒ÷ –1 ñó ò + [3 ôù (ù – 1) ïw÷ –2 + 9 ô This is a special case of equation 4.1.6.25 with  =     .

39.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

+ (ô

ïƒ÷

+ õ )ñó ò ò ó +

ôƒõ ƒï ÷ ñ

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

The substitution ¨ = 

 +  ¨2

 +   ¨ = 0. 40.

= 0.

= cos  £   ,

ï 2÷ ]ñ

2

= 0.

= sin  £   .

2

= exp  − £ −   ,

= exp  £ −   .

2

leads to a second-order linear equation of the form 2.1.2.7:

ï 2 ñ óò ò ó ò ò ó ó

– 2( ô ï 2 + 6)ñó ò ò ó + ô ( ô ï 2 + 4)ñ = 0. Particular solutions: 1 =  −1 J 2 ® 1 J 2   £   , ; 1 J 2 (H ) are the modified Bessel functions.

J ;

= 

2

J   £   , where

−1 2

1 2

® 1J 2 (H

) and

4.1.2-4. Other equations. 41.

ñ óò ò ó ò ò ó ó

+ ô 3 ñ ó ò ò ó ò ó + ô 2 ñ ó ò ò ó + ô 1 ñ ó ò + ô 0 ñ = 0. A fourth-order constant coefficient linear equation. For  . leads to a third-order equation. Let  0 ≠ 0 and ( Œ ) = Œ characteristic polynomial.

.

0 4

= 0, the substitution ¨ ( ) = 
 +  3 Π3 +  2 Π2 +  1 Π+  0 be the

.

1 b . Let be factorizable, so that ( Œ ) = ( Œ − Œ 1 )( Œ − Œ 2 )( Œ − Œ 3 )( Œ − Œ 4 ), where Œ 1 , Œ 2 , Œ 3 , and Œ 4 are real numbers. The following cases are possible: a) Œ© are all different, then =L b) Œ

1

= Œ 2; Œ

3

and Œ

4

1

 

1

=Œ

2

=Œ

1

1

=Œ

2

=Œ

2

 

2



+L

3

 

+L

2

 ) 

1



+L

3

3

 

3

≠ Œ 4 , then

3

= (L d) Œ

+L



+L

4

+L

1

2

 + L 3  2 ) 

1



+L



+L

4

4

= (L

.

1

1



;

 

+L

1

2

 + L 3

2

+L

 3 ) 

4

4

4

1

1







;

;

. and Œ

2

are real numbers, and

≠ Œ 2 , then =L

b) Œ

4

 

= Π4 , then

3

2 b . Let ( Œ ) = ( Œ − Œ 1 )( Œ − Œ 2 )( Œ 2 + 2  1 Œ +  0 ), where Œ  21 −  0 < 0. The following cases are possible: a) Œ

 

are different and not equal to Π1 , then = (L

c) Œ



1

1

 

1



+L

2

 

2



+

−



1



[L

3

cos(“/ ) + L

sin(“/ )],

“ = ü  0 −  21 ;

sin(“/ )],

“ = ü  0 −  21 .

4

= Π2 , then = (L

1

+L

2

 ) 

1



+

−



1



[L

3

cos(“/ ) + L

4

Page 645

646

FOURTH-ORDER DIFFERENTIAL EQUATIONS

.

3 b . Let us assume that ( Œ ) = ( Œ 2 + 2  1 Œ +  0 )( Œ ý 12 − ý 0 < 0. The following cases are possible:

2

+ 2ý 1 Œ + ý 0 ), where 

2 1

−

0

< 0 and

a) (  1 − ý 1 )2 + (  0 − ý 0 )2 ≠ 0, then = where “ =

−



1



[L

1

cos(“/ ) + L

 sin(“/ )] +  −: 1 [ L

2

 0 −  21 , @ = ý 0 − ý 12 ; ü b)  1 = ý 1 ,  0 = ý 0 , then  =  −  1 [( L 1 + L 2  ) cos(/ “  ) + (L

42.

ñóò ò ó ò ò ó ó

43.

4

ï ñóò ò ó ò ó

+ 4ô

− 6 Œ

=

ñóò ò ó ò ò ó ó

2

+ (ô

+ 3

ï

+ 6ô

74

© =1

45.

46.

47.

2

ï ñóò ò ó

+ õ )ñó ò ò ó ò

+L

+

4

 ) sin(“/ )],

ô 4ï 4ñ

+ [ õ ( ô + ö ) ï + ö ]ñ„ó ò ò ó +

ó 1

= ,

2

=

“ = ü  0 −  21 .

= 0.

−





õ 2 ö ï ñ óò

–

õ 2 ^ö ñ

= 0.

.

= ô ï ÷ ñ ó ò ò ó ò ó + õÿñ ó ò – ôƒõ ï ÷ ñ . Particular solutions:  = exp( Œ   ) ( , = 1, 2, 3), where the Œ  are roots of the cubic equation Œ 3 −  = 0.

ñ óò ò ó ò ò ó ó

+ ô ï ÷ +3 ñ ó ò ò ó ò ó – 3 ô ï ÷ +2 ñ ó ò ò ó + 6 ô ï ÷ +1 ñ ó ò – 6 ô ï ÷ ñ = 0. Particular solutions: 1 =  , 2 =  2 , 3 =  3 . The substitution ¨ =  3 


 −3 2 

 +6 
 −6 leads to a first-order linear equation: ¨u
+   +3 ¨ = 0.

ñ óò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

+

ô ƒï ÷ ñ óò ò ó ò ó

+

õ ïƒú

+1

ñóò ò ó

– 2õ

ïƒú ñóò

+ 2õ

ïƒú

–1

ñ

= 0.

Particular solutions: 1 =  , 2 =  . The substitution ¨ =  2 

 − 2 
 + 2 leads to a second-order linear equation:  ¨ 

 + (   +1 − 2)¨ 
+ ' + +2 ¨ = 0. + ô ïƒ÷ ñó ò ò ó ò ó + õ ïƒú ñó ò ò ó + ô„ö 1 b . Particular solutions with ( > 0:

ñóò ò ó ò ò ó ó

2

ïƒ÷ ñóò 1

+ ö ( õ ïƒú – ö )ñ = 0. = cos   £ (  , 2 = sin   £ (  .

1

= exp  − £ − (  ,

+ ô ï ÷ ñ ó ò ò ó ò ó + ( õ ï ú + ö )ñ ó ò ò ó + ô„ö ï ÷ ñ ó ò + õ^ö ï 1 b . Particular solutions with ( > 0: 1 = cos  £ (  ,

ñ óò ò ó ò ò ó ó

The substitution ¨ = 

 + (

2

= exp   £ − (  .

¨ 

 +   2 ¨ 
+( ' + − ( )¨ = 0. leads to a second-order equation: u

2 b . Particular solutions with ( < 0:

50.

3 3

3

= 0.

The substitution ¨ = 

 + (

49.

ï ñóò

+ 4ô

2 2

2 b . Particular solutions with ( < 0: 48.

sin( @! )],

L·© exp( Œ© − 12   2 ), where the Œ"© are roots of the biquadratic equation

Particular solutions: 44.

4

ü

Solution:

Œ

cos( @! ) + L

3

ú ñ

2

£ −(  , 1 = exp  − 

= 0. = sin  £ (  . 2

= exp  £ − (  .

¨ 

 +   2 ¨ 
+ ' +*¨ = 0. leads to a second-order linear equation: u

+ 4ñó ò ò ó ò ó + ô ï ñ = 0. The substitution ¨ ( ) =  leads to a constant coefficient linear equation of the form 4.1.2.1: ¨2



 +  ¨ = 0.

ï ñóò ò ó ò ò ó ó ï ñóò ò ó ò ò ó ó

– 4ùñó ò ò ó ò

ó

ô ï ñ

= 0, ù = 1, 2, 3, uuu Solution: =  4  +3 ( −3 c )  ( −3 ¨ ), where c = x x  and ¨ = ¨ ( ) is the general solution of a linear constant coefficient equation of the form 4.1.2.1: ¨p



  +  ¨ = 0. +

Page 646

647

4.1. LINEAR EQUATIONS

51.

ï 2 ñ óò ò ó ò ò ó ó

+ 6 ï ñó ò ò ó ò ó + 6ñó ò ò ó – ô 2 ñ = 0. Equation of transverse vibrations of a pointed bar. 1 V Solution: = £ L 1 ø 1  2 £   + L 2 n 1  2 £   + L

®

 2 £   + L 4 ; 1  2 £   W , where ø 1 ( H ) and n 1 ( H ) are the Bessel functions, and ® 1 ( H ) and ; 1 ( H ) are the modified Bessel 

3 1

functions.

52.

53.

54. 55.

56.

57.

58.

ï 2 ñ óò ò ó ò ò ó ó

+ 2( ô + 2)ï ñ ó ò ò ó ò ó + ( ô + 1)( ô + 2)ñ ó ò ò ó – õ 4 ñ = 0. Solution: =  −  J 2 V L 1 ø   2  £   + L 2 n   2  £   + L 3 ®   2  £   + L 4 ;   2  £   W , where ø  (H ) and n  (H ) are the Bessel functions, and ®  (H ) and ;  (H ) are the modified Bessel functions.

ï 2 ñ óò ò ó ò ò ó ó

+ 8 ï ñó ò ò ó ò ó + 12ñó ò ò ó + ô ï 2 ñ = 0. The substitution ¨ ( ) =  2 leads to a constant coefficient linear equation of the form 4.1.2.1: ¨2



 +  ¨ = 0.

ï 2 ñ óò ò ó ò ò ó ó

+ 8 ï ñó ò ò ó ò ó + 12ñó ò ò ó = ô ï 3 ñ + õ . ¨ 



 =  ¨ +  . The substitution ¨ ( ) =  2 leads to an equation of the form 4.1.2.3: u

ï 2 ñ óò ò ó ò ò ó ó

+ ô ï ñó ò ò ó ò ó + ( õ ïƒ÷ +1 + ö )ñó ò ò ó + ( ô – 4) õ ïƒ÷ ñó ò + õ ( ö – 2 ô + 6) ïw÷ –1 ñ = 0. The substitution ¨ ( ) =  2 

 + (  − 4) 
 + ( ( − 2  + 6) leads to a second-order equation of the form 2.1.2.7: ¨3

 + '  −1 ¨ = 0.

ï 3 ñ óò ò ó ò ò ó ó

+ 2 ï 2 ñó ò ò ó ò ó – ï ñó ò ò ó + ñó ò – ô 4 ï 3 ñ = 0. Solution: = L 1 ø 0 (  ) + L 2 n 0 (  ) + L 3 ® 0 (  ) + L 4 ; 0 (  ), where ø 0 ( H ) and n 0 ( H ) are the Bessel functions, and ® 0 ( H ) and ; 0 ( H ) are the modified Bessel functions.

ï 4 ñ óò ò ó ò ò ó ó

+
3 ï 3 ñ ó ò ò ó ò ó +
2 ï 2 ñ ó ò ò ó +
1 ï ñ ó ò +
0 ñ = 0. The Euler equation. The substitution a = ln | | leads to a constant coefficient linear equation of the form 4.1.2.41: g

g
g
g + ( # 3 − 6)
g
g
g + (11 − 3# 3 + # 2 )
g
g + (2 # 3 − # 2 + # 1 − 6)
g + # 0 = 0.

ï 4 ñ óò ò ó ò ò ó ó

+ 2 ï 3 ñó ò ò ó ò ó – (2 ô 2 + 1) ï 2 ñó ò ò ó + (2 ô 2 + 1) ï ñó ò – V õ 4 ï 4 – ô 2 ( ô 2 – 4)WNñ = 0. This equation governs free transverse vibration modes of a thin round elastic plate. The equation arises from separation of variables in the two-dimensional equation ¢e¢ ¨ −  4 ¨ = 0, where ¢ is the Laplace operator written in the polar coordinate system, with  being the polar radius. Solution: = L 1 ø  ( ' ) + L 2 n  ( ' ) + L 3 ®  ( ' ) + L 4 ;  ( ' ), where ø  ( H ) and n  ( H ) are the Bessel functions, and ®  ( H ) and ;  ( H ) are the modified Bessel functions. In applications, one usually sets  = B , where B = 0, 1, 2, 

^_ 59.

The solution is specified by Popov (1998).

ï 4 ñ óò ò ó ò ò ó ó

– 2ù (ù + 1) ï 2 ñ ó ò ò ó + 4ù (ù + 1)ï ñ ó ò + [ ô ï 4 + ù (ù + 1)(ù + 3)(ù – 2)]ñ = 0. Here, B is a positive integer and  ≠ 0 (for  = 0, we have the Euler equation 4.1.2.57). Solution: equation Œ

60.

ï 4 ñ óò ò ó ò ò ó ó

4

74 . =  − LlÁ exp( Œ"Á ) Á ( ), where the Œ Á are four different roots of the Á

+  = 0, and

+ 2(2 – ù )ï

3

.

=1

ñ óò ò ó ò ó

Á ( ) is some definite polynomial of degree ≤ 4B . + (1 – ù )(2 – ù ) ï

2

ñ óò ò ó

–ô

4

ï 2÷ ñ

= 0.

Solution: = £  [ L 1 ø 1 J (ˆ ) + L 2 n 1 J (ˆ ) + L 3 ® 1 J (ˆ ) + L 4 ; 1 J  (ˆ )], where ˆ = 2(  B ) ´J 2; ø Á (ˆ ) and n Á (ˆ ) are the Bessel functions, and ®Á (ˆ ) and ; Á (ˆ ) are the modified Bessel functions.

Page 647

648 61.

62.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ï 4 ñ óò ò ó ò ò ó ó

+6 ï 3 ñó ò ò ó ò ó +[4 ï 4 +(7– ô 2 – õ 2 ) ï 2 ]ñó ò ò ó + ï (16 ï 2 +1– ô 2 – õ 2 )ñó ò +(8 ï 2 + ô 2 õ 2 )ñ = 0. Solution for  ≠ 0: = L 1 ø ( ) ø Á ( ) + L 2 ø ( ) n Á ( ) + L 3 n  ( ) ø Á ( ) + L 4 n  ( ) n Á ( ), where ø ( ) and n  ( ) are the Bessel functions; “ = 12 (  +  ) and @ = 12 (  −  ).

ï 8 ñ óò ò ó ò ò ó ó

+ 4 ï 7 ñó ò ò ó ò ó = ôñ . The substitution ¨ ( ) =  leads to an equation of the form 4.1.2.8: 

8

¨2



 =  ¨ .

4.1.3. Equations Containing Exponential and Hyperbolic Functions 4.1.3-1. Equations with exponential functions.

ó

ñóò ò ó ò ò ó ó

ó

1.

+ ô 3 ñó ò + õD ( ô 2 – õD )ñ The substitution ¨ = 

 +  
 +    2.1.3.10: ¨3

 −  ¨2
+ (  2 −    )¨

2.

ñ óò ò ó ò ò ó ó

+

ô  ó ñ óò

+ ô

ó

 ñ

=

= 0. leads to a second-order linear equation of the form = 0.



õ ó

.

Integrating yields a third-order linear equation: 3.

ñóò ò ó ò ò ó ó

+

ô ó ñóò

Particular solution:

ó

ñóò ò ó ò ò ó ó

ó

– ( ôƒõ 0



=

+



.

4.

+ 2 ô ñó ò + ô ( 2   The substitution ¨ = 

 +   ¨2

 −    ¨ = 0.

5.

ñóò ò ó ò ò ó ó

+ ( ô

ó

õ 3 )ñ óò

+

Particular solution: 6. 7.

0

ñ óò ò ó ò ò ó ó

+

ô  ó ñ óò ò ó

0

   +L .

ó

óñ

= 0. = 0.

óñ

= 0.

– 3 ô

óñ

= 0.

2

= (  +  ) . 3

= (  +  ) .

– õ ( ô



ó

+ õ )ñ = 0.

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ò ó ó

−1

= 0.

– 2 ô

The substitution ¨ = 

 −  ¨2

 + (    +  )¨ = 0. 10.

.

= '“

– ô 2  )ñ = 0. leads to a second-order linear equation of the form 2.1.3.1:

– ô =  +  .

ñóò ò ó ò ò ó ó + (ô ï + õ )  ó ñ óò

Particular solution: 9.

−

ñóò ò ó ò ò ó ó + (ô ï + õ )  ó ñ óò

Particular solution: 8.



0 =  

0

ó

ôƒõ ó ñ

+

ñóò ò ó ò ò ó ó + (ô ï + õ )  ó ñ óò

Particular solution:

õ 4 )ñ






 +   

ó

= exp  − £   , = cos   £ −   ,

2

= exp   £   .

= sin   £ −   .

leads to a second-order linear equation of the form 2.1.3.2:

ó

+ ( ô + õ )ñó ò ò ó + ôƒõ ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   , 2 b . Particular solutions with  < 0: The substitution ¨ = 

 +  ¨2

 +    ¨ = 0.

2

1

2

= exp  − £ −   ,

= sin   £   . 2

= exp   £ −   .

leads to a second-order linear equation of the form 2.1.3.1:

Page 648

649

4.1. LINEAR EQUATIONS

11.

ñ óò ò ó ò ò ó ó

+ 10 ô

ñóò ò ó ò ò ó ó

+

ó

 ñ óò ò ó

+ 10 ô

ó

 ñ óò

+ (3 ô

2

 

ó

+ 9ô

 

2 2



This is a special case of equation 4.1.6.25 with  ( ) =   . 12.

13.

ôñóò ò ó ò ó

+

õ ó ñóò

Particular solution:

 . 0 = 

ñ óò ò ó ò ò ó ó

+

ô  ó ñ óò ò ó ò ó

=

Particular solutions: ý 3 −  = 0. 14.

15.

16.

17.

+

ô ó ñóò ò ó ò ó

+ ( õ

+

ô ï 3   ó ñóò ò ó ò ó

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

20.

 ñ

.

õ ó ñ óò ò ó

ó

 ñ óò

+ ô„ö

+ ö ( õ

ó

– ö )ñ = 0.

1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ ( 2 b . Particular solutions with ( < 0: 1 = exp  − £ − (  , 2 = exp  The substitution ¨ = 

 + ( leads to a second-order equation: ¨u

 + 

ó



ô„ö ó ñóò

+ ö )ñó ò ò ó +

+

õ^ö  ó ñ

ï  ó ñóò

– 6 ô

. £ −(  .    ¨2
+(    − ( )¨ = 0.

= 0.

1 b . Particular solutions with ( > 0: 1 = cos  £ (  , 2 = sin  £ (  . 2 b . Particular solutions with ( < 0: 1 = exp  − £ − (  , 2 = exp  £ − (  .  ¨
  +    ¨ = 0. The substitution ¨ = 

 + ( leads to a second-order equation: ¨u

 +   2

ï 2   ó ñóò ò ó

– 3ô

+ 6ô

óñ

Particular solutions: 1 =  , 2 =  , 3 =  . The substitution ¨ =   leads to a first-order linear equation: ¨u
+  3  ¨ = 0.

ï ñóò ò ó ò ò ó ó

ï  ó ñóò

+ô

ó

( ô

+ õ )ñó ò ò ó ò ò ó

ó

=

(ô

ïƒú

+

õ ó

ô ó ñ ó 0

óò ò ó ò ò ó ó

=

.

= 

0

+ ö )ñó ò ò ó ò ò ó

+ õ )ñ



ó

=  .

0

Particular solution:

ó (ô ï ú 

2

– [ ô ( ï + 1) -

Particular solution: 19.

= 0.

ó

– ôƒõ

= 0.

:ž¼  ( , = 1, 2, 3), where the ý are roots of the cubic equation  = 

+

+

−

õÿñ óò

ô  ó ñ óò ò ó ò ó

ñ óò ò ó ò ò ó ó

Particular solution: 18.

ôƒõ ó ñ

+

ó )ñ



= 0. 3




 −3

2



 +6 
−6

+ ï + 4]ñ = 0.

+ .

õ ó ñ

, þ = 1, 2, 3.  =  + +   + ( . =

õÿñ

þ

,

= 0, 1, 2, 3.

   + +  − . 0 = 

Particular solution:

3

ïƒ÷ ) ñ óò ò ó

+ ô [ õ exp(

ïƒ÷

21.

ñóò ò ó ò ò ó ó

+ õ exp(

) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  exp( Œ   ).

22.

ñóò ò ó ò ò ó ó

+ [ ô + õ exp(

ïw÷ )]ñ óò ò ó

+ ôƒõ exp(

ïƒ÷ ) ñ

= 0.

This is a special case of equation 4.1.6.21 with  ( ) =  exp( Π  ).

4.1.3-2. Equations with hyperbolic functions. 23.

ñóò ò ó ò ò ó ó

+

ô

sinh ÷ (

Particular solution:

ï ) ñ óò 0

=

+ õ [ ô sinh ÷ ( −





ï

)–

õ 3 ]ñ

= 0.

.

Page 649

650 24.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ò ó ó

+ [ ô sinh ÷ (

ï

Particular solution:

)+ 0

=

õ 3 ]ñ óò 



−

+

ôƒõ

sinh ÷ (

.

25.

+ ( ô ï + õ ) sinh ÷ ( ï )ñó ò – ô sinh ÷ ( Particular solution: 0 =  +  .

26.

ñ óò ò ó ò ò ó ó

+ õ ) sinh ÷ (

ï

+ (ô

ñ óò ò ó ò ò ó ó

0

29.

0

ï )ñ

= 0.

ï )ñ ó ò

– 3 ô sinh ÷ (

ï )ñ

= 0.

3

= (  +  ) .

+ õ sinh ÷ ( ï ) ñ ó ò ò ó + ô [ õ sinh ÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  sinh  ( Œ  ).

ñ óò ò ó ò ò ó ó

+ [ ô + õ sinh ÷ ( ï )]ñó ò ò ó + ôƒõ sinh ÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  sinh  ( Œ  ).

ñóò ò ó ò ò ó ó

( ô ïƒú + õ sinh ï )ñ„ó ò ò ó ò ò ó ó = õ sinh ï ñ , + +  sinh  . Particular solution: 0 = 

31.

ñóò ò ó ò ò ó ó

+

ô

cosh÷ (

Particular solution:

ñóò ò ó ò ò ó ó

+ [ ô cosh÷ (

Particular solution:

ï ) ñ óò ï

0

=

)+

+ õ [ ô cosh÷ ( −





.

õ 3 ]ñ óò

0 =   −



+

ôƒõ

ï

þ

õ 3 ]ñ

= 0.

cosh÷ (

ï )ñ

= 0.

.

+ ( ô ï + õ ) cosh ÷ ( ï )ñ ó ò – ô cosh÷ ( Particular solution: 0 =  +  .

34.

ñóò ò ó ò ò ó ó

+ õ ) cosh ÷ (

ï

+ (ô

ñóò ò ó ò ò ó ó

+ (ô

ï

38. 39. 40.

0

0

= 0.

ï )ñ óò

– 2 ô cosh÷ (

ï )ñ

= 0.

ï )ñ óò

– 3 ô cosh÷ (

ï )ñ

= 0.

2

= (  +  ) .

+ õ ) cosh ÷ (

Particular solution:

37.

ï )ñ

ñ óò ò ó ò ò ó ó

Particular solution:

36.

= 1, 2, 3.

)–

33.

35.

= 0.

– 2 ô sinh ÷ (

30.

32.

= 0.

ï )ñ ó ò

= (  +  )2 .

+ õ ) sinh ÷ (

ï

+ (ô

Particular solution: 28.

ï )ñ

ñóò ò ó ò ò ó ó

Particular solution: 27.

ï )ñ

3

= (  +  ) .

+ õ cosh÷ ( ï ) ñó ò ò ó + ô [ õ cosh÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  cosh  ( Œ  ).

ñóò ò ó ò ò ó ó

+ [ ô + õ cosh÷ ( ï )]ñ ó ò ò ó + ôƒõ cosh÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  cosh  ( Œ  ).

ñ óò ò ó ò ò ó ó

( ô ïƒú + õ cosh ï )ñó ò ò ó ò ò ó ó = õ cosh ï ñ , Particular solution: 0 =  + +  cosh  .

þ

= 1, 2, 3.

= ñ + ô (ñó ò cosh ï – ñ sinh ï ).
The substitution ¨ =  cosh  − sinh  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

= ñ + ô (ñó ò sinh ï – ñ cosh ï ). The substitution ¨ = 
sinh  − cosh  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

Page 650

651

4.1. LINEAR EQUATIONS

41.

ñóò ò ó ò ò ó ó

tanh÷ (

ô

+

ï ) ñ óò

Particular solution: 42.

ñóò ò ó ò ò ó ó

+ [ ô tanh÷ (

ï

0

=

)+

+ õ [ ô tanh ÷ (





−

.

õ 3 ]ñ óò 

0 =   −

Particular solution:

+

ôƒõ

ï

) – õ 3 ]ñ = 0.

tanh÷ (

+ ( ô ï + õ ) tanh ÷ ( ï )ñó ò – ô tanh÷ ( Particular solution: 0 =  +  .

44.

ñóò ò ó ò ò ó ó

+ õ ) tanh ÷ (

Particular solution: 45.

ñóò ò ó ò ò ó ó

0

0

= 0.

ï )ñ óò

– 2 ô tanh ÷ (

ï )ñ

= 0.

ï )ñ óò

– 3 ô tanh ÷ (

ï )ñ

= 0.

2

= (  +  ) .

+ õ ) tanh ÷ (

ï

+ (ô

Particular solution:

46.

ï )ñ

ñóò ò ó ò ò ó ó

ï

= 0.

.

43.

+ (ô

ï )ñ

3

= (  +  ) .

+ õ tanh÷ ( ï ) ñó ò ò ó + ô [ õ tanh÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  tanh  ( Œ  ).

ñóò ò ó ò ò ó ó

47.

+ [ ô + õ tanh÷ ( ï )]ñó ò ò ó + ôƒõ tanh÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  tanh  ( Œ  ).

48.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

coth÷ (

ô

+

Particular solution: 49.

ñóò ò ó ò ò ó ó

+ [ ô coth÷ (

Particular solution: 50. 51.

+ õ [ ô coth÷ (

ï ) ñ óò ï

0

=

)+ 0

=





−

.

õ 3 ]ñ óò 

−



ï

)–

õ 3 ]ñ

= 0.

+ ôƒõ coth÷ (

ï )ñ

= 0.

.

+ õ coth÷ ( ï ) ñó ò ò ó + ô [ õ coth÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  coth  ( Œ  ).

ñóò ò ó ò ò ó ó

+ [ ô + õ coth÷ ( ï )]ñ ó ò ò ó + ôƒõ coth÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  coth  ( Œ  ).

ñ óò ò ó ò ò ó ó

4.1.4. Equations Containing Logarithmic Functions 1.

ñ óò ò ó ò ò ó ó

+

ô

ln 

ï ñ óò

Particular solution:

– ( ôƒõ ln  0

=





ï

+

õ 4 )ñ

.

2.

+ ( ô ï + õ ) ln  ( ï )ñó ò – ô ln  ( Particular solution: 0 =  +  .

3.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

ï

+ õ ) ln  (

+ (ô

Particular solution: 4.

ñóò ò ó ò ò ó ó

+ (ô

ï

+ õ ) ln  (

Particular solution: 5.

ï 2 ñ óò ò ó ò ò ó ó

0

0

= 0.

ï )ñ

= 0.

ï )ñ óò

– 2 ô ln  (

ï )ñ

= 0.

ï )ñ óò

– 3 ô ln  (

ï )ñ

= 0.

2

= (  +  ) . 3

= (  +  ) .

+ 2 ô ï ñó ò – ô [1 + ô ï 2 ln2 ( õ ï )]ñ = 0. ¨ 

 −  ln( ' )¨ = 0. The substitution ¨ = 

 +  ln( ' ) leads to a second-order equation: u

Page 651

652 6. 7. 8.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ï ) ñ óò ò ó

ñóò ò ó ò ò ó ó

+ õ ln  (

ñ óò ò ó ò ò ó ó

+ [ ô + õ ln  (

ñóò ò ó ò ò ó ó

+ ln  (

ï )]ñ óò ò ó

ï )(ï 2 ñ óò ò ó

ñóò ò ó ò ò ó ó

+ ln  (

1

ï )(ï 2 ñ óò ò ó

ñóò ò ó ò ò ó ó

ô ï

+

2

1

– 2ï

ï )ñ

ln  (

ôƒõ

0

ñóò

= ,

2

– 4ï

= 0.

+ 2ñ ) = 0.

=  2.

ñóò

2

= ,

ï )ñ óò ò ó

ln  (

Particular solution: 11.

+

This is a special case of equation 4.1.6.21 with  ( ) =  ln  ( Π ).

Particular solutions: 10.

) – ô ]ñ = 0.

This is a special case of equation 4.1.6.20 with  ( ) =  ln  ( Π ).

Particular solutions: 9.

ï

+ ô [ õ ln  (

+ 6ñ ) = 0. 2

=  3.

ï )ñ

– 2 ô ln  (

2

= .

+ ôñó ò ò ó ò ó + õ ln  ( ï )ñó ò + ôƒõ ln  (  Particular solution: 0 =  −  .

ñóò ò ó ò ò ó ó

= 0.

ï )ñ

= 0.

4.1.5. Equations Containing Trigonometric Functions 4.1.5-1. Equations with sine and cosine. 1.

ñóò ò ó ò ò ó ó

+

ô

sin ÷ (

Particular solution: 2.

ñóò ò ó ò ò ó ó

+ õ [ ô sin ÷ (

ï ) ñ óò

+ [ ô sin÷ (

Particular solution:

ï

0

=

)+ 0





−

=



)–

õ 3 ]ñ

= 0.

sin ÷ (

ï )ñ

= 0.

.

õ 3 ]ñ óò −

ï



+

ôƒõ

.

3.

+ ( ô ï + õ ) sin ÷ ( ï )ñ ó ò – ô sin÷ ( Particular solution: 0 =  +  .

4.

ñóò ò ó ò ò ó ó

ñ óò ò ó ò ò ó ó

ï

+ õ ) sin ÷ (

+ (ô

Particular solution: 5.

ñóò ò ó ò ò ó ó

+ (ô

ï

7. 8.

9.

0

= 0.

ï )ñ óò

– 2 ô sin ÷ (

ï )ñ

= 0.

ï )ñ óò

– 3 ô sin ÷ (

ï )ñ

= 0.

2

= (  +  ) .

+ õ ) sin ÷ (

Particular solution: 6.

0

ï )ñ

3

= (  +  ) .

+ ô sin ÷ ( ï ) ñ ó ò ò ó + õ [ ô sin ÷ ( ï ) – õ ]ñ = 0. ¨ 

 + [  sin ( Œ  ) −  ]¨ = 0. The substitution ¨ = 

 +  leads to a second-order equation: u

ñ óò ò ó ò ò ó ó

+ [ ô + õ sin ÷ ( ï )]ñó ò ò ó + ôƒõ sin÷ ( ï ) ñ = 0. ¨ 

 +  sin ( Œ  ) ¨ = 0. The substitution ¨ = 

 +  leads to a second-order linear equation: u

ñóò ò ó ò ò ó ó

= ô sin÷ ( ï ) ñó ò ò ó ò ó + õÿñó ò – ôƒõ sin ÷ ( ï ) ñ .  Particular solutions:  =  :ž¼ ( , = 1, 2, 3), where the ý  are roots of the cubic equation 3 ý −  = 0.

ñóò ò ó ò ò ó ó

+ ô sin ÷ ( ï ) ( ï 3 ñó ò ò ó ò ó – 3 ï 2 ñó ò ò ó + 6 ï ñó ò – 6ñ ) = 0. This is a special case of equation 4.1.6.33 with  ( ) =  sin  ( Œ  ).

ñóò ò ó ò ò ó ó

Page 652

4.1. LINEAR EQUATIONS

10. 11.

ï 2 ñ óò ò ó ò ò ó ó

sin ÷ (

ô

+

) (ï

ï

( ô sin ï + õ )ñ„ó ò ò ó ò ò ó

ó

ï

ó

=

ï ú

(ô

+ õ sin ï )ñ

óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

cos÷ (

ô

+

ñóò ò ó ò ò ó ó

+ [ ô cos÷ (

16.

ñ óò ò ó ò ò ó ó ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

ï

19.

20. 21.

cos÷ (

ô

+ [ ô + õ cos÷ (

ñóò ò ó ò ò ó ó

=

23.

24.

+





cos÷ (

ô

ñóò ò ó ò ò ó ó

+

cos÷ (

ô

ï

= 2, 3.

)–

õ 3 ]ñ

= 0.

cos÷ (

ï )ñ

= 0.

.

ôƒõ

.

ï )ñ ó ò

– ô cos÷ (

ï )ñ óò

– 2 ô cos÷ (

ï )ñ

2

– 3 ô cos÷ (

ï )ñ ó ò

3

= (  +  ) .

+ õ [ ô cos÷ (

ï )]ñ óò ò ó

The substitution ¨ = 

 + 

= 0.

ï )ñ

= 0.

ï )ñ

= 0.

ï

) – õ ]ñ = 0. ¨ 

 + [  cos ( Œ  ) −  ]¨ = 0. leads to a second-order equation: u + ôƒõ cos÷ (

ï )ñ

= 0.

¨ 

 +  cos ( Œ  ) ¨ = 0. leads to a second-order linear equation: u

ï ) ñ óò ò ó ò ó

ï

) (ï

3

cos÷ (

ï

) (ï

óò ò ó ò ò ó ó

=

+

– ôƒõ cos÷ (

õÿñóò

ï )ñ

. :ž¼  ( , = 1, 2, 3), where the ý are roots of the cubic equation =    – 3ï

ñóò ò ó ò ó

2

ñóò ò ó

+ 6ï

ñóò

– 6ñ ) = 0.

This is a special case of equation 4.1.6.33 with  ( ) =  cos  ( Π ).

ï 2 ñ óò ò ó ò ò ó ó

ô

– 4ï

ñóò + 6ñ ) = 0. The substitution ‡ =  2 

 −4 
 +6 leads to a second-order equation: ‡

 +  cos ( Œ  )‡ = 0. +

( ô cos ï + õ )ñ

Particular solution: 25.



The substitution ¨ = 

 + 

Particular solutions: ý 3 −  = 0. 22.

−

ï ) ñ óò ò ó

ñ óò ò ó ò ò ó ó

þ

,

= (  +  ) .

0

+

ñ

=  +  .

0

ñóò ò ó ò ò ó ó



õ 3 ]ñ óò

=

0

Particular solution:

.

sin ï

−

+ õ ) cos÷ (

ï

ñ

+ õ [ ô cos÷ (

+ õ ) cos÷ (

+ (ô + (ô

õ

+ õ ) cos÷ (

ï

+ (ô

Particular solution: 18.

=

=

0

Particular solution: 17.

sin ï

)+

Particular solution:

.

  + +  sin  . 0 = 

0

ï

ñ

õ

ï ) ñ óò

Particular solution: 15.

+ 6ñ ) = 0.

=  +  sin  .

0

Particular solution: 14.

ñóò

=  sin  +  .

0

+ õ sin ï )ñ„ó ò ò ó ò ò ó

(ô

sin ï

ô

=

Particular solution: 13.

– 4ï

ñóò ò ó

The substitution ‡ =  

 −4 
 +6 leads to a second-order equation: ‡

 +  sin ( Œ  )‡ = 0. 2

Particular solution: 12.

2

653

(ô

ï

+ õ cos ï )ñ„ó ò ò ó ò ò ó

Particular solution:

ô

ó

0

= 0

2

ñóò ò ó

cos ï

ñ

.

=  cos  +  .

õ

cos ï

ñ

.

=  +  cos  .

Page 653

654 26. 27.

28.

FOURTH-ORDER DIFFERENTIAL EQUATIONS ( ô ï ú + õ cos ï )ñ ó ò ò ó ò ò ó Particular solution:

ó 0

= õ cos ï ñ , =  + +  cos  .

þ

= 2, 3.

+ 2 ôƒõ cos( õ ï ) ñ„ó ò – ô [ õ 2 sin( õ ï ) + ô sin2 ( õ ï )]ñ = 0. The substitution ¨ = 

 +  sin( ' ) leads to a second-order linear equation of the form 2.1.6.2: ¨ 

 −  sin( ' )¨ = 0.

ñóò ò ó ò ò ó ó

sin4 ï

+ sin2 ï (sin2 ï – 3) ñ ó ò ò ó + sin ï cos ï (2 sin2 ï + 3) ñó ò + ( ô 4 sin4 ï – 3)ñ = 0. Equation of a loaded rigid spherical shell. If  4 = 1 − Œ 2 , the equation can be rewritten as

ñ óò ò ó ò ò ó ó

+ 2 sin3 ï cos ï

ñ óò ò ó ò ó

2

LL( ) − Œ

where L ≡ M

= 0,

This equation falls into two second-order equations: L( ) + Œ

M

2



2

L( ) − Œ

= 0,

+ cot M

M



− cot2  .

= 0,

which differ only in the sign of the parameter Œ . The transformation ˆ = sin 2  , ¨ =  sin  reduces the latter equations to the hypergeometric equations 2.1.2.171:

ˆ (ˆ − 1)¨ ªŠ

Š + ( 25 ˆ − 2)¨ Š
+ 14 (1 TeŒ )¨ = 0.

29. 30.

= ñ + ô (ñó ò sin ï – ñ cos ï ). The substitution ¨ = 
sin  − cos  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

= ñ + ô (ñó ò cos ï + ñ sin ï ).
The substitution ¨ =  cos  + sin  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

4.1.5-2. Equations with tangent and cotangent. 31.

+ ôñó ò + ( ô tan ï – 1)ñ = 0. Particular solution: 0 = cos  .

ñóò ò ó ò ò ó ó

32.

+ ( ô ï + õ ) tan÷ ( ï )ñó ò – ô tan÷ ( Particular solution: 0 =    + .

33.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

+ õ ) tan÷ (

ï

+ (ô

Particular solution: 34.

ñ óò ò ó ò ò ó ó

+ (ô

ï

0

Particular solution: 35.

ñóò ò ó ò ò ó ó

+

ô

tan÷ (

0

ï ) ñ óò

Particular solution: 36.

ñ óò ò ó ò ò ó ó

+ [ ô tan÷ (

Particular solution: 37.

ï

0

ï )ñ

= 0.

ï )ñ ó ò

– 3 ô tan÷ (

ï )ñ

= 0.

)–

õ 3 ]ñ

= 0.

+ ôƒõ tan÷ (

ï )ñ

= 0.

2

3

+ õ [ ô tan÷ (

)+ 0

– 2 ô tan÷ (

= (  +  ) . =





−

.

õ 3 ]ñ ó ò

=

= 0.

ï )ñ ó ò

= (  +  ) .

+ õ ) tan÷ (

ï )ñ

−





ï

.

+ õ tan÷ ( ï ) ñó ò ò ó + ô [ õ tan÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  tan  ( Œ  ).

ñóò ò ó ò ò ó ó

Page 654

655

4.1. LINEAR EQUATIONS

38. 39.

40.

+ [ ô + õ tan÷ ( ï )]ñ ó ò ò ó + ôƒõ tan÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  tan  ( Œ  ).

ñ óò ò ó ò ò ó ó

= ô tan÷ ( ï ) ñ ó ò ò ó ò ó + õÿñ ó ò – ôƒõ tan÷ ( ï ) ñ .  Particular solutions:  =  :ž¼ ( , = 1, 2, 3), where the ý  are roots of the cubic equation ý 3 −  = 0.

ñ óò ò ó ò ò ó ó

+ ôñó ò – (1 + ô cot ï )ñ = 0. Particular solution: 0 = sin  .

ñóò ò ó ò ò ó ó

41.

+ ( ô ï + õ ) cot÷ ( ï )ñó ò – ô cot÷ ( Particular solution: 0 =  +  .

42.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

+ õ ) cot÷ (

ï

+ (ô

Particular solution: 43.

ñóò ò ó ò ò ó ó

ï

+ (ô

Particular solution: 44.

ñóò ò ó ò ò ó ó

+

ô

cot÷ (

Particular solution: 45.

ñóò ò ó ò ò ó ó

+ [ ô cot÷ (

Particular solution:

46. 47. 48.

ï

– 2 ô cot÷ (

ï )ñ

= 0.

ï )ñ óò

– 3 ô cot÷ (

ï )ñ

= 0.

2

= (  +  ) .

0

= (  +  )3 .

ï ) ñ óò 0

+ õ [ ô cot÷ (

0





−

=

)+

= 0.

ï )ñ óò

0

+ õ ) cot÷ (

ï )ñ

.

õ 3 ]ñ óò −

=





ï

) – õ 3 ]ñ = 0.

+ ôƒõ cot÷ (

ï )ñ

= 0.

.

+ õ cot÷ ( ï ) ñó ò ò ó + ô [ õ cot÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  cot  ( Œ  ).

ñóò ò ó ò ò ó ó

+ [ ô + õ cot÷ ( ï )]ñó ò ò ó + ôƒõ cot÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  cot  ( Œ  ).

ñóò ò ó ò ò ó ó

= ô cot÷ ( ï ) ñó ò ò ó ò ó + õÿñó ò – ôƒõ cot÷ ( ï ) ñ .  Particular solutions:  =  :ž¼ ( , = 1, 2, 3), where the ý  are roots of the cubic equation 3 ý −  = 0.

ñóò ò ó ò ò ó ó

4.1.6. Equations Containing Arbitrary Functions 4.1.6-1. Equations of the form  4 ( )



 +  1 ( )
 +  0 ( ) = ( ). 1.

2.

3.

= ! ( ï )ñ . The transformation  = a

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

ô ï öï

=I ! O

+õ + ý P (ö

−1

ï

= ‡"a

,

−3

leads to an equation of the same form: ‡ g

g
g
g = a

−8

 (1 a )‡ .

ñ

. + ý )8  +  The transformation H = ,‡ = leads to a simpler equation: ‡ ‘'

‘'

‘'‘ = ¢ (˜ + ( (˜ + )3 M M where ¢ =  − ˜( .

−4

 ( H )‡ ,

M

– ! ó ò ò ó ò ò ó ó ñ = 0, ! = ! ( ï ). Particular solution: 0 =  ( ).

!„ñ óò ò ó ò ò ó ó

Page 655

656

FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.

ñóò ò ó ò ò ó ó

+ !„ñó ò – ô ( ! + ô 3 )ñ = 0,  Particular solution: 0 =   .

5.

ñóò ò ó ò ò ó ó

= ! (ï ).

!

+ ( ! + ô 3 )ñó ò + ô!„ñ = 0,  Particular solution: 0 =  −  .

!

= ! (ï ).

6.

+ ( ô ï + õ ) ! ( ï )ñ„ó ò – ô! ( ï )ñ = 0. Particular solution: 0 =  +  .

7.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

+ õ ) ! ( ï )ñ

ï

+ (ô

Particular solution: 8.

9.

ñ óò ò ó ò ò ó ó

+ (ô

ï

0

+ õ ) ! ( ï )ñ

óò

– 2 ô! ( ï )ñ = 0.

óò

– 3 ô! ( ï )ñ = 0.

= (  +  )2 .

Particular solution: 0 = (  +  )3 . The substitution H = (  +  )
 −3  linear equation: H 


 + (  +  )  ( )H = 0.

ñ óò ò ó ò ò ó ó

+ ! ( ï )ñ

óò

+

! ó ò ( ï )ñ

= " ( ï ).




 +  ( ) = Xw ( )  + L . M

Integrating yields a third-order linear equation: 10. 11. 12.

13. 14. 15. 16.

leads to a third-order

+ 2 !„óÈò ñó ò + ( !„ó ò ò ó – ! 2 )ñ = 0, ! = ! (ï ). ¨ 

 −  ( )¨ = 0. The substitution ¨ = 

 +  ( ) leads to a second-order linear equation: u

ñóò ò ó ò ò ó ó

ï ñóò ò ó ò ò ó ó

+ ï ! ( ï )ñó ò – [( ï + 1) ! ( ï ) + ï + 4]ñ = 0.  Particular solution: 0 =   .

+ ! ( ï )ñ ó ò + " ( ï )ñ + % ( ï ) = 0. The transformation  = a −1 , = ¨ a −3 leads to an equation of the same form:

ñ óò ò ó ò ò ó ó

¨ g

g
g
g − a −6  (1 a )¨ g
+ V 3a −7  (1 a ) + a −8 (1 a )W ¨ + a −5 % (1 a ) = 0.

= ñ + ! ( ï )(ñ ó ò cosh ï – ñ sinh ï ). The substitution ¨ = 
cosh  − sinh  leads to a third-order linear equation.

ñ óò ò ó ò ò ó ó

= ñ + ! ( ï )(ñó ò sinh ï – ñ cosh ï ). The substitution ¨ = 
sinh  − cosh  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

= ñ + ! ( ï )(ñó ò sin ï – ñ cos ï ).
The substitution ¨ =  sin  − cos  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

= ñ + ! ( ï )(ñ ó ò cos ï + ñ sin ï ). The substitution ¨ = 
cos  + sin  leads to a third-order linear equation.

ñ óò ò ó ò ò ó ó

17.

+ !„ñó ò + ( ! tan ï – 1)ñ = 0, Particular solution: 0 = cos  .

18.

ñ óò ò ó ò ò ó ó

+ !„ñ ó ò – (1 + Particular solution:

19.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

=

r óò ò óò ò ó ó r ñ

!

cot ï )ñ = 0, 0 = sin  .

+ ! ( ï ) Oñó ò –

Y
 The substitution ¨ = 
− Y

r óò r ñ P

!

= ! ( ï ).

!

= ! ( ï ).

,

r

=

r (ï

).

leads to a third-order linear equation.

Page 656

657

4.1. LINEAR EQUATIONS

4.1.6-2. Equations of the form  4 ( )



 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ). 20.

ñóò ò ó ò ò ó ó

+

!„ñóò ò ó

+ ô ( ! – ô )ñ = 0,

1 b . Particular solutions with  > 0:

1

= cos   £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

ñóò ò ó ò ò ó ó

!

The substitution ¨ = 

 +  21.

23.

2

+ ( ! + ô )ñó ò ò ó + ô!„ñ = 0,

1

= cos  £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

+ ! ( ï )( ï

2

ñóò ò ó ò ò ó ó

+ ! ( ï )( ï

2

2

= ! (ï ).

1 b . Particular solutions with  > 0:

ñóò ò ó ò ò ó ó

= sin   £   . = exp   £ −   .

¨ 

 + (  −  )¨ = 0. leads to a second-order linear equation: u

The substitution ¨ = 

 +  22.

= ! ( ï ).

!

2

= sin  £   . 2

= exp   £ −   .

¨ 

 +  ( )¨ = 0. leads to a second-order linear equation: u

ñóò ò ó

– 2ï

ñóò

+ 2ñ ) = 0.

ñóò ò ó

– 4ï

ñóò

+ 6ñ ) = 0.

Particular solutions: 1 =  , 2 =  2 . The substitution H =  2 

 − 2 
 + 2 leads to a second-order linear equation: "H 

 − 2 H 
+  3  ( ) H = 0. Particular solutions: 1 =  , 2 =  3 . The substitution ¨ =  2 

 − 4 
 + 6 leads to a ¨ 

 +  2  ( )¨ = 0. second-order linear equation: u 2

24.

ñóò ò ó ò ò ó ó

+ (ô

ï

2

+

õï

+ ö ) ! ( ï )ñó ò ò ó – 2 ô! ( ï )ñ = 0. 0

= 

+ 10 !

ó ò ñ óò

Particular solution: 25.

ñ óò ò ó ò ò ó ó

+ 10 !„ñ

óò ò ó

Solution:

2

+ ' + ( .

+ (3 ! =L

1

¨

óò òó 3 1

+ 9!

+L

2

¨

2

2 1

)ñ = 0,

¨

2

+L

3

= ! (ï ).

! ¨ ¨ 1

2 2

+L

4

¨ 3, 2

where ¨ 1 and ¨ 2 are nontrivial linearly independent solutions of the second-order linear equation: ¨3

 +  ¨ = 0. 26.

ñóò ò ó ò ò ó ó

+ ( ! + " )ñó ò ò ó + 2 !„óò ñó ò + ( !„ó ò ò ó +

The substitution ¨ = 

 + 

!" )ñ

= ! ( ï ),

!

= 0,

= " ( ï ).

"

¨ 

 + ¨ = 0. leads to a second-order linear equation: u

4.1.6-3. Other equations. 27.

28.

óò ò ó ò ó

+ï

Particular solution:

0

ñ óò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

+ ! ( ï )ñ

+ ! ( ï )ñó ò ò ó ò

ó

ñ óò ò ó ò ò ó ó

+

!„ñ óò ò ó ò ó

+ "Iñ

– 2" ( ï )ñ = 0.

=  2.

– 2ô

Particular solutions: 29.

" ( ï )ñ ó ò

ñóò ò ó

–ô

+ ô!„ñ

óò

2

−  , 1 = 

óò ò ó

2

! (ï )ñ óò

 . 2 = 

+

ô 4ñ

= 0.

+ ô (" – ô )ñ = 0,

!

1 b . Particular solutions with  > 0:

1

= cos  £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

The substitution ¨ = 

 + 

2

= ! ( ï ),

"

= sin  £   . 2

= " ( ï ).

= exp   £ −   .

¨ 

 +  2 ¨ 
+( −  )¨ = 0. leads to a second-order linear equation: u

Page 657

658 30.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ò ó ó

+ !„ñ ó ò ò ó ò ó + (" + ô )ñ ó ò ò ó + ô!„ñ 1 b . Particular solutions with  > 0:

óò

+ ô"Iñ = 0, £  , 1 = cos  

+ ! ( ï )ñó ò ò ó ò ó + " ( ï )ñó ò ò ó + ï Particular solution: 0 =  .

32.

ñóò ò ó ò ò ó ó

33.

34.

35.

ñóò ò ó ò ò ó ó

ó

+ " ( ï )( ï

2

ñóò ò ó

% (ï )ñ óò

– % ( ï )ñ = 0.

– 2ï

+ 2ñ ) = 0.

ñóò

+ ! ( ï )( ï 3 ñó ò ò ó ò ó – 3 ï 2 ñó ò ò ó + 6 ï ñó ò – 6ñ ) = 0. Particular solutions: 1 =  , 2 =  2 , 3 =  3 . The substitution ¨ =  3 


 −3 2 

 +6 
 −6 leads to a first-order linear equation: ¨u
+  3  ( )¨ = 0.

= ! ( ï )ñó ò ò ó ò ó + ôñó ò – ô! ( ï )ñ .  Particular solutions:  =  ¼ ( , = 1, 2, 3), where the Œ  are roots of the cubic equation Œ 3 −  = 0.

ñóò ò ó ò ò ó ó

= ( ! – ô )ñó ò ò ó ò ó + ( ô! – õ )ñó ò ò ó + ( õ! – ö )ñó ò + ö!„ñ , ! = ! ( ï ).  ¼ Particular solutions:  =  ( , = 1, 2, 3), where the Œ  are roots of the cubic equation Œ 3 + Œ 2 + Œ + ( = 0.

ñóò ò ó ò ò ó ó

37.

ñ óò ò ó ò ò ó ó

40.

ñóò ò ó ò ò ó ó

+ ( ! 3 + ô )ñ ó ò ò ó ò Particular solution:

ï ñóò ò ó ò ò ó ó

ó 0

+ (! 2 +  =  − .

ô! 3 )ñ óò ò ó

+ (!

2

1

ï "Iñóò – ô 2I" ñ

+

ô! 2 )ñ óò

!

= 0,

+ ô!

1

ñ

= 0,

= ! ( ï ),

!÷

"

= " ( ï ).

! ÷ (ï

=

).

+ 4ñó ò ò ó ò ó + ô ï ñ = ! ( ï ). The substitution ¨ ( ) =  leads to a constant coefficient nonhomogeneous linear equation: ¨2



 +  ¨ =  ( ).

ï 2 ñ óò ò ó ò ò ó ó

+ ô ï ñó ò ò ó ò ó + ( ï 2 ! + õ )ñó ò ò ó + ( ô – 4) ï !„ñó ò + ( õ – 2 ô + 6) !„ñ = 0, ! = ! (ï ). ¨ 
2 

 The substitution =  + (  − 4)  + (  − 2  + 6) leads to a second-order linear equation: ¨2

 +  ¨ = 0.

ï 4 ñ óò ò ó ò ò ó ó

+

ô ï 3 ñ óò ò ó ò ó

+

Particular solution: 41.

= exp   £ −   .

ñóò ò ó ò ò ó ó

+ ( ! + ô )ñó ò ò ó ò ó + ( ô! + " + ô ï " )ñó ò ò ó + ô  Particular solutions: 1 =  , 2 =  −  .

39.

2

Particular solutions: 1 =  , 2 =  . The substitution H =  2 

 − 2 
 + 2 leads to a second-order linear equation: "H 

 + [  ( ) − 2] H 
+  3 ( ) H = 0. 2

36.

38.

= ! ( ï ), " = " ( ï ). = sin   £   .

¨ 

 +  2 ¨ 
+ ¨ = 0. leads to a second-order linear equation: u

31.

+ ! ( ï )ñó ò ò ó ò

2

£ −  , 1 = exp  − 

2 b . Particular solutions with  < 0: The substitution ¨ = 

 + 

!

ñ óò ò ó ò ò ó ó

+ 6 !„ñ

óò ò ó ò ó

Solution:

0

ï ! (ï )ñ óò

=

3−

.

+ ( ô – 3) ! (ï )ñ = 0.

+ (4 ! ó ò + 11 ! 2 + 10" )ñ + 3(2 !„óò " + 5 !"Ió ò + 6 ! =L

1

¨

3 1

+L

2

óò ò ó

¨

2

"

+ (! ó ò ò ó + 7! ! ó ò + 6! + "Ió ò ò ó + 3" 2 )ñ = 0,

2 1

¨

2

+L

3

¨ ¨ 1

2 2

+L

4

3

+ 30 !" + 10" ! = ! (ï ),

"

ó ò )ñ ó ò

= " ( ï ).

¨ 3, 2

where ¨ 1 and ¨ 2 are nontrivial linearly independent solutions of the second-order linear equation: ¨3

 +  ¨2
+ ¨ = 0. 42.

( !„ñó ò ò ó )ò òó

ó

= 0,

!

= ! ( ï ).

Equation of transverse vibrations of a bar. Solution:

=L

1+L

2 + X 

  −a (L 0  (a )

3

+L

4

a) a. M

Page 658

659

4.2. NONLINEAR EQUATIONS

4.2. Nonlinear Equations 4.2.1. Equations Containing Power Functions 4.2.1-1. Equations of the form 



 =  ( , ). 1.

ñ óò ò ó ò ò ó ó

=


ñ

)

–5 3

.

Multiply both sides of the equation by 5 J 3 and differentiate the resulting expression with respect to  to obtain  + 5 




  = 0. 3! (5) Integrating this equation three times, we arrive at the chain of equalities:





 + 2 


 − 


 − 2( 


3! 3! 3!





 − ( 

 )2 = 2 L 2 , 

 = 2 L 2  + L 1 , )2 = L 2  2 + L 1  + L 0 ,

(1) (2) (3)

where L 0 , L 1 , and L 2 are arbitrary constants. By eliminating the highest derivatives from (1)–(3) with the help of the original equation, we obtain a first-order equation: (2

.

.

. 
− 3 
)2 = 9( L

2 1

− 4L

0

V 9( L

2.

ñóò ò ó ò ò ó ó

=


ñ ú

2 1

− 4L

0

−2

.

3

. = (  ¨ )3 J

= L 2  2 + L 1  + L 0 . The substitution where the integration of which finally yields:

ù

2

L 2)

L 2 ) + 54 # ¨ − 2¨

J

3 −1 2

W

¨

M¨

.

.

+ 54 # 2

J ,

4 3

leads to a separable equation,

AçX

M

3

.

=L

3.

+ .

This is a special case of equation 4.2.6.1 with  (¨ ) = # 1 b . By integrating, we obtain 2# 2 



 − ( 

 )2 = where L is an arbitrary constant ( ¡ second-order equation:

+1

¡

+

+1

+

4 L , 3

≠ −1). The substitution ¨ ( ) = (
 )3 J

+ ¨

= O 3#  2 +2 ¡

+1

+L

¨ P

2

leads to a

J .

−5 3

The value L = 0 corresponds to the Emden–Fowler equation, whose integrable cases are specified in Section 2.3 for some values of (to those cases there correspond three-parameter ¡ families of particular solutions of the original equation). 2 b . Particular solution: 3.

ñóò ò ó ò ò ó ó = ô ï –3ú

–5

ñ ú

ñ óò ò ó ò ò ó ó

=

ô ï

–

3

ú

+5

Q

8(

¡

.

The transformation  = a 4.

=

−1

,

=a

−3

+ 1)( + 3)(3 ¡ ¡ # ( − 1)4

¡

+ 1) +

S

1 −1

 +L 

4 1−

+ .

¨ (a ) leads to an equation of the form 4.2.1.2: p ¨ g

g
g
g =  ¨2+ .

ñ ú

. This is a special case of equation 4.2.6.5 with  (¨ ) =  ¨p+ . 2

Page 659

660 5.

FOURTH-ORDER DIFFERENTIAL EQUATIONS =ô ï ÷ ñ ú . Generalized homogeneous equation.

ñ óò ò ó ò ò ó ó

1 b . The transformation a =   2 b . The transformation  = H H −  −3 + −5 ¨2+ .

6.

+4

−1

,

+

−1

=H


 leads to a third-order equation.

, ‡ = −3

¨ ( H ) leads to an equation of the same form: ¨u‘'

‘'

‘'‘ =

= ( ôñ + õ ï  )ú , ü = 0, 1, 2, 3. ¨ 



 =  +2¨2+ . The substitution  ¨ =  + '  leads to an equation of the form 4.2.1.2: u

ñ óò ò ó ò ò ó ó

7.

ï 3ú

( ô ï + õ )4 ñ ó ò ò ó ò ò ó ó = ö^ñ ú . This is a special case of equation 4.2.6.10 with  (¨ ) = ( ¨p+ .

8.

= (ô ï 2 + õ ï + ö ) 2 ñ ú . This is a special case of equation 4.2.6.12 with  (¨ ) = ¨p+ .

+1

ñóò ò ó ò ò ó ó

–

3

ú

+5

4.2.1-2. Equations of the form 



 =  ( , , 
 ). 9.

ñóò ò ó ò ò ó ó

=

ôñ ÷ ñóò

+

õ ï

.




 =

By integrating, we find



B +1



+1

+



, +1

 

+1

+ L . For  = 0, the order of this

equation can be reduced by one with the help of the substitution ¨ ( ) =
 . 10.

11.

12.

13.

14.

= ô ï – 4 (ï ñ óò The transformation ¨ g

g
g − 5¨ g

g + 6¨ g
=

ñ óò ò ó ò ò ó ó

– ñ )÷ . a = ln | |, ¨  ¨  .

=  
 −

leads to a third-order autonomous equation:

= ô ïƒ÷ ( ï ñó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =   ¨  . The substitution ¨ =  ¨ 
 ) 

 =   ¨  . leads to a third-order generalized homogeneous equation: (p

ñóò ò ó ò ò ó ó


−

= ô ïƒ÷ ( ï ñó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =   ¨  . The substitution ¨ =  
−2 leads to a third-order generalized homogeneous equation:  ¨p


 − ¨2

 =   +2 ¨  .

ñóò ò ó ò ò ó ó

= ô ïƒ÷ ( ï ñó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =   ¨  . The substitution ¨ =  leads to a third-order generalized homogeneous equation: ¨p


 =   +1 ¨  .

ñóò ò ó ò ò ó ó


−3

= ô 4 ñ + õ ïƒ÷ (ñó ò – ôñ )  . The substitution ‡ = 
−  leads to a third-order equation: ‡


 + ‡

 +  2 ‡
 +  3 ‡ = '  ‡  .

ñóò ò ó ò ò ó ó

4.2.1-3. Equations of the form 



 =  ( , , 
 , 

 ). 15.

+ ôñ ó ò ò ó = õÿñ ÷ + ö . This is a special case of equation 4.2.6.33 with  ( ) =   + ( .

16.

ñ óò ò ó ò ò ó ó

ñ óò ò ó ò ò ó ó

–

5 2

ôñ óò ò ó

+

õÿñ –5 ) 3 . 2  ¨ ˆ = , (ˆ ) = ˆ 3 J 2

9 16

The transformation 4.2.1.1: ¨3Šª

Šª

ŠªŠ =  −2  ¨

ô 2ñ

=

J .

−5 3

leads to an autonomous equation of the form

Page 660

661

4.2. NONLINEAR EQUATIONS

17.

ñóò ò ó ò ò ó ó

+ ôñó ò ò ó + õÿñ = ö^ññó ò ò ó – ö (ñó ò )2 + ü . 1 b . Particular solution: =L where the constants L

sinh( L

1

L 2 , L 3 , and L

1,

4

4 4

L 2 2

( (L

4

 )+L

2

cosh( L

4

 ) + L 3,

are related by two constraints

+ (  − (L 2 1)

−L

2 4

L

2 4

+  = 0,

3

+ , = 0.

L

3)

− L

2 b . Particular solution: =L where the constants L

sin( L

1

L 2 , L 3 , and L

1,

4 4

L 2 1

( (L 18.

ñ óò ò ó ò ò ó ó

+

ôññ óò ò ó

+

õÿñ óò ò ó

Particular solution: 19.

– ô (ñ

=L

1

)2 +

exp( L

2

 )+L

cos( L

4

 ) + L 3,

are related by two constraints

− (  − (L 2 2)

+L

ö^ñ óò

 )−

2

2 4

L

3)

2 4

+  = 0,

3

− , = 0.

L

+ L

= 0.

L

= ôñ 2 ñó ò ò ó – ôñ (ñó ò )2 + õÿñ . 1 b . Particular solution: = L 1 exp( L

3 2

+ L

L

2

+(

2

.

ñóò ò ó ò ò ó ó

where the constants L

L 2 , and L

1,

2 b . Particular solution: where the constants L

L 2 , and L

1,

1

2

3

 )+L

exp(− L

2

3

 ),

are related by the constraint L

2

=L

3 b . There are also solutions 20.

óò

4

4

cos( L

3

 )+L

2

sin( L

3

ü

  +L

and

− 4 L

4 3

+  (L

1

L 2L

−  = 0.

2 3

 ),

are related by the constraint L

= A"

4 3

2 1

+L

2 2)

L

2 3

−  = 0.

= 0.

+ ô (ñó ò )÷ ñó ò ò ó = õÿñ  + ö . This is a special case of equation 4.2.6.34 with  (‡ ) = ‡  and ( ) =   + ( .

ñóò ò ó ò ò ó ó

21.

= ô ï –2 ( ï ñó ò – ñ )÷ ñó ò ò ó . This is a special case of equation 4.2.6.35 with  (¨ ) =  ¨  .

22.

ññ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = 0. 1 b . Particular solutions: =L =L =L =L

1

 + L 2,

 + L 2 )−3 J 2 , 1 exp( L 3  ) + L 2 exp(− L 3  ), 1 cos( L 3  ) + L 2 sin( L 3  ). 1(

2 b . Integrating the equation twice, we arrive at a second-order equation:

! 

 − ( 
)2 = L  + L . 1 2 The substitution H = L

1

 +L

2

leads to a generalized homogeneous equation.

Page 661

662 23.

24.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 + ôñ + Particular solutions:

õ

= 0.

=L

1

exp( Π ) + L

=L

1

sin( Π ) + L

2

exp(− Œ  ) −   ,

cos( Œ  ) −   ,

2

Π= ( 2  Π= ( 2 

)1 J 4 ; )1 J 4 .

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 + ôñó ò ò ó = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 −(
 )2 +  L 1  + L 2.

=

2 b . Particular solutions:

 ) + L 2 exp(− L 3  ) − L 3−2 , = L 1 sin( L 3  ) + L 2 cos( L 3  ) + L 3−2 , = L 1  + L 2. =L

25.

1

exp( L

3

ññ óò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = ô [ññ ó ò ò ó – (ñ ó ò )2 ] + õ . 1 b . The substitution ¨ ( ) = !

 − (
 )2 leads to a second-order constant coefficient linear equation of the form 2.1.9.1: ¨ 

 =  ¨ +  . 2 b . Particular solutions:

  −

=L

1

exp ÝL

=L

1

exp  £   −

=L

1

sin  Π  + L

2

4 L

2

 

1

L

4 L 2

2 2 1

exp  − L

2

 

if  ≠ 0,

 − £   + L

cos  Π  ,

Œ

2

=

if  > 0,

2

 (L

2 1



+L

2 2)

if  > 0,

£ − sin   £ −  + L 1  + L 2 if  < 0,  < 0,  = A"   +L 1 if  > 0. ü ññóò ò ó ò ò ó ó – (ñóò ò ó )2 = ô V ññóò ò ó – (ñóò )2 W + õ ï + ö . The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨p

 =  ¨ + '  + ( . =

26.

27.

ññóò ò ó ò ò ó ó

– 16 (ñó ò ò ó )2 = Particular solution:

ô ï

2

õï

+

+ö .

L 1  4 + 16 L 2  3 + 12 L 3  2 + L 4  + L 5 , where the constants L 1 , L 2 , L 3 , L 4 , and L 5 are related by three constraints 1 24

=

1 3

L 1 L 3 − 16 L L 1 L 4 − 13 L 2 L L 1 L 5 − 16 L

28.

29.

ñóò ò ó ò ò ó ó

= ô 2 ñ + õ (ñó ò ò ó The substitution ¨ = 2.9.1.1: ¨3

 =  ¨ + 

2 2

= ,

3

=,

2 3

=(.

+ ôñ )  . 

 +  leads to a second-order autonomous equation of the form ¨ .

÷

= ôñ V ññ ó ò ò ó – (ñ ó ò )2 W . This is a special case of equation 4.2.6.42 with  (¨ ) = 0 and (¨ ) =  ¨  .

ñ óò ò ó ò ò ó ó

Page 662

663

4.2. NONLINEAR EQUATIONS

4.2.1-4. Equations of the form 



 =  ( , , 
 , 

 , 


 ). 30.

ñ óò ò ó ò ò ó ó

+ ô 3 ñ óò ò ó ò ó + ô Particular solutions:

2

ñ óò ò ó

+ ô 1 ñ ó ò + ô 0 ñ = õÿññ ó ò ò ó – õ (ñ ó ò )2 + ü . = L exp( Œ   ) + , −1 0 , where L is an arbitrary constant and Œ = Œ  4

are roots of the algebraic equation Π31.

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

=

õ ïƒ÷

+  3Œ

3

+ O

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

, Œ  0P

−

2

+  1Π+ 

0

= 0.

.

Integrating, we arrive at a third-order equation: 32.

2

1 


 +  ! 

 −  ( 
)2 = 2



B +1

 

+1

+L .

– ôñó ò ñóò ò ó = 0.

This equation arises in hydrodynamics. 1 b . Particular solutions:

=L

 + L 2, = L 1 exp( L 2  ) −  = 6(  + L 1 )−1 . 1

−1

L 2,

2 b . Integrating, we arrive at a third-order autonomous equation:


 +  !

 −  (
 )2 = L .

^_ 33.

34. 35.

Reference: A. D. Polyanin and V. F. Zaitsev (2002).

ñ óò ò ó ò ò ó ó + ôññ óò ò ó ò ó – ôñ Particular solutions:

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

óò ñ óò ò ó

=

õÿñ

.

=L

1

exp( Π ) + L

=L

1

sin( Π ) + L

+ õ (ñó ò )÷

ñóò ò ó

=

ö ï

2 2

exp(− Œ  ),

cos( Π ),

+ý .

This is a special case of equation 4.2.6.50 with  (‡ ) = '‡  and ( ) = (˜  + .

M

ñóò ò ó ò ò ó ó = ô ñ ÷ ñ óò ñóò ò ó ò ó

.

This is a special case of equation 4.2.6.51 with  ( ) =  36. 37. 38.

39.

40.

Π=  1J 4, Π=  1J 4.

ï ñóò ò ó ò ò ó ó

+ 4ñó ò ò ó ò

ó

=

ï ñóò ò ó ò ò ó ó

+ 4ñó ò ò ó ò

ó

= ô (ï

ñ )

.

ï ñóò ò ó ò ò ó ó

+ 2ñó ò ò ó ò

ó

= ô (ï

ñóò

– ñ ) .

ï ñóò ò ó ò ò ó ó

+ ( ô + 3)ñó ò ò ó ò

=

õ ïƒ÷ (ï ñ óò

 .

ô ï –5 ) 3 ñ –5 ) 3 .

The substitution ¨ ( ) =  The substitution ¨ ( ) = 

¨ 



 =  ¨ leads to an equation of the form 4.2.1.1: I

J .

−5 3

¨ 



 =  ¨  . leads to an equation of the form 4.2.1.2: I

The substitution ¨ ( ) =  
− leads to a third-order equation: ¨u


 =  ¨  (Section 3.2 presents its solutions for , = − 72 , − 52 , −2, − 43 , − 67 , − 12 , 0, and 1).

ó

The substitution ¨ =  
+  ¨ 


 = '  ¨  .

ï 2 ñ óò ò ó ò ò ó ó

+ 8ï

ñóò ò ó ò ó

2

ôñ )

.

leads to a third-order generalized homogeneous equation:

+ 12ñó ò ò ó =

The substitution ¨ ( ) = 

+

ô ï

) ñ

)

–10 3 –5 3

. ¨ 



 =  ¨ leads to an equation of the form 4.2.1.1: u

J .

−5 3

Page 663

664 41.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ï 4 ñ óò ò ó ò ò ó ó

+ 6ï

3

+ 7ï

ñóò ò ó ò ó

2

ñóò ò ó

ï ñóò

+

=

ôñ –5 ) 3 .

The substitution a = ln | | leads to an equation of the form 4.2.1.1:



 = 

J .

−5 3

42.

ññ óò ò ó ò ò ó ó = ôñ óò ñ óò ò ó ò ó . Having integrated this equation, we obtain the third-order equation


 = L solvable cases are specified in Subsection 3.2.2.

43.

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

=

ô ƒï ÷ ñ

2

.

44.

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

ôñóò

.

+ 3(ñ

óò ò ó

=

   B +1




 = O

Integrating yields a third-order linear equation:

.

P

óò ñ óò ò ó ò ó

ô ï ÷

ññ óò ò ó ò ò ó ó

+ 4ñ

. This is a special case of equation 4.2.6.58 with  ( ) =   .

46.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

ôñ

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

ôñ ÷

ññ óò ò ó ò ò ó ó

+

48.

+L

Integrating yields a third-order constant coefficient nonhomogeneous linear equation: /


 = L − .

45.

47.

+1

 , whose

The substitution ¨ = The substitution ¨ = 2 3

2

ñ óò ñ óò ò ó ò ó

2

)2 =

)

–10 3

.

¨ 



 = 2  ¨ leads to an equation of the form 4.2.1.1: u

J .

−5 3

.

¨ 



 = 2  ¨ ´J 2 . leads to an equation of the form 4.2.1.2: u

– 13 (ñ

óò ò ó

)2 = ô .

This is a special case of equation 4.2.6.59 with  = 23 and  ( ) =  . Integrating the equation twice, we arrive at a second-order equation of the form 2.8.1.54:

  3! 

 − 2( 
)2 = 32  49.

ññóò ò ó ò ò ó ó

+

3 2

ñóò ñóò ò ó ò ó

2

+L

1

 + L 2.

+ 12 (ñó ò ò ó )2 = ô .

Integrating the equation twice, we arrive at a second-order equation of the form 2.8.1.53:

! 

 − 1 ( 
)2 = 1   4 2 50.

ññóò ò ó ò ò ó ó

+

3 2

ñóò ñóò ò ó ò ó

+ 12 (ñó ò ò ó )2 = ( ô

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

=

ô ƒï ÷ ñ ñóò ò ó ò ó

+ õ )ñ

.

Integrating yields a third-order linear equation: 52.

ï ññóò ò ó ò ò ó ó

– ï (ñó ò ò ó )2 = ô .

)

–1 2

+L

1

 + L 2.

.

2 = ( g
) leads to a constant coefficient linear equation:

The transformation  =  (a ), 2 (5) g =  +  . 51.

ï

2




 = L exp O



B +1

 

+1

P

.

Integrating the equation twice, we arrive at a second-order equation:

53.

ï ññóò ò ó ò ò ó ó

ï ñóò ñóò ò ó ò ó

! 

 − ( 
)2 =  ln | | + L  + L . 1 2

+ ôññó ò ò ó ò ó . Integrating yields a third-order linear equation of the form 3.1.2.7:


 = L3  . =

Page 664

665

4.2. NONLINEAR EQUATIONS

54.

ñ 3 ñ óò ò ó ò ò ó ó

= 4ñ 2 ñó ò ñó ò ò ó ò ó + 3ñ 2 (ñó ò ò ó )2 – 6(ñó ò )4 . This is a special case of equation 4.2.6.67 with  ≡ 0. Solution in parametric form:

 =A X 55.

56.

2ˆ

ü

+L

ñóò ò óñóò ò ó ò ò ó ó

= ô (ñó ò ò ó ò ó )2 .

Solution:

=

a

L

+L L 0+L 0

 M

4

2

ˆ +L

+L

1

=L

3,

ˆ

exp ”A X

4

2ˆ

ü 3−2

 + ( L 2 + L 3  ) 1−  1  + L 2 exp( L 3  ) 1



4

ˆ M

+L

2

ˆ +L

1

•

.

if  ≠ 1, if  = 1.

– 12 (ñ ó ò ò ó ò ó )2 =  ( ï ñ ó ò – ñ ) + ñ Differentiating with respect to  yields

ñ óò ò ó ñ óò ò ó ò ò ó ó

óò

+ .

 − ƒ… − ý W = 0. 

 V (5)

(1)

Equating the second factor in (1) with zero and integrating it, we find the solution: =ƒ



6

6!



+ý

5

5!

+L

4



4

+L

3

3



+L

2

2



+L

1

 + L 0.

The constants L  and parameters ƒ , ý , and = are related by the constraint 48 L

2

L

4

− 18 L

2 3

= − ƒsL

+ ýhL

0

1

+= ,

obtained by means of substituting the solution into the original equation. The other solution, which corresponds to setting the first factor in (1) to zero, is given by: =L

57.

¶

1

¶  + L 0,

where

ƒ L

¶

0

= ôñ  ñó ò (ñó ò ò ó ò ó )s . This is a special case of equation 4.2.6.72 with  ( ) =  ° = 1, see equation 4.2.1.42. The first integral has the form:À

−ý L

ñóò ò ó ò ò ó ó

For L

1   +1 = L ( 


 )1− − 1−° , +1   +1 = L ln | 


 | − À , +1 1


(  )1− −  ln | | = L 1−° = 0, equality (1) is changing to the equation




 = ¬

 (1 − ° ) ® , +1

1

− = = 0.

À

 and (¨ ) = ¨ . For , = −1 and

if , ≠ −1, ° ≠ 1;

(1)

if , ≠ −1, ° = 1;

(2)

if , = −1, ° ≠ 1.

(3)

À 1 1−

¶



À +1 1−

,

which is discussed in Subsection 3.2.2 (the solutions given there generate 3-parametric families of particular solutions of the original equation for , = (1 − ° )ý − 1, where ý = − 72 , − 52 , −2, − 43 , − 67 , − 12 , 0, and 1). 58.

ô 1 (ñ ó ò ò ó ò ò ó ó

)2 + ( ô

ñ ó ò ò ó ò ó + ô 3 ñ ó ò ò ó + 6 ô 4 ñ + ô 5 ï + ô 6 )ñ ó ò ò ó ò ò ó ó + õ 1 (ñó ò ò ó ò ó )2 + ( õ 2 ï + õ 3 )ñó ò ò ó ò ó – ô 4 (ñó ò ò ó )2 + õ 4  ñ óò ò ó 2

+õ There are particular solutions of the form = L 1  4 + L 2  3 + L 3  2 + L five constants L 1 , L 2 , L 3 , L 4 , and L 5 are related by three constraints.

+ õ 6 ï + õ 7 = 0. 4  + L 5 , where the

5

ï

2

Page 665

666

FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.2.2. Equations Containing Exponential Functions 4.2.2-1. Equations of the form 



 =  ( , ). 1.

ñóò ò ó ò ò ó ó

=

ñóò ò ó ò ò ó ó

=

ô]

+õ .

This is a special case of equation 4.2.6.1 with  ( ) =   +  . 2.

3.

ô] +_ ó

+õ .

+ ( ý Œ )

¨ = The substitution ? ¨2



 =   +  .

ñóò ò ó ò ò ó ó

=

ñóò ò ó ò ò ó ó

=

ô ï

–4

]

leads to an autonomous equation of the form 4.2.6.1:

.

This is a special case of equation 4.2.6.2 with  ( ) =    . The substitution a = ln | | leads to an autonomous equation. 4. 5. 6.

7. 8.

ô ï   ]

. This is a special case of equation 4.2.6.15 with  (¨ ) =  ¨ and

ñ óò ò ó ò ò ó ó

ô  ó ñ ÷

=

.

This is a special case of equation 4.2.6.14 with  (¨ ) =  ¨ and

ñ óò ò ó ò ò ó ó

=

ô

= , + 4.

¡

exp( Ė + 

The substitution ? ¨ = ¨2



 =   +  .

ñóò ò ó ò ò ó ó

= ô (ñ +

õ ó

ñóò ò ó ò ò ó ó

= ô (ñ +

õ ó )ú

)–5 )

ï

)+õ . + ( ý Œ )

¡

= B − 1.

2

3

–

õ ó

2

leads to an autonomous equation of the form 4.2.6.1:

.

 ¨ 



 =  ¨ The substitution ¨ = +   leads to an equation of the form 4.2.1.1: u –

õ ó

J .

−5 3

.

 ¨ 



 =  ¨2+ . The substitution ¨ = +   leads to an equation of the form 4.2.1.2: u

4.2.2-2. Other equations. 9.

ñ óò ò ó ò ò ó ó

=

ô ] ñ óò

+

õ _ ó

.

Integrating yields a third-order equation: 10. 11. 12. 13.

ñóò ò ó ò ò ó ó = ô 4 ñ + õ ó (ñ óò




 = Œ



  + ý



  : +L .

– ôñ )  .

 This is a special case of equation 4.2.6.27 with  ( , ¨ ) =   ¨  .

ñóò ò ó ò ò ó ó

=

õ ó (ï ñ óò

– ñ ) .

ñóò ò ó ò ò ó ó

=

õ ó (ï ñ óò

– 2ñ )  .

ñóò ò ó ò ò ó ó

=

õ ó (ï ñ óò

– 3ñ )  .

 This is a special case of equation 4.2.6.23 with  ( , ¨ ) =   ¨  .  This is a special case of equation 4.2.6.24 with  ( , ¨ ) =   ¨  .  This is a special case of equation 4.2.6.25 with  ( , ¨ ) =   ¨  .

Page 666

667

4.2. NONLINEAR EQUATIONS

14.

ó

ññ óò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = ô  . 1 b . Integrating the equation twice, we arrive at a second-order equation: ! 

 − ( 
)2 = Œ −2    + L  + L . 1 2 For L 1 = L 2 = 0, it is an equation of the form 2.8.3.47. 2 b . Particular solution:

15.

= L exp( Π ) +

ó

ññóò ò ó ò ò ó ó

 . L Π4 p

– (ñó ò ò ó )2 – ôñó ò + õ = 0. 1 b . Integrating yields a third-order equation: !"


 − 
 

 −  2 b . Particular solutions: LpŒ −  = L exp( Œ  ) + , 4

+ Œ

−1





=L .

LpŒ   = exp( Œ  ) + L exp(− Œ  ) − 3 . 2 Œ Œ

ó

16.

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 – ôñó ò ò ó + õ = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: ! 

 − ( 
)2 −   + L 1  + L 2 + Œ −2  = 0. LpŒ 2 −  2 b . Particular solution: = L exp( Œ  ) + . 4

17.

ññóò ò ó ò ò ó ó

= ô [ññó ò ò ó – ( ñ óò ) ] + õ ó 2

2 b . Particular solution: 18.

= L exp( Π ) +

ó

ññóò ò ó ò ò ó ó

1b .

 
 2b .

19.

LpŒ

– (ñó ò ò ó ) . ¨ !


2 1 b . The substitution ( ) =  − (  ) leads to a second-order constant coefficient linear  equation of the form 2.1.9.1: ¨u

 =  ¨ +   . 2

LpŒ

Œ



2( 2

− )

.

– ôñó ò ò ó ò ó – (ñó ò ò ó )2 + õ = 0. Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 + Œ −2  = 0. Particular solutions: LpŒ 3 −  = L exp( Œ  ) + , 4

 = 2 Œ

ñ óò ò ó ò ò ó ó

LpŒ

3

exp( Œ  ) + L exp(− Œ  ) −

= ô ] ñ ó ò ñ ó ò ò ó ò ó . This is a special case of equation 4.2.6.51 with  ( ) =   .

 Œ

.

ó

20.

= ( ô]¦ñó ò + õ _ )ñó ò ò ó ò ó .  This is a special case of equation 5.2.6.58 with B = 4,  ( ) =   , and ( ) =   : .

21.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

– 4 ƒñó ò ò ó ò

ó

+6

2

ñóò ò ó

The substitution ¨ ( ) =  − 

–4



3

ñóò

+

4

ñ

=

ô

exp 

ï  ñ –5 ) 3 .

8 3

¨ 



 =  ¨ leads to an equation of the form 4.2.1.1: u

ó

J .

−5 3

22.

– 4 ƒñ ó ò ò ó ò ó + 6 2 ñ ó ò ò ó – 4 3 ñ ó ò + 4 ñ = ô  (1–ú ) ñ ú .  ¨ 



 =  ¨2+ . The substitution ¨ ( ) =  −  leads to an equation of the form 4.2.1.2: u

23.

ññ óò ò ó ò ò ó ó

+ 4ñ ó ò ñ Solution: 2 = L

24.

ññ óò ò ó ò ò ó ó

ñ óò ò ó ò ò ó ó

óò ò ó ò ó

+ 3(ñ 3 3 + L 2 

óò ò ó

ó

)2 = ô  . 2 + L 1  + L 0 + 2 Œ

−4

  .

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô ] + õ . This is a special case of equation 4.2.6.60 with  ( ) =   +  .

Page 667

668

FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.2.3. Equations Containing Hyperbolic Functions 4.2.3-1. Equations with hyperbolic sine. 1. 2.

3.

4. 5.

= ô sinhú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  sinh ( Œ

ñóò ò ó ò ò ó ó

= ô sinh( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  sinh( Œ ¨ ) +  .

ñ óò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô (ñ + õ sinh ï )2 – õ sinh ï . The substitution ¨ = +  sinh  leads to an autonomous equation of the form 4.2.6.1: ¨ 



 =  ¨ 2 .

ñóò ò ó ò ò ó ó

= ô ï – 4 sinh ú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  sinh ( Œ

ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

=

ô

sinh( ƒñ )ñ

óò

+ õ sinh(

ï

).

Integrating yields a third-order equation: 6. 7. 8. 9. 10.

Œ



cosh( Œ

)+

ý



cosh(ý± ) + L .

= ô 4 ñ + õ sinh( ï )(ñ ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  sinh( Œ  )¨  .

= õ sinh( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  sinh( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ sinh( ï )( ï ñ ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  sinh( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ sinh( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  sinh( Œ  )¨  .

ñóò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = ô sinh( ï ). 1 b . Integrating the equation twice, we arrive at a second-order equation: !

 − (
 )2 = Œ −2 sinh( Œ  ) + L 1  + L 2 .

ññ óò ò ó ò ò ó ó

= L sinh( Π ) +

 . LpΠ4

– (ñó ò ò ó )2 – ôñó ò + õ sinh( ï ) = 0. 1 b . Integrating yields a third-order equation: !"


 − 
 

 − 

ññóò ò ó ò ò ó ó

2 b . Particular solution: 12.




 =

).

ñ óò ò ó ò ò ó ó

2 b . Particular solution: 11.

)+ .

=

 Œ ( 2 − L

Œ

2 6)

+ Œ

−1

cosh( Π ) = L .

V LpΠ3 sinh( Π ) +  cosh( Π )W + L .

– (ñó ò ò ó )2 – ôñó ò ò ó + õ sinh( ï ) = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 + Œ −2 sinh( Œ  ) = 0.

ññóò ò ó ò ò ó ó

2 b . Particular solution:

= L sinh( Π ) +

LpŒ 2 −  . LpŒ 4

Page 668

669

4.2. NONLINEAR EQUATIONS

13.

14.

– (ñó ò ò ó )2 = ô [ññó ò ò ó – (ñó ò )2 ] + õ sinh( ï ) + ö . The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨p

 =  ¨ +  sinh( Œ  ) + ( .

ññóò ò ó ò ò ó ó

– ôñ ó ò ò ó ò ó – (ñ ó ò ò ó )2 + õ sinh( ï ) = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  
 + L 1  + L 2 + Œ −2 sinh( Œ  ) = 0.

ññ óò ò ó ò ò ó ó

2 b . Particular solution: 15.

16.

ñóò ò ó ò ò ó ó

=

 Œ 3( 2 − L

Œ

2 2)

V LpΠsinh( Π ) +  cosh( Π )W + L .

sinh  ( ƒñ )ñóÈ ò ñóò ò ó ò ó .

ô

=

This is a special case of equation 4.2.6.51 with  ( ) =  sinh  ( Œ

).

– ñó ò ñóò ò ó ò ó = ô sinh( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  sinh( Œ  ). Integrating yields a 


third-order linear equation:  = cosh( Π ) + L3S .

ññóò ò ó ò ò ó ó

Q Œ

17. 18.

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô sinh( ï ). Solution: 2 = L 3  3 + L 2  2 + L 1  + L 0 + 2 Œ

ññ óò ò ó ò ò ó ó ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

sinh( Π ).

−4

sinh  ( ƒñ ) + õ .

ô

This is a special case of equation 4.2.6.60 with  ( ) =  sinh  ( Œ

)+ .

4.2.3-2. Equations with hyperbolic cosine. 19. 20.

21.

22. 23.

= ô coshú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  cosh ( Œ

ñ óò ò ó ò ò ó ó

= ô cosh( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  cosh( Œ ¨ ) +  .

ñ óò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô (ñ + õ cosh ï )2 – õ cosh ï . The substitution ¨ = +  cosh  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  ¨ 2 .

ñ óò ò ó ò ò ó ó

= ô ï – 4 coshú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  cosh ( Œ

ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

=

ô

cosh( ƒñ )ñ

óò

+ õ cosh(

ï

).

Integrating yields a third-order equation: 24. 25.

)+ .




 = Œ



sinh( Œ

)+

ý

).



sinh(ý± ) + L .

= ô 4 ñ + õ cosh( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  cosh( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cosh( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  cosh( Œ  )¨  .

ñóò ò ó ò ò ó ó

Page 669

670 26.

27.

28.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ò ó ó

=

õ

cosh(

ï )(ï ñ óò

– 2ñ )  .

ñóò ò ó ò ò ó ó

=

õ

cosh(

ï )(ï „ñ óò

– 3ñ )  .

This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  cosh( Œ  )¨  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  cosh( Œ  )¨  .

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 =

ô

ññ óò ò ó ò ò ó ó

óò ò ó

– (ñ

óò

)2 – ôñ

32.

34.

) = 0.

1 b . Integrating yields a third-order equation: !"


 − 
 

 − 

ññóò ò ó ò ò ó ó

 Œ ( 2 − L

=

+ Œ

−1

sinh( Π ) = L .

V LpΠ3 cosh( Π ) +  sinh( Π )W + L .

Œ

2 6)

ï

– (ñó ò ò ó )2 – ôñó ò ò ó + õ cosh(

) = 0.

1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 + Œ −2 cosh( Œ  ) = 0.

ññ óò ò ó ò ò ó ó

– (ñ

óò ò ó

ññ óò ò ó ò ò ó ó

– ôñ

óò ò ó ò ó

LpŒ 2 −  . LpŒ 4

= L cosh( Π ) +

óò ò ó

)2 = ô [ññ

óò

– (ñ

)2 ] + õ cosh(

ï

)+ö .

The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨p

 =  ¨ +  cosh( Œ  ) + ( .

óò ò ó

– (ñ

)2 + õ cosh(

ï

) = 0.

1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  
 + L 1  + L 2 + Œ −2 cosh( Œ  ) = 0. 2 b . Particular solution:

33.

ï

+ õ cosh(

2 b . Particular solution: 31.

 . LpΠ4

= L cosh( Π ) +

2 b . Particular solution: 30.

).

1 b . Integrating the equation twice, we arrive at a second-order equation: !

 − (
 )2 = Œ −2 cosh( Œ  ) + L 1  + L 2 . 2 b . Particular solution:

29.

ï

cosh(

ñóò ò ó ò ò ó ó

ô

=

=

Œ



 −L

3( 2

Œ

2 2)

V LpΠcosh( Π ) +  sinh( Π )W + L .

cosh  ( ƒñ )ñóÈ ò ñóò ò ó ò ó .

This is a special case of equation 4.2.6.51 with  ( ) =  cosh  ( Œ

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

=

ô

cosh(

ï )ñ

2

).

.

This is a special case of equation 4.2.6.57 with  ( ) =  cosh( Π ). Integrating yields a  third-order linear equation: 


 = sinh( Π ) + L3S .

Q Œ

35.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

Solution: 36.

ññóò ò ó ò ò ó ó

2

=L

3

+ 4ñóÈ ò ñóò ò ó ò

ó 

ó

3

+ 3(ñó ò ò ó )2 =

ô

+L

 +L

2



2

+L

1

+ 3(ñó ò ò ó )2 =

ô

cosh( 0

ï

).

+ 2 Œ

−4

cosh( Π ).

cosh  ( ƒñ ) + õ .

This is a special case of equation 4.2.6.60 with  ( ) =  cosh  ( Œ

)+ .

Page 670

671

4.2. NONLINEAR EQUATIONS

4.2.3-3. Equations with hyperbolic tangent. 37. 38.

39. 40. 41. 42. 43. 44.

= ô tanhú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  tanh ( Œ

ñóò ò ó ò ò ó ó

= ô tanh( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  tanh( Œ ¨ ) +  .

)+ .

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô ï – 4 tanhú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  tanh ( Œ

ñóò ò ó ò ò ó ó

= ô tanh( ƒñ )ñ„ó ò + õ tanh( ï ). This is a special case of equation 4.2.6.21 with  ( ) =  tanh( Œ

ñóò ò ó ò ò ó ó

). ) and ( ) =  tanh(ý± ).

= ô 4 ñ + õ tanh( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  tanh( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ tanh( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  tanh( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ tanh( ï )( ï ñ ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  tanh( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ tanh( ï )( ï ñ ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  tanh( Œ  )¨  .

ñ óò ò ó ò ò ó ó

45.

= ñ + ô (ñ ó ò – ñ tanh ï )  . This is a special case of equation 5.2.6.32 with  ( , ‡ ) = ‡  and Y ( ) = cosh  .

46.

ñ óò ò ó ò ò ó ó

ñ óò ò ó ò ò ó ó

=

ô

tanh ( ƒñ )ñ

óò ñ óò ò ó ò ó

.

This is a special case of equation 4.2.6.51 with  ( ) =  tanh  ( Œ

).

47.

– ñ ó ò ñ ó ò ò ó ò ó = ô tanh( ï )ñ 2 . This is a special case of equation 4.2.6.57 with  ( ) =  tanh( Œ  ).

48.

ññ óò ò ó ò ò ó ó

+ 4ñ

óò ñ óò ò ó ò ó

+ 3(ñ

óò ò ó

)2 =

ô

tanh  (

ññ óò ò ó ò ò ó ó

+ 4ñ

óò ñ óò ò ó ò ó

+ 3(ñ

óò ò ó

)2 =

ô

tanh  ( ƒñ ) + õ .

49.

ññ óò ò ó ò ò ó ó

ï

)+ õ .

This is a special case of equation 4.2.6.58 with  ( ) =  tanh  ( Œ  ) +  . This is a special case of equation 4.2.6.60 with  ( ) =  tanh  ( Œ

)+ .

4.2.3-4. Equations with hyperbolic cotangent. 50. 51.

= ô cothú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  coth ( Œ

ñóò ò ó ò ò ó ó

= ô coth( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  coth( Œ ¨ ) +  .

)+ .

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

Page 671

672 52. 53. 54. 55. 56. 57.

FOURTH-ORDER DIFFERENTIAL EQUATIONS = ô ï – 4 cothú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  coth ( Œ

ñóò ò ó ò ò ó ó

).

= ô coth( ƒñ )ñ ó ò + õ coth( ï ). This is a special case of equation 4.2.6.21 with  ( ) =  coth( Œ

ñ óò ò ó ò ò ó ó

) and ( ) =  coth(ý± ).

= ô 4 ñ + õ coth( ï )(ñ ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  coth( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ coth( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  coth( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ coth( ï )( ï ñ„ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  coth( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ coth( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  coth( Œ  )¨  .

ñóò ò ó ò ò ó ó

58.

= ñ + ô (ñó ò – ñ coth ï )  . This is a special case of equation 5.2.6.32 with  ( , ‡ ) = ‡  and Y ( ) = sinh  .

59.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

=

ô

coth ( ƒñ )ñó ò ñóò ò ó ò ó .

This is a special case of equation 4.2.6.51 with  ( ) =  coth  ( Œ

).

60.

– ñó ò ñóò ò ó ò ó = ô coth( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  coth( Œ  ).

61.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

ô

coth  (

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

ô

coth  ( ƒñ ) + õ .

62.

ññóò ò ó ò ò ó ó

ï

)+ õ .

This is a special case of equation 4.2.6.58 with  ( ) =  coth  ( Œ  ) +  . This is a special case of equation 4.2.6.60 with  ( ) =  coth  ( Œ

)+ .

4.2.4. Equations Containing Logarithmic Functions 4.2.4-1. Equations of the form 



 =  ( , ). 1. 2.

3.

4.

= ô lnú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  ln ( Œ

ñóò ò ó ò ò ó ó

= ô ln( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨ 



 =  ln( Œ ¨ ) +  .

)+ .

ñóò ò ó ò ò ó ó

= ô ln( ƒñ +  ï The substitution ¨ = ¨2



 =  ln( Œ ¨ ) +  .

ñ óò ò ó ò ò ó ó

2

)+õ . + ( ý Œ )

leads to an autonomous equation of the form 4.2.6.1:

2

leads to an autonomous equation of the form 4.2.6.1:

= ô ï – 4 ln ú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  ln ( Œ

ñóò ò ó ò ò ó ó

).

Page 672

673

4.2. NONLINEAR EQUATIONS

5. 6. 7. 8. 9. 10.

= ôñ ( ï + þ ln ñ ). This is a special case of equation 4.2.6.14 with  (¨ ) =  ln ¨ .

ñ óò ò ó ò ò ó ó

= ô ï – 4 ( ƒñ + þ ln ï ). This is a special case of equation 4.2.6.15 with  (¨ ) =  ln ¨ .

ñóò ò ó ò ò ó ó

= ô ï –3 (ln ñ – ln ï ). This is a special case of equation 4.2.6.3 with  (¨ ) =  ln ¨ .

ñóò ò ó ò ò ó ó

= ô ï –5 (ln ñ – 3 ln ï ). This is a special case of equation 4.2.6.4 with  (¨ ) =  ln ¨ .

ñóò ò ó ò ò ó ó

= ô ï –5 ) 2 (2 ln ñ – 3 ln ï ). This is a special case of equation 4.2.6.5 with  (¨ ) = 2  ln ¨ .

ñóò ò ó ò ò ó ó

= ô ï ÷ –4 (ln ñ – ù ln ï ). This is a special case of equation 4.2.6.6 with  (¨ ) =  ln ¨ and , = −B .

ñ óò ò ó ò ò ó ó

4.2.4-2. Other equations. 11. 12. 13. 14. 15. 16. 17.

= ô 4 ñ + õ ln( ï )(ñ ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  ln( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ ln( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  ln( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ ln( ï )( ï ñ„ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  ln( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ ln( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  ln( Œ  )¨  .

ñóò ò ó ò ò ó ó

– (ñó ò ò ó )2 = ô ln( ï ). This is a special case of equation 4.2.6.36 with  ( ) =  ln( Œ  ).

ññóò ò ó ò ò ó ó

ï ñ óò ò ó ò ò ó ó

+ 4ñ ó ò ò ó ò ó = ô (ln ï + ln ñ ). ¨ 



 =  ln ¨ . The substitution ¨ ( ) =  leads to an equation of the form 4.2.6.1: I

ñóò ò ó ò ò ó ó = ô ln( ƒñ )ñóÈò ñóò ò ó ò ó . This is a special case of equation 4.2.6.51 with  ( ) =  ln( Œ

).

18.

– ñó ò ñóò ò ó ò ó = ô ln( ï )ñ 2. Integrating yields a third-order linear equation:

19.

+ 4ñóÈ ò ñóò ò ó ò ó + 3(ñóò ò ó )2 = ô lnú ( ï ) + õ . + This is a special case of equation 4.2.6.58 with  ( ) =  ln ( Œ  ) +  .

20.

ññóò ò ó ò ò ó ó




 = V  ln( Œ  ) −  + L3W .

ññóò ò ó ò ò ó ó

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô ln ú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.60 with  ( ) =  ln ( Œ

ññ óò ò ó ò ò ó ó

)+  .

Page 673

674

FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.2.5. Equations Containing Trigonometric Functions 4.2.5-1. Equations with sine. 1. 2.

3.

4. 5.

= ô sinú ( ƒñ ) + õ . This is a special case of equation 4.2.6.1 with  ( ) =  sin + ( Œ

ñóò ò ó ò ò ó ó

= ô sin( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  sin( Œ ¨ ) +  .

ñ óò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô (ñ + õ sin ï )2 – õ sin ï . The substitution ¨ = +  sin  leads to an autonomous equation of the form 4.2.6.1: ¨ 



 =  ¨ 2 .

ñóò ò ó ò ò ó ó

= ô ï – 4 sin ú ( ƒñ ). This is a special case of equation 4.2.6.2 with  ( ) =  sin + ( Œ

ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

=

ô

sin( ƒñ )ñ

óò

+ õ sin(

ï

).

7. 8. 9. 10.

= ô 4 ñ + õ sin( ï )(ñ ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  sin( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ sin( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  sin( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ sin( ï )( ï ñ ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  sin( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ sin( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  sin( Œ  )¨  .

ñóò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = ô sin( ï ). 1 b . Integrating the equation twice, we arrive at a second-order equation: !

 − (
 )2 = − Œ −2 sin( Œ  ) + L 1  + L 2 .

ññ óò ò ó ò ò ó ó

2 b . Particular solution: 11.

 . LpΠ4

= L sin( Π ) +

– (ñó ò ò ó )2 – ôñó ò + õ sin( ï ) = 0. 1 b . Integrating yields a third-order equation: !"


 − 
 

 − 

ññóò ò ó ò ò ó ó

2 b . Particular solution: 12.

).

  


 = − cos( Œ ) − cos(ý± ) + L . Œ ý

Integrating yields a third-order equation: 6.

)+ .

=−

 Π( 2 + L

Œ

2 6)

− Œ

−1

cos( Π ) = L .

V  cos( Π ) + LpΠ3 sin( Π )W + L .

– (ñó ò ò ó )2 – ôñó ò ò ó + õ sin( ï ) = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 − Œ −2 sin( Œ  ) = 0.

ññóò ò ó ò ò ó ó

2 b . Particular solution:

= L sin( Œ  ) −

 + LpΠ2 . LpΠ4

Page 674

675

4.2. NONLINEAR EQUATIONS

13.

14.

– (ñó ò ò ó )2 = ô [ññó ò ò ó – (ñó ò )2 ] + õ sin( ï ) + ö . The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨ 

 =  ¨ +  sin( Œ  ) + ( .

ññóò ò ó ò ò ó ó

– ôñó ò ò ó ò ó – (ñó ò ò ó )2 + õ sin( ï ) = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  
 + L 1  + L 2 − Œ −2 sin( Œ  ) = 0.

ññóò ò ó ò ò ó ó

2 b . Particular solution: 15. 16.

ñóò ò ó ò ò ó ó

=

 Π3( 2 + L

18.

V  cos( Œ  ) − LpŒ sin( Œ  )W + L .

sin  ( ƒñ )ñóÈ ò ñóò ò ó ò ó .

ô

=

This is a special case of equation 4.2.6.51 with  ( ) =  sin  ( Œ

).

– ñó ò ñóò ò ó ò ó = ô sin( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  sin( Œ  ). Integrating yields a third order linear equation: 


 = L − cos( Œ  )S .

ññóò ò ó ò ò ó ó

Q

17.

Œ

2 2)

Œ

+ 4ñóÈ ò ñóò ò ó ò ó + 3(ñóò ò ó )2 = ô sin( ï ). Solution: 2 = L 3  3 + L 2  2 + L 1  + L 0 + 2 Œ

ññóò ò ó ò ò ó ó

sin( Π ).

−4

+ 4ñóÈ ò ñóò ò ó ò ó + 3(ñóò ò ó )2 = ô sin  ( ƒñ ) + õ . This is a special case of equation 4.2.6.60 with  ( ) =  sin  ( Œ

ññóò ò ó ò ò ó ó

)+ .

4.2.5-2. Equations with cosine. 19. 20.

21.

22. 23.

ñóò ò ó ò ò ó ó

cosú ( ƒñ ) + õ .

ô

=

This is a special case of equation 4.2.6.1 with  ( ) =  cos + ( Œ = ô cos( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  cos( Œ ¨ ) +  .

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô (ñ + õ cos ï )2 – õ cos ï . The substitution ¨ = +  cos  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  ¨ 2 .

ñ óò ò ó ò ò ó ó

= ô ï – 4 cosú ( ƒñ ). This is a special case of equation 4.2.6.2 with  ( ) =  cos + ( Œ

ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

=

ô

cos( ƒñ )ñ

óò

+ õ cos(

ï

).

Integrating yields a third-order equation: 24. 25.

)+ .




 = Œ



sin( Œ

)+

ý

).



sin(ý± ) + L .

= ô 4 ñ + õ cos( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  cos( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cos( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  cos( Œ  )¨  .

ñóò ò ó ò ò ó ó

Page 675

676 26.

27.

28.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ò ó ó

=

õ

cos(

ï )(ï ñ óò

– 2ñ )  .

ñóò ò ó ò ò ó ó

=

õ

cos(

ï )(ï „ñ óò

– 3ñ )  .

This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  cos( Œ  )¨  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  cos( Œ  )¨  .

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 =

ô

ññ óò ò ó ò ò ó ó

– (ñ

óò ò ó

óò

)2 – ôñ

32.

34.

) = 0.

1 b . Integrating yields a third-order equation: !"


 − 
 

 − 

ññóò ò ó ò ò ó ó

=

 Π( 2 + L

Œ

2 6)

ï

– (ñó ò ò ó )2 – ôñó ò ò ó + õ cos(

+ Œ

−1

sin( Π ) = L .

V  sin( Œ  ) − LpŒ 3 cos( Œ  )W + L .

) = 0.

1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 − Œ −2 cos( Œ  ) = 0.

LpΠ2 +  . LpΠ4

= L cos( Œ  ) −

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 = ô [ññó ò ò ó – (ñó ò )2 ] + õ cos(

ññóò ò ó ò ò ó ó

– ôñó ò ò ó ò

ï

)+ö .

The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨p

 =  ¨ +  cos( Œ  ) + ( .

ó

– (ñó ò ò ó )2 + õ cos(

ï

) = 0.

1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  
 + L 1  + L 2 − Œ −2 cos( Œ  ) = 0. 2 b . Particular solution:

33.

ï

+ õ cos(

2 b . Particular solution: 31.

 . LpΠ4

= L cos( Π ) +

2 b . Particular solution: 30.

).

1 b . Integrating the equation twice, we arrive at a second-order equation: ! 

 − ( 
)2 = − Œ −2 cos( Œ  ) + L 1  + L 2 . 2 b . Particular solution:

29.

ï

cos(

ñ óò ò ó ò ò ó ó

ô

=

cos ( ƒñ )ñ

=−

óò ñ óò ò ó ò ó

 Π3 ( 2 + L

Œ

2 2)

V LpΠcos( Π ) +  sin( Π )W + L .

.

This is a special case of equation 4.2.6.51 with  ( ) =  cos  ( Œ

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

=

ô

cos(

ï )ñ

2

).

.

This is a special case of equation 4.2.6.57 with  ( ) =  cos( Π ). Integrating yields a third order linear equation: 


 = sin( Π ) + L3S .

Q Œ

35.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

Solution: 36.

ññóò ò ó ò ò ó ó

2

+ 4ñó ò

=L

3

ó 

ñóò ò ó ò ó

3

+ 3(ñó ò ò ó )2 =

ô

+L

 +L

2



2

+L

1

+ 3(ñó ò ò ó )2 =

ô

cos( 0

ï

).

+ 2 Œ

−4

cos( Π ).

cos  ( ƒñ ) + õ .

This is a special case of equation 4.2.6.60 with  ( ) =  cos  ( Œ

)+ .

Page 676

677

4.2. NONLINEAR EQUATIONS

4.2.5-3. Equations with tangent. 37. 38.

39. 40. 41. 42. 43. 44.

= ô tanú ( ƒñ ) + õ . This is a special case of equation 4.2.6.1 with  ( ) =  tan + ( Œ

ñ óò ò ó ò ò ó ó

= ô tan( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  tan( Œ ¨ ) +  .

)+ .

ñ óò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô ï – 4 tanú ( ƒñ ). This is a special case of equation 4.2.6.2 with  ( ) =  tan + ( Œ

ñóò ò ó ò ò ó ó

= ô tan( ƒñ )ñ„ó ò + õ tan( ï ). This is a special case of equation 4.2.6.21 with  ( ) =  tan( Œ

ñóò ò ó ò ò ó ó

). ) and ( ) =  tan(ý± ).

= ô 4 ñ + õ tan( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  tan( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ tan( ï )( ï ñ ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  tan( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ tan( ï )( ï ñ„ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  tan( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ tan( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  tan( Œ  )¨  .

ñóò ò ó ò ò ó ó

45.

= ñ + ô (ñó ò + ñ tan ï )  . This is a special case of equation 5.2.6.32 with  ( , ‡ ) = ‡  and Y ( ) = cos  .

46.

= ô tan ( ƒñ )ñóÈ ò ñóò ò ó ò ó . This is a special case of equation 4.2.6.51 with  ( ) =  tan  ( Œ

47.

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

).

– ñó ò ñóò ò ó ò ó = ô tan( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  tan( Œ  ).

ññóò ò ó ò ò ó ó

48.

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô tan  ( ï ) + õ . This is a special case of equation 4.2.6.58 with  ( ) =  tan  ( Œ  ) +  .

49.

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô tan  ( ƒñ ) + õ . This is a special case of equation 4.2.6.60 with  ( ) =  tan  ( Œ

ññ óò ò ó ò ò ó ó ññ óò ò ó ò ò ó ó

)+ .

4.2.5-4. Equations with cotangent. 50. 51.

= ô cotú ( ƒñ ) + õ . This is a special case of equation 4.2.6.1 with  ( ) =  cot + ( Œ

ñ óò ò ó ò ò ó ó

= ô cot( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  cot( Œ ¨ ) +  .

)+ .

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

Page 677

678 52. 53. 54. 55. 56. 57.

FOURTH-ORDER DIFFERENTIAL EQUATIONS = ô ï – 4 cotú ( ƒñ ). This is a special case of equation 4.2.6.2 with  ( ) =  cot + ( Œ

ñóò ò ó ò ò ó ó

).

= ô cot( ƒñ )ñó ò + õ cot( ï ). This is a special case of equation 4.2.6.21 with  ( ) =  cot( Œ

ñóò ò ó ò ò ó ó

) and ( ) =  cot(ý± ).

= ô 4 ñ + õ cot( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  cot( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cot( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  cot( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cot( ï )( ï ñ„ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  cot( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cot( ï )( ï ñ ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  cot( Œ  )¨  .

ñ óò ò ó ò ò ó ó

58.

= ñ + ô (ñó ò – ñ cot ï )  . This is a special case of equation 5.2.6.32 with  ( , ‡ ) = ‡  and Y ( ) = sin  .

59.

= ô cot ( ƒñ )ñóÈ ò ñóò ò ó ò ó . This is a special case of equation 4.2.6.51 with  ( ) =  cot  ( Œ

60.

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

).

– ñó ò ñóò ò ó ò ó = ô cot( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  cot( Œ  ).

ññóò ò ó ò ò ó ó

61.

+ 4ñóÈ ò ñóò ò ó ò ó + 3(ñóò ò ó )2 = ô cot ( ï ) + õ . This is a special case of equation 4.2.6.58 with  ( ) =  cot  ( Œ  ) +  .

62.

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô cot ( ƒñ ) + õ . This is a special case of equation 4.2.6.60 with  ( ) =  cot  ( Œ

ññóò ò ó ò ò ó ó ññ óò ò ó ò ò ó ó

)+ .

4.2.6. Equations Containing Arbitrary Functions 4.2.6-1. Equations of the form 



 =  ( , ). 1.

ñóò ò ó ò ò ó ó

= ! (ñ ).

Autonomous equation. By integrating, we obtain 2 



 − ( 

 )2 = 2 X

substitution ¨ ( ) = | 
|3 J 2.

3.

2

leads to a second-order equation: ¨

=

= ï – 4 ! (ñ ). This is a special case of equation 4.2.6.55 with  leads to an autonomous equation.



ñ óò ò ó ò ò ó ó

1

=

2

=

= ï –3 ! (ñ ï ). Homogeneous equation. The transformation a = ln  , ¨ equation of the form 4.2.6.79.

ñóò ò ó ò ò ó ó

3

3 2

Q

 ( )

Xv ( ) M

M

+ 2 L . The

+ L2S ¨

J .

−5 3

= 0. The substitution a = ln | |

=  

leads to an autonomous

Page 678

679

4.2. NONLINEAR EQUATIONS

4.

5.

6.

= ï –5 ! (ñ ï –3 ). The transformation  = a ¨2g

g
g
g =  (¨ ).

ñóò ò ó ò ò ó ó

−1

=¨ a

,

−3

= ï –5 ) 2 ! (ñ ï –3 ) 2 ). g The transformation  =  , =  3 J 2 ¨ ¨2g

g
g
g − 5 ¨2g

g = − 9 ¨ +  (¨ ). 2 16

leads to an autonomous equation of the form 4.2.6.1:

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.33:

= ï –  –4 ! ( ï  ñ ). Generalized homogeneous equation.

ñ óò ò ó ò ò ó ó

1 b . The transformation a = ln  , H =  

leads to an autonomous equation. 2 b . The transformation H =   , ¨ =  
  leads to a third-order equation.

7.

8.

9. 10.

= ñ ï – 4 ! ( ï ñ ú ). Generalized homogeneous equation. The transformation H =   + , ¨ third-order equation.

ñóò ò ó ò ò ó ó

= ! (ñ + ô 3 ï 3 + ô 2 ï 2 + ô 1 ï + ô 0 ). The substitution ¨ = +  3  3 +  2  2 +  1  +  ¨2



 =  (¨ ).

ñ óò ò ó ò ò ó ó

= ! (ñ + ô ï 4 ). The substitution ¨ = + 

ñóò ò ó ò ò ó ó ï (ô ï

+ õ )4 ñó ò ò ó ò ò ó

ó

= ! (ñ

ï

4 –3

0

leads to an equation of the form 4.2.6.1:

).

leads to an autonomous equation of the form

3

ô ï  ï

11.

ñóò ò ó ò ò ó ó

= (ô

ï

+ õÿñ + ö . + ñ + • This is a special case of equation 5.2.6.19 with B = 4.

12.

ñ óò ò ó ò ò ó ó

= (ô

ï

2

õÿñ

+

+ ö )–3 !e”

õï


 leads to a

leads to an equation of the form 4.2.6.1: ¨u



 =  (¨ ) + 24  .

 +  ¨ The transformation ˆ = ln û û, =  û  û 4.2.6.79. û û +

=

+ ö )–5 ) 2 !e”

ñ

(ô ï

2



õï

+

+ ö )3 )

2

•

.

, ¨ = M + ' + ( (  2 + ' + ( )3 J equation of the form 4.2.6.33 with respect to ¨ = ¨ (ˆ ): 1 b . The transformation ˆ = X



2

¨ Šª

ªŠ

ŠªŠ − 5 ¢ ¨ ªŠ

Š + 2

9 16

¢ 2 ¨ =  (¨ ),

where

2

leads to an autonomous

¢ =  2 − 4 !( .

Therefore, having integrated the latter equation, we obtain

¨ Š
¨ Šª

ªŠ
Š − 1 (¨ Šª

Š )2 − 5 ¢ (¨ Š
)2 = − 9 ¢ 2 ¨ 2 4 32 3J The substitution H (¨ ) = |¨3Š
|

H ?

? =

15 8

2

3 2

Q

9 − 32 ¢

2

¨

2

where

.

3 2

= 

.


)


 − 2

1 2

.

(

 )2 +

+ ' + ( , ¨ =

.

 (¨ ) ¨ + L . M

+ Xv (¨ ) ¨ + L2S´H

2 b . The first integral of the original equation has the form:

. ( 
 −

+X

leads to a second-order equation:

J +

−1 3

¢çH

2

J .

1 2

.

M

J .

−5 3



 

 + 3  !

 − 2  (
 )2 = X| (¨ ) ¨ + L , M

−3 2

Page 679

680 13.

14.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ó

ñ óò ò ó ò ò ó ó

ó

=   ! (ñ –  ).  This is a special case of equation 4.2.6.47 with  =  = ( = 0. The substitution ¨ ( ) =  −  leads to an autonomous equation.

ó

= ñ! (  ñ ú ).  The transformation H = ž  + , ¨ ( H ) = 
  leads to a third-order equation.

ñóò ò ó ò ò ó ó

15.

= ï – 4 ! ( ïƒú ] ). The transformation H =  +  , ¨ ( H ) =  
 leads to a third-order equation.

16.

ñ óò ò ó ò ò ó ó

17.

18.

19.

20.

ñóò ò ó ò ò ó ó

ó

ó

= ! (ñ + ô ) – ô .  The substitution ¨ = +   leads to an equation of the form 4.2.6.1: ¨ 



 =  (¨ ).

= ! (ñ + ô cosh ï ) – ô cosh ï . The substitution ¨ = +  cosh  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  (¨ ).

ñóò ò ó ò ò ó ó

= ! (ñ + ô sinh ï ) – ô sinh ï . The substitution ¨ = +  sinh  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  (¨ ).

ñ óò ò ó ò ò ó ó

= ! (ñ + ô cos ï ) – ô cos ï . The substitution ¨ = +  cos  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  (¨ ).

ñóò ò ó ò ò ó ó

= ! (ñ + ô sin ï ) – ô sin ï . The substitution ¨ = +  sin  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  (¨ ).

ñóò ò ó ò ò ó ó

4.2.6-2. Equations of the form 



 =  ( , , 
 ). 21.

22.

23. 24.

ñóò ò ó ò ò ó ó

= ! (ñ )ñó ò + " ( ï ).

By integrating, we find 


 = X  ( ) + X ( )  + L . For ( ) ≡ 0, the order of this M M equation can reduced by one with the help of the substitution ¨ ( ) =
 . = ï – 4 ! ( ï ñó ò – ñ ). The transformation a = ln | |, ¨ ¨2g

g
g − 5¨2g

g + 6¨2g
=  (¨ ).

ñóò ò ó ò ò ó ó

= ! ( ï , ï ñó ò – ñ ). The substitution ¨ =  
−

ñóò ò ó ò ò ó ó

=  
 −

leads to a third-order autonomous equation:

¨ 
 )

 =  ( , ¨ ). leads to a third-order equation: (u

= ! ( ï , ï ñó ò – 2ñ ). ¨ 


 − ¨2

 =  2  ( , ¨ ). The substitution ¨ =  
− 2 leads to a third-order equation:  u

ñóò ò ó ò ò ó ó

25.

= ! ( ï , ï ñó ò – 3ñ ). ¨ 


 =   ( , ¨ ). The substitution ¨ =  
− 3 leads to a third-order equation: u

26.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

=

ñ ï

–4

! (ï ñ óò ¦ ñ

The transformation H = 

).


 , ¨ = 

2



  leads to a second-order equation.

Page 680

681

4.2. NONLINEAR EQUATIONS

27.

28. 29. 30.

= ô 4 ñ + ! ( ï , ñó ò – ôñ ). ¨ 


 +  2 ¨ 

 +  2 2 ¨ 
+  3 ¨ = The substitution ¨ = 
−  leads to a third-order equation: u ¨  ( , ).

ñóò ò ó ò ò ó ó

= ! ( ï , ñó ò sinh ï – ñ cosh ï ) + ñ . The substitution ¨ = 
sinh  − cosh  leads to a third-order equation.

ñóò ò ó ò ò ó ó

= ! ( ï , ñó ò cosh ï – ñ sinh ï ) + ñ . The substitution ¨ = 
cosh  − sinh  leads to a third-order equation.

ñóò ò ó ò ò ó ó

= ! ( ï , ñó ò sin ï – ñ cos ï ) + ñ . The substitution ¨ = 
sin  − cos  leads to a third-order equation.

ñóò ò ó ò ò ó ó

31.

= ! ( ï , ñó ò cos ï + ñ sin ï ) + ñ . The substitution ¨ = 
cos  + sin  leads to a third-order equation.

32.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

=

r óò ò óò ò ó ó r ñ

+I ! O

ï , ñ óò



Y  The substitution ¨ =  − Y

–

r óò r ñ P

r

,

r (ï

=

).

leads to a third-order equation.

4.2.6-3. Equations of the form 



 =  ( , , 
 , 

 ). 33.

ñóò ò ó ò ò ó ó

+

ôñóò ò ó

= ! (ñ ).

Having integrated this equation, we obtain 2 



 − ( 

 )2 +  ( 
)2 = 2 XZ ( )

where L is an arbitrary constant. The substitution ¨ ( ) = |"
 |3 J

equation:

34.

¨

= −3 ¨ 4 

J +

−1 3

3 2

Q

X  ( ) M

+ L2S ¨

M

+ 2L ,

leads to a second-order

J .

−5 3

ñóò ò ó ò ò ó ó + ! (ñóò )ñóò ò ó = " (ñ ). Having integrated this equation, we obtain a third-order autonomous equation: 2 



 − ( 

 )2 + 2  ( 
) = 2 X¥ ( )

M

+ 2L ,

where

The substitution ¨ ( ) = 
leads to a second-order equation. 35.

2

= ï –2 ! ( ï ñ ó ò – ñ )ñ ó ò ò ó . The transformation a = ln | |, ¨ =  
 −

 (‡ ) = Xȇ/ (‡ ) ‡ . M

ñ óò ò ó ò ò ó ó

leads to a third-order equation:

¨ g

g
g − 5¨ g

g + 6¨ g
=  (¨ )¨ g
. Integrating it, we obtain a second-order autonomous equation:

¨ g

g − 5¨ g
+ 6¨ = X  (¨ ) ¨ + L . M ¨ 1 ¨2
The substitution H ( ) = 5 g leads to an Abel equation of the second kind: HH ?
− H =

1 25

Q

−6¨ + X

 (¨ ) ¨ + L2S M

(see Subsection 1.3.1).

Page 681

682 36.

FOURTH-ORDER DIFFERENTIAL EQUATIONS – (ñó ò ò ó )2 = ! ( ï ). Integrating the equation twice, we arrive at a second-order equation:

ññóò ò ó ò ò ó ó

! 

 − ( 
)2 = X 37.

38.

39.



( − a )  (a ) a + L

M

0

ñóò ò ó ò ò ó ó

= ô 2 ñ + ! (ñó ò ò ó The substitution ¨ = 2.9.1.1: ¨ 

 =  ¨ + 

+ ôñ ). 

 +  ¨( ).

leads to a second-order autonomous equation of the form

ñóò ò ó ò ò ó ó = ! (ñ , ñóò ò ó ). The substitution ¨ ( ) = A (
 )2 leads to a third-order equation: 2   , A 12 ¨2
 . 

= ô 2 ñ + ! ( ï , ñó ò ò ó + The substitution ¨ = 

 + 

41.

ñóò ò ó ò ò ó ó

= ñ!  ññó ò ò ó – ñó ò 2  . This is a special case of equation 4.2.6.42.

42.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

= ñ ó ò ò ó !u ññ ó ò ò ó – ñ 1 b . Particular solution: where the constants L

ôñ

óò 2 

4 3

+ ñ"*

ññ óò ò ó

–ñ

=L

exp( L

3

L

45.

46.

= ïƒú ! ( ï 2 ñó ò ò ó The substitution ¨ =  + +3  (¨ ). For = ¡ ?
second kind: HH − H

ñóò ò ó ò ò ó ó



2

−L

L 2 , and L

+L

– 2ï

2 3

 (− L

ñóò

2

L

2

exp(− L

3

 ),

are related by the constraint

1

2 1

óò 2  .

 )+L 2 3)

 (4 L 1 L 2 L

=L

1, 4 3

1

2 3

2 b . Particular solution: where the constants L

¨*¨u


+ 1 2 ¨
2 ¨

= 2    

). ¨ 

 =  ¨ +  ( , ¨ ). leads to a second-order equation: u

L 2 , and L

1,

L

44.

 + L 2.

– (ñó ò ò ó )2 = ! ( ï )[ññó ò ò ó – (ñó ò )2 ] + " ( ï ). This is a special case of equation 4.2.6.93. The substitution ¨ ( ) = ! 

 − ( 
)2 leads to a second-order linear equation: ¨u

 =  ( )¨ + ( ).

ññóò ò ó ò ò ó ó

40.

43.

1

cos( L

3

− (4 L

 )+L

2

1

L 2 L 32 ) = 0.

sin( L

3

 ),

are related by the constraint 2 3

−L

2 2

L

2 3)

− (− L

2 1

L

2 3

−L

2 2

L

2 3)

= 0.

+ 2ñ ).



 − 2 
+ 2 leads to a second-order equation:  ¨ 

 − 2¨ 
= −4, the substitution H (¨ ) = 13  ¨2
leads to an Abel equation of the = 19  (¨ ) (see Subsection 1.3.1). 2

= ! ( ï , ï ñó ò – ñ , ñó ò ò ó ). The substitution ¨ ( ) =  
−

ñóò ò ó ò ò ó ó

leads to a third-order equation.

= ! ( ï , ï 2 ñó ò ò ó – 2 ï ñó ò + 2ñ ). ¨ 

 − 2¨2
= The substitution ¨ =  2 

 − 2 
 + 2 leads to a second-order equation:  u ¨ 3   ( , ).

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

=

ñóò !IO

ñ óò ò ó , ñ óò ñ óò

–ñ

ñ óò ò ó ñ óò

P

.

Particular solution: = L 1 exp( L 2  )+ L 3 , where L 1 is an arbitrary constant and the constants L 2 and L 2 are related by the constraint L 23 =  ( L 2 , − L 2 L 3 ).

Page 682

683

4.2. NONLINEAR EQUATIONS

4.2.6-4. Equations of the form 



 =  ( , , 
 , 

 , 


 ). 47.

ñóò ò ó ò ò ó ó

+

ôñóò ò ó ò ó

õÿñóò ò ó

+

+

ö^ñóò

=

ó

 ! (ñ  –

ó

).

 The substitution ¨ ( ) =  −  leads to an autonomous equation: ¨ 



 + (4 Œ +  )¨ 


 + (6 Œ 2 + 3  Œ +  )¨ 

 + (4 Œ 3 + 3 Œ 2 + 2 Œ + ( )¨ 
+ ( Œ 4 + Œ 3 + Œ

2

+ (Œ )¨ =  (¨ ),

which can be reduced to a third-order equation by means of the substitution H (¨ ) = ¨I
. For  = −4 Œ and ( = 8 Œ 3 − 2 Œ , the above equation coincides, up to notation, with equation 4.2.6.33 and can be reduced to a second-order equation. 48.

ñ óò ò ó ò ò ó ó

+

ôññ óò ò ó ò ó

= ! ( ï ).

Integrating, we arrive at a third-order equation: 49.

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

– ôñó ò ñóò ò ó = ! (ï ).

Integrating, we arrive at a third-order equation: 50.

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

+ ! (ñó ò )ñó ò ò ó = " ( ï ).




 +  ! 

 − 12  ( 
)2 = X  ( )  + L . M 


 +  ! 

 −  ( 
)2 = Xv ( )  + L . M

Integrating, we arrive at a third-order equation:




 +  ! 

 − 12  ( 
)2 +  ( 
) = X ( )  + L , M 51.

ñ óò ò ó ò ò ó ó

óò ñ óò ò ó ò ó

= ! (ñ )ñ

where

 (‡ ) = X  (‡ ) ‡ . M

.

Integrating, we arrive at a third-order autonomous equation of the form 3.5.1.1: L exp X¯ ( ) S .

Q

52. 53. 54.

55.

M

ï ñóò ò ó ò ò ó ó

+ 4ñó ò ò ó ò

ï ñóò ò ó ò ò ó ó

+ ( ô + 3)ñó ò ò ó ò

ó

= ! (ï

The substitution ¨ ( ) = 

ñ

).

¨ 



 =  (¨ ). leads to an equation of the form 4.2.6.1: I

= ! (ï , ï

ó

The substitution ¨ =  
+ 

ï 2 ñ óò ò ó ò ò ó ó

+ 8ï

ñóò ò ó ò ó

ï 4 ñ óò ò ó ò ò ó ó

+

ñóò

ô 3 ï 3 ñ óò ò ó ò ó

+

+ ôñ ).

¨ 


 =  ( , ¨ ). leads to a third-order equation: u

+ 12ñó ò ò ó = ! ( ï

The substitution ¨ ( ) =  ¨2



 =  (¨ ).




 =

2

2

ñ

).

leads to an autonomous equation of the form 4.2.6.1:

ô 2 ï 2 ñ óò ò ó

+ô

1

ï ñóò

= ! (ñ ).

The substitution a = ln | | leads to an autonomous equation:











(1) g g g g + (  3 − 6) g g g + (11 − 3 3 +  2 ) g g + (2  3 −  2 +  1 − 6) g =  ( ), the order of which can be lowered with the help of the substitution ¨ ( ) =
g . For  3 = 6 and  1 =  2 − 6, equation (1) coincides, up to notation, with equation 4.2.6.33 and can be reduced to a second-order equation.

56.

ï 4 ñ óò ò ó ò ò ó ó

+

ô ï 3 ñ óò ò ó ò ó

+

õ ï 2 ñ óò ò ó

+ö

ï ñóò

+ sñ =

ï – ! (ñ ï

). The transformation a = ln  , ¨ =   leads to an autonomous equation of the form 4.2.6.79.

Page 683

684 57.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ññóò ò ó ò ò ó ó

– ñó ò

= ! ( ï )ñ 2 .

ñóò ò ó ò ó




 = X  ( )  + L2S . Q M

Integrating yields a third-order linear equation: 58.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

Solution: 59.

2

=L

3

+ 3(ñó ò ò ó )2 = ! ( ï ).

ó 

3

+L

2



2

+L

1

 +L



0

+ 13 X  ( − a )3  (a ) a . M 0

+ ôñ ó ò ñ ó ò ò ó ò ó + ( ô – 1)(ñ ó ò ò ó )2 = ! ( ï ). Integrating the equation two times, we obtain a second-order equation:

ññ óò ò ó ò ò ó ó



! 

 +  − 2 ( 
)2 = L  + L 1

0

2

+ X  ( − a )  (a ) a . M 0

60.

ññ óò ò ó ò ò ó ó

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ! (ñ ). ¨ 



 = 2   A £ ¨  . The substitution ¨ = 2 leads to an equation of the form 4.2.6.1: u

61.

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

= ! ( ï )ññó ò ò ó ò ó .

ó




 = L exp X  ( )  S . Q M

Integrating yields a third-order linear equation: 62.

ññóò ò ó ò ò ó ó

+ ñó ò ñóò ò ó ò

= ! ( ï )ññó ò ò ó ò ó .

ó




Integrating yields a third-order equation of the form 3.5.1.2: !   = L exp X¯ ( )  S .

! = ! (ï

ññóò ò ó ò ò ó ó

" = " (ï

Q

M

63.

+ ( ! – 1)ñóÈ ò ñóò ò ó ò ó + !"Iññóò + "Ióò ñ 2 = 0, ), ). The functions that solve the third-order linear equation "


 + ( ) = 0 are solutions of the given equation.

64.

ññóò ò ó ò ò ó ó

+ (4ñó ò +

+ 3(ñó ò ò ó )2 + 3 !„ñó ò ñóò ò ó + " (ï ) = 0,

!„ñ )ñ óò ò ó ò ó

!

= ! ( ï ).



The substitution ¨ = (! 
) leads to a first-order linear equation: ¨u
+  ¨ + = 0. Solution:  2 = L 2  2 + L 1  + L 0 + X  (  − a )2 ¨ ( a ) a , M 0 ¨ −Ž ( ) ( ) Ž where ( ) =  L −X  ( )  S ,  ( ) = X¯ ( )  ;  0 is an arbitrary number. Q 3 M M

65.

ññ óò ò ó ò ò ó ó

+ (4ñ ó ò + !„ñ )ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 + (3 !„ñ ó ò + "Iñ )ñ ó ò ò ó + " (ñ ó ò )2 + %„ññ ó ò + s = 0. Here,  =  ( ), = ( ), % = % ( ), ° = ° ( ). The substitution ¨ = !"
 leads to a third-order nonhomogeneous linear equation: ¨u


 +  ¨2

 + ¨2
+ % ¨ + ° = 0.

66.

(ñ +

ô ï

+ õ )ñ

óò ò ó ò ò ó ó

óò

+ 4(ñ

Solution: ( +  +  )2 = L 67.

ññ óò ò ó ò ò ó ó

= 4ñ

óò ñ óò ò ó ò ó

3

óò ò ó

+ 3(ñ

The transformation ˆ =

+ ô )ñ




3

óò ò ó ò ó

+L

)2 – 6

2

(ñ

óò ò ó

+ 3(ñ +L

ñ

2

óò

)4 2



 , ¨ = −O

)2 = ! ( ï ).

+ V ññ




2



1 3 0 + 3 X  ( − a )  (a ) a . M 0

1 + L

óò ò ó

– (ñ

ñó óò )2 W+!IO ò ñ P

.

leads to a second-order linear equation with

P respect to ¨ 2 : (¨ 2 )Šª

Š = 24ˆ 2 + 2  (ˆ ). Integrating it, we obtain Š ¨ 2 = L ˆ + L + 2ˆ 4 + 2 X (ˆ − a )  (a ) a . 2 1 Š M 0

Page 684

685

4.2. NONLINEAR EQUATIONS

Taking into account that ˆ 
= ¨ , 
 = ˆ , 
Š = ˆ  ¨ , we find the solution in parametric form:

 =X where ¨ = A 68.

Q

L 2ˆ + L

1

M¨

4

+ 2ˆ

ˆ

+L

=L

3,

Š

+ 2 X Š (ˆ − a )  (a ) aÝS M 0

4

exp O X

J

1 2

ˆ

¨M

ˆ

P

,

.

– 2ññó ò ñó ò ò ó ò ó + ! ( ï )ñ 2 ñó ò ò ó ò ó + 2(ñó ò )2 ñó ò ò ó – ! ( ï )ññó ò ñó ò ò ó + 2 !„ó ò ( ï )ñ 2 ñó ò ò ó + 2 ! ( ï )(ñó ò )3 + [ ! 2 ( ï ) – 2 !„ó ò ( ï )]ñ (ñó ò )2 +

ñ 2 ñ óò ò ó ò ò ó ó

!„ó ò ò ó (ï )ñ 2 ñ óò

= 0.

The solution satisfies the second-order linear equation "

 +  ( )
 − H ( , L 1 , L 2 ) = 0, where H = H ( , L 1 , L 2 ) is the Weierstrass elliptic function determined by the second-order autonomous equation H 

 + H 2 = 0. 69.

ñ 2 ñ óò ò ó ò ò ó ó

– 2ññó ò ñóò ò ó ò ó + ! (ï )ñ 2 ñóò ò ó ò ó + 2(ñóò )2 ñóò ò ó – ! (ï )ññóò ñóò ò ó + 2 !„ó ò ( ï )ñ 2 ñó ò ò ó + 2 ! ( ï )(ñó ò )3 + [ ! 2 ( ï ) – 2 !„ó ò ( ï )]ñ (ñó ò )2 +

!„ó ò ò ó (ï )ñ 2 ñ óò

=


ï ñ

3

.

The solution satisfies the second-order linear equation "

 +  ( )
 − H ( , L 1 , L 2 ) = 0, where H = H ( , L 1 , L 2 ) is the solution of the first Painlev´e transcendent H 

 + H 2 = #* . 70.

ñ 2 ñ óò ò ó ò ò ó ó – 2ññóò ñóò ò ó ò ó + [ô + ! (ï )]ñ 2 ñóò ò ó ò ó – ñ (ñóò ò ó )2 – [3ô + ! (ï )]ñ ñ„óÈò ñóò ò ó + 2(ñó ò )2ñó ò ò ó + [ ô! ( ï ) + " (ï )]ñ 2ñó ò ò ó + 2 ô (ñó ò )3 – ô! ( ï )ñ (ñó ò )2 +  ô " (ï )ñ 2 ñóò

= % ( ï )ñ 3.

The solution satisfies the second-order linear equation "

 +  
 − H ( , L 1 , L 2 ) = 0, where H = H ( , L 1 , L 2 ) is the solution of the second-order linear equation H 

 +  ( ) H 
+ ( ) H = % ( ). 71.

72.

ñóò ò óñóò ò ó ò ò ó ó

– 3(ñó ò ò ó ò ó )2 = ! ( ï

ñóò

– ñ )(ñó ò ò ó )5 .

The Legendre transformation  = ‡ g
, ‡ g

g
g
g = −  (‡ ).

ñóò ò ó ò ò ó ó

= aև g
− ‡ leads to an equation of the form 4.2.6.1:

= ! (ñ )ñó ò " (ñóò ò ó ò ó ).

Integrating yields a third-order autonomous equation:

¨ M ¨ = Xv ( ) ( ) M X

+L ,

where

¨ = 


 ,

the order of which can be lowered by means of the substitution H ( ) =
 . 73.

ï ñóò ò ó ò ò ó ó

+ 2ñó ò ò ó ò

ó

= (ï

ñóò ò ó

The substitution ¨ ( ) = 

)–5 !IO

75.

ï 2 ñ óò ò ó ò ò ó ó

+ 2ï

ñóò ò ó ò ó

= ! (ï

−5 2

2

P

.

leads to a third-order equation of the form 3.5.2.27:

¨2
J O  , £ ¨ P

¨ 


 = ¨ 74.

ü


−

ï ñ óò ò ó ï ñ óò – ñ

ñóò ò ó

– 2ï

ñóò

where

+ 2 ñ )" ( ï

2

ñóò ò ó ò ó

 (ˆ ) = ˆ

−5

 (ˆ ).

).

The substitution ¨ ( ) =  2 

 − 2 
 + 2 leads to a second-order equation of the form ¨ 
). 2.9.4.36: ¨3

 =  (¨ ) (2

ñóò ò ó ò ò ó ó

= ! ( ï )" ( ï

ñóò ò ó ò ó

– 3ï

2

ñóò ò ó

+ 6ï

ñóò – 6ñ ). The substitution ¨ ( ) =  3 


 − 3 2 

 + 6 
 − 6 leads to a first-order separable equation: ¨2
=  3  ( ) (¨ ). 3

Page 685

686

FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.2.6-5. Other equations. 76.

ññóò ò ó ò ò ó ó

– 16 (ñó ò ò ó )2 = Particular solution: where the constants L

ï 2 ! 1 (ñ óò ò ó ò ò ó ó 1,

4

! 2 (ñ óò ò ó ò ò ó ó

+ 16 L

2

L 2 , L 3 , L 4 , and L

5

=

1 24

)+ï

L 1

1 3



3

ññóò ò ó ò ò ó ó

– 16 (ñó ò ò ó )2 =

ñóò ò ó ò ó! 1 (ñ óò ò ó ò ò ó ó

Particular solution:

+ 12 L

3



! 3 (ñ óò ò ó ò ò ó ó 2

+L

4

).

 + L 5,

are related by three constraints

L 1 L 3 − 16 L L 1 L 4 − 13 L 2 L L 1 L 5 − 16 L 77.

)+

2 2

=  1(L

1 ),

3

=  2(L

1 ),

2 3

=  3(L

1 ).

) + ñó ò ò ó! 2 (ñó ò ò ó ò ò ó

ó) ï 2 + ! 3 (ñ ó ò ò ó ò ò ó ó

)+

ï ! 4 (ñ ó ò ò ó ò ò ó ó

)+

! 5 (ñ ó ò ò ó ò ò ó ó

).

L 1  4 + 16 L 2  3 + 12 L 3  2 + L 4  + L 5 , where the constants L 1 , L 2 , L 3 , L 4 , and L 5 are related by three constraints =

1 3

1 24

L 1 L 3 − 16 L L 1 L 4 − 13 L 2 L L 1 L 5 − 16 L

2 2

1 2

L 1  2 ( L 1 ) +  3 ( L 1 ), 3 = L 1  1 ( L 1 ) + L 2  2 ( L 1 ) +  4 ( L 1 ), 2 3 = L 2  1 ( L 1 ) + L 3  2 ( L 1 ) +  5 ( L 1 ). =

78.

= H ( ï , ñó ò , ñó ò ò ó , ñó ò ò ó ò ó ). ¨ 


 =  ( , ¨ , 2 ¨ 
, 2 ¨ 

 ). The substitution ¨ ( ) = 
leads to a third-order equation: u

79.

ñóò ò ó ò ò ó ó

80.

81.

82.

83.

ñóò ò ó ò ò ó ó

= H (ñ , ñó ò , ñó ò ò ó , ñó ò ò ó ò ó ). Autonomous equation. The substitution ¨ ( ) = ("
 )2 leads to a third-order equation: = ï –3 H (ñ ï , ñó ò , ï ñó ò ò ó , ï 2 ñó ò ò ó ò ó ). Homogeneous equation. The transformation a = ln  , ¨ equation of the form 4.2.6.79.

ñóò ò ó ò ò ó ó

1 2

£ ¨ ¨

). 

=  

leads to an autonomous

= ï –  –4 H ( ï ñ , ï +1 ñó ò , ï +2 ñó ò ò ó , ï +3 ñó ò ò ó ò ó ). Generalized homogeneous equation. The transformation a = ln  , ¨ autonomous equation of the form 4.2.6.79.

ñóò ò ó ò ò ó ó

=  

leads to an

= H ( ï , ï ñ ó ò – ñ , ñ ó ò ò ó , ñ ó ò ò ó ò ó ). This is a special case of equation 5.2.6.78 with B = 4. The substitution ¨ = 
 − a third-order equation.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó = H (ï , ï ñóò  The substitution ¨ = ¨2

  .

84.

¨*¨


+ 1 ¨
¨

= 2  ( , A £ ¨ , 1 ¨
, A 2   2   

– 2ñ , ñó ò ò ó ò ó ). =  
− 2 leads to a third-order equation:






=  ( , ¨ , ), where

= H ( ï , ï 2 ñó ò ò ó – 2 ï ñó ò + 2ñ , ñó ò ò ó ò ó ). The substitution ¨ ( ) =  2 

 − 2 
 + 2 leads to a second-order equation: (  ( , ¨ ,  −2 ¨2
).

ñóò ò ó ò ò ó ó

leads to

−2

¨2
)
 =

Page 686

687

4.2. NONLINEAR EQUATIONS

85.

ñóò ò ó ò ò ó ó

= ñHwO

ñ óò ñ

ñ óò ò ó ñ

,




The transformation ˆ =

¨ 86.

ñóò ò ó ò ò ó ó

2

=

ñ óò ò ó ò ó ñ

,

.

P



 
 , ¨ = −O

¨ Šª

Š + ¨ (¨ Š
)2 + 4ˆ ¨*¨ Š
+ 3¨

ñ ï

–4

HwO ï  ñ ú

ï ñ óò ñ

,

2

P

+ 6ˆ 2 ¨ + ˆ

ï 2 ñ óò ò ó ñ

,

2

leads to a second-order equation: 4

ï 3 ñ óò ò ó ò ó ñ

,

=  (ˆ , ¨ + ˆ 2 , ¨*¨ Š
+ 3ˆ ¨ + ˆ 3 ).

P

.

 
 Generalized homogeneous equation. The transformation a =   + , H = third-order equation. 87.

ñóò ò ó ò ò ó ó

=

ñ ï

–4

HwO

ï ñ óò ñ

,

The transformation H = 88.

ñ óò ò ó ò ò ó ó

=

ñ óò ò ó ñ óò HwO ó , ñ óò ñ ò

ï 2 ñ óò ò ó ñ

 


ï 3 ñ óò ò ó ò ó ñ

,

ñ óò ò ó ñ óò

–ñ

,

ñ óò ò ó ò ó ñ óò

.

P

 2 

 , ¨ =

leads to a

leads to a second-order equation. .

P

Autonomous equation. This is a special case of equation 4.2.6.79. Particular solution: = L 1 exp( L 2  ) + L 3 , where L 1 is an arbitrary constant and the constants L L 23 =  ( L 2 , − L 2 L 3 , L 22 ). 89.

ó

ó

ó

ó

2

and L

2

are related by the constraint

ó

=  – ^ HÊm ^ ñ ,  ^ ñ ó ò ,  ^ ñ ó ò ò ó ,  ^ ñ ó ò ò ó ò ó  .  Equation invariant under “translation–dilatation” transformation. The substitution ‡ =  9 leads to an autonomous equation of the form 4.2.6.79.

ñ óò ò ó ò ò ó ó

90.

= ï – 4 H ( ïƒú  ^ ] , ï ñó ò , ï 2 ñó ò ò ó , ï 3 ñó ò ò ó ò ó ). The transformation H =  +  9  , ¨ =  
 leads to a third-order equation.

91.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

= ñHwO+

^ ó ñ ú

,

ñ óò ñ óò ò ó , ñ ñ

,

ñ óò ò ó ò ó ñ

P

.

The transformation H =  9 + , ¨ = 
  leads to a third-order equation. 92.

H (ñ óò ò ó¦ñ , ñ ñóò ò ó

– ñó ò 2 , ñó ò ò ó ò ó¦ñó ò , ñó ò ò ó ò ò ó ó¦ñ ) = 0. Autonomous equation. This is a special case of equation 4.2.6.79. 1 b . Particular solution:

 ), where the constants L 1 , L 2 , L 2 are related by the constraint   L 32 , 4 L 1 L 2 L 32 , L 32 , L 34  = 0. 2 b . Particular solution: = L 1 cos( L 3  ) + L 2 sin( L 3  ), where L 1 , L 2 , and L 2 are related by the constraint   − L 32 , −( L 12 + L 22 ) L 32 , − L 32 , L 34  = 0. 93.

=L

1

exp( L

3

 )+L

2

exp(− L

3

H  ï , ñ ñóò ò ó

– (ñó ò )2 , ññó ò ò ó ò ó – ñó ò ñóò ò ó , ññóò ò ó ò ò ó ó – (ñóò ò ó )2  = 0. ¨ !


¨ 

 ) = 0. The substitution ( ) =  − (  )2 leads to a second-order equation:  ( , ¨ , ¨u
, 2

Page 687

688 94.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

HwO

ñ óò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó , ñ ñ óò ñ óò

– ñó ò ò ó ò

ó

= 0.

P

A solution of this equation is any function that solves the third-order linear equation:




 = L where the constants L 95.

H ( ï , ñ óò ò ó

1

and L

+ ôñ , ñó ò ò ó ò ò ó ó – ô The substitution ¨ = 

 + 

2 2

1

+L

2,

are related by the constraint  ( L

1, −

L 2 ) = 0.

ñ , ñ óò ò ó ò ò ó ó

+ ôñó ò ò ó ) = 0. leads to a second-order equation:  ( , ¨ , ¨u

 −  ¨ , ¨2

 ) = 0.

Page 688

Chapter 5

Higher-Order Differential Equations 5.1. Linear Equations 5.1.1. Preliminary Remarks In this Chapter, we denote higher derivatives by ( 

)



to mean M

.   M 1 b . The general solution of a homogeneous linear equation of the B th-order   ( ) (  ) +   −1 ( ) (  −1) + ))) +  1 ( ) 
+  0 ( ) = 0 has the form:

=L

1 1(

 )+L

2 2(

(1)

 ) + ))) + L   ( ).

(2)

Here, 1 ( ), 2 ( ),  ,  ( ) make up a fundamental set of solutions (the  are linearly independent solutions;  1 0); L 1 , L 2 ,  , L  are arbitrary constants. 2 b . Let

0

=

0(

 ) be a nontrivial particular solution of equation (1). Then the substitution =

0(

 )ù

H ( )  M

leads to a linear equation of the (B − 1)st-order for H ( ). Let 1 = 1 ( ) and 2 = 2 ( ) be two nontrivial linearly independent particular solutions of equation (1) with ≡ 0. Then the substitution =

1

X

2

¨

 − M

2

X

1

¨

leads to a linear equation of the (B − 2)nd-order for ¨ = ¨ ( ).

M



3 b . Some additional information about higher-order linear equations can be found in Subsections 0.4.1–0.4.3.

5.1.2. Equations Containing Power Functions 5.1.2-1. Equations of the form   ( ) (  ) +  0 ( ) = ( ). 1.

ñ ó(6) + ôñ

= 0. 1 b . Solution for  = 0: =L 2 b . Solution for  = , =L

1

cos ,! + L

2

6

1

+L

2

2

 + L 3

+L

4



3

+L

5



4

+L

6

 5.

> 0:

sin ,! + cos 

1 2

,!   L + sin 

3 1 2

cosh ˆ + L

,!   L

5

4

sinh ˆ 

cosh ˆ + L

6

sinh ˆ  ,

where

ˆ =

2

3 2

,! .

Page 689

690

HIGHER-ORDER DIFFERENTIAL EQUATIONS 3 b . Solution for  = − , =L

2.

ñ ó(2÷

1

cosh ,! + L

= ô 2÷ Solution: )

6

< 0:

sinh ,! + cosh 

2

1 2

,!   L

+ sinh 

ñ

1

  ½   + L 2  − +

9

,

=ô ï Solution: )

ñ

+õ , =

where 4.

ñ ó( ÷

½ Á

LlÁ = =0

,!   L

,

−1



=1

where Y  =  cos  ,  =  sin  ; L B B constants.

ñ ó( ÷

cos ˆ + L 5

4

sin ˆ 

cos ˆ + L

6

sin ˆ  ,

where

ˆ =

2

3 2

,! .

. =L

3.

3

1 2

ô ½ Á 

=0



1,

9 9  t ¼ ( #  cos  + $  sin  ), L 2 , #  , $  ( , = 1, 2,  , B − 1) are arbitrary

> 0.

a  +1 S a, LlÁ ´ÁtX ÷ exp ´Áa − Q 0  (B + 1) M

´Á = exp O

2 @!™  , B +1P

and ™ 2 = −1.

=ô ï _ ñ . For specific ý , see equations 5.1.2.2, 5.1.2.3 (with  = 0), 5.1.2.5–5.1.2.9, and 5.1.2.10 (with  = 0). )

1 b . Let B ≥ 2, ý > −B , and (B + ý )( ° + 1) ≠ 1, 2,  , B − 1, where ° = 0, 1,  Then the ¾ ¾ ¾ that can be represented as: equation has B solutions ( ) = 

−1

« 

,1+

: J ,(:

+ −1)

: +  , J  

 = 1, 2,  , B .

(1)

Here, «  , + , ( H ) is a Mittag-Leffler type special function defined by: -

½ «  , + ,- ( H ) = 1 + ÷   H  ,  =1 (2) » À −1 »  À −1  B ( ° + 0 ) + 1 1 w ¡  = = , [B ( ° + 0 ) + 1]  [B ( ° + 0 ) + B ]  B ( ° + 0 + 1) + 1  =0 w =0 ¡ ¡ ¡ where (ˆ ) is the gamma function, 0 is an arbitrary number, and > 0. w ¡ If ý ≥ 0, solutions (1) are linearly independent. Series expansions of (1) are convenient for small  . 2 b . Let B ≥ 2, ý < −B , and (B + ý )( ° + 1) ≠ −1, −2,  , −(B − 1), where ° = 0, 1,  Then ¾ ¾ question the equation in has B solutions ¾ that can be represented as: ( ) =  −1 «  ,−1−: J ,−1−(: + ) J   (−1)   : +   ,  = 1, 2,  , B , (3) where «  , + , ( H ) is the Mittag-Leffler type special function defined by (2). If ý ≤ −2B , solutions (3) are linearly independent. Series expansions of (3) are convenient for large  . 3 b . The transformation  = a −1 , = ¨ a 1−  leads to an equation of similar form: ¨ (g  ) =  (−1)  +1 a −2  −: ¨ . ^_ Reference: M. Saigo and A. A. Kilbas (2000).

Page 690

691

5.1. LINEAR EQUATIONS

5.

6.

ï 2 ÷ ñ ó( ÷

)

ïƒ÷ ñ ó(2÷

)

= ôñ . The transformation  = a ¨ (g  ) = (−1)   ¨ .

ôñ

= Solution:

−1

= ¨ a

,

1−



leads to a constant coefficient linear equation:

.

½ V ´  J 2 = L   =1

1

®

 2ý  £   + L  2 | ;   2ý  £   W ,

where ®  ( H ) and | ;  (H ) are the modified Bessel functions; ý 1 , ý 2 ,  , ý  are roots of the equation ý  = £  . 7.

ï 3÷ ñ ó(2÷

)

=

ôñ

.

The transformation  = a 8.

ïƒ÷ +1 ) 2 ñ ó(2÷

+1)

=

Solution:

ôñ

−1

=¨ a

,

1−2

 leads to an equation of the form 5.1.2.6: a  ¨ (2 g  )= ¨ .

.

2 J ½ L  Vø £   + ™ ø  +1 J 2  2ý  £   W , −  −1 J 2  2 ý   =0 where ø + ( H ) are the Bessel functions; ý 0 , ý 1 ,   , ý 2  are roots of the equation ý 2  ™ 2 = −1.



(2 +1) 4

=

9.

10.

11.

ï 3÷ +3 ) 2 ñ ó(2÷

= ôñ . The transformation  = a a  +1 J 2 ¨ (2 g  +1) = −  ¨ .

ï

,

= ¨ a

−2

 leads to a linear equation of the form 5.1.2.8:

−1

,

= ¨ a

1−

 leads to a linear equation of the form 5.1.2.3:

ñ ó÷

12.

13.

(ô

+1 ( )

= (ö

ï

+ ý )ñ . (˜ + The transformation ˆ = M , ¨ =  +  (  +  )  ¨ (Š  ) = ¢ −  ˆ ¨ , where ¢ = ˜( −  .

(ô ï

14.

−1

)

+ õ )2 ÷

(ô

= − ™ ;

+1)

= ôñ + õ ïƒ÷ . The transformation  = a ¨ (g  ) = (−1)  ( a ¨ +  ).

ï 2÷ +1 ñ ó(÷

+1

+ õ )÷ (ö ï

+ ý ) ÷ ñ ó( ÷

õï

ñ ó( ÷

)

leads to an equation of the form 5.1.2.3:

M

üIñ

=

−1

.

 +  ¨ 1 b . The transformation ˆ = ln , = leads to a constant coefficient linear (˜ + ( (˜ + )  −1 M M equation.   +  ¨ 2 b . The transformation = , = leads to the Euler equation 5.1.2.39: (˜ + ( (˜ + )  −1 M ¨ (û  ) = , ¢ −  ¨ , where ¢ =  −M ˜( . M

ï

2

+

+ ö )÷

)

=

üIñ

.

The transformation ˆ = X linear equation.



(ô

)

ï

+ õ )÷ (ö

ï

+ ý )3 ÷

ñ ó(2÷

=

2

M



+ ' + (

üIñ

, ¨ = ( 

2

+ ' + ( )



1− 2

leads to a constant coefficient

.

 +  ¨ The transformation ˆ = , = (˜ + ( (˜ + )2  M M  ) ¨ ˆ ¨ (2 −2  Š =, ¢ , where ¢ =  − ˜( . M

−1

leads to an equation of the form 5.1.2.6:

Page 691

692 15.

HIGHER-ORDER DIFFERENTIAL EQUATIONS + õ )÷

ï

(ô

+ ý )3 ÷

) (ö ï

+1 2

) ñ ó ÷

+3 2 (2 +1)

=

üIñ

.

 +  ¨ , = (˜ + ( (˜ + )2  M M  +1) ¨ ˆ +1 J 2 ¨ (2 −2  −1 Š =, ¢ , where ¢ =  − ˜( . M

The transformation ˆ =

leads to an equation of the form 5.1.2.8:

5.1.2-2. Equations of the form   ( ) (  ) +  1 ( )
 +  0 ( ) = ( ). 16. 17. 18.

19. 20.

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

) + ô ï +1 ñó ò

ñ ó( ÷

)

ñ ó( ÷

)

+ ô ï ñó ò + ôƒü ï –1 ñ = 0. Integrating yields an (B − 1)st-order linear equation:

23.

+  

=L .

+ ô ( ü + ù ) ï ñ

−1)

+  

+1

H = 0.

= 0. = ¨ a 1−  leads to an equation of the form 5.1.2.16: The transformation  = a −1 , ¨ (g  ) + 'a Á ¨2g
+ @!a Á −1 ¨ = 0, where  =  (−1)  +1, @ = 1 − , − 2B .

+ ( ô ï + õ ) ï ñó ò – ô ï ñ = 0. Particular solution: 0 =  +  .

ï

+ õ ) ï

+ (ô

ñóò

ñ ó( ÷

)

ï

+ (ô

+ õ ) ï

ñ ó÷

ñóò

ñ ó( ÷

+ õ ) ï  ñ óò

ï ñ

= 0.

ï ñ

= 0.

= (  +  )2 .

– 3ô

= (  +  )3 .

0

– ô (ù – 1) ï ñ = 0. Particular solution: 0 = (  +  )  −1 . The substitution H = (  +  )
 −  (B − 1) leads to an (B − 1)st-order linear equation: H (  −1) + (  +  )  H = 0. ( )

+ (ô ï

– 2ô 0

Particular solution:

22.

−1)

+ ô ï +1 ñó ò – ô (ù – 1) ï ñ = 0. The substitution H =  
− (B − 1) leads to an (B − 1)st-order equation: H ( 

Particular solution: 21.

( 

+ ô ï +1 ñó ò – ôþ Particular solution: linear equation: )

ï ñ

0

= 0, þ = 1, 2, uuu , ù – 1. =  + . The substitution H =  
−

c  −+

−1

O

H ( + ) +   H = 0, öP

where

¡

leads to an (B − 1)st-order

c = M .  M

5.1.2-3. Other equations. 24.

25.

ñ ó(2÷

= ô ÷ ñ + õ ï  (ñ ó ò ò ó – ôñ ). This is a special case of equation 5.1.6.18 with  ( ) = '  . The substitution ¨ = 

 −    −2) +  ¨ (2  −4) + ) )) +   −1 ¨ = '  ¨ . leads to an (2B − 2)nd-order linear equation: ¨ (2

ñ ó( ÷

)

+ ô ÷ –1 ñ ó(÷ –1) + ­­­ + ô 1 ñó ò + ô 0 ñ = 0. Constant coefficient homogeneous linear equation. To solve this equation, determine the B roots of the characteristic equation: Œ  +   −1 Œ  −1 + ))) +  1 Œ +  0 = 0. If the roots Œ 1 , Œ 2 ,  , Œ  are all real and different, then the general solution of the original equation is: = L 1 exp( Œ 1  ) + L 2 exp( Œ 2  ) + ))) + L  exp( Œ   ). The general case, which involves the cases of multiple and/or complex roots of the characteristic equation, is discussed in Subsection 0.4.1. )

Page 692

693

5.1. LINEAR EQUATIONS

ñ ó( ÷

26.

)

ô ï  ñ ó(ú

+

Particular solution:

ñ ó( ÷

27.

)

ï 

+ (ô

0

÷ – ú ) ñ ó( ú

–õ

Particular solution:

ñ ó( ÷

28.

)

ôñ ó(÷

+

–1)

ï ñ ó( ÷

)

– ùþxñ

ó( ÷

õ ÷ )ñ

.

ú ï ñ

– ôƒõ





= 0.

ïƒú ñ

+ ôƒõ

ï ñ

= 0.

. = 0.

−  . 0 = 

+ô

–1)



)

õ ïƒú ñóò

+

+



= =

0

Particular solution: 29.

ú¹ï

– ( ôƒõ

)

ù

= 0,

Solution:

=  (+

= 2, 3, 4,



+1) −1

O

1−

uuu

 M 3P M

þ

,

+

(

1−

= 1, 2, 3,

uuu

 ¨ ),

where ¨ is the general solution of the constant coefficient linear equation: ¨ (  ) +  ¨ = 0. 30. 31.

ï ñ ó( ÷

)

ï ñ ó( ÷

)

ï ñ ó( ÷

)

+ ùñ

ó( ÷

–1)

+ ùñ

ó( ÷

–1)

=

ô ï ñ

+õ .

ô ï 2ñ

+õ . leads to an equation of the form 5.1.2.3: ¨ ( 

The substitution ¨ =  =

The substitution ¨ = 

32.

+ (ù – þ

– 1)ñ

ï ñ ó( ÷

)

+

ô ï  ñ ó(ú

ï ñ ó( ÷

)

=

÷ 7 –1 

=0

[( ô


Here, #  = 1, #

 0

–1)

ï

– (ô

)

Particular solution: 34.

ó( ÷

0

+1

+

ô ï  ñ óò

–


+



ï –1 ñ

– ôþ

+ . 0 = 

Particular solution: 33.

leads to a constant coefficient linear equation: ¨ ( 

ôþ ï –1 + ï

)ï +




+1 ]

ñ ó( 

= ¨ + .

=  ¨ +  .

= 0.

+ ù )ñ = 0.

=  .



)

)

)

.

= 0;  and #3Á are arbitrary numbers ( @ = 1, 2,  , B − 1).

 7 −1 #3Á Á =0

Œ Á . Let the roots Œ 1 , Œ 2 ,  , Œ  −1 of the algebraic equation  ( Œ ) = 0 be all different, and  ( ) ≠ 0. Then the solution is as follows:     
( ) S . = L 1   1 + L 2   2 + ))) + L  −1   n −1 + L     −  Q  ( ) Denote  ( Œ ) =

35.

7÷



=0

(ô



ï

+

õ  ) ñ ó( 

)

+1

= 0.

The Laplace equation. Particular solutions:

where

.

(a ) =

:ž¼  =X ¼ 9 7

Á

7 =0

Áa Á , Æ (a ) =

Á

=0

.

1 exp a + X (a ) Q

Æ (a ) . (a ) M Ýa S M a ,

Áa Á ; ƒ  and ý  are found from the condition

exp O a + X

:ž¼ Æ (a ) . (a ) M a P û ¼ = 0. û9 û

In many cases, the path of integration has to be chosen on the complex plane.

Page 693

694 36. 37. 38. 39.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ ó( ÷

)

+ 2ù ï ñ ó(÷ –1) + ù (ù – 1)ñ ó(÷ –2) = ô ï 2 ñ + õ . The substitution ¨ =  2 leads to a constant coefficient linear equation: ¨ ( 

ï 2 ñ ó( ÷

)

+ 2ù ï ñ ó(÷ –1) + ù (ù – 1)ñ ó(÷ –2) = ô ï 3 ñ + õ . The substitution ¨ =  2 leads to an equation of the form 5.1.2.3: ¨ ( 

ï (ï

+ þ ) ñ ó( ÷ ) + ï ( ô Particular solution:

ô ÷ ƒï ÷ ñ ó(÷

ï

0

– ï – ù )ñ  =  .

ó( ú

)

– ô (ï + þ

) ï

ñ

)

= ¨ + .

)

=  ¨ +  .

= 0.

+ ô ÷ –1 ïƒ÷ –1 ñ ó(÷ –1) + ­­­ + ô 1 ï ñó ò + ô 0 ñ = 0. Euler equation. If all roots Œ  ( , = 1, 2,   , B ) of the algebraic equation )

½  Á

ÁžŒ ( Œ − 1)  ( Œ − @ + 1) = − 

0

=1

are different, the general solution of the original differential equation is given by: =L

1|

 |  1 + L 2 | |  2 + ))) + L  | | Dn .

In the general case, the substitution a = ln | | leads to a constant coefficient linear equation of the form 5.1.2.25:

½

40. 41. 42. 43. 44. 45. 46.

Á

ï 2÷ +1 ñ ó(÷

)

ï 2÷ +1 ñ ó(÷

)

Ážc ( c − 1)  ( c − @ + 1) = − 

0

,

where

=1

c = M .  M

+ ù ï 2÷ ñ ó(÷ –1) = ô ï ñ . The substitution ¨ =  leads to an equation of the form 5.1.2.5:  2  ¨ (  + ù ï 2÷ ñ ó(÷ –1) = ôñ . The substitution ¨ =  leads to an equation of the form 5.1.2.10:  2 

ïƒ÷ ñ ó(2÷

+ 2ù ïƒ÷ –1 ñ ó(2÷ The substitution ¨ =  )

ï 3÷ ñ ó(2÷

+1

¨ (  ) =  ¨ .

= ôñ .   leads to an equation of the form 5.1.2.6:   ¨ (2

–1)

+ 2ù ï 3÷ –1 ñ ó(2÷ –1) = ôñ .   The substitution ¨ =  leads to an equation of the form 5.1.2.7:  3  ¨ (2

ï ÷ +1 ñ ó(2÷

7

+1)

ï 3÷ +3 ) 2 ñ ó(2÷

)

= ¨ .

J ¨ (2 

+1 2

+ (2ù + 1) ï 3÷ +1 ) 2 ñ ó(2÷ ) = ôñ . The substitution ¨ =  leads to an equation of the form 5.1.2.9:  3 

+1)

= ¨ .

+1)

J ¨ (2 

+3 2

ï )ñ ó(÷ ) + g ÷ –2 (ï )ñ ó(÷ –1) + ­­­ + g 1 (ï )ñ óò ò ó . Here, the Á ( ) are polynomials of degree ≤ @ , as:

= ¨ .

)

)

ï ñ . + (2ù + 1) ï ÷ ñ ó(2÷ ) = ô The substitution ¨ =  leads to an equation of the form 5.1.2.8:  

g ÷

= ¨ .

)

+1)

= ¨ .

+ ( ô 1 ï + õ 1 )ñ ó ò – þ ô 1 ñ = 0. is a positive integer,  1 ≠ 0. ¡ A particular solution of this equation is the polynomial of degree that can be written

–1 (

½+

. 1  + O− [ ®´ − + −1 (  0 =  1P  =0  Á +1 Á with @ ≠ −1. where c = M , ®´ =  @ +1 ^_ M Reference: E. Kamke (1977).

−1

. c  + ))) + 1 c

¡

2

+  1 c )]   + ,

Page 694

695

5.1. LINEAR EQUATIONS

47.

[ ô ÷ ïƒ÷ + g ÷ –1 ( ï )]ñ ó(÷ ) + ­­­ + [ ô 1 ï + g 0 ( ï )]ñó ò + . Here, the Á ( ) are polynomials of degree ≤ @ . Assume that for some integer ≥ 0:

= 0.

¡

½ Á

ô 0ñ

L + Á @ ! Á = 0,

=0

where

! L +Á = ¡ , @ ! ( − @ )! ¡

and is the least of the numbers satisfying this condition. Then there exists a solution in the ¡ form of a polynomial of degree such that no polynomial of a smaller degree satisfies the ¡ original equation.

^_ 48.

Reference: E. Kamke (1977).

V ï g ( S ) – i ( S )WUñ = 0,

. Here,

S

. =

≡

ï ý ý ï

.

( H ) and Æ = Æ ( H ) are arbitrary polynomials of degree  and  , respectively. . Suppose Æ ( H + 1) = ( H + 1) Æ 1( H + 1), where the polynomial Æ 1 ( H + 1) is such that ( H ) and Æ 1 ( H + 1) do not have common factors. Then the original equation admits a formal solution in the power series form: 0

^_

½

= ÷



=0

#    ,

where

.

#  +1 (B ) = . #  Æ (B + 1)

Reference: H. Bateman and A. Erd´elyi (1953, Vol. 1).

5.1.3. Equations Containing Exponential and Hyperbolic Functions 5.1.3-1. Equations with exponential functions.

ó ñóò

1.

ñ ó( ÷

)

+ ( ô ï + õ )  Particular solution:

2.

ñ ó( ÷

)

ï

+ (ô

ó

 ñ óò

+ õ )

Particular solution: 3.

ñ ó( ÷

)

ï

+ (ô

ó ñóò

+ õ ) 

Particular solution: 4.

5. 6.

7.

ñ ó( ÷

)

ñ ó( ÷

)

ó

– ô ñ = 0. 0 =  +  . – 2 ô 0

= 0. 2

= (  +  ) . – 3 ô

0

ó

ó

 ñ

óñ

= 0. 3

= (  +  ) .

ó

+ ( ô ï + õ )   ñ ó ò – (ù – 1) ô  ñ = 0. Particular solution: 0 = (  +  )  −1 . The substitution H = (  +  )
 −  (B − 1) leads to  an (B − 1)st-order linear equation: H (  −1) + (  +  )  H = 0.

ó

ó

+ ô ï  ñó ò – ôþ ñ = 0, Particular solution: 0 =  + .

ñ ó(2÷

þ

= 1, 2,

uuu , ù

– 1.

ó

= ô ÷ ñ + õ (ñó ò ò ó – ôñ ).  This is a special case of equation 5.1.6.18 with  ( ) =    . The substitution ¨ = 

 −     −2) +  ¨ (2  −4) + ) )) +   −1 ¨ =   ¨ . leads to an (2B − 2)nd-order linear equation: ¨ (2

ñ ó( ÷

)

)

+ ( ô



ó

–õ

÷ – ú ) ñ ó( ú

Particular solution:

0

=

)





– ôƒõ

ú óñ

= 0.

.

Page 695

696 8. 9.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó( ÷

ó ñóò

+ ôñ ó(÷ –1) + õ Particular solution: )

ñ ó(÷ ) + ô  ó ñ ó(ú

)

0

– ( ôƒõ

Particular solution: 10.

ñ ó( ÷

)

÷ 7 –1

=



=0

ó

 +1  

(


0

+ ôƒõ  =  − .

ú ó 



=

+

óñ

= 0.

õ ÷ )ñ

= 0.

.

õ
 +1 –
 )ñ ó( ) .

+

Here, #  = 1, # 0 = 0;  and #  are arbitrary numbers ( , = 1, 2,  , B − 1). Ù , where the “ + are roots of the polynomial equation Particular solutions: + =  

 7 −1 

11. 12. 13. 14.

=0

# 

ï ñ ó( ÷

)

ï ñ ó( ÷

)

+1

“  = 0.

ó

+ ô ï   ñ ó( ú Particular solution:

– [ô (ï + þ  0 =   .

)

ó( ÷

+ (ù – þ – 1)ñ Particular solution:

–1)

+

ó

+ þ )ñ ó(÷ ) + ï ( ô – ï – ù )ñ  Particular solution: 0 =   .

ó

( ô ïƒú + õ + ö )ñ ó(÷ Particular solution:

)

ó

+ ï + ù ]ñ = 0.

ô ï  ó ñóò

+ . 0 = 

ï (ï

ó 

)

ó( ú

)

– ô (ï + þ

ó

= õ ñ , þ = 0, 1,  + +  +( . 0 = 

15.

( ô ïƒú  + õ )ñ ó(÷ ) = (–1)÷ õÿñ , þ = 0, 1, + − Particular solution: 0 =    +  .

16.

ODô

ó

+

÷ 7 –1 ï  (÷ õ Pñó 

)

=

=0

Particular solution:

óñ

– ôþ

0

ô ó ñ 

= 

= 0. ) 

uuu , ù uuu , ù

óñ

= 0.

– 1. – 1.

.

+

7

−1



=0

  .

5.1.3-2. Equations with hyperbolic functions. 17. 18.

ñ ó(2÷

)

=

ô ÷ ñ

+ õ sinh  (

ñ ó( ÷

)

+

ô

sinh 

ï ñ ó( ú

ñ ó( ÷

)

+ ( ô sinh 

ï

–

Particular solution: 20. 21.

– ôñ ).

ú

ï

This is a special case of equation 5.1.6.18 with  ( ) =  sinh  ( Π ). Particular solution:

19.

ï )(ñ óò ò ó

– ( ôƒõ

)

0

=





õ ÷ – ú ) ñ ó( ú 0

=





sinh  . )

– ôƒõ

ú

+

õ ÷ )ñ

.

)

ï ñ ó( ÷

+ ô ï sinh  ï Particular solution: )

ñ ó( ú

0

– [ô (ï + þ  =  . )

ï ñ

sinh 

+ ( ô ï + õ ) sinh ú ( ï )ñ ó ò – ô sinhú ( Particular solution: 0 =  +  .

ñ ó( ÷

= 0.

) sinh 

ï )ñ ï

= 0. = 0.

+ ï + ù ]ñ = 0.

Page 696

697

5.1. LINEAR EQUATIONS

22.

23.

ñ ó(2÷

)

ô ÷ ñ

=

ñ ó( ÷

)

cosh 

ô

+

ï ñ ó( ú

ñ ó( ÷

)

ï

+ ( ô cosh 

–

ñ ó( ÷

)

26.

ï ñ ó( ÷

)

ô ï

+

cosh 

27.

28.

29.

ñ ó(2÷

)

ñ ó(2÷

)

ï ñ ó( ú

33.

34.

= 0.

ï )ñ óò

– ô coshú (

ï )ñ

= 0.

– [ô (ï + þ

)

ï

) cosh 



+ ï + ù ]ñ = 0.

=

ñ

+ ô (ñó ò sinh ï – ñ cosh ï ).

The substitution ¨ = 
cosh  − sinh  leads to an (2B − 1)st-order linear equation. The substitution ¨ = 
sinh  − cosh  leads to an (2B − 1)st-order linear equation.

ñ ó( ÷

)

tanh 

ô

+

ï ñ ó( ú

ñ ó( ÷

)

ï

+ ( ô tanh 

ñ ó( ÷

)

ï ñ ó( ÷

)

+

ô ï

0

ñ ó( ÷

)

)

+

ô

coth

ï ñ ó( ú



)

ñ ó( ÷

)

+ (ô

ï

ï ñ ó( ÷

)

+

ô ï

0

coth

ï ñ ó( ú

Particular solution:

0

= 0.

ï )ñ

= 0.

ï

) tanh 

+

ï

+ ù ]ñ = 0.

=  .

ú

– ( ôƒõ





=

=

+ õ ) coth ú ( 0

– ô tanhú (





ï

coth

÷ – ú ) ñ ó( ú

–õ

ï ñ

tanh

– [ô (ï + þ

ñ ó( ÷

ï

ú

– ôƒõ

)

ï )ñ óò

0

+ ( ô coth

= 0.

.

Particular solution: )

õ ÷ )ñ

=  +  .

0

Particular solution:

+

.



=

0

ï

tanh 

õ ÷ – ú ) ñ ó( ú

ï ñ ó( ú

tanh





=

+ õ ) tanh ú (

ï

+ (ô

ú

– ( ôƒõ

) 0

–

Particular solution: 36.

ï ñ

cosh 

.

+ ô (ñó ò cosh ï – ñ sinh ï ).

Particular solution: 35.

ú

– ôƒõ

ñ

Particular solution: 32.



)

=

Particular solution: 31.

= 0.

=  .

0

Particular solution: 30.

õ ÷ )ñ

=  +  .

0

Particular solution:

+

.



=

ï

cosh

õ ÷ – ú ) ñ ó( ú 0

Particular solution:





=

+ õ ) cosh ú (

ï

+ (ô

ú

– ( ôƒõ

) 0

Particular solution: 25.

– ôñ ).

This is a special case of equation 5.1.6.18 with  ( ) =  cosh  ( Π ).

Particular solution: 24.

ï )(ñ óò ò ó

+ õ cosh  (

+

õ ÷ )ñ

= 0.

.



)

– ôƒõ

ú

ï ñ

= 0.

ï )ñ

= 0.

coth

.

ï ) ñ óò

– ô cothú (

=  +  . )

– [ô (ï + þ



) coth 

ï

+ ï + ù ]ñ = 0.

=  .

Page 697

698

HIGHER-ORDER DIFFERENTIAL EQUATIONS

5.1.4. Equations Containing Logarithmic Functions 1. 2.

ñ ó(2÷

= ô ÷ ñ + õ ln ï (ñ ó ò ò ó – ôñ ). This is a special case of equation 5.1.6.18 with  ( ) =  ln  .

ñ ó( ÷

)

)

ï ñ ó( ú

ln 

ô

+

Particular solution: 3.

ñ ó( ÷

)

+ ( ô ln 

ï

–õ

– ( ôƒõ

)

0

5. 6.

ñ ó( ÷

)

ñ ó( ÷

)

0

+ ( ô ï + õ ) ln  ( Particular solution:

ñ ó( ÷

)

ï

+ õ ) ln  (

+ (ô

ñ ó( ÷

)

ï

+ õ ) ln  (

+ (ô

ñ ó( ÷

)

ï

+ õ ) ln  (

+ (ô

10.

ñ ó( ÷

0

.

ú

– ôƒõ

)

õ ÷ )ñ

= 0.

ï ñ

ln 

.

= 0.

– 3 ô ln  (

ï )ñ

= 0.

3

= (  +  ) .

– ô (ù – 1) ln  (

 0 = (  +  )

+ ô ï ln  ( ï )ñó ò – ôþ ln  ( Particular solution: 0 =  + . )

ï ñ ó( ÷

+ ô ï ln  ( ï )ñ Particular solution: )

ó( ú

0

= 0.

ï )ñ

= (  +  ) .

ï )ñ ó ò

= 0.

– 2 ô ln  ( 2

ï )ñ óò

= 0.

ï )ñ

ï )ñ

– ô ln  ( =   + .

ï )ñ óò

0

Particular solution:

9.

=

+

ï )ñ óò

0

Particular solution: 8.





ï

+ ôñ ó(÷ –1) + õ ln  ( ï )ñó ò + ôƒõ ln  (  Particular solution: 0 =  −  .

Particular solution: 7.



ln 



=

÷ – ú ) ñ ó( ú

Particular solution:

4.

ú

−1

þ

= 0,

ï

) ln  (

)

= 0.

.

ï )ñ

– [ô (ï + þ  =  .

ï )ñ

= 1, 2,

uuu , ù

– 1.

) + ï + ù ]ñ = 0.

5.1.5. Equations Containing Trigonometric Functions 5.1.5-1. Equations with sine and cosine. 1.

ñ ó( ÷

)

+

ô

sin ,ï

ñ ó( ú

Particular solution: 2.

ñ ó( ÷

4. 5.

ú

– ( ôƒõ 0





=



sin ,ï +



ú

+ ôñ ó(÷ –1) + õ sin ú ( ï )ñ ó ò +  Particular solution: 0 =  −  .

ôƒõ

+ ( ô sin 

ï

–

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

õ ÷ – ú ) ñ ó( ú

. – ôƒõ

)

Particular solution: 3.

)

0

=

)

.

õ ÷ )ñ sin 

ï

+ õ ) sin ú (

Particular solution:

0

ï )ñ óò

– 2 ô sin ú ( 2

= (  +  ) .

ï ñ

sinú (

+ ( ô ï + õ ) sin ú ( ï )ñó ò – ô sinú ( Particular solution: 0 =  +  . + (ô

= 0.

ï )ñ ï )ñ

= 0.

ï )ñ

= 0.

= 0. = 0.

Page 698

699

5.1. LINEAR EQUATIONS

6.

ñ ó( ÷

)

ï

+ õ ) sin ú (

+ (ô

Particular solution: 7. 8.

ñ ó(2÷

)

=

ô ÷ ñ

+

ô ï

0

ï ñ ó( ÷

)

ïƒú

(ô

ó( ÷

0 )



ñ ó( ÷

)

+

cos ,ï

ô

ñ ó( ú

ñ ó( ÷

)

ï

+ ( ô cos 

–

ñ ó( ÷

)

+

)

ôñ ó(÷ ï

–1)

0

ñ ó( ÷

)

ï

0

Particular solution: 16.

ñ ó( ÷

)

+ (ô

ï

0

+ õ ) cosú (

Particular solution: 17. 18.

ñ ó(2÷

)

=

ô ÷ ñ

+

ô ï

0

þ

ú



=

ï

sin 

7

−1



=0

® ù ñ

= 0, 1,

uuu , ù

– 1.

.

  .

cos )ï +



1 2

+

+ ù ]ñ = 0.

õ ÷ )ñ

= 0.

. )



– ôƒõ

ú

+

ôƒõ

ï ñ

cos

= 0.

.

ï )ñ óò

cosú (

ï )ñ óò

– ô cosú (

ï )ñ

ï )ñ óò

– 2 ô cosú (

=  +  .

2

= (  +  ) .

ï )ñ óò

– 3 ô cosú ( 3

ï )ñ

= 0.

= 0.

ï )ñ

= 0.

ï )ñ

= 0.

= (  +  ) .

ï )(ñ óò ò ó

– ôñ ).

This is a special case of equation 5.1.6.18 with  ( ) =  cos  ( Π ).

ï ñ ó( ÷

)

ï ) ñ ó( ú

cos (

ï ú

(ô

+ õ cos ï )ñ

ó( ÷

Particular solution: 20.

ô

=  sin  +

+ õ cos  (

Particular solution: 19.

=

−  . 0 = 

+ õ ) cosú (

+ (ô

)

+ õ cosú (

Particular solution: 15.

ï

sin 

– ( ôƒõ

+ õ ) cosú (

+ (ô

õ

õ ÷ – ú ) ñ ó( ú

Particular solution:

ñ ó( ÷

=  .

0 =  

Particular solution:

ï

)+

+ 12 ® ù  ñ , +   +  sin  . 0 = 

=

0 )

ï

) sin  (



=0

Particular solution:

14.

= 0.

– ôñ ).

– [ô (ï + þ

)

÷ 7 –1 ï  (÷ õ Pñó

Particular solution:

13.

3

ï )(ñ óò ò ó

ï ) ñ ó( ú

sin  (

+ õ sin ï )ñ

ODô sin ï +

10.

12.

ï )ñ

This is a special case of equation 5.1.6.18 with  ( ) =  sin  ( Π ).

Particular solution:

11.

– 3 ô sin ú (

= (  +  ) .

+ õ sin  (

Particular solution: 9.

ï )ñ óò

O

ô

cos ï +

0 )

)

õ

) cos  (

ï

+ 12 ® ù  ñ , +   +  cos  . 0 = 

=

=0

Particular solution:



=  .

÷ 7 –1 ï  (÷ õ Pñó 

– [ô (ï + þ

0

cos 

)

=

ô

cos 

=  sin  +

ï

7

−1



=0

+

1 2

® ù ñ

) + ï + ù ]ñ = 0.

ï

þ

= 0, 1,

uuu , ù

– 1.

.

  .

Page 699

700 21. 22.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

)

= (–1)÷ ñ + ô (ñó ò sin ï – ñ cos ï ). The substitution ¨ = 
sin  − cos  leads to an (2B − 1)st-order linear equation.

ñ ó(2÷

)

= (–1)÷ ñ + ô (ñó ò cos ï + ñ sin ï ). The substitution ¨ = 
cos  + sin  leads to an (2B − 1)st-order linear equation.

5.1.5-2. Equations with tangent and cotangent. 23.

ñ ó( ÷

)

tan 

ô

+

ï ñ ó( ú

Particular solution: 24.

ñ ó( ÷

)

0

26. 27.

30.

ôƒõ

tanú (

)

ñ ó( ÷

)

ñ ó( ÷

)

0

33. 34.

ï

+ õ ) tan ú (

+ (ô

ñ ó( ÷

)

0

+ õ ) tan ú (

ï

+ (ô

0

ñ ó( ÷

)

+

ô

cot)ï

ñ ó( ú

ñ ó( ÷

)

+ ( ô cot,ï –

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

)

ï )ñ

ï )ñ ï )ñ

= 0.

ï )ñ ó ò

– 3 ô tanú (

ï )ñ

= 0.

2

= (  +  ) . 3

= (  +  ) .

ú

cot,ï +

=





õ ÷ – ú ) ñ ó( ú 0 =  

.



)

– ôƒõ

ú

+ õ ) cotú (

ñ ó( ÷

)

+ (ô

ï

ï ñ ó( ÷

0

+ õ ) cotú (

0

õ ÷ )ñ

cot)ï

ñ

ï )ñ

ï )ñ

= 0.

= 0.

– 2 ô cotú (

ï )ñ

= 0.

ï )ñ óò

– 3 ô cotú (

ï )ñ

= 0.

= (  +  ) . 3

= (  +  ) .

+ ù ]ñ = 0.

= 0.

ï )ñ ó ò

2

ï

)+

= 0.

.

+ ô ï cot ( ï )ñ ó(ú ) – [ ô ( ï + þ  Particular solution: 0 =   . )

ï

) tan  (

+ ( ô ï + õ ) cotú ( ï )ñó ò – ô cotú ( Particular solution: 0 =  +  .

ï

= 0.

= 0.

+ ôñ ó(÷ –1) + õ cotú ( ï )ñó ò + ôƒõ cotú (  Particular solution: 0 =  −  .

+ (ô

= 0.

– 2 ô tanú (

– ( ôƒõ 0

ñ

ï )ñ óò

)

Particular solution:

36.

.

+ ô ï tan ( ï )ñ ó(ú ) – [ ô ( ï + þ  Particular solution: 0 =   .

ï ñ ó( ÷

Particular solution: 35.

=

)

+ ( ô ï + õ ) tan ú ( ï )ñó ò – ô tanú ( Particular solution: 0 =  +  .

Particular solution:

32.

= 0.

+ ôñ ó(÷ –1) + õ tanú ( ï )ñó ò +  Particular solution: 0 =  −  .

ñ ó( ÷

Particular solution: 31.

õ ÷ )ñ

tan ,ï

Particular solution:

29.

.

+

ú

Particular solution: 28.





÷ – ú ) ñ ó( ú

Particular solution: 25.



=

ï

tan



– ôƒõ

)

+ ( ô tan)ï – õ

ú

– ( ôƒõ

) cot  (

ï

)+

ï

+ ù ]ñ = 0.

Page 700

701

5.1. LINEAR EQUATIONS

5.1.6. Equations Containing Arbitrary Functions 5.1.6-1. Equations of the form   ( ) (  ) +  1 ( ) 
+  0 ( ) = ( ). 1.

ñ ó( ÷

)

= ! ( ï ).

=

Solution: 2.

ñ ó( ÷

)

= ! ( ï )ñ .

½

−1

Á

=0

The transformation  = a (−1)  a −2   (1 a )¨ . 3.

4. 5.

7.

)

!„ñ ó ÷

= (ö

( )

–! ó÷ ñ

= 0, Particular solution:

„! ñ ó ÷

(2 +1)

+! ó ÷

(2 +1)

ñ ó( ÷

)

ñ ó( ÷

)

½ 2 

ñ

10.

0

= " ( ï ),

12.

!

(−1)    (2  − 

=0



=

is an arbitrary number.

0

leads to an equation of similar form: ¨ (g 

)

=

leads to a simpler equation: ¨ (Š 

)

=

( (˜ + ) 

M

−1

= ! (ï ).

  = X ( )  + L . M

) ( )

+ ( ô ï + õ ) ! ( ï )ñ„ó ò – ô! ( ï )ñ = 0. Particular solution: 0 =    + .

+ õ ) ! ( ï )ñ„ó ò – 2 ô! ( ï )ñ = 0.

ï

+ (ô

ñ ó( ÷

)

+ (ô

ï

ñ ó( ÷

)

ñ ó( ÷

)

0

= (  +  )2 .

0

= (  +  )3 .

+ õ ) ! ( ï )ñ„ó ò – 3 ô! ( ï )ñ = 0.

+ ( ô ï + õ ) ! ( ï )ñ„ó ò – (ù – 1) ô! (ï )ñ = 0. Particular solution: 0 = (  +  )  −1 . The substitution H = (  +  )
 −  (B − 1) leads to an (B − 1)st-order linear equation: H (  −1) + (  +  )  ( )H = 0. + ï ! ( ï )ñó ò – þ¯! ( ï )ñ = 0, þ = 1, 2, uuu , + Particular solution: 0 =  . The substitution H =  equation:

c  −+

11.

1−

M

= ! ( ï ). =  ( ).

!

( )

Particular solution:

9.

= ¨ a

,

ô ï öï

! O + ý )–2÷ I

ï

Particular solution: 8.

−1

+õ ñ . +ý P  +  ¨ The transformation ˆ = , (˜ + ¨ − M ¢  (ˆ ) , where ¢ =  − ˜( .

ñ ó( ÷

First integral: 6.

 ( − a )  −1  (a ) a , where  (B − 1)! M 0

LlÁ Á + X 

ñ ó( ÷

−1

O

H ( + ) +  ( ) H = 0, öP

ù

– 1.


−

where

¡

leads to an (B − 1)st-order

c = M .  M

+ ! ( ï )ñó ò + " ( ï )ñ + % ( ï ) = 0. The transformation  = a −1 , = ¨ a 1−  leads to an equation of similar form:

ñ ó( ÷

)

¨ (g  ) + (−1)  a −2  ¦ −a 2  (1 a )¨ g
+ V (B − 1)aª (1 a ) + (1 a )W ¨ + a 

)

+ ! ( ï )ñó ò +

!„ó ò (ï )ñ

= " ( ï ).

Integrating yields an (B − 1)st-order linear equation:



( −1)

−1

% (1 a ) § = 0.

+  ( ) = X ( )  + L .

M

Page 701

702 13.

14.

15.

16.

17.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

)

=

ñ

+ ! ( ï )(ñó ò cosh ï – ñ sinh ï ).

=

ñ

+ ! ( ï )(ñó ò sinh ï – ñ cosh ï ).


The substitution ¨ =  cosh  − sinh  leads to an (2B − 1)st-order linear equation. The substitution ¨ = 
sinh  − cosh  leads to an (2B − 1)st-order linear equation. = (–1)÷

ñ

+ ! ( ï )(ñ

= (–1)÷

ñ

+ ! ( ï )(ñó ò cos ï + ñ sin ï ).

sin ï – ñ cos ï ).

óò

The substitution ¨ = 
sin  − cos  leads to an (2B − 1)st-order linear equation. The substitution ¨ = 
cos  + sin  leads to an (2B − 1)st-order linear equation.

ñ ó( ÷

)

r (ó ÷ ) r ñ

=

r óò r ñ P

+ ! ( ï ) OUñó ò –



Y  The substitution ¨ =  −

r

,

=

r (ï

).

leads to an (B − 1)st-order linear equation.

Y

5.1.6-2. Other equations. 18.

ñ ó(2÷

)

=

ô ÷ ñ

+ ! ( ï )(ñó ò ò ó – ôñ ).

The substitution ¨ = 

 − 

leads to an (2B − 2)nd-order linear equation:

¨ (2  19.

ñ ó( ÷

)

+ ! ( ï )( ï

2

– 2ï

ñ óò ò ó

ñ óò

  +  ¨ (2

−2)

−4)

+ ))) +  

−1

¨ =  ( )¨ .

+ 2ñ ) = 0.

Particular solutions: 1 =  , 2 =  2 . The substitution H =  linear equation of the (B − 2)nd-order. 20.

21.

22.

ñ ó( ÷

+2)

ñ ó(2÷

)

+ ! ( ï )[ ï

2

ñ óò ò ó

24.

=

ô 2ñ

+ ! ( ï )[ñ

ó( ÷

)

ôñ

+

The substitution ¨ = (  ) + 

ñ ó( ÷

)

+ ! ( ï )ñ

ó( ú

)

– [ô

ñ ó( ÷

)

+ (! – ô

÷ – ú ) ñ ó( ú

+ ù (ù + 1)ñ ] = 0.

÷

].

leads to an B th-order linear equation: ¨ ( 

+ô

ú ! (ï )]ñ

–ô

ú !„ñ

 =  .

0 )

Particular solution:

ñ ó( ÷

+ ô!„ñ = 0,

)

+

ôñ ó(÷

–1)

+

!„ñóò

ñ ó( ÷

)

+ ! ( ï )ñ

ó( ÷

–1)

−  . 0 = 

+ " ( ï )ñ

ó( ÷

The substitution ¨ ( ) = (  ( )¨ + % ( ) = 0.

–2) −2)

)

= [  (  ) +  ]¨ .

= 0.

!

= 0,

 . 0 = 

Particular solution: 25.



 − 2 
+ 2 leads to a

The substitution ¨ ( ) =  2 

 − 2B/ 
 + B (B + 1) leads to an B th-order linear equation: ¨ (  ) +  2  ( )¨ = 0.

Particular solution: 23.

ï ñ óò

– 2ù

2

!

= ! ( ï ).

= ! (ï ).

+ % ( ï ) = 0.

¨ 

 +  ( )2 ¨ 
+ leads to a second-order linear equation: u

Page 702

703

5.1. LINEAR EQUATIONS

26.

ñ ó( ÷

)

ô ÷ –1 ñ ó(÷

+

–1)

+

­­­

+ô

ñóò

1

+

= ! ( ï ).

ô 0ñ

Constant coefficient nonhomogeneous linear equation. The general solution of this equation is the sum of the general solution of the corresponding homogeneous equation (see 5.1.2.25) and any particular solution of the nonhomogeneous equation. If the roots Π1 , Π2 ,  , Π of the characteristic equation

Π + 

−1

Π

−1

+ ))) +  1 Π+ 

0

=0

are all different, the original equation has the general solution:

  ½ LlÁ   ¡ + . 
 (¡ Œ"Á ) X  ( ) −  ¡  M  Á =1 

½ 

=

Á

=1

(with complex roots, the real part should be taken). Paragraph 0.4.1-2 lists the forms of particular solutions corresponding to some special forms of the right-hand side function of the nonhomogeneous linear equation. 27.

ñ ó( ÷

+ ! (ï )

)

÷ 7 –1

Here, L +  =



=0

(–1) 

ü ! = ÷  –1 ï ÷ – –1 ñ ó(÷ – –1) = 0.

!

¡ are binomial coefficients. , ! ( − , )! ¡ Particular solutions: + =  + , where = 1, 2,  , B − 1. ¡  7 −1 The substitution H = (−1)  , ! L   −1   −  −1 (  −  −1) leads to a first-order linear equation:  =0 H 
+   −1  ( ) H = 0. Having solved this equation, we obtain an (B − 1)st-order linear equation of the form 5.1.6.34 for ( ). 28.

ñ ó( ÷

)

÷ 7 –1

=



(ô

=0

–ô

 +1 !

 ) ñ ó( 

Here,  =  ( );   = 1,  Particular solutions:

)

.

= 0;   are arbitrary numbers ( , = 1, 2,  , B − 1). ¼  ( , = 1, 2,  , B − 1), where the Œ are roots of the  =     7 −1  polynomial equation   +1 Œ = 0.  =0 29. 30.

ï ñ ó( ÷

)

ï ñ ó( ÷

)

+ ( ô + ù – 1)ñ

–1)

+

ï !„ñ ó(ú

– [( ï + þ

)

ï ñ ó( ÷

)

+ ùñ

ó( ÷

–1)

=

0

ï

1–2

÷ !

33.

ï 2 ñ ó( ÷

+2)

+

 ï ñ ó( ÷

)ñ

ó( ÷

+1)

+ ñ

ñ óò

+

ôñ

).

leads to an (B − 1)st-order linear equation: ¨ ( 

) ! + ï + ù ]ñ = 0,

!



=  .

The transformation a =  (−1)  [  (a )¨ + (a )]. 32.

= ! ( ï )( ï

The substitution ¨ =  
+  Particular solution:

31.

ó( ÷

0

(1 

ï )ñ

−1

, ¨

ó( ÷

)

÷ " (1  ï

+ï =

– –1



+ ! ( ï )[ ï

2−

2

= ! ( ï ).

−1)

=  ( )¨ .

).

 leads to an B th-order linear equation: ¨ (g 

ñóò ò ó

+ (  – 2ù ) ï

ñóò

+ ( – ù + ù

2

)

=

+ ù )ñ ] = 0.

The substitution ¨ ( ) =  2 

 + ( ƒ − 2B ) 
 + (ý − ƒ…B + B 2 + B ) leads to an B th-order linear equation: ¨ (  ) +  ( )¨ = 0.

ï (ï

+þ

)

+ ï ( ! – ï – ù )ñ

Particular solution:

0



=  .

ó( ú

)

– (ï + þ

) !„ñ = 0,

!

= ! ( ï ).

Page 703

704 34.

35. 36.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ïƒ÷ ñ ó(÷

)

+ õ ÷ –1 ïƒ÷ –1 ñ ó(÷ –1) + ­­­ + õ 1 ï ñó ò + õ 0 ñ = ! ( ï ). g Nonhomogeneous Euler equation. The substitution  =   (  ≠ 0) leads to a constant coefficient nonhomogeneous linear equation of the form 5.1.6.26.

ïƒ÷ ñ ó(÷

)

ïƒ÷ ñ ó(÷

)

+ (ù – þ – 1)ïw÷ –1 ñ ó(÷ Particular solution: 0 =  + .

ïƒú ñ ó(÷

37.

)

=

÷ 7 –1 ïƒú 

=0

)

(ô

[

polynomial equation

39.

– þ¯!„ñ = 0,

= ! ( ï ).

!

÷

– (ù ! =

D (

–ô

 +1 !

Here,  =  ( );   = 1,  Particular solutions:

38.

ï !„ñóò

+

+ þ ! = ú ! )ñ = 0. D + 1) w Here,  =  ( ), L   = are binomial coefficients, and (  ) is the gamma w B ! (  − B + 1) w function. Particular solution: 0 =   . +

ïƒú !„ñ ó(ú

–1)

7

−1



=0



)+ô



 

+1

Π = 0.

ï

cos ï ñ ó(÷ ) + cos ï ! ( ï )ñ ó(ú ) – V cos  Particular solution: 0 = cos  .

7÷



=2

!  ( ï ) ñ ó( 

)

= " ( ï )( ï

Particular solution: equation.

7÷

41.

 =ú

+1

!  ( ï ) ñ ó( 

)

7÷



=3

!  ( ï ) ñ ó( 

)

0

= " ( ï )( ï

ï

+

1 2

® ù 

+ ! ( ï ) sin 

+

1 2

® ù 

+ ! ( ï ) cos 

ï

+

ï

+

® þ  WUñ

= 0.

® þ  Wñ

= 0.

1 2

ñóò

– þxñ ),

þ

= 1, 2,

leads to an (B − 1)st-order linear

uuu , ù

=  + . The substitution ¨ ( ) =  
 − 2

1 2

– ñ ).

=  . The substitution ¨ ( ) =  
 −

= " ( ï )( ï

Particular solution: linear equation. 42.

0

ñóò

ñ ó(  ) .

= 0; and   are arbitrary numbers ( , = 1, 2,  , B − 1). ¡ ¼  ( , = 1, 2,  , B − 1), where the Œ are roots of the   = 

0

sin ï ñ ó(÷ ) + sin ï ! ( ï )ñ ó(ú ) – V sin  Particular solution: 0 = sin  .

40.

+1 ]

– 2ï

ñ óò ò ó

ñ óò

¡

– 1. leads to an (B − 1)st-order

+ 2ñ ).

Particular solutions: 1 =  , 2 =  2 . The substitution ¨ ( ) =  2 

 − 2 
 + 2 leads to an (B − 2)nd-order linear equation. 43.



7÷

=4

!  ( ï ) ñ ó( 

)

= " ( ï )( ï

3

ñóò ò ó ò ó

– 3ï

2

ñóò ò ó

+ 6ï

ñóò

– 6ñ ).

Particular solutions: 1 =  , 2 =  2 , 3 =  3 . The substitution ¨ ( ) =  3 


 − 3 2 

 + 6 
− 6 leads to an (B − 3)rd-order linear equation. 44.

7÷

 =ú

+1

!  ( ï ) ñ ó( 

)

+ " (ï )



7ú

=0

(–1) 

ü ! = ú  ïƒú – ñ ó(ú – ) = 0.

À ! À binomial ¡ are coefficients. , ! ( − , )! ¡ =  , where ° = 1, 2,  , Particular solutions:

Here, L +  =

¡

.

Page 704

705

5.2. NONLINEAR EQUATIONS

The substitution H =

7  =+ 7÷

45.



=0

+1

(!



  ( ) c  − + – ô!



+1 )

−1

7+ 

=0

(−1)  , ! L +   +

−



)

leads to an (B −

  − + H 
 + ( ) H = 0, where c =  . M M

¡

)th-order linear equation:

ñ ó( ) = 0.

Here,   =   ( ) ( , = 1, 2,  , B );    Particular solution: 0 =   . 46.

 ( + − 

7 ÷ ï [!  + (ü – þ )! 

+1 ]

=0

+1

≡

0

≡ 0.

+1

≡

0

≡ 0.

ñ ó( ) = 0.

Here,   =   ( ) ( , = 1, 2,  , B );   Particular solution: 0 =  + .

5.2. Nonlinear Equations 5.2.1. Equations Containing Power Functions 5.2.1-1. Fifth- and sixth-order equations. 1.

ñ ó(5) = ôññ óò ò ó ò ò ó ó

– ô (ñ

óò ò ó

)2 +

õï

+ö .

This is a special case of equation 5.2.6.1 with  ( ) =  +  . 2.

ññ ó(5) = ô ï

+õ .

This is a special case of equation 5.2.6.17 with B = 2 and  ( ) =  +  . 3.

ññ ó(5) = ôñóÈò ñóò ò ó ò ò ó ó

.

1 b . For  ≠ −1, integrating the equation two times, we arrive at a third-order autonomous equation: 



 − 12 (

 )2 = L 1  +1 + L 2 . The substitution ¨ ( ) = 12 (
 )2 leads to a second-order equation: ¨*¨

− 1 (¨
)2 = 1 L  +1 + 1 L . 4 2 1 2 2





For  = 1, this is a solvable equation of the form 2.8.1.53. 2 b . For  = −1, integrating the equation two times, we arrive at a third-order autonomous equation: 



 − 12 ( 

 )2 = L 1 ln | | + L 2 . 3 b . Particular solution: 4.

=L

1

3ññ

ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

= 0.

2ññ

ó(5) + 5ñóÈò ñóò ò ó ò ò ó ó

+ 5ñó ò ò óÈñó ò ò ó ò



3

+L

2



2

+L

3

 + L 4.

This is a special case of equation 5.2.1.3 with  = − 35 . Integrating the equation three times, we arrive at a second-order equation: 3!

 − 2(
 )2 = L 1  2 + L 2  + L 3 . The substitution = ¨ 3 leads to a solvable equation of the form 2.8.1.5: ¨p

 = 19 ( L 1  2 + L 2  + L 3 )¨ −5 .

5.

ó

= 0.

This is a special case of equation 5.2.6.4 with  = 52 and  ( ) = 0. Integrating the equation three times, we arrive at a second-order equation of the form 2.8.1.53: !

 − 14 (
 )2 = L 1  2 + L 2  + L 3.

Page 705

706

HIGHER-ORDER DIFFERENTIAL EQUATIONS

6.

+ (3 ô – 5)ñ„ó ò ò óÈñó ò ò ó ò ó = 0. Integrating the equation three times, we obtain a second-order equation: !

 + 12 (  −3)(
 )2 = L 1  2 + L 2  + L 3.

7.

ññ ó(5) + ôñóò ñóò ò ó ò ò ó ó

8. 9. 10.

ññ ó(5) + ôñóò ñóò ò ó ò ò ó ó

+ õÿñó ò ò óñó ò ò ó ò ó = 0. Integrating yields a fourth-order equation: !"



 + (  − 1)
 


 + 12 (1 −  +  )( 

 )2 = L . + 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô ï ÷ . This is a special case of equation 5.2.6.2 with  ( ) =   .

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ôñ ÷ . This is a special case of equation 5.2.6.3 with  ( ) = 

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

 .

ñ ó(6) =
ñ –7 ) 5 .

 + 7
 (6)  = 0. Multiplying by 7 J 5 and differentiating with respect to  , we obtain 5! (7) Having integrated this equation three times, we arrive at the chain of equations: 5! 5! 5!

 + 2 
(5)  − 2 

 



 + ( 


 )2 = 2 L 2 , (5)  − 3 




 + 

 


 = 2 L 2  + L 1 , 



 − 8 



 + 92 ( 

 )2 = L 2  2 + L 1  + L 0 , (6)

(1) (2) (3)

where L 0 , L 1 , and L 2 are arbitrary constants. Eliminating the highest derivatives from (1)–(3), with the aid of the original equation, we obtain a third-order equation that can be reduced to a second-order equation (see equation 5.2.1.12 with B = 3). 11.

+ 10(ñó ò ò ó ò ó )2 = ô ïƒ÷ . This is a special case of equation 5.2.6.6 with  ( ) =   .

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

  =  ( , ). ( )

5.2.1-2. Equations of the form 12.

ñ ó(2÷

)

=


ñ

1+2 1–2

÷

÷

.



2 +1 2 −1



Multiply both sides by

and differentiate with respect to  . As a result, we obtain

  (2B − 1)! (2

+1)


  + (2B + 1)  (2

)

= 0.

Three integrals containing arbitrary constants L 0 , L 1 , and L 2 are presented in 5.2.6.62, where one should let  ≡ 0. Eliminating the highest derivatives from those integrals and the original equation, one can always obtain an (2B − 3)rd-order equation. With the aid of the transformation

 M. ,

a =X

¨ =

.

1−2 2



,

where

.

=L

2



2

+L

1

 + L 0,

this equation can be reduced to the autonomous form 5.2.6.77. Therefore, the substitution H (¨ ) = ¨2g
finally leads to an (2B − 4)th-order equation with respect to H = H (¨ ). 13.

ñ ó(2÷

= ôñ  + õ , ü ≠ –1. This is a special case of equation 5.2.6.8. Integrating yields an (2B − 1)st-order equation: )

½

+

−1

(−1) =1

+

+ (

  +

) (2 −

)

+

1  2  (−1) ÑV (  ) W = − 2 , +1

+1

−

+L ,

where L is an arbitrary constant. Furthermore, the order of the obtained autonomous equation can be reduced by one by the substitution ¨ ( ) = 
 .

Page 706

707

5.2. NONLINEAR EQUATIONS

14. 15.

ñ ó( ÷

)

= ô ï –÷ ñ ú . This is a special case of equation 5.2.6.11 with  ( ) =  "+ .

ñ ó( ÷

)

=ô ï  ñ ú . 1 b . The transformation  = a −1 , ¨ (g  ) = (−1)  #*a −  −(  −1) + −  −1 ¨2+ . 2 b . The transformation ˆ =   + 

16.

ññ ó(2÷

+

= a −1

1−

 ¨ (a ) leads to an equation of the same form:

, H =


 leads to an (B − 1)st-order equation.

= ô ï + õ . This is a special case of equation 5.2.6.17 with  ( ) =   +  .

17.

= ïƒú –÷¿ú –÷ –1 ( ôñ + õ ïƒ÷ –1 )ú . This is a special case of equation 5.2.6.13 with  (¨ ) = (  ¨ +  ) + .

18.

ñ ó(2÷

19.

20. 21.

ñ ó( ÷

+1)

)

)

=

ï ú

–2

÷¿ú

2

÷

–2 –1

Oiôñ + õ

ñ ó( ÷

24. 25.

26.

.

= ( ôñ + õ ï )ú ; ü = 1, 2, uuu , ù – 1. ¨ The substitution  =  + '  leads to an autonomous equation of the form 5.2.6.8: ¨ (  ) =  +2¨2+ (see also 5.2.1.12 and 5.2.1.13 with  = 0). )

ú –¿÷ ú –÷

2 = (ô ï 2 + õ ï + ö ) ñ ú . This is a special case of equation 5.2.6.22 with  (¨ ) = ¨ + .

ñ ó( ÷

)

ñ ó( ÷

)

–1

= ( ô ï + õ )–÷ ( ö ï + ý )ú –÷¿ú –1 ñ ú . This is a special case of equation 5.2.6.21 with  (¨ ) = ¨p+ .

ñ ó( ÷

(  ) =  ( , , 
 , 

 ).

= ôñ  ñ ó ò + õ ï ú . This is a special case of equation 5.2.6.34 with  ( ) =  )

an (B − 1)st-order equation: 23.

P

This is a special case of equation 5.2.6.14 with  (¨ ) = (  ¨ +  ) + .

5.2.1-3. Equations of the form 22.

ï 2÷ 2–1 ú

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

( 

−1)

=

  , +1

+1

+

¡



 and ( ) = ' + . Integrating yields  + +1 + L .

+1

= ô ÷ ñ + õ (ñó ò – ôñ )  . This is a special case of equation 5.2.6.38 with  ( , ¨ ) =  ¨  . The substitution ¨ = 
−  leads to an (B − 1)st-order autonomous equation: ¨ (  −1) +  ¨ (  −2) + ))) +   −1 ¨ =  ¨  . = ô ïƒú –÷ ñ 1–ú (ñó ò )ú . This is a special case of equation 5.2.6.37 with  (¨ ) =  ¨p+ .

= ô ïƒú ( ï ñó ò – ñ )  . This is a special case of equation 5.2.6.39 with  ( , ¨ ) =  +2¨  . The substitution ¨ =  leads to an (B − 1)st-order equation.

= ô ï ( ï ñó ò – þxñ )A .   Here, is a positive integer and B ≥ + 1. The substitution ¨ ¡ ¡ + ¨ (B − 1)st-order equation:  (  − −1) =   - , where = ¨ ( + )  .

=  "
 −

¡


−

leads to an

Page 707

708 27.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

+ ôñó ò ò ó + õÿñ = ö^ññó ò ò ó – ö (ñó ò )2 + ü . 1 b . Particular solution: )

=L where the constants L

1,

1

sinh( L

4

 )+L

2

cosh( L

4

 ) + L 3,

L 2 , L 3 , and L 4 are related by two constraints L 42  + (  − (L 3 ) L 42 +  = 0, ( ( L 22 − L 12 ) L 42 − L 3 + , = 0.

2 b . Particular solution: =L where the constants L

28.

ñ ó( ÷

)

+

ôññ óò ò ó

– ô (ñ

Particular solution: 29.

30.

31.

ñ ó(2÷

1,

óò

1

sin( L

4

 )+L

2

cos( L

4

 ) + L 3,

L 2 , L 3 , and L 4 are related by two constraints L 42  − (  − (L 3 ) L 42 +  = 0, ( ( L 12 + L 22 ) L 42 + L 3 − , = 0.

)2 +

õÿñ óò ò ó

=L

1

ö^ñ óò

+

exp( L

2

 )−

= 0.

L 2 −1 + L 2 + ( . L 2

= ô ÷ ñ + õ (ñ ó ò ò ó – ôñ )  . This is a special case of equation 5.2.6.50 with  ( , ¨ ) =  ¨  . The substitution ¨ = 

 −    −2) +  ¨ (2  −4) + ))) +   −1 ¨ =  ¨  . leads to an (2B − 2)nd-order autonomous equation: ¨ (2

ñ ó( ÷

)

= ô ïƒú ( ï ñó ò – ñ )  (ñó ò ò ó )A . The substitution ¨ ( ) =  
− leads to an (B − 1)st-order equation: )

ñó ÷

(2 )

M

 = ôñ  ññó ò ò ó – ñó ò  2

¨ 
 −2 2 =  + −- ¨  (¨ 
)- . M  −2 O  P 

. This is a special case of equation 5.2.6.52 with  (¨ ) = 0 and (¨ ) =  ¨  .

5.2.1-4. Other equations. 32.

ñ ó( ÷

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 . 1 b . Integrating the equation two times, we obtain an (B − 2)nd-order equation: )



( −2)




=  !   −  (  )2 + L

1

 + L 2.

2 b . Particular solutions:

 ) + L 2 exp(− L 3  ) +  −1 L 3 −4 = L 1 sin( L 3  ) + L 2 cos( L 3  ) + (−1) ´J 2  −1 L 3 −4 = L 1 exp( L 2  ) +  −1 L 2 −4 = L 1 + L 2 =L

33.

ñ ó( ÷

1

exp( L

3

if B is even number, if B is even number, if B is odd number, if B ≥ 2 is any number.

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ ï + ü . This is a special case of equation 5.2.6.54 with  ( ) = ' + , . Integrating the equation two times, we obtain an (B −2)nd-order equation: (  −2) =  !

 −  (
 )2 + 16 ' 3 + 12 ,! 2 + L 1  + L 2 . )

Page 708

709

5.2. NONLINEAR EQUATIONS

34.

35.

ñ ó(2÷

)

ô 2ñ

+ õ [ñ

ô ï ñ ó( ÷

–1)

ôññ ó(÷

–1)

ó( ÷

–1)

=

ó( ÷

)

+

ôñ ]

.

The substitution ¨ = (  ) + 

ñ ó( ÷

)

ñ ó( ÷

)

+

leads to an B th-order autonomous equation: ¨ (  ) =  ¨ +  ¨  .

+ 2 õÿññó ò + ôƒõ

ï ñ

2

+

ö^ñ

+

öï

= 0.

The functions that solve the (B − 1)st-order autonomous equation solutions of the original equation. 36.

+

õÿñóò

+

+ ôƒõÿñ

2



( −1)

= −

2

− (  are

= 0.

The functions that solve the (B − 1)st-order constant coefficient nonhomogeneous linear equation (  −1) +  = − (  are solutions of the original equation. 37.

38.

39.

40.

ï ñ ó( ÷

)

ï ñ ó( ÷

)

+ ùñ

ô ïƒú ñ ú

=

.

This is a special case of equation 5.2.6.59 with  (¨ ) =  ¨p+ .

ó( ÷

= õ (ï

ñóò

+ ù (ù – 1)ñ

ó( ÷

+ ( ô + ù – 1)ñ

–1)

+ ôñ )  .

This is a special case of equation 5.2.6.60 with  ( , ¨ ) =  ¨  . The substitution ¨ =  leads to an (B − 1)st-order autonomous equation: ¨ (  −1) =  ¨  .

ï 2 ñ ó( ÷

)

ññ ó(2÷

+1)

+ 2ù

ï ñ ó( ÷

–1)

–2)

=

ô ï 2ú ñ ú


+ 

.

This is a special case of equation 5.2.6.61 with  (¨ ) =  ¨ + . =

ôñóÈò ñ ó(2÷

)

.

The equation admits two different (with  ≠ −1) first integrals:

  =L (2 )

¶

,

1

 −1 ! (2  ) + (  + 1) ½ (−1) + ( + ) (2  − + ) + 1 (−1)  (  + 1) V (  ) W 2 = L ¶ , 2 2 + =1 ¶ ¶ where L 1 and L 2 are arbitrary constants. Eliminating the highest derivative from the first integrals, we arrive at an (2B − 1)st-order autonomous equation: ½

where L

1

=−

L

+ ¶ 1

(−1) +

=1

+

, L

=

2

L

  +

+ 12 (−1) ÑV (  ) W

2



(

) (2 −

2

. The order of the obtained equation next can be lowered by

)

=L

1

+1

+L

2,

¶

 +1 the standard substitution ¨ ( ) = 
 .

ó(2÷

+1)

ó( ÷

–1)

ñóò ñ ó(÷

–1)

41.

(2ù – 1)ññ

42.

ññ ó(÷

)

ññ ó(÷

)

43.

 +1

−1

+ (2ù + 1)ñ„óÈ òñ

ó(2÷

)

=

ô ïƒú

.

This is a special case of equation 5.2.6.62 with  ( ) =  + . – ñó òñ

=

ôñ

2

.

Integrating yields an (B − 1)st-order linear equation: (  −1) = (  + L ) . The transformation H =  + L  brings it to an equation of the form 5.1.2.3 with  = 0. =

+ ôñó ò .

Integrating yields an (B − 1)st-order constant coefficient nonhomogeneous linear equation: (  −1) = L −  .

Page 709

710 44.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

÷ ½



ô  ñ ó( ) = õÿññóò ò ó

=0

– õ (ñó ò )2 + ü . = L ´

Particular solutions: the algebraic equation  45. 46.

ï ññ ó(÷

0

7



=0



+ ,

−1 0 ,

where L

is an arbitrary constant and Πare roots of

  Œ  = , Œ 2 .

= ( ï ñó ò + ôñ )ñ ó(÷ –1) . Integrating yields an (B − 1)st-order linear equation of the form 5.1.2.4: )

(ñ + ô ïƒú –1 )ñ ó(÷ ) – ñ ó(ú ) ñ ó(÷ –ú ) + õ ïƒú –1 ñ ó(ú ) = 0, ù > The functions that solve the (B − )th-order linear equation

¡

 + ( −

)

=L

+ ( L +  )

+

þ



( −1)

= L3  .

.

−1

are solutions of the original equation. 47.

ñ ó( ÷

ñ ó÷

–2) ( )

=

Solution: 48. 49.

ñ ó( ÷

)

ñ ó( ÷

)

ô V a ñ ó( ÷

–1) 2

W .

+L L 0+L

L

=

 + ))) + L  1  + ))) + L 

0

1

ú

 

−3

= ôñ  ñó ò V ñ ó(÷ –1) W . This is a special case of equation 5.2.6.73 with  ( ) =  , (¨ ) =  ¨ + .

= ô ïƒú 1 ñ ú 2 (ñó ò )ú 3 ­­­ (ñ ó(÷ –1) )ú±° +1 . Generalized homogeneous equation. The transformation ˆ =  

Π=B +

1

¡

−

¡

3

−2

¡

4

− ))) − (B − 1) 

¡

leads to an (B − 1)st-order equation. 50.

O

÷ ñ ýýï P

7

–1

(ñ

óò

)=

ô ï

½÷

2

ú

–1

(–1)ú

=1

ñ ó(ú ) ñ ó(2÷ –ú

“ =

+1 ,

¡

2

+

,¨ ¡

3

if  ≠ 1, if  = 1.

=

+ ))) +


 , where ¡ 

+1

− 1,

+õ . 2 = ( g
) leads to a constant coefficient linear equation:

The transformation  =  (a ), 2 (g  +1) =  +  . 51.

1

+ ( L  −2 + L  −1  )  −2+ 1−   −3 + L  −2 exp( L  −1  ) −3  −3

)

+ (–1)÷ V ñ

ó( ÷

W + (ñó ò )2 =

) 2

ôñ

2

+

õÿñ

+ö .

Differentiating both sides with respect to  and dividing by
 , we arrive at a constant   ) − 2 Œ 

 + 2  +  = 0. coefficient linear equation: 2 (2 52.

2

ú

½÷

–1 =1

(–1)ú

ñ ó(ú ) ñ ó(2÷ –ú

)

+ (–1)÷ V ñ

ó( ÷

W =  (ï

) 2

ñóò

– ñ ) + ñó ò + .

Differentiating both sides of the equation with respect to  , we have



 V 2 (2  

−1)

− ƒ… − ý/W = 0.

(1)

Equate the second factor to zero to obtain: =

2½  −2 ƒ… 2  ý± 2  −1 + + L   . 2(2B )! 2(2B − 1)!  =0

Page 710

711

5.2. NONLINEAR EQUATIONS

The integration constants L  and parameters ƒ , ý , and = are related by 2

½

−1

+

(−1) +

=2

! (2B −

¡

)! L + L

¡

 + + (−1)  (B !)2 L  2 = ýhL

2 −

1

− ƒsL

0

+= ,

which is obtained as a result of substituting the above solution into the original equation. In there is¶ a solution corresponding to equating the first factor in (1) to zero: ¶ ¶ addition, ¶ = L 1  + L 0 , where ý L 1 − ƒ L 0 + = = 0. 53.

½÷

2

–1

ú

(–1)ú

ñ ó(ú ) ñ ó(2÷ –ú

=1

)

+ (–1)÷ V ñ

ó( ÷

W + s(ñó ò ò ó )2 =  ( ï

ñóò

) 2

– ñ ) + ñó ò + ,

ù

≥ 3.

For the case ° = 0, see equation 5.2.1.52. Let now ° ≠ 0. Differentiating the equation with respect to  , we have 

 V 2 (2   −1) + 2 ° 


 − ƒ… − ý/W = 0. Equate the second factor to zero and integrate to obtain:

ƒ…

4

ý±

3

+ L 2  2 + L 1  + L 0 + XXX ¨    , M M M 48 ° 12 ° where ¨ = ¨ ( ) is the general solution of a linear constant coefficient linear equation of the   −4) + ° ¨ = 0. The constants of integration are related by the constraint that form 5.1.2.2: ¨ (2 results from substituting the obtained solution ¶ into¶ the original equation. In addition, there is the solution = L 1  + L 0 , where the constants of integration are ¶ ¶ related by ý L 1 − ƒ L 0 + = = 0. =

54.

÷

½

ú

=1

ô ú

¦ 2

ú½

–1



=1

)

+ƒ

(–1)¦ñ

+

ó( ) ñ ó(2ú –

+ (–1)ú

)

Vñ

ó( ú

W § = ñ

) 2

2

+ 2ñ + .

Differentiating with respect to  , we arrive at a constant coefficient linear equation:

7

+

=1

 +

 +

(2

+ ý = 0.

5.2.2. Equations Containing Exponential Functions 5.2.2-1. Fifth- and sixth-order equations. 1.

ó

ñ ó(5) = ôññóò ò ó ò ò ó ó

– ô (ñó ò ò ó )2 + õ .  1 b . This is a special case of equation 5.2.6.1 with  ( ) =   . Integrating the equation two  times, we obtain a third-order equation: 


 =  !

 −  (
 )2 + L 1  + L 2 + Œ −2  . 2 b . Particular solutions: = L exp( Œ  ) + =

2.

ñ ó(5) = ô]¦ñóò ñóò ò ó ò ò ó ó

5

exp( Œ  ) + L exp(− Œ  ) −

Π

.

. Integrating yields a fourth-order autonomous equation of the form 5.2.6.8 with B = 4:  L exp O   .

Œ

3.

2Œ



LpŒ 5 −  , LpŒ 4

ññ ó(5) = ô ó





 =

P

+õ .  This is a special case of equation 5.2.6.17 with B = 2 and  ( ) =   +  .

Page 711

712 4.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó 2

Solution: 5.

=L

4

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó



+ 10ñó ò ò óÈñó ò ò ó ò +L

4

3



3

ó

+L

=

+ 10ñó ò ò óÈñó ò ò ó ò

2

ó



2

ô ó +L

1

ô]

=

.

 +L

+ 2 Œ

0

  .

−5

.

This is a special case of equation 5.2.6.3 with  ( ) =   . 6.

ñ ó(6) = ô]

+õ .

This is a special case of equation 5.2.6.8 with B = 6 and  ( ) =   +  . 7.

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

+ 10(ñó ò ò ó ò ó )2 =

ô ó

.



This is a special case of equation 5.2.6.6 with  ( ) =   . 5.2.2-2. Equations of the form 8.

ñ ó( ÷

)

ñ ó( ÷

)

=

ô] +_ ó

ñ ó( ÷

)

=

ô ï –÷  ]

ñ ó( ÷

)

ô ]

=

  =  ( , ). ( )

+õ .

This is a special case of equation 5.2.6.8 with  ( ) =   +  . 9.

10.

+õ . ¨ = + (ý Œ ) leads This is a special case of equation 5.2.6.9 with = 1. The substitution ? ¡ ¨ to an autonomous equation of the form 5.2.6.8: (  ) =   +  . .

This is a special case of equation 5.2.6.11 with  ( ) =   . 11.

12.

ô ï   ^ ]

=

.

This is a special case of equation 5.2.6.26 with  (¨ ) =  ¨ and

ñ ó( ÷

)

ó
B ^ ñ ú

=

.

This is a special case of equation 5.2.2.23 with 13.

ññ ó(2÷

+1)

=

ô ó

+õ .

=

¡

¡

1

and

This is a special case of equation 5.2.6.17 with  ( ) =   14.

ñ ó( ÷

)

=

ô

= , +B .

¡

exp( Ė + 

ïƒú

)+ õ ,

¨ = The substitution ? ¨ (  ) =   +  .

+ ( ý Œ ) +

þ

= 1, 2,

uuu , ù



2

¡

=

¡

3

= ))) =

= 0. ¡ 

+ .

– 1.

leads to an autonomous equation of the form 5.2.6.8:

5.2.2-3. Other equations. 15.

ñ ó( ÷

)

=

ô ] ñ óò

+

õ _ ó

.

Integrating yields an (B − 1)st-order equation: 16.

ñ ó( ÷

)

=

ôññ óò ò ó ò ò ó ó

– ô (ñ

óò ò ó

)2 +

õ  ó

.



( −1)

=

Œ



  + ý



  : +L .



1 b . This is a special case of equation 5.2.6.54 with  ( ) =   . Integrating the equation two  times, we obtain an (B − 2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 + Œ −2  .

Page 712

713

5.2. NONLINEAR EQUATIONS

2 b . Particular solutions:

LpŒ  −  LpŒ 4

= L exp( Π ) +

Œ  exp( Œ  ) + L exp(− Œ  ) −  2Œ  

= 17.

18. 19.

õ ó (ñ óò

ñ ó( ÷

)

=

ô ÷ ñ

ñ ó( ÷

)

=

õ ó (ï ñ óò

+

(B is any number), −4

(B is odd number).

– ôñ )  .

 This is a special case of equation 5.2.6.38 with  ( , ¨ ) =    ¨  . – ñ ) .

 This is a special case of equation 5.2.6.39 with  ( , ¨ ) =   ¨  . (2 ù – 1)ññ

ó(2÷

ó(2÷

+ (2ù + 1)ñ„óÈ òñ

+1)

)

ô ó

=

.



This is a special case of equation 5.2.6.62 with  ( ) =   .

20.

ñ ó( ÷

)

ñ ó( ÷

)

= ( ô

ñ ó( ÷

)

=

ô]Èñóò V ñ ó(÷

ñ ó( ÷

)

=

ó
B ^ ñ ú

=

ô]Èñóò ñ ó(÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =   . 21. 22.

23.

] ñ óò

õ _ ó )ñ ó(÷

+

–1)

.

 This is a special case of equation 5.2.6.58 with  ( ) =   and ( ) =   : . –1)

W

ú

.

This is a special case of equation 5.2.6.73 with  ( ) =   and (¨ ) = ¨2+ . 1

(ñó ò )ú

2

uuu  ñ ó(÷



–1)

ú±°

.

 The substitution ¨ ( ) =  : , where ý = equation of the form 5.2.6.77.

24.

ñ ó( ÷

)

=


B ^ ] ïƒú

(ñó ò )ú

(ñó ò ò ó )ú

uuu  ñ ó(÷

1

¡

–1)



+

ƒ

2

¡

+ ))) +

2

3

, leads to an autonomous

ú±°

. The transformation H =  ‡  9  , ¨ =  
 , where . = B + ¡ leads to an (B − 1)st-order equation. 1

−1 ¡ 

1−

¡

2 −2

¡

3 −3

¡

4−

))) −(B −1)  , ¡

5.2.3. Equations Containing Hyperbolic Functions 5.2.3-1. Equations with hyperbolic sine. 1.

ñ ó(5) = ôññóò ò ó ò ò ó ó

– ô (ñó ò ò ó )2 + õ sinh(

ï

).

1 b . This is a special case of equation 5.2.6.1 with  ( ) =  sinh( Π ). Integrating the equation two times, we obtain a third-order equation: "


 =  !

 −  (
 )2 + L 1  + L 2 + Œ −2 sinh( Œ  ). 2 b . Particular solution: 2.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó Solution:

3.

2

=L

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

4



=

Œ 4(Œ

+ 10ñó ò ò óÈñó ò ò ó ò 4

+L

+ 10ñ

3



3

ó

+L

óò ò ó ñ óò ò ó ò ó



−  2L

2

ô

= 2



+L

ô

1

V L sinh( Π ) + Πcosh( Π )W + L .

ï

).

 +L

0

sinh(

2

=

2)

sinh ú (

ï

+ 2 Œ )+õ .

−5

cosh( Π ).

+

This is a special case of equation 5.2.6.2 with  ( ) =  sinh ( Π ) +  .

Page 713

714 4. 5. 6. 7. 8. 9.

HIGHER-ORDER DIFFERENTIAL EQUATIONS + 10ñó ò ò óÈñó ò ò ó ò ó = ô sinh ú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.3 with  ( ) =  sinh ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñó ò ò ó ò ó )2 = ô sinhú ( ï ). + This is a special case of equation 5.2.6.6 with  ( ) =  sinh ( Œ  ).

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

= ô sinh ú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.8 with  ( ) =  sinh ( Œ

ñ ó( ÷

)

= ô ï –÷ sinhú ( ƒñ ). + This is a special case of equation 5.2.6.11 with  ( ) =  sinh ( Œ

ñ ó( ÷

)+ .

)

).

= ô sinh ú ( ï ) + õ . + This is a special case of equation 5.2.6.17 with  ( ) =  sinh ( Œ  ) +  .

ññ ó(2÷ ñ ó( ÷

+1)

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ sinh( ï ). 1 b . This is a special case of equation 5.2.6.54 with  ( ) =  sinh( Œ  ). Integrating the equation two times, we obtain an (B − 2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 + Œ −2 sinh( Œ  ). )

2 b . Particular solutions:

LpŒ  −  LpŒ 4  V L sinh( Œ  ) + Œ  = 2  −4 Œ −  2L 2Œ 4 = L sinh( Œ  ) +

10. 11. 12.

if B is even number, −4

cosh( Π )W + L

if B is odd number.

ò ñ ó(2÷ ) = ô sinhú ( ï ) + õ . (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ + This is a special case of equation 5.2.6.62 with  ( ) =  sinh ( Œ  ) +  . ñ ó( ÷

)

=

ô

sinh  ( ƒñ )ñóÈ òñ

ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  sinh  ( Œ

ññ ó(÷

).

– ñó ò ñ ó(÷ –1) = ô sinh( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  sinh( Œ  ). Integrating yields an  (B − 1)st-order linear equation: (  −1) = cosh( Œ  ) + L3S . )

Q Œ

5.2.3-2. Equations with hyperbolic cosine. 13.

– ô (ñó ò ò ó )2 + õ cosh( ï ). 1 b . This is a special case of equation 5.2.6.1 with  ( ) =  cosh( Œ  ). Integrating the equation twice, we obtain a third-order equation: 


 =  !

 −  (
 )2 + L 1  + L 2 + Œ −2 cosh( Œ  ).

ñ ó(5) = ôññóò ò ó ò ò ó ó

2 b . Particular solution: 14.

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó Solution:

15.

2

+ 10ñ = L 4 4 + L

=

Œ 4(Œ

2



−  2L

2)

V L cosh( Π ) + Πsinh( Π )W + L .

cosh( ï ). 3 + L 2  2 + L 1  + L 0 + 2 Œ 3

óò ò ó ñ óò ò ó ò ó

=

ô

−5

sinh( Π ).

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô coshú ( ï ) + õ . + This is a special case of equation 5.2.6.2 with  ( ) =  cosh ( Œ  ) +  .

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

Page 714

715

5.2. NONLINEAR EQUATIONS

16.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò

ó

coshú ( ƒñ ) + õ .

ô

=

+

This is a special case of equation 5.2.6.3 with  ( ) =  cosh ( Π17.

ññ ó(6) + 6ñ óò ñ ó(5) + 15ñ óò ò ó ñ óò ò ó ò ò ó ó

óò ò ó ò ó

+ 10(ñ

coshú (

ô

)2 =

+

ï

)+ .

).

This is a special case of equation 5.2.6.6 with  ( ) =  cosh ( Π ). 18.

ñ ó( ÷

)

ñ ó( ÷

)

coshú ( ƒñ ) + õ .

ô

=

+

This is a special case of equation 5.2.6.8 with  ( ) =  cosh ( Π19.

coshú ( ƒñ ).

ô ï –÷

=

+

This is a special case of equation 5.2.6.11 with  ( ) =  cosh ( Π20.

ññ ó(2÷

+1)

=

coshú (

ô

ï

)+õ .

)+ . ).

+

This is a special case of equation 5.2.6.17 with  ( ) =  cosh ( Π ) +  . 21.

ñ ó( ÷

)

ôññóò ò ó ò ò ó ó

=

ï

– ô (ñó ò ò ó )2 + õ cosh(

).

1 b . This is a special case of equation 5.2.6.54 with  ( ) =  cosh( Œ  ). Integrating the equation twice, we obtain an (B −2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 + Œ −2 cosh( Œ  ). 2 b . Particular solutions: = L cosh( Œ  ) + = 22.

(2 ù – 1)ññ

Π

2 −4

ó(2÷



−  2L

2

Œ

LpŒ  −  LpŒ 4 V L cosh( Œ  ) + Œ  4

óò ñ ó(2÷

+ (2ù + 1)ñ

+1)

)

ô

=

if B is even number, −4

sinh( Π )W + L

coshú (

ï

)+õ .

if B is odd number.

+

This is a special case of equation 5.2.6.62 with  ( ) =  cosh ( Π ) +  .

23.

24.

ñ ó( ÷

)

=

ô

cosh  ( ƒñ )ñóÈ òñ

ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  cosh  ( Œ

ññ ó(÷

)

–ñ

ó ò ñ ó( ÷

–1)

=

ô

cosh(

ï )ñ

2

).

.

This is a special case of equation 5.2.6.64 with  ( ) =  cosh( Œ  ). Integrating yields an  (B − 1)st-order linear equation: (  −1) = sinh( Œ  ) + L3S .

Q Œ

5.2.3-3. Equations with hyperbolic tangent. 25.

ñ ó(5) = ôññóò ò ó ò ò ó ó

– ô (ñó ò ò ó )2 + õ tanh(

ï

)+ö .

This is a special case of equation 5.2.6.1 with  ( ) =  tanh( Π ) + ( . 26.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò

ó

=

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò

ó

=

ô

tanhú (

ô

tanhú ( ƒñ ) + õ .

ï

)+ õ .

+

This is a special case of equation 5.2.6.2 with  ( ) =  tanh ( Π ) +  . 27.

+

This is a special case of equation 5.2.6.3 with  ( ) =  tanh ( Π28.

ññ ó(6) + 6ñ óò ñ ó(5) + 15ñ óò ò ó ñ óò ò ó ò ò ó ó

+ 10(ñ

óò ò ó ò ó

)2 =

ô

tanhú (

+

ï

)+ .

).

This is a special case of equation 5.2.6.6 with  ( ) =  tanh ( Π ).

Page 715

716 29. 30. 31. 32. 33. 34. 35. 36.

HIGHER-ORDER DIFFERENTIAL EQUATIONS = ô tanhú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.8 with  ( ) =  tanh ( Œ

ñ ó( ÷

)

= ô ï –÷ tanhú ( ƒñ ). + This is a special case of equation 5.2.6.11 with  ( ) =  tanh ( Œ

ñ ó( ÷

)+ .

)

).

= ô tanhú ( ï ) + õ . + This is a special case of equation 5.2.6.17 with  ( ) =  tanh ( Œ  ) +  .

ññ ó(2÷

+1)

ñ ó(2÷

)

= ñ + ô (ñó ò – ñ tanh ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = cosh  .

ñ ó(2÷

+1)

= ñ tanh ï + ô (ñ„ó ò – ñ tanh ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = cosh  .

(2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„ó ò ñ ó(2÷ ) = ô tanhú ( ï ) + õ . + This is a special case of equation 5.2.6.62 with  ( ) =  tanh ( Œ  ) +  .

ñ ó( ÷

)

=

ô

tanh ( ƒñ )ñ

ó ò ñ ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  tanh  ( Œ

ññ ó(÷

).

– ñ ó ò ñ ó(÷ –1) = ô tanh( ï )ñ 2 . This is a special case of equation 5.2.6.64 with  ( ) =  tanh( Œ  ). )

5.2.3-4. Equations with hyperbolic cotangent. 37. 38. 39. 40. 41. 42. 43. 44.

– ô (ñó ò ò ó )2 + õ coth( ï ) + ö . This is a special case of equation 5.2.6.1 with  ( ) =  coth( Œ  ) + ( .

ñ ó(5) = ôññóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô cothú ( ï ) + õ . + This is a special case of equation 5.2.6.2 with  ( ) =  coth ( Œ  ) +  .

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô cothú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.3 with  ( ) =  coth ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñó ò ò ó ò ó )2 = ô cothú ( ï ). + This is a special case of equation 5.2.6.6 with  ( ) =  coth ( Œ  ).

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

= ô cothú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.8 with  ( ) =  coth ( Œ

ñ ó( ÷

)

= ô ï –÷ cothú ( ƒñ ). + This is a special case of equation 5.2.6.11 with  ( ) =  coth ( Œ

ñ ó( ÷

)+ .

)

).

= ô cothú ( ï ) + õ . + This is a special case of equation 5.2.6.17 with  ( ) =  coth ( Œ  ) +  .

ññ ó(2÷ ñ ó(2÷

+1)

= ñ + ô (ñó ò – ñ coth ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = sinh  . )

Page 716

717

5.2. NONLINEAR EQUATIONS

45. 46. 47.

48.

ñ ó(2÷

= ñ coth ï + ô (ñ„ó ò – ñ coth ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = sinh  . +1)

(2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ ò ñ ó(2÷ ) = ô cothú ( ï ) + õ . + This is a special case of equation 5.2.6.62 with  ( ) =  coth ( Œ  ) +  .

ñ ó( ÷

)

=

ô

coth ( ƒñ )ñóÈ òñ

ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  coth  ( Œ

ññ ó(÷

).

– ñó ò ñ ó(÷ –1) = ô coth( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  coth( Œ  ). )

5.2.4. Equations Containing Logarithmic Functions 5.2.4-1. Equations of the form 1. 2. 3. 4. 5. 6. 7.

  =  ( , ). ( )

= ô lnú ( õÿñ ) + ö . + This is a special case of equation 5.2.6.8 with  ( ) =  ln (  ) + ( .

ñ ó( ÷

)

= ô lnú ( õ ï ). + This is a special case of equation 5.2.6.17 with  ( ) =  ln ( ' ).

ññ ó(2÷ ñ ó( ÷

)

ñ ó( ÷

)

+1)

= ñ (  ï + þ ln ñ + õ ). This is a special case of equation 5.2.6.25 with  (¨ ) = ln ¨ +  . = ï –÷ ( ñ + þ ln ï + õ ). This is a special case of equation 5.2.6.26 with  (¨ ) = ln ¨ +  .

= ô ï –÷ lnú ( õÿñ ). + This is a special case of equation 5.2.6.11 with  ( ) =  ln (  ).

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

= ô ï –÷ –1 [ln ñ + (1 – ù ) ln ï ]. This is a special case of equation 5.2.6.13 with  (¨ ) =  ln ¨ . = ô ï –÷ –  (ln ñ + ü ln ï ). This is a special case of equation 5.2.6.15 with  (¨ ) =  ln ¨ .

8.

= ôñ ï –÷ (þ ln ñ + ü ln ï ). This is a special case of equation 5.2.6.16 with  (¨ ) =  ln ¨ .

9.

ñ ó(2÷

= ô ï – 2 [2 ln ñ + (1 – 2ù ) ln ï ]. This is a special case of equation 5.2.6.14 with  (¨ ) = 2  ln ¨ .

10.

= ( ô ï 2 + ö ) 2 [2 ln ñ + (1 – ù ) ln( ô ï 2 + ö )]. This is a special case of equation 5.2.6.22 with  = 0 and  (¨ ) = 2 ln ¨ .

11.

÷

2 +1

)

ñ ó( ÷

)

ñ ó( ÷

)

–

÷

+1

ó

= õ ^ (ln ñ –  ï ). This is a special case of equation 5.2.6.24 with  (¨ ) =  ln ¨ .

Page 717

718

HIGHER-ORDER DIFFERENTIAL EQUATIONS

5.2.4-2. Other equations. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

= ôñ ó ò ln ñ + õ ln ï . This is a special case of equation 5.2.6.34 with  ( ) =  ln and ( ) =  ln  .

= ô ÷ ñ + õ ln ï (ñ ó ò – ôñ )  . This is a special case of equation 5.2.6.38 with  ( , ¨ ) =  ¨  ln  .

= ô ln ï ( ï ñó ò – ñ )  . This is a special case of equation 5.2.6.39 with  ( , ¨ ) =  ¨  ln  . = ô ln ï ( ï ñó ò – 2ñ )  . This is a special case of equation 5.2.6.40 with

ñ ó( ÷

= 2 and  ( , ¨ ) =  ¨  ln  .

¡

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ ln ï + ö . This is a special case of equation 5.2.6.54 with  ( ) =  ln  + ( . )

ï ñ ó( ÷

+ ùñ ó(÷ –1) = ô ln ï + ô ln ñ . This is a special case of equation 5.2.6.59 with  (¨ ) =  ln ¨ . )

ï 2 ñ ó( ÷

+ 2ù ï ñ ó(÷ –1) + ù (ù – 1)ñ ó(÷ –2) = 2 ô ln ï + ô ln ñ . This is a special case of equation 5.2.6.61 with  (¨ ) =  ln ¨ .

ññ ó(÷

)

– ñó ò ñ ó(÷ –1) = ôñ 2 ln ï . This is a special case of equation 5.2.6.64 with  (¨ ) =  ln  . )

ò ñ ó(2÷ ) = ô lnú ( õ ï ) + ö . (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ + This is a special case of equation 5.2.6.62 with  ( ) =  ln ( ' ) + ( . ñ ó( ÷

)

=

ô

ln  ( õÿñ ) ñó òñ

ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  ln  (  ).

= ôñ ú ñó ò ln ñ ó(÷ –1) . This is a special case of equation 5.2.6.73 with  ( ) =  "+ and (¨ ) = ln ¨ .

ñ ó( ÷

)

5.2.5. Equations Containing Trigonometric Functions 5.2.5-1. Equations with sine. 1.

– ô (ñó ò ò ó )2 + õ sin( ï ). 1 b . This is a special case of equation 5.2.6.1 with  ( ) =  sin( Œ  ). Integrating the equation twice, we obtain a third-order equation: 


 =  !

 −  (
 )2 + L 1  + L 2 − Œ −2 sin( Œ  ).

ñ ó(5) = ôññóò ò ó ò ò ó ó

2 b . Particular solution: 2.

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó Solution:

3.

2

+ 10ñ ó ò ò ó = L 4 4 + L 3

=−

Œ

ñ óò ò ó ò ó

3

+L

 L

4( 2

= 2

2

ô



2

+ Π2)

sin( +L

ï

1 + L

V L sin( Π ) + Πcos( Π )W + L . ). 0

− 2 Œ

−5

cos( Π ).

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô sin ú ( ï ) + õ . This is a special case of equation 5.2.6.2 with  ( ) =  sin + ( Œ  ) +  .

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

Page 718

719

5.2. NONLINEAR EQUATIONS

4. 5. 6. 7. 8. 9.

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô sin ú ( ƒñ ) + õ . This is a special case of equation 5.2.6.3 with  ( ) =  sin + ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñ ó ò ò ó ò ó )2 = ô sinú ( ï ). This is a special case of equation 5.2.6.6 with  ( ) =  sin + ( Œ  ).

ññ ó(6) + 6ñ óò ñ ó(5) + 15ñ óò ò ó ñ óò ò ó ò ò ó ó ñ ó( ÷

)

= ô sinú ( ƒñ ) + õ . This is a special case of equation 5.2.6.8 with  ( ) =  sin + ( Œ

ñ ó( ÷

)

= ô ï –÷ sinú ( ƒñ ). This is a special case of equation 5.2.6.11 with  ( ) =  sin + ( Œ

ññ ó(2÷

)+ . ).

= ô sin ú ( ï ) + õ . This is a special case of equation 5.2.6.17 with  ( ) =  sin + ( Œ  ) +  .

ñ ó( ÷

+1)

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ sin( ï ). 1 b . This is a special case of equation 5.2.6.54 with  ( ) =  sin( Œ  ). Integrating the equation two times, we obtain an (B − 2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 − Œ −2 sin( Œ  ). )

2 b . Particular solutions: = L sin( Œ  ) + =− 10. 11. 12.

Π2

−4



+  2L

(−1) ´J 2 p L Œ  − 2

Œ

if B is even number,

LpŒ 4 V L sin( Œ  ) + (−1)  4

−1 2

Π

−4

cos( Π )W + L

if B is odd number.

(2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ ò ñ ó(2÷ ) = ô sinú ( ï ) + õ . This is a special case of equation 5.2.6.62 with  ( ) =  sin + ( Œ  ) +  .

ñ ó( ÷

= ô sin  ( ƒñ )ñ ó ò ñ ó(÷ –1) . This is a special case of equation 5.2.6.57 with  ( ) =  sin  ( Œ )

ññ ó(÷

).

– ñó ò ñ ó(÷ –1) = ô sin( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  sin( Œ  ). Integrating yields an  (B − 1)st-order linear equation: (  −1) = L − cos( Œ  )S . )

Q

Œ

5.2.5-2. Equations with cosine. 13.

– ô (ñó ò ò ó )2 + õ cos( ï ). 1 b . This is a special case of equation 5.2.6.1 with  ( ) =  cos( Œ  ). Integrating the equation twice, we obtain a third-order equation: 


 =  !

 −  (
 )2 + L 1  + L 2 − Œ −2 cos( Œ  ).

ñ ó(5) = ôññóò ò ó ò ò ó ó

2 b . Particular solution: 14.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó Solution:

15.

2

=−

Π4 ( 2 L

+ 10ñó ò ò óÈñó ò ò ó ò = L 4 4 + L 3 3 + L

ó

= 2

2

ô



2

+ Π2)

cos( +L

ï

1 + L

V L cos( Œ  ) − Œ sin( Œ  )W + L . ). 0

+ 2 Œ

−5

sin( Π ).

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô cosú ( ï ) + õ . This is a special case of equation 5.2.6.2 with  ( ) =  cos + ( Œ  ) +  .

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

Page 719

720 16. 17. 18. 19. 20. 21.

HIGHER-ORDER DIFFERENTIAL EQUATIONS + 10ñó ò ò óÈñó ò ò ó ò ó = ô cosú ( ƒñ ) + õ . This is a special case of equation 5.2.6.3 with  ( ) =  cos + ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñ ó ò ò ó ò ó )2 = ô cosú ( ï ). This is a special case of equation 5.2.6.6 with  ( ) =  cos + ( Œ  ).

ññ ó(6) + 6ñ óò ñ ó(5) + 15ñ óò ò ó ñ óò ò ó ò ò ó ó ñ ó( ÷

)

= ô cosú ( ƒñ ) + õ . This is a special case of equation 5.2.6.8 with  ( ) =  cos + ( Œ

ñ ó( ÷

)

= ô ï –÷ cosú ( ƒñ ). This is a special case of equation 5.2.6.11 with  ( ) =  cos + ( Œ

ññ ó(2÷

)+ . ).

= ô cosú ( ï ) + õ . This is a special case of equation 5.2.6.17 with  ( ) =  cos + ( Œ  ) +  .

ñ ó( ÷

+1)

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ cos( ï ). 1 b . This is a special case of equation 5.2.6.54 with  ( ) =  cos( Œ  ). Integrating the equation two times, we obtain an (B − 2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 − Œ −2 cos( Œ  ). )

2 b . Particular solutions: = L cos( Œ  ) + =− 22. 23. 24.

Π

2 −4



+ L 2

(−1) ´J 2 p L Œ  −

Œ

2 4

LpŒ 4 V L cos( Œ  ) + (−1) 

if B is even number, +1 2

Π

−4

sin( Π )W + L

if B is odd number.

ò ñ ó(2÷ ) = ô cosú ( ï ) + õ . (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ This is a special case of equation 5.2.6.62 with  ( ) =  cos + ( Œ  ) +  . ñ ó( ÷

= ô cos  ( ƒñ )ñ ó ò ñ ó(÷ –1) . This is a special case of equation 5.2.6.57 with  ( ) =  cos  ( Œ )

ññ ó(÷

).

– ñó ò ñ ó(÷ –1) = ô cos( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  cos( Œ  ). Integrating yields an  (B − 1)st-order linear equation: (  −1) = sin( Œ  ) + L3S . )

Q Œ

5.2.5-3. Equations with tangent. 25. 26. 27. 28.

– ô (ñó ò ò ó )2 + õ tan( ï ) + ö . This is a special case of equation 5.2.6.1 with  ( ) =  tan( Œ  ) + ( .

ñ ó(5) = ôññóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô tanú ( ï ) + õ . This is a special case of equation 5.2.6.2 with  ( ) =  tan + ( Œ  ) +  .

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô tanú ( ƒñ ) + õ . This is a special case of equation 5.2.6.3 with  ( ) =  tan + ( Œ

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

)+ .

+ 10(ñó ò ò ó ò ó )2 = ô tanú ( ï ). This is a special case of equation 5.2.6.6 with  ( ) =  tan + ( Œ  ).

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

Page 720

721

5.2. NONLINEAR EQUATIONS

29. 30. 31. 32. 33. 34. 35. 36.

ñ ó( ÷

)

= ô tanú ( ƒñ ) + õ . This is a special case of equation 5.2.6.8 with  ( ) =  tan + ( Œ

ñ ó( ÷

)

= ô ï –÷ tanú ( ƒñ ). This is a special case of equation 5.2.6.11 with  ( ) =  tan + ( Œ

ññ ó(2÷

)+ . ).

= ô tanú ( ï ) + õ . This is a special case of equation 5.2.6.17 with  ( ) =  tan + ( Œ  ) +  . +1)

ñ ó(2÷

)

= (–1)÷ ñ + ô (ñó ò + ñ tan ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = cos  .

ñ ó(2÷

+1)

= (–1)÷ +1 ñ tan ï + ô (ñó ò + ñ tan ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = cos  .

(2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ ò ñ ó(2÷ ) = ô tanú ( ï ) + õ . This is a special case of equation 5.2.6.62 with  ( ) =  tan + ( Œ  ) +  .

ñ ó( ÷

= ô tan ( ƒñ )ñóÈ ò ñ ó(÷ –1) . This is a special case of equation 5.2.6.57 with  ( ) =  tan  ( Œ )

ññ ó(÷

).

– ñó ò ñ ó(÷ –1) = ô tan( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  tan( Œ  ). )

5.2.5-4. Equations with cotangent. 37. 38. 39. 40. 41. 42. 43. 44.

– ô (ñó ò ò ó )2 + õ cot( ï ) + ö . This is a special case of equation 5.2.6.1 with  ( ) =  cot( Œ  ) + ( .

ñ ó(5) = ôññóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô cotú ( ï ) + õ . This is a special case of equation 5.2.6.2 with  ( ) =  cot + ( Œ  ) +  .

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô cotú ( ƒñ ) + õ . This is a special case of equation 5.2.6.3 with  ( ) =  cot + ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñó ò ò ó ò ó )2 = ô cotú ( ï ). This is a special case of equation 5.2.6.6 with  ( ) =  cot + ( Œ  ).

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó ñ ó( ÷

)

= ô cotú ( ƒñ ) + õ . This is a special case of equation 5.2.6.8 with  ( ) =  cot + ( Œ

ñ ó( ÷

)

= ô ï –÷ cotú ( ƒñ ). This is a special case of equation 5.2.6.11 with  ( ) =  cot + ( Œ

ññ ó(2÷

)+ . ).

= ô cotú ( ï ) + õ . This is a special case of equation 5.2.6.17 with  ( ) =  cot + ( Œ  ) +  .

ñ ó(2÷

+1)

= (–1)÷ ñ + ô (ñ ó ò – ñ cot ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = sin  . )

Page 721

722 45.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

= (–1)÷

+1)

This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = sin  .

46.

(2 ù – 1)ññ

47.

ñ ó( ÷

48.

cot ï + ô (ñó ò – ñ cot ï )  .

ñ

ó(2÷

+ (2ù + 1)ñ

+1)

óò ñ ó(2÷

)

cotú (

ô

=

ï

)+ õ .

This is a special case of equation 5.2.6.62 with  ( ) =  cot + ( Π ) +  . )

=

ó( ÷

cot ( ƒñ )ñóÈ òñ

ô

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  cot  ( Œ

ññ ó(÷

)

ó ò ñ ó( ÷

–ñ

–1)

=

ô

cot(

ï )ñ

2

).

.

This is a special case of equation 5.2.6.64 with  ( ) =  cot( Π ).

5.2.6. Equations Containing Arbitrary Functions 5.2.6-1. Fifth- and sixth-order equations. 1.

ñ ó(5) = ôññ óò ò ó ò ò ó ó

)2 + ! ( ï ).

óò ò ó

– ô (ñ

Integrating the equation two times, we obtain a third-order equation:






 =  ! 

 −  ( 
)2 + L 1  + L 2.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

2 + X  ( − a )  (a ) a , M 0

3.

4.

=L

ó

4 3 2 4 + L 3 + L 2 + L 1 + L

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò

The substitution ¨ = 2 l A £ ¨  .

ññ ó(5) + ôñóò ñóò ò ó ò ò ó ó

2

0

is an arbitrary number.

= ! ( ï ).

+ 10ñó ò ò óÈñó ò ò ó ò

Solution: 2

where 

0+

ó



1 X (  − a )4  ( a ) a , 12  0 M

where 

0

is an arbitrary number.

= ! (ñ ).

leads to an autonomous equation of the form 5.2.6.8:

+ (3 ô – 5)ñ„ó ò ò óÈñó ò ò ó ò

¨ (5)  =

= ! ( ï ).

ó

Integrating the equation three times, we obtain a second-order equation:

 ! 

 +  − 3 ( 
)2 = L  2 + L  + L + 1 X ( − a )2  (a ) a , 2 1 0 2 2  0 M 5.

( ô + ñ )ñ

ó(5) + õÿñóò ñóò ò ó ò ò ó ó

+ ö^ñó ò ò ó

where 

0

is an arbitrary number.

= ! ( ï ).

ñóò ò ó ò ó

Integrating yields a fourth-order equation: 1 (  + ) 



 + (  − 1) 



 + (1 −  + ( )( 

 )2 = X¯ ( )  + L . 2 M 6.

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó 2

Solution: 7.

ñ ó(6) = (ô ï

2

+

=L

õï

5

5

+L

+ ö )–7 )

4 2

4

+ 10(ñó ò ò ó ò ó )2 = ! ( ï ).

+L

!  ñ (ô ï

3

3 2

+L

+

õï

2

2

+L

1 + L

0+



1 X (  − a )5  ( a ) a . 60  0 M

+ ö )–5 ) 2  .

This is a special case of equation 5.2.6.22 with B = 6.

Page 722

723

5.2. NONLINEAR EQUATIONS

(  ) =  ( , ).

5.2.6-2. Equations of the form 8.

ñ ó( ÷

= ! (ñ ).

)

Autonomous equation. This is a special case of equation 5.2.6.77.

1 b . The substitution ¨ ( ) = 
 leads to an (B − 1)st-order equation. 2 b . For even B = 2 , the first integral of the equation is:

¡

+½

−1



 + −  ) + 12 (−1) + V ( + ) W 2 + X  ( )

(−1)  ( 

) (2

=1

M

=L .

Furthermore, the order of the obtained equation can be reduced by one by the substitution ¨ ( ) = 
 . 9.

10.

11.

ñ ó( ÷

ï ú

= ! (ñ + ô

)

The substitution ¨ ¨ (  ) =  (¨ ).

ñ ó( ÷

=u ! 

)

ñ

+

þ

),

+  +

=

ô ÷ ï ÷

+

= 0, 1,

ô ÷ –1 ï ÷

)

ñ ó( ÷

)

ï –÷ ! (ñ

=

– 1.

leads to an autonomous equation of the form 5.2.6.8: –1

+

The substitution ¨ = +     +   form 5.2.6.8: ¨ (  ) =   B ! +  (¨ ).

ñ ó( ÷

uuu , ù

uuu

 −1 

+

ô

−1

. + ))) +  0

0

leads to an autonomous equation of the

).

The substitution a = ln | | leads to an autonomous equation of the form 5.2.6.77. 12.

ï 1–÷ ! (ñ  ï

=

).

Homogeneous equation. This is a special case of equation 5.2.6.83. The transformation a = ln  , ¨ =   leads to an autonomous equation of the form 5.2.6.77. 13.

ñ ó( ÷

)

ï –÷ –1 ! (ï 1–÷ ñ

=

).

The transformation  = a ¨ (g  ) = (−1)   (¨ ). 14.

ñ ó(2÷

)

=

ï

–

÷

2 +1 2

!IO ï

1–2 2

÷

−1

ñ g

=a

,

P

1−

 ¨ leads to an autonomous equation of the form 5.2.6.8:

.



2 −1

The transformation  =  , =  2 ¨ (a ) leads to an autonomous equation of the form 5.2.6.68, whose order can be reduced by two. 15.

ñ ó( ÷

)

=

ï –÷ –  ! (ñ ï 

).

This is a special case of equation 5.2.6.86. 1 b . The transformation a = ln  , H =   leads to an autonomous equation of the form 5.2.6.77. 2 b . The transformation H =   , ¨ = 

16.

ñ ó( ÷

)

=

ñ ï –÷ ! (ï  ñ ú


 leads to an (B − 1)st-order equation.

).

This is a special case of equation 5.2.6.89. The transformation a =   + , ¨ to an (B − 1)st-order equation.

=


 leads

Page 723

724 17.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ññ ó(2÷

+1)

= ! ( ï ).

Integrating yields a 2B th-order equation: 2

½

−1

(−1)

+

+

+ (

ñ ó( ÷

)

= ! ( ï , ñ ).

=H

1−

)

= (ô

ï

+

+ ö )1–÷

õÿñ

2

= 2 XZ ( )  + L ,

M

 ¨ ( H ) leads to an equation of the same form: ¨ (‘  ) =

ô ï  ï

+ õÿñ + ñ 1 b . For ý − ˜ƒ = 0, the substitution the form 5.2.6.8.

ñ ó( ÷

+ (−1) rV (  ) W

is used.

The transformation  = H −1 , (−1)  H −  −1  ( H −1 , H 1−  ¨ ). 19.

)

=0

 ≡ where the notation (0) 18.

  +

) (2 −

! O

+ö . + P  ¨ =  + 

+ ( leads to an autonomous equation of

2 b . For ý − ˜ƒ ≠ 0, the transformation

¨ = − , 0

H =  −  0, where 

0

and

0

are the constants which are determined by solving the linear algebraic system



0

+

+ ( = 0,

0

ƒ…

0

+ý

0

+ = = 0,

leads to a homogeneous equation of the form 5.2.6.12:

¨ (‘  ) = H 20.

ñ ó( ÷

)

= (ô

1

ï

+

õ 1ñ

1−

ö

+

¨

 O

1)

1–

,

HIP

 (ˆ ) = ( + 'ˆ )1−  *O

where

ï ÷ !e” ô 2 ô 3ï 

Let the following condition hold: ûû 

+ +

õ 2ñ

+ö +ö

õ 3ñ 

1

1

(

2 3

 + 'ˆ . ƒ + ý±ˆÑP

.

•

1

( 2 ûû = 0. û 3 3 (3û û û û û For  2  3 −  3  2 ≠ 0, the transformation 

2

2

¨ = − , 0

H =  −  0, where 

0

and

0

are the constants determined by the linear algebraic system

 2

+

0

+(

2 0

2

 3

= 0,

0

+

3 0

+(

3

= 0,

leads to a homogeneous equation of the form 5.2.6.12:

21.

(ô

ï

+ õ )÷ (ö

¨ (‘  ) = H

1−

 O

ï

ó( ÷

)

+ ý )ñ

¨

,

HIP

=e ! ”

(ö

 (ˆ ) = ( 1 +  1 ˆ )1−  *O

where

ï

 +  The transformation ˆ = ln û û ˜(  + M û the form 5.2.6.77.

ñ

+ ý )÷

û

û

û,

–1

¨ =

•

 2 +  2ˆ .  3 +  3Ñ ˆ P

. ( (˜ + ) 

M

−1

leads to an autonomous equation of

Page 724

725

5.2. NONLINEAR EQUATIONS

22.

ñ ó( ÷

)

ï

= (ô

2

õï

+

+ö )

–

÷

!IOiñ (ô ï

1+ 2

2

õï

+

+ö )

÷

1– 2

1 b . The transformation

a =X





M

+ ' + (

2

P

.

¨ = ( 

,

leads to an autonomous equation with respect to ¨ by the substitution H (¨ ) = ¨ g
.

2

+ ' + ( )



1− 2

(1)

= ¨ (a ), which admits reduction of order

2 b . Let B = 2 be an even integer ( = 1, 2, 3,  ). In this case, transformation (1) yields ¡ ¡ an equation of the form 5.2.6.68, whose order can be reduced by two.

.

Setting


. by ¨  =

+ ' + ( , = ¨ . 
+ 1 − 2¡ O 2

= 

2

1+2 + − 2

.

O

.

.

+

2




−1 2

, we obtain

P

1−2 ¡ 2


+

and multiplying both sides of the original equation

.




 + (2

P

)

=  (¨ )¨ 
.

Integrating both sides of this equality with respect to  (the left-hand side is integrated by parts), we have

+½

−2



 +

(−1)  j ( 

 + (−1) +

−1− )

) (2

−1

X j ( +

+

−1) (

+1)

=0

where

 j (  ) = M  O  M

.

1−2 ¡ 2


+

.




=

P

.



( +1)

+ O, −

¡

+

M

 = X  (¨ ) ¨ + L , M

1 2P

.

(2)


(  ) + , ( , − 2 ) (  ¡

−1)

(remember that B = 2 ). It can be shown that the integrand on the left-hand side of (2) is a ¡ total differential. Finally, we arrive at the first integral:

+½ 

−2

(−1) 

=0

Q

.



+ (−1) +

( +1)

−1

1 2

¦

+, −

.

¡

.

+ 12 

V ( + ) W 2 −

1 2


(  ) + , ( , − 2 ) (  −1) S (2  + −1−  ) ¡ 
( + −1) ( + ) +  (1 − 2 ) ( + −2) ( + ) + 12  ¡ ¡

.

= Xv (¨ ) ¨ + L . 23.

ñ ó( ÷

)

=

ñ

1+ 1–

M

÷

÷ !IOiñ (ô ï

1 b . Setting  (‡ ) = ‡ 



2

õï

+

+1 −1 1 (

+ö )

÷

1– 2

P

2

V ( +

−1) 2

W §

.

‡ ), we have equation 5.2.6.22 with the function  1 (instead of  ). ‘ 2 b . The transformation  = H −1 , = H 1−  ¨ ( H ) leads to an equation of similar form: ¨ (  ) = 1−  1+  (−1)  ¨ 1−  *O ¨ ( (H 2 + ˜H +  ) 2 . P 24. 25. 26.

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

ó



ó

=  ^ ! (ñ – ^ ).  The substitution ¨ ( ) =  − 9 leads to an autonomous equation of the form 5.2.6.77.

ó

= ñ! (  ^ ñ ú ).  The transformation H =  9 + , ¨ ( H ) = 
  leads to an (B − 1)st-order equation.

= ï –÷ ! ( ïƒú  ^ ] ). The transformation H =  +  9  , ¨ ( H ) =  
 leads to an (B − 1)st-order equation.

Page 725

726 27.

28.

29.

30.

31.

32.

33.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó( ÷

ó

÷  ó

= ! (ñ + ô ) – ô The substitution ¨ ( ) = ¨ (  ) =  (¨ ). )

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

+1)

ñ ó(2÷

+1)

+  ´



leads to an autonomous equation of the form 5.2.6.8:

= ! (ñ + ô cosh ï ) – ô cosh ï . The substitution ¨ ( ) = +  cosh  leads to an autonomous equation of the form 5.2.6.8: ¨ (2  ) =  (¨ ).

= ! (ñ + ô sinh ï ) – ô sinh ï . The substitution ¨ ( ) = +  sinh  leads to an autonomous equation of the form 5.2.6.8: ¨ (2  ) =  (¨ ).

= ! (ñ + ô cosh ï ) – ô sinh ï . The substitution ¨ ( ) = +  cosh  leads to an autonomous equation of the form 5.2.6.8: ¨ (2  +1) =  (¨ ). = ! (ñ + ô sinh ï ) – ô cosh ï . The substitution ¨ ( ) = +  sinh  leads to an autonomous equation of the form 5.2.6.8: ¨ (2  +1) =  (¨ ).

ñ ó( ÷

)

= ! (ñ + ô cos ï ) – ô cos  ï + The substitution ¨ ( ) = +  cos  ¨ (  ) =  (¨ ).

ñ ó( ÷

)

= ! (ñ + ô sin ï ) – ô sin  ï + The substitution ¨ ( ) = +  sin  ¨ (  ) =  (¨ ).

ñ ó( ÷

)

= ! (ñ )ñ

óò

1 2

® ù 

. leads to an autonomous equation of the form 5.2.6.8:

® ù 

1 2

. leads to an autonomous equation of the form 5.2.6.8:

  =  ( , , 
 ). ( )

5.2.6-3. Equations of the form 34.

.

+ " ( ï ).

Integrating yields an (B − 1)st-order equation: 35. 36.

37.

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

39.

= X¯ ( )

M

+ Xw ( )  + L .

M

= ! ( ï , ñó ò ). The substitution ¨ ( ) = 
leads to an (B − 1)st-order equation: ¨ ( 

−1)

=  ( , ¨ ).

= ! (ñ , ñó ò ). Autonomous equation. This is a special case of equation 5.2.6.77. The substitution ¨ ( ) = 
leads to an (B − 1)st-order equation. =

ñ ï –÷ ! (ï ñ óò ¦ñ

).

The transformation H = 

38.



( −1)


 , ¨ = 

2



  leads to an (B − 2)nd-order equation.

= ô ÷ ñ + ! ( ï , ñó ò – ôñ ). The substitution ¨ = 
−  leads to an (B − 1)st-order equation:

ñ ó( ÷

)

= ! (ï , ï

¨ ( 

ñ óò

−1)

+  ¨ ( 

−2)

+ ))) +  

−1

¨ =  ( , ¨ ).

– ñ ).

The substitution ¨ = 


−

leads to an (B − 1)st-order equation:

M

¨ 
 −2 2 =  ( , ¨ ). M  −2 O   P

Page 726

727

5.2. NONLINEAR EQUATIONS

40.

41.

42.

ñ ó( ÷

)

!  ï , ï ñ óò

=

– þxñ  .

  Here, is a positive integer and B ≥ + 1. The substitution ¨ =  "
 − ¡ ¡ ¡ ¨ (  − + −1) (B − 1)st-order equation: =  ( , ), where = ¨ ( + )  . ïƒ÷ ñ ó(÷

= ! (ï , ï

ñóò

+

ôñ

!  ï , g ú ñóò

–

g úò ñ 

)

44.

45.

46.

47.

ñ ó( ÷

)

=

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

)

g ú

,

.


− +


+ ! (ï , ñ

=

ñ

+ ! ( ï , ñó ò sinh ï – ñ cosh ï ).

½ú



=0

ô  ü ï –1 , ù

>

þ

.

The substitution ¨ = 
sinh  − cosh  leads to an (2B − 1)st-order equation. = (–1)÷

ñ

+ ! ( ï , ñó ò sin ï – ñ cos ï ).

= (–1)÷

ñ

+ ! ( ï , ñó ò cos ï + ñ sin ï ).

The substitution ¨ = 
sin  − cos  leads to an (2B − 1)st-order equation. The substitution ¨ = 
cos  + sin  leads to an (2B − 1)st-order equation.

ñ ó÷

( )

=

r (ó ÷ ) r ñ

+ !IO ï ,  ñ óò

–

ñ ó( ÷

= ! (ï , ï

)

ñóò

r óò r ñ P

,

r

=

r (ï

).

leads to an (B − 1)st-order equation.

  =  ( , , 
 , 

 ). ( )

– ñ , ñó ò ò ó ).

This is a special case of equation 5.2.6.78. The substitution ¨ ( ) = 
 − (B − 1)st-order equation.

ñ ó( ÷

= ! (ï , ï

)

2

ñóò ò ó

– 2ï

ñóò

+ 2ñ ).

This is a special case of equation 5.2.6.81. The substitution ¨ ( ) =  to an (B − 2)nd-order equation.

ñ ó(2÷

)

=

ô ÷ ñ

The substitution ¨ = 

 − 

ñ ó(2÷

)

=

2

leads to an



 − 2 
 + 2 leads

+ ! ( ï , ñó ò ò ó – ôñ ).

¨ (2  51.

=


The substitution ¨ =  cosh  − sinh  leads to an (2B − 1)st-order equation.

5.2.6-4. Equations of the form

50.

=0

g úò

,

cosh ï – ñ sinh ï ).

Y

49.



ô  ï

leads to an (B − 1)st-order equation.

ñ

óò

½ú

=

=

Y
 The substitution ¨ = 
−

48.

.

Here, (  )  =  (  + 1)  ( + B − 1) is the Pochhammer symbol. The substitution ¨ = 
 +  leads to an (B − 1)st-order equation.

. The substitution ¨ = + 43.

ñ

) – ( ô )÷

leads to an

ñ!  ñ ñóò ò ó

leads to an (2B − 2)nd-order equation: −2)

  +  ¨ (2

−4)

+ ))) +  

−1

¨ =  ( , ¨ ).

– ñó ò 2  .

This is a special case of equation 5.2.6.52.

Page 727

728 52.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

= ñó ò ò ó'!  ññó ò ò ó – ñó ò 2  + ñ"  1 b . Particular solution: =L )

where the constants L

1

– ñó ò 2  .

exp( L

3

 )+L

2

exp(− L

3

 ),

L 2 , and L 2 are related by the constraint L 32  − L 32  (4 L 1 L 2 L 32 ) − (4 L 1 L 2 L 32 ) = 0.

1,

2 b . Particular solution: where the constants L

ññóò ò ó

=L

1

cos( L

3

 )+L

2

sin( L

3

 ),

L 2 , and L 2 are related by the constraint (−1)  L 32  + L 32  (− L 12 L 32 − L 22 L 32 ) − (− L 12 L 32 − L

53.

ñ ó( ÷

)

=

1,

ñ óò ò ó ñ óò !IO ó , ñ óò ñ ò

ñ óò ò ó ñ óò

–ñ

2 2

2 3)

L

= 0.

.

P

Particular solution: = L 1 exp( L 2  )+ L 3 , where L 1 is an arbitrary constant and the constants L 2 and L 2 are related by the constraint L 2 −1 =  ( L 2 , − L 2 L 3 ). 5.2.6-5. Equations of the form  ( , ) (  ) + ( , , 
 ) (  54.

ñ ó( ÷

−2)

.

= ôññ ó ò ò ó ò ò ó ó – ô (ñ ó ò ò ó )2 + ! ( ï ). Integrating the equation two times, we obtain an (B − 2)nd-order equation:



56.

= %   , , 
 ,  , ( 

)

( −2)

55.

−1)

ñ ó(2÷




=  !   −  (  )2 + L

1



 +L

2

+ X  ( − a )  (a ) a , M 0

where 

= ô 2 ñ + !u ï , ñ ó(÷ ) + ôñ  . The substitution ¨ = (  ) +  leads to an B th-order equation: ¨ ( 

ñ ó( ÷

0

is an arbitrary number.

)

=  ¨ +  ( , ¨ ).

)

= !u ñ ó(÷ –2)  . Having set ‡ ( ) = (  the form: )

−2)

 =X

M

, we obtain a second-order equation ‡

 =  (‡ ), whose solution has

‡

+L

2,

Y (‡ ) = A L Q

where

1

+2X

J

1 2

 (‡ ) ‡ S M

.

Y (‡ ) Expressing ‡ in terms of  and integrating the resulting relation B − 2 times, we find . Solution in parametric form:

57.

ñ ó( ÷

 ‡  =X k M , 2 Y (‡ )

=X

‡

1

k 3 Y (‡ 1 ) X k M



1 4

‡

n n

−4 −1

‡ M  Y (‡ 

−3

X −3 ) k

n n

−3

‡ 

−2 ‡  −2 . M Y (‡  −2 )

= ! (ñ )ñ ó ò ñ ó(÷ –1) . Integrating yields an (B − 1)st-order autonomous equation of the form 5.2.6.8:



ñ ó( ÷

)

= V ! (ñ )ñó ò + " ( ï )WNñ

=  ( ),

ó( ÷

–1)

where

ï ñ ó( ÷

 ( ) = L exp X  ( ) S . Q M

.

Integrating yields an (B − 1)st-order equation: 59.

2

'X k Y (‡ 2 ) M

)

( −1)

58.





( −1)

= L exp X

Q

 ( ) M

+ X ( )  S .

M

+ ùñ ó(÷ –1) = ! ( ï ñ ). The substitution ¨ ( ) =  leads to an autonomous equation of the form 5.2.6.8: ¨ (  ) =  (¨ ). )

Page 728

729

5.2. NONLINEAR EQUATIONS

60. 61. 62.

ï ñ ó( ÷

+ ( ô + ù – 1)ñ ó(÷ –1) = ! ( ï , ï ñó ò + ôñ ). The substitution ¨ =  
+  leads to an (B − 1)st-order equation: ¨ (  )

ï 2 ñ ó( ÷

−1)

=  ( , ¨ ).

+ 2ù ï ñ ó(÷ –1) + ù (ù – 1)ñ ó(÷ –2) = ! ( ï 2 ñ ). The substitution ¨ ( ) =  2 leads to an autonomous equation of the form 5.2.6.8: ¨ (  ) =  (¨ ). )

(2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ ó ò ñ ó(2÷ ) = ! ( ï ). Having integrated the equation, we obtain

  )+2 (2B − 1)! (2

½



−1

(−1) 



   + (−1) 

+1 ( ) (2 − )

+1

=1

V (  ) W 2 = X  ( )  + 2 L 2 . M

The second integration leads to an (2B − 1)st-order equation:

½

−1



(2B − 1 − 2 , )(−1)  ( 

 

 = 2L 2  + L

) (2 −1− )



+ X  ( − a )  (a ) a . M 0

1

=0

The third integration leads to an (2B − 2)nd-order equation:

½

−2



( , + 1)(2B − , − 1)(−1)  ( 

 + 1 (−1) 

 

) (2 −2− )

2

=0

=L 63.

2

2

ññ ó(÷

64.

)

ó( ÷

– ñó òñ

–1)

½

−1



(−1) 



   + (−1) 

+1 ( ) (2 − )

+1

ññ ó ÷

( )

65.

= ! ( ï )ñ 2 .

= ñó ò ñ ó÷

( –1)

+ ! ( ï ) ñ ñ ó( ÷

–1)

.

Integrating yields an (B − 1)st-order linear equation:

68.

+L

V ( 

−1) 2

1 + L

W

0+

ññ ó ÷



1 X (  − a )2  ( a ) a . 2  0 M

V (  ) W 2 = Xv ( )

=1

Integrating yields an (B − 1)st-order linear equation:

67.

B

(2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ ò ñ ó(2÷ ) = ! (ñ )ñóò + " (ï ). Integrating yields an (B − 1)st-order equation:

  )+2 (2B − 1)! (2

66.

−1 2

+ ( ! – 1)ñ ó ò ñ ó ÷

+ XÈ ( )  + L .

M

M

( −1)



= X¯ ( )  + L2S .



= L exp X¯ ( )  S .

( −1)

Q

M

Q

! = ! (ï

" = " (ï

M

+ !"Iññ ó ò + " ó ò ñ = 0, ), ). This equation is solved by the functions that are solutions of the (B − 1)st-order linear equation (  −1) + ( ) = 0. ( )

( –1)

2

[ñ + ! (ï )]ñ ó(÷ ) = [ñ ó ò + ! ó ò ( ï )]ñ ó(÷ –1) + ô! ( ï )ñ ó ò – ô! ó ò ( ï )ñ . Integrating yields an (B − 1)st-order constant coefficient nonhomogeneous linear equation: (  −1) −L = ( L −  )  ( ). There is also the trivial solution = 0.

½

ú

÷

=1

ô ú ñ ó(2ú

)

= ! (ñ ).

The first integral has the form:

½ 

+

 + =1

™

+½

Á

−1

(−1) =1

Á ( Á

1  + − Á ) + (−1) + V ( + ) W

) (2

2

2

¢

+X

 ( ) M

=L ,

where L is an arbitrary constant. Furthermore, the order of the obtained equation can be reduced by one by the substitution ¨ ( ) = 
 .

Page 729

730

HIGHER-ORDER DIFFERENTIAL EQUATIONS

÷ ½

69.

ú

ô ú ïƒú ñ ó(ú

=1

= ! (ñ ).

)

The substitution a = ln | | leads to an autonomous equation of the form 5.2.6.77.

½

ñ

70.

÷

ú

=0

ô ú ñ ó(2ú

= ! ( ï ).

+1)

Integrating yields a 2B th-order equation:

½  +

™

 + =0

2

+½

−1

Á

(−1)

Á ( Á

 + − Á ) + (−1) + V ( + ) W

2

) (2

=0

¢

=2X

 ( )  + L , M

 stands for . where (0) 71.

½

ú

÷

ô ú ñ ó(ú ) ñ ó(2÷

=0

+1–

ú

)

= ! ( ï ).

The first integral has the form: 2 where # + =

7+ 

=0

½

−1

+

( +

# +

  +

) (2 −

)

(−1) +

=0 +

   =  + − +

−1 +

+ #  V (  ) W

 +

−2 −

2

=2X

 ( )  + L , M 7

))) . If the condition #  = 2 +

−1

(−1) 

−1+

=0

+ # +

is satisfied, the obtained equation can be integrated two times more (see, in particular, equation 5.2.6.62).

 −1)  .   =    , , 
 ,  , (? ( )

5.2.6-6. Equations of the form 72.

ñ ó( ÷

= !  ñ ó(÷ –1)  . Having set ‡ ( ) = (  )

relation  = X

−1)

‡

, we obtain a first-order equation ‡
 =  (‡ ). Further, find ‡ from the

+ L 1 . Then the (B − 1)-fold integration yields . M  (‡ )

Solution in parametric form:

73.

ñ ó( ÷

 ‡  =X k M , 1  (‡ )

=X



‡

1

k 2  (‡ 1 ) X k M



1 3

‡

2

'X k  (‡ 2 ) M

= ! (ñ )ñó ò "  ñ ó(÷ –1)  . Integrating yields an (B − 1)st-order equation: )

X

¨ M ¨ = X¯ ( ) ( ) M

+L ,

n n

−3 −1

where

‡ M   (‡ 

−2

X −2 ) k

¨ = ( 

n n

−1)

−2

‡ 

‡ M  −1 .  (‡  −1 ) −1

.

Furthermore, the order of this equation can be reduced by one by the substitution H ( ) = /
 . 74.

ñ ó( ÷

= V ! (ñ )ñó ò + " ( ï )W²% (ñ ó(÷ –1) ). Integrating yields an (B − 1)st-order equation: )

ù 75.

ñ ó( ÷

¨ M ¨ =ù % ( )

 ( ) M

+ù

( )  + L , M

¨ = ( 

−1)

.

= !  ï , ñ ó(÷ –2) , ñ ó(÷ –1)  . ¨ 

 =  ( , ¨ , ¨2
). The substitution ¨ ( ) = (  −2) leads to a second-order equation: u )

Page 730

731

5.2. NONLINEAR EQUATIONS ( )  5.2.6-7. Equations of the general form    , , 
 ,  ,  = 0.

76.

H  ï , ñ óò , ñ óò ò ó , u uu , ñ ó(÷ ) 

= 0.

explicitly. Hence, the substitution ¨ ( ) =
 leads to an

The equation does not depend on (B − 1)st-order equation:

   , ¨ , ¨ 
,  , ¨ ( 

77.

78.

H  ñ , ñ óò , ñ óò ò ó , u uu , ñ ó(÷ ) 

Autonomous equation. It does not depend on  explicitly. The substitution ¨ ( ) = /
 leads to an (B − 1)st-order equation. The derivatives of the original equation and the transformed one are related by

H  ï , ï ñ óò



 = ¨*¨
, 




 = ¨

2

¨

+ ¨ ( ¨
)2 ,  

ñóò ò ó , ñ óò ò ó ò ó , u uu , ñ ó(÷ )  = 0. The substitution ¨ ( ) =  
 −  leads to  an (B

−1)


 . 

−2)

− 1)st-order equation: 

 = 0,

= ¨ 
 .

where

– 2ñ ,

H  ï , ï ñ óò

– þxñ ,

ñ ó( ú

+1)

The substitution ¨ =  


ñ ó( ú − ¡  ,

+2)

,

H  ï , ï 2 ñ óò ò ó

– 2ï

ñóò



(–1)  =0

, !(

¡

= ¨ 

  .

where

ù

= 0,

≥

þ

+ 1.

leads to an (B − 1)st-order equation:  −1)

ñóò ò ó ò ó , u uu , ñ ó(÷ 
2 

   − 2  + 2

ü ! = ú  ï ú –  ñ ó( ú –  ) = H ( ï , ñ ó( ú

Here, L +  =

 = 0,

+ 2ñ ,

The substitution ¨ ( ) = 

½ú

−3)

uuu , ñ ó(÷ ) 

  , ¨ , , 
,  ,  (  82.

( )

ñ óò ò ó ò ó , ñ óò ò ó ò ò ó ó , u uu , ñ ó(÷ )  = 0. The substitution ¨ =  
−  2  leads to an (B − 1)st-order equation:  HÊ ï , ï ñ óò

   , ¨ , , 
,  ,  (  − + 81.

  = ¨  ( 

 ,

–ñ ,

   , ¨ , , 
,  ,  ( 

80.

 = 0.

= 0.

   , ¨ , , 
,  ,  ( 

79.

−1)

)

 = 0,

= ¨ ( +

where

)

 .

 = 0.

leads to an (B − 2)nd-order equation: 

−3)

 = 0,

+1)

,

where

uuu , ñ ó(÷

)

=

−2

¨ 
.

).

! are binomial coefficients. − , )!

¡ 7+ The substitution ¨ ( ) = (−1)  , ! L +   + 

=0

−

 ( + −  ) leads to an (B − )th-order equation; ¡

the derivatives on the right-hand side are calculated in consecutive manner using the formula ( + +1) =  − +3¨2
. 83.

HwO ï

ñ , ñ óò , ï ñ óò ò ó , u uu , ïƒ÷ –1 ñ ó(÷

)

P

= 0.

Homogeneous equation. The transformation a = ln  , ¨ equation of the form 5.2.6.77.

=  

leads to an autonomous

Page 731

732 84.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ô ï  ï

+ õÿñ + ö , ñ ó ò , uuu , ( ô ï + õÿñ + ö )÷ –1 ñ ó(÷ ) = 0. P + ñ + ¨ 1 b . For ý − ˜ƒ = 0 , the substitution  =  +  + ( leads to an autonomous equation of the form 5.2.6.77.

HwO

2 b . For ý − ˜ƒ ≠ 0, the transformation

¨ = − , 0

H =  −  0, where 

and

0

0

are the constants determined by the linear algebraic system



0

+

0

+ ( = 0,

ƒ…

0

+ý

0

+ = = 0,

−1

H 

leads to a homogeneous equation of the form 5.2.6.83:

 O 85.

HwO

ô 1ï

+ +

ô 2ï

õ 1ñ õ 2ñ

+ö +ö

1

,

2

¨ H  + u ¨ ‘
¨u  ¨ H , ,  , (  +  H ) ƒ +ý u

ñóò , u uu , (ô 3 ï 

Let the following condition hold: ûû 

+

õ 3ñ

1



+ ö 3 )÷ 1

(

ñ ó÷

–1 ( )

−1

¨ (‘ 

)

= 0.

P

= 0.

P

1

( 2 ûû = 0. û 3 3 (3û û û û û For  1  2 −  2  1 ≠ 0, the transformation H =  −  0, ¨ = − 0,

where 

and

0

0



2

2

are the constants determined by the linear algebraic system

 1

0

+

1 0

+(

1

= 0,

 2

0

+

2 0

+(

2

H 

−1

= 0,

leads to a homogeneous equation of the form 5.2.6.83:

 O 86.

87.

¨ H  1 +  1u ¨ ‘
¨u  ¨ H , ,  , (  3 +  3 H )  2 +  2u

−1

H  ï  ñ , ï +1 ñ óò , u uu , ï +÷ ñ ó(÷ ) 

)

P

= 0. Generalized homogeneous equation. The transformation a = ln  , ¨ autonomous equation of the form 5.2.6.77.

H ”

ï ñ óò ñ

ï 2 ñ óò ò ó ï ÷ ñ ó( ÷ ,  u uu , ñ ñ

,

)

•

Hv”Èñóò

ñ óò ò ó ñ óò

–ñ

,

ñ óò ò ó ñ óò

,

= 0.

=  

leads to an

= 0.

Generalized homogeneous equation. The transformation H =  an (B − 2)nd-order equation. 88.

¨ (‘ 

ñ óò ò ó ò ó ñ ó( ÷ ,  u uu , ñ óò ñ óò

)

•


 , ¨ = 

2



  leads to

= 0.

Autonomous equation. Particular solution: = L 1 exp( L 2  ) + L 3 , where L 1 is an arbitrary constant and the constants L 2 and L 3 are related by  (− L 2 L 3 , L 2 , L 22 ,  , L 2 −1 ) = 0. 89.

H ” ï ñ ú

,

ï ñ óò ñ

,

ï 2 ñ óò ò ó ï ÷ ñ ó( ÷ ,  u uu , ñ ñ

)

•

= 0.

Generalized homogeneous equation. The transformation a =   + , H =  (B − 1)st-order equation.


 leads to an

Page 732

5.2. NONLINEAR EQUATIONS

90.

Hv”

ñ ó( ÷ ñ óò

)

ñ ó( ÷ ñ óò

ñ

,

)

–ñ

ó( ÷

–1)

733

= 0.

•

A solution of this equation is any function that satisfies the (B − 1)st-order constant coefficient linear equation (  −1) = L 1 + L 2 , where the constants L 1 and L 2 are related by the constraint  ( L 1 , − L 2 ) = 0. 91.

Hv”

ñ ó(÷ ) ï 1– ñ ñ , ñ ó(  ) ñ

ó(÷ ) ï 1– (÷ – ) ñó • – ó(  )

ù

= 0,

>ü .

A solution of this equation is any function that satisfies the (B − , )th-order linear equation (  −  ) = L 1 + L 2   −1 , where the constants L 1 and L 2 are related by  ( L 1 , − L 2 ) = 0. 92. 93.

H  ï , ñ ó( ÷

– ñ , ñ ó(ú ) – ñ  = 0. The substitution ¨ = 
− reduces the order of the equation by one.

H  ñ ó( ÷

)

– ñ ,ñ

ó(2÷

–ñ ,ñ

ó(2÷

ó( ÷

)

ñ , ñ ó(2÷

)

–ñ

 = 0.  The substitution ‡ =  − leads to an B th-order autonomous equation   ‡ , ‡ (  )+ ‡ , ‡ (  )  = 0. )

)

)

( )

94.

H  ï , ñ ó( ÷

)

+

ôñ , ñ ó(2÷

)

–ô

The substitution ‡ =   +  ( )

95.

96.

2

+

ôñ ó(÷

)

 = 0.

 )  = 0. leads to an B th-order equation    , ‡ , ‡ (  ) − ‡ , ‡ (v

ó ó ó ó HÊp ^ ñ ,  ^ ñ óò ,  ^ ñ óò ò ó , u uu ,  ^ ñ ó(÷ ) 

= 0.  Equation invariant under “translation–dilatation” transformation. The substitution ‡ =  9 leads to an autonomous equation of the form 5.2.6.77.

ó Hv”- ^ ñ ú

,

ñ óò ñ

,

ñ óò ò ó ñ ó( ÷ ,  u uu , ñ ñ

)

•

= 0.

Equation invariant under “translation–dilatation” transformation. The transformation  H =  9 + , ¨ = 
  leads to an (B − 1)st-order equation. See also Paragraph 0.5.2-7. 97.

H  ïƒú  ^ ] , ï ñ óò , ï 2 ñ óò ò ó , u uu , ƒï ÷ ñ ó(÷ ) 

= 0. Equation invariant under “dilatation–translation” transformation. The transformation H =  +  9  , ¨ =  
leads to an (B − 1)st-order equation. See also Paragraph 0.5.2-8.

Page 733

Supplements S.1. Elementary Functions and Their Properties ]

Throughout Section S.1 it is assumed that B is a positive integer unless otherwise specified.

S.1.1. Trigonometric Functions S.1.1-1. Simplest relations. sin2  + cos2  = 1, sin(− ) = − sin  , sin  tan  = , cos  tan(− ) = − tan  , 1 1 + tan2  = , cos2 

tan  cot  = 1, cos(− ) = cos  , cos  cot  = , sin  cot(− ) = − cot  , 1 1 + cot2  = . sin2 

S.1.1-2. Relations between trigonometric functions of single argument. tan  1 sin  = A £ 1 − cos2  = A £ =A £ , 2 1 + tan  1 + cot2  1 cot  cos  = A £ 1 − sin2  = A £ =A £ , 2 1 + tan  1 + cot2  £ 1 − cos2  1 sin  tan  = A £ =A = , 2 cos  cot  1 − sin  cot  = A

£ 1 − sin2  cos  1 =A £ = . 2 sin  tan  1 − cos 

S.1.1-3. Reduction formulas. ) = (−1)  sin  , 2B + 1 sin OuA = A (−1)  cos  , 2  P tan(pAIB ) = tan  ,  2B + 1 tan OpA = − cot  , 2  P

) = (−1)  cos  , 2B + 1 cos OpA = T (−1)  sin  , 2  P cot(pAIB ) = cot  ,  2B + 1 cot OpA = − tan  , 2  P

sin OuA}

cos OpA}

sin(uACB



4P

=

£ 2

(sin pA cos  ),

2 tan pA 1 tan OpA} = , 4P 1 T tan 

cos(pAIB



£ 2

(cos uT sin  ), 2 cot uT 1 cot OpA} = . 4P 1 A cot  4P

=

Page 735

736

SUPPLEMENTS

S.1.1-4. Addition and subtraction of trigonometric functions. sin  + sin = 2 sin O sin  cos  cos  sin2 

 +

cos O

 −

, 2 P 2 P  −  + − sin = 2 sin O cos O , 2 P 2 P  +  − + cos = 2 cos O cos O , 2 P 2 P  +  − − cos = −2 sin O sin O , 2 P 2 P − sin2 = cos2 − cos2  = sin( + ) sin( − ),

sin2  − cos2

= − cos( + ) cos( − ), sin(pA ) sin( AI ) tan pA tan = , cot pA cot = , cos  cos sin  sin  cos  +  sin  = & sin( + Y ) = & cos( − j ).

Here, & = £ 

2

+  2 , sin Y =  & , cos Y =  & , sin j =  & , and cos j =  & .

S.1.1-5. Products of trigonometric functions. sin  sin = 12 [cos( − ) − cos( + )],

cos  cos = 12 [cos( − ) + cos( + )], sin  cos = 12 [sin( − ) + sin( + )].

S.1.1-6. Powers of trigonometric functions. cos2  =

1 2

cos 2 + 12 ,

cos3  =

1 4

cos 3 +

3 4

cos  ,

sin3  = − 14 sin 3 +

cos4  =

1 8

cos 4 +

1 2

cos 2 + 38 ,

sin4  =

1 8

cos5  =

1 16

sin5  =

1 16

cos 5 +

5 16

cos2   =

sin2  = − 12 cos 2 + 12 ,

cos 3 + 1



22 −1

½

5 8



−1 =0

cos  ,

cos 4 − sin 5 −

3 4 1 2

sin  ,

cos 2 + 38 ,

5 16

sin 3 +

5 8

sin  ,

1 L 2  cos[2(B − , ) ] + 2  L 2  , 2

 1 ½ cos2  +1  = 2  L 2  +1 cos[(2B − 2 , + 1) ], 2  =0  −1 1 ½ 1 2 sin  = 2  −1 (−1)  −  L 2  cos[2(B − , ) ] + 2  L 2  , 2 2  =0  ½ 1 sin2  +1  = 2  (−1)  −  L 2  +1 sin[(2B − 2 , + 1) ]. 2  =0 Here, L +  =

!

are binomial coefficients (0! = 1). ¡ , ! ( − , )! ¡

Page 736

737

S.1. ELEMENTARY FUNCTIONS AND THEIR PROPERTIES

S.1.1-7. Addition formulas. sin(uA tan(pA

) = cos  cos T sin  sin , 1 T tan  tan )= . tan pA tan

cos(uA

) = sin  cos A cos  sin , tan uA tan )= , 1 T tan  tan

cot(uA

S.1.1-8. Trigonometric functions of multiple arguments. cos 2 = 2 cos2  − 1 = 1 − 2 sin2  ,

sin 2 = 2 sin  cos  ,

cos 3 = −3 cos  + 4 cos3  ,

sin 3 = 3 sin  − 4 sin3  ,

cos 4 = 1 − 8 cos2  + 8 cos4  ,

sin 4 = 4 cos  (sin  − 2 sin3  ),

cos 5 = 5 cos  − 20 cos3  + 16 cos5  ,

sin 5 = 5 sin  − 20 sin3  + 16 sin5  ,

cos(2B/ ) = 1 +

½  

(−1) 

B 2 (B

=1

cos[(2B +1) ] = cos 

™

1+

½ 

(−1)  =1

½

sin(2 B/ ) = 2B cos w¬ sin  +



sin[(2B +1) ] = (2B +1) ™ sin  + tan 2 =

2 tan  , 1 − tan2 

− 1)  [B 2 − ( , − 1)2 ]  4 sin2   , (2 , )!

2

[(2B +1)2 −1][(2B +1)2 −32 ]  [(2B +1)2 −(2 , −1)2] sin2  ˆ¢ , (2 , )!

(−4) 

(B

(−1) 

[(2B +1)2−1][(2B +1)2−32]  [(2B +1)2−(2 , −1)2] sin2  (2 , +1)!

2

− 1)(B

=1

½  

=1

tan 3 =

− 22 )  (B (2 , − 1)!

2

3 tan  − tan3  , 1 − 3 tan2 

2

tan 4 =

− , 2)

sin2 

−1

 ® , +1

^¢ ,

4 tan  − 4 tan3  . 1 − 6 tan2  + tan4 

S.1.1-9. Trigonometric functions of half argument.



1 − cos   1 + cos  , cos2 = , 2 2 2  sin  1 − cos   sin  1 + cos  tan = = , cot = = , 2 1 + cos  sin  2 1 − cos  sin     2 tan 2 1 − tan2 2 2 tan 2 sin  =  , cos  =  , tan  =  . 2 1 + tan 2 1 + tan2 2 1 − tan2 2 sin2

2

=

S.1.1-10. Euler and de Moivre formulas. Relationship with hyperbolic functions.

   + © =   (cos  + ™ sin  ), (cos  + ™ sin  )  = cos(B/ ) + ™ sin(B/ ), ™ 2 = −1, sin(™Ý ) = ™ sinh  , cos(™Ý ) = cosh  , tan(™Ý ) = ™ tanh  , cot(ݙ  ) = −™ coth  . S.1.1-11. Differentiation formulas.

M

sin 

M



= cos  ,

M

cos 

M



= − sin  ,

M

tan 

M



=

1 , cos2 

M

cot 

M



=−

1 . sin2 

Page 737

738

SUPPLEMENTS

S.1.1-12. Expansion into power series.



cos  = 1 − sin  =  −



2



+

2! 3



4



4!

−

5



6

6!

+ )))

(| | < F ),

7

+ ))) 5! 7!  3 2 5 17 7 + + + ))) tan  =  + 3 15 315 1   3 2 5 cot  = − − − − )))  3 45 945 +

3!

−

(| | < F ), (| | < 2),



(| | < ).



S.1.2. Hyperbolic Functions S.1.2-1. Definitions. sinh  =





− 2

−

 ,

cosh  =





+ 2

−

 ,





− − tanh  =   ,  + −







+ − coth  =   .  − −



S.1.2-2. Simplest relations. cosh2  − sinh2  = 1, sinh(− ) = − sinh  , sinh  tanh  = , cosh  tanh(− ) = − tanh  , 1 , 1 − tanh2  = cosh2 

tanh  coth  = 1, cosh(−  ) = cosh  , cosh  coth  = , sinh  coth(−  ) = − coth  , 1 . coth2  − 1 = sinh2 

S.1.2-3. Relations between hyperbolic functions of single argument ( ≥ 0). tanh 

sinh  = ü cosh2  − 1 = cosh  = ü sinh2  + 1 =

ü coth  = ü

ü 1 − tanh  1 ü

1 − tanh2 

1

= =

2 ü coth  − 1 coth 

ü

coth2  − 1

, ,

cosh2  − 1 1 = ü = , 2 cosh  coth  sinh  + 1 sinh 

tanh  =

2

sinh2  + 1 = sinh 

cosh  2

ü cosh  − 1

=

1 . tanh 

S.1.2-4. Addition formulas. sinh(uA

) = sinh  cosh A sinh cosh  , tanh pA tanh tanh(uA ) = , 1 A tanh  tanh

cosh(pA coth(uA

) = cosh  cosh A sinh  sinh , coth  coth A 1 )= . coth A coth 

Page 738

739

S.1. ELEMENTARY FUNCTIONS AND THEIR PROPERTIES

S.1.2-5. Addition and subtraction of hyperbolic functions. sinh uA sinh = 2 sinh O

pA 

cosh  + cosh = 2 cosh O cosh  − cosh = 2 sinh O sinh2  − sinh2 2



= cosh2 

uT

cosh O 2 P 2 P +  − cosh O 2 P 2 +  − sinh O 2 P 2 P 2 − cosh = sinh(

, ,

P ,

+ ) sinh( − ),

2

sinh  + cosh

= cosh( + ) cosh( − ), sinh(uA ) sinh(pA ) tanh pA tanh = , coth pA coth = A . cosh  cosh sinh  sinh

S.1.2-6. Products of hyperbolic functions. sinh  sinh = 12 [cosh( + ) − cosh( − )],

cosh  cosh = 12 [cosh( + ) + cosh( − )], sinh  cosh = 12 [sinh( + ) + sinh( − )]. S.1.2-7. Powers of hyperbolic functions. cosh2  = cosh3  = cosh4  = cosh5  =

Here, L + 

sinh2  =

1 2 cosh 2 1 4 cosh 3 1 8 cosh 4 1 16 cosh 5

 + 12 ,

1 2 cosh 2 1 4 sinh 3 1 8 cosh 4 1 16 sinh 5

 − 12 ,

 + 34 cosh  , sinh3  =  − 34 sinh  ,  + 12 cosh 2 + 38 , sinh4  =  − 12 cosh 2 + 38 ,  + 165 cosh 3 + 58 cosh  , sinh5  =  − 165 sinh 3 + 58 sinh  ,  −1 1 ½ 1 cosh2   = 2  −1 L 2  cosh[2(B − , ) ] + 2  L 2  , 2 2  =0  ½ 1 cosh2  +1  = 2  L 2  +1 cosh[(2B − 2 , + 1) ], 2  =0  −1 ½ 1 (−1)  2 sinh  = 2  −1 (−1)  L 2  cosh[2(B − , ) ] + 2  L 2  , 2 2  =0  1 ½ sinh2  +1  = 2  (−1)  L 2  +1 sinh[(2B − 2 , + 1) ]. 2  =0

are binomial coefficients.

S.1.2-8. Hyperbolic functions of multiple arguments. cosh 2 = 2 cosh2  − 1,

sinh 2 = 2 sinh  cosh  ,

cosh 3 = −3 cosh  + 4 cosh  ,

sinh 3 = 3 sinh  + 4 sinh3  ,

cosh 4 = 1 − 8 cosh2  + 8 cosh4  ,

sinh 4  = 4 cosh  (sinh  + 2 sinh3  ),

cosh 5 = 5 cosh  − 20 cosh3  + 16 cosh5  ,

sinh 5  = 5 sinh  + 20 sinh3  + 16 sinh5  .

3

Page 739

740

SUPPLEMENTS cosh(B/ ) = 2 

−1

[ ´J B ½

cosh   +

2



½

=0

J

[( −1) 2]

sinh( B/ ) = sinh  Here, L + 



(−1)  +1  −2  L 2 , + 1  −  −2

2]

2 − 

−1

=0

L   −



(cosh  ) 

 )

−2 −1

−2 −2

−1 (cosh





−2 −2

,

.

are binomial coefficients and [# ] stands for the integer part of a number # .

S.1.2-9. Relationship with trigonometric functions. sinh(™Ý ) = ™ sin  ,

cosh(™Ý ) = cos  ,

tanh(™Ý ) = ™ tan  ,

coth(™Ý ) = −™ cot  ,

™ 2 = −1.

S.1.2-10. Differentiation formulas.

M

sinh 

M



= cosh  ,

M

cosh 

M



= sinh  ,

tanh 

M

M

=



1 , cosh2 

M

coth 

M



=−

1 . sinh2 

S.1.2-11. Expansion into power series.



cosh  = 1 + sinh  =  +



2

2! 3



+



4

4! 5



+



6

6!

+ )))

7

(| | < F ),

+ ))) 5! 7!  3 2 5 17 7 tanh  =  − + − + ))) 3 15 315  3 2 5 1  + − ))) coth  = + −  3 45 945 3!

+

+

(| | < F ), (| | < 2),



(| | < ).



S.1.3. Inverse Trigonometric Functions S.1.3-1. Definitions and some properties. sin(arcsin  ) =  , tan(arctan  ) =  ,

cos(arccos  ) =  , cot(arccot  ) =  .

Principal values of inverse trigonometric functions are defined by the inequalities: − z2 ≤ arcsin  ≤ − z2 < arctan  <

z2 , z2 ,

0 ≤ arccos  ≤  0 < arccot  <



(−1 ≤  ≤ 1), (− F <  < F ).

Page 740

741

S.1. ELEMENTARY FUNCTIONS AND THEIR PROPERTIES

S.1.3-2. Simplest formulas. arcsin(− ) = − arcsin  , arctan(− ) = − arctan  , arcsin(sin  ) =

™

 − 2B  − + 2(B + 1)

arccos(cos  ) =

™

 − 2B  − + 2(B + 1)

arctan(tan  ) =  − B arccot(cot  ) =  − B

arccos(− ) = − arccos  ,  arccot(−  ) = − arccot  ,



if 2B − z 2 ≤  ≤ 2B + z 2 ,   if (2B + 1) − z 2 ≤  ≤ 2(B + 1) +







if 2B ≤  ≤ (2B + 1) ,   if (2B + 1) ≤  ≤ 2(B + 1) ,



B

if if

 



B



 z + 2,

− z2



z2 ,

<


S.1.3-3. Relations between inverse trigonometric functions.






arccos £ 1−  2 − arccos £ 1− 












arcsin  =






arctan £ 1−  £ 1−  arccot










arctan  =

z2 ,

arcsin  +arccos  =




























arcsin £ 1+  2 1 arccos £ 1+  2 1 − arccos £ 1+  1 arccot








2

if 0 ≤  ≤ 1, if −1 ≤  ≤ 0, if −1 <  < 1,

2 2

arctan  +arccot  =

−



arccos  =








1 arcsin £ 1+ 






if  ≤ 0, if  > 0;

if −1 <  < 1;

2

if  > 0,

2


  −arcsin £ 1 1+  =





1

arctan 
 1 +arctan  

if 0 ≤  ≤ 1, if −1 ≤  ≤ 0, if 0 <  ≤ 1,



arccot £ 1− 






arccot 

2







if  ≥ 0, 2

arcsin £ 1−  2 £ 1−   −arcsin £ 1−  2 arctan






if −1 ≤  < 0; for any  ,

z2 ;

2

if  < 0, if  > 0, if  < 0.

S.1.3-4. Addition and subtraction of inverse trigonometric functions. arcsin  + arcsin = arcsin  

1−

ü

arccos pA arccos = A arccos V 

T

2

+ £ 1−

2



(1 −  2 )(1 − 2 ) W  + arctan  + arctan = arctan for  1−  − arctan  − arctan = arctan for  1+

ü

2

for



for

pA

+

2

≤ 1,

≥ 0,

< 1, > −1.

S.1.3-5. Differentiation formulas. 1

1

, M arcsin  = £  1− 2 M

M

1

M arctan  =  1+

2

, M arccos  = − £  1− 2 M 1

,

M

. M arccot  = −  1+ 2

Page 741

742

SUPPLEMENTS

S.1.3-6. Expansion into power series. arcsin  =  + arctan  =  −

1  3 1×3  5 1×3×5  7 + + + ))) 2 3 2×4 5 2×4×6 7



3



5



(| | < 1),

7

+ ))) (| | ≤ 1), 5 7 1 1 (| | > 1). arctan  =  − + 3 − 5 + ))) 2  3 5 The expansions for arccos  and arccot  can be obtained with the aid of the formulas arccos  = z − arcsin  and arccot  = z − arctan  . 2 2 3 1

+

−

S.1.4. Inverse Hyperbolic Functions S.1.4-1. Relationships with logarithmic functions. arcsinh  = ln   + £ 

2

+1 ,

arccosh  = ln  + £ 

2

−1 ,

1 1+ , ln 2 1− 1 1+ , arccoth  = ln 2  −1 arctanh(− ) = −arctanh  , arccoth(− ) = −arccoth  . arctanh  =

arcsinh(− ) = −arcsinh  , arccosh(− ) = arccosh  ,

S.1.4-2. Relations between inverse hyperbolic functions. arcsinh  = arccosh £ 

2

+ 1 = arctanh £ 

arccosh  = arcsinh £ 

2

− 1 = arctanh



arctanh  = arcsinh £ 1−

2



+1 £  2−1 2



1 = arccosh £ 1−

, ,

2

= arccoth



1

.

S.1.4-3. Addition and subtraction of inverse hyperbolic functions.

£ 1+

, V arccosh pA arccosh = arccosh  A ( 2 − 1)( 2 − 1) W , ü arcsinh pA arccosh = arcsinh V  A ( 2 + 1)( 2 − 1) W , ü pA  A 1 arctanh pA arctanh = arctanh , arctanh pA arccoth = arctanh . 1 AC A  I arcsinh pA arcsinh = arcsinh  

ü

1+

2

A

2

S.1.4-4. Differentiation formulas.

M

M arcsinh  = £   M

1 2

1

M arctanh  =  1−

+1 2

M arccosh  = £   M

, (

2

< 1),

M

1 2

1

M arccoth  =  1−

−1 2

, (

2

> 1).

Page 742

743

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.1.4-5. Expansion into power series. arcsinh  =  − arctanh  =  +

^_

1  3 1×3  5 1×3×5  7 + − + ))) 2 3 2×4 5 2×4×6 7



3

3

+



5

5

+



7

7

+ )))

(| | < 1), (| | < 1).

References for Section S.1: H. B. Dwight (1961), M. Abramowitz and I. A. Stegun (1964), G. A. Korn and T. M. Korn (1968), W. H. Beyer (1991).

S.2. Special Functions and Their Properties ]

Throughout Section S.2 it is assumed that B is a positive integer unless otherwise specified.

S.2.1. Some Symbols and Coefficients S.2.1-1. Factorials. Definitions and some properties: 0! = 1! = 1, B ! = 1 × 2 × 3  (B − 1)B , (2B )!! = 2 × 4 × 6  (2B − 2)(2B ) = 2  B !,

B = 2, 3,  ,

2 (2B + 1)!! = 1 × 3 × 5  (2B − 1)(2B + 1) = £

B !! =

™

(2 , )!! if B = 2 , , (2 , + 1)!! if B = 2 , + 1,

+1



w

OB +

3 , 2P

0!! = 1.

S.2.1-2. Binomial coefficients. Definition:

B ! L  = , where , = 1,  , B , , !(B − , )! (−  )   (  − 1)   (  − , + 1) = , L   = (−1)  , ! , !

where

, = 1, 2, 

General case: (  + 1) L  = w , (  + 1) (  −  + 1) Properties:

w

w

where

w

( ) is the gamma function.

L   = 0 for , = −1, −2,  or , > B ,   − +1 +1 +1   =  + 1 L   −1 =  + 1 L   , L   + L   = L   +1 , (−1)   (2B − 1)!!  L 2  = (−1)  , −1 J 2 = 22  (2B )!! (−1)  −1  −1 (−1)  −1 (2B − 3)!!  L = , 2  −2 1J 2 = B 22  −1 B (2B − 2)!! 2  +1  −4  −1 L  , L  −2  L 42  +1 , 2  +1 J 2 = (−1) 2 2  +1 J 2 = 2 2 22  +1 1J 2 ´J 2 = 2 L (  −1) J 2 . = , L      L 2 

L

L

L

L

L

L 0 = 1,

Page 743

744

SUPPLEMENTS

S.2.1-3. Pochhammer symbol. Definition and some properties ( , = 1, 2,  ): ( + B ) (1 −  ) = (−1)  w , ( ) (1 −  − B ) w w (B + , − 1)! , (  )0 = 1, (  )  +  = (  )  (  + B )  , (B )  = (B − 1)! ( − B ) (−1)  = , where  ≠ 1,  , B ; (  )−  = w ( ) (1 −  )  w (2B )! (2B + 1)! (1)  = B !, (1 2)  = 2−2  , (3 2)  = 2−2  , B ! B ! (  ) +  + ´ (  )2  ( )  ( + , ) (  + , ) ´ = , ( + B ) = , ( + B )  = . ¡ ( )+  ( ) ( ) (  )  =  (  + 1)  (  + B − 1) =

w

S.2.2. Error Functions and Exponential Integral S.2.2-1. Error function and complementary error function. Definitions: erf  = £



2

ù



exp(−a 2 ) a ,

erfc  = 1 − erf  = £

0

M

Expansion of erf  into series in powers of  as þD erf  = £

2 ½

 

÷ (−1)  =0

2

ùÈ ÷ exp(−a 2 ) M a .



0:

½ 2  2  +1 2   2  +1 = £ . exp  − 2  ÷ , ! (2 , + 1) (2 , + 1)!! =0  

Asymptotic expansion of erfc  as þDGF : erfc  = £

1



exp  − 2  ¬ 6

½

−1

(−1)

+

=0

1 +  2  + + [þ | |−2 6  2 + +1

−1

® , 8

= 1, 2, 

S.2.2-2. Exponential integral. Definition: Ei( ) = ù

 −

 g a aRM

−

÷

Ei( ) = lim ª ” ù +0

Other integral representations:



−

÷

ª

 g  g  a +ù a aÊM a M • ª R

 sin a + a cos a a  2+a 2 M 0   sin a − a cos a a Ei(− ) =  − ùÈ÷  2+a 2 M 0 g Ei(− ) = − ù ÷  − ln a a M 1 Ei(− ) = − 

−

ùÈ÷

for

 < 0,

for

 > 0.

for

 > 0,

for

 < 0,

for

 > 0.

Page 744

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

Expansion into series in powers of  as þD






Ei(


 z ) =





0:

 

½

+ ln(− ) + ÷



½

z

745

+ ln  + ÷



if  < 0,

, ×, !   , ×, !

=1

=1

if  > 0,

where z = 0.5772  is the Euler constant. Asymptotic expansion as EDGF : Ei(− ) = 

−

 ½ 

( , − 1)!

(−1)  =1

B ! ²  <  . 

+²  ,

 

S.2.2-3. Logarithmic integral.





Definition: li( ) =




 ù

M

a

= Ei(ln  )

ln a

0

+0 ” ª lim ù

1− ª

M

0

For small  ,

a

ln a li( ) ≈

Asymptotic expansion as ED

if 0 <  < 1,



+ù

a

M

1+ ª

if  > 1.

ln a•



ln(1  )

.

1:

½

li( ) = z + ln |ln  | + ÷



=1

ln   . , ×, !

S.2.3. Gamma and Beta Functions S.2.3-1. Gamma function. The gamma function, ( H ), is an analytic function of the complex argument H everywhere, except w for the points H = 0, −1, −2,  For Re H > 0,

w

(H ) = ù ÷ a

‘

−1 −

0



g a. M

For −(B + 1) < Re H < −B , where B = 0, 1, 2,  ,

g ½ (H ) = ù ÷ ¬  − w 0 + −

Euler formula:

 =0

‘ (−1) + ®a ! ¡

‘ B !B ( H ) = lim  w H ( H + 1)  ( H + B ) ÷

−1

M

a.

( H ≠ 0, −1, −2,  ).

Simplest properties:

w

( H + 1) = H

w

(H ),

w

(B + 1) = B !,

w

(1) = (2) = 1.

w

Page 745

746

SUPPLEMENTS Symmetry formulas: ( H ) (− H ) = −

w

w

Multiple argument formulas:

w



w



w

(3 H ) =

33



sin( H )



,



‘

−1

‘

J

(H ) O H +

w

w

−1 2

2

w



1 , 2P 1 3P

( H ) OH +

w

(B±H ) = (2 )(1−  ) J 2 B 

 J »

‘

−1 2



w

( H ) (1 − H ) =

w

1 1 O −H =  . +H 'P w 2 P cos( H ) 2

O

22 (2 H ) = £

w

,

H sin( H )

Fractional values of the argument:

−1

w

 =0 w

OH +

2 , 3P

OH +

, . BpP

£ 1 1 OB + =   (2B − 1)!!, =£ , w 2P 2 w 2P  1 1 2 £ O− = −2 £ , O − B = (−1)   . w 2P  w 2 P (2B − 1)!! Asymptotic expansion (Stirling ‘ ‘ formula): O

w

(H ) = £ 2 



−

J V1+

−1 2

H

1 −1 12

H

+

1 −2 288

H

+ [ (H

−3

)W

(|arg | H < ).



S.2.3-2. Logarithmic derivative of the gamma function. Definition:

j (H ) = M Functional relations:

ln ( H )

M

w

H

=


‘ ( H ) . (H )

w w

1

j ( H ) − j (1 + H ) = − , H j ( H ) − j (1 − H ) = − cot( H ),





1

j ( H ) − j (− H ) = − cot( H ) − ,   H jç 12 + H  − jç 12 − H  = tan( H ),   + −1 1 ½ , j ( H ) = ln + j\OH + . ¡ ¡ P  =0 ¡ ¡ Integral representations (Re H > 0): ‘ j ( H ) = ùÈ÷ V  −g − (1 + a )− Wa −1 a , M 0 ‘ j ( H ) = ln H + ùÈ÷ V a −1 − (1 −  −g )−1 W  −g a , M 0 ‘ 1 −1 1−a j ( H ) = −z + ù a, 1−a M 0 where z = −j (1) = 0.5772  is the Euler constant. Values for integer argument:

j (1) = −z ,

j ( B ) = −z +

½ 

−1

,

−1

(B = 2, 3,  ).

=1

Page 746

747

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.3-3. Beta function. Definition:

1

$ ( , ) = ù

0

 a

−1

(1 − a ) 

−1

a, M

where Re  > 0 and Re > 0. Relationship with the gamma function:

$ ( , ) = w

( ) ( ) w . ( + )

w

S.2.4. Incomplete Gamma and Beta Functions S.2.4-1. Incomplete gamma function. Definitions (integral representations):

= (ƒ ,  ) = ù



 −g a 9

0

(ƒ ,  ) = È ù  ÷ 

w Recurrence formulas:

Asymptotic expansions as þD

w

−

−1

ga9

a, M

−1

M

Re ƒ > 0,

a = ( ƒ ) − = ( ƒ ,  ).

w

 = ( ƒ + 1,  ) = ƒ±= (ƒ ,  ) −  9  − ,  ( ƒ + 1,  ) = ƒ ( ƒ ,  ) +  9  − .

w

0:

½ (−1)   9 +  = (ƒ ,  ) = ÷ ,  =0 B ! ( ƒ + B ) ½ (−1)   9 + (ƒ ,  ) = (ƒ ) − ÷ . w w  =0 B ! ( ƒ + B ) Asymptotic expansions as þDGF :

= (ƒ ,  ) = (ƒ ) −  9

−1 −



w

w

(ƒ ,  ) =  9

−1 −





¬ 6

½ +

−1 =0



¬ 6 +

½

−1

(1 − ƒ ) + (− ) +

=0

(1 − ƒ ) + (− ) +

+ [  | |−  ® ,

6

+ [  | |−  ®

 − 32 < arg  < 32  .



6

Integral functions related to the gamma function: erf  = £

1



=2O

1 2 , , 2 P

1

erfc  = £



w

O

1 2 , , 2 P

Ei(− ) = − (0,  ).

w

S.2.4-2. Incomplete beta function. Definition: where Re  > 0 and Re  > 0.

$  ( ,  ) = ù

 0

aK

−1

(1 − a ) &

−1

M

a,

Page 747

748

SUPPLEMENTS

S.2.5. Bessel Functions S.2.5-1. Definitions and basic formulas. The Bessel function of the first kind, ø Á ( ), and the Bessel function of the second kind, n…Á ( ) (also called the Neumann function), are solutions of the Bessel equation



 +  
+ (

2



2

− @ 2) = 0

and are defined by the formulas:

Á ½ (−1)  ( 2) +2  ø Á ( ) cos  @ − ø − Á ( ) . Áø ( ) = ÷ , n Á ( ) = (1) sin @  =0 , ! w ( @ + , + 1)  The formula for n/Á ( ) is valid for @ ≠ 0, A 1, A 2,  (the cases @ ≠ 0, A 1, A 2,  are discussed in the following). The general solution of the Bessel equation has the form called the cylindrical function. Some relations:

:

:

M

Á ( ) = L

1

ø

Á ( ) + L 2 n Á ( ) and is

:

2 @ Á ( ) =  [ Á −1 ( ) + Á +1 ( )], : : 1 : @ : Á ( ) = [ Á −1 ( ) − Á +1 ( )] = A Á ( ) − Á³1 1 ( )S , 2 Q 

: 

:

M : : : : M [ Á Á ( )] =  Á Á −1 ( ), M [ − Á Á ( )] = − − Á Á +1 ( ),   M  M  1 1 Á Á Á Á ” [ ø Á ( )] =  −  ø Á −  ( ), ” [ − ø Á ( )] = (−1)   − −  ø Á +  ( ), M M  2•  *• M M ø − ( ) = (−1) ø ( ), n − ( ) = (−1) n  ( ), B = 0, 1, 2,  1 2,

S.2.5-2. Bessel functions for @ = A"BçA

ø

J  )=

ø

1 2(

J  )=

2



ø

3 2(

2

J  )=

™





ø

2

J  )=

− −1 2 (



n 1 J 2 ( ) = − n 







 J  ) = (−1)

+1 2 (

™

J



B

B

 2

ø



P

 J

 B

=0

½

P n



½ ´J

=0

J  )=

2]

=0

where [ # ] is the integer part of a number # .

1

” − cos  − sin  ,  •

(−1)  (B + 2 , + 1)! (2 , + 1)! (B − 2 , − 1)! (2 )2 

J  )= n −



¢

+1

,

(−1)  (B + 2 , )! (2 , )! (B − 2 , )! (2 )2 

−1 2 (

J  ),



2

(−1)  (B + 2 , + 1)! (2 , + 1)! (B − 2 , − 1)! (2 )2 

P  J

(−1)  (B + 2 , )! (2 , )! (B − 2 , )! (2 )2 

2]

[

 2

cos  ,



−3 2 (

=0

[( −1) 2]

− −1 2 (



½ ´J

[

2 P

 ø

,

•

½

 2

2

 )=

[( −1) 2]

cos  , +1

B

cos O +

− sin O + 2

−1 2 (

sin O −

+ cos O −



sin  − cos 



2

+1 2 (



1

”





ø

sin  ,



where B = 0, 1, 2, 



2



¢

,

sin  ,

 J  ) = (−1) ø 

−1 2 (

+1

J  ),

+1 2 (

Page 748

749

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.5-3. Bessel functions for @ = A"B , where B = 0, 1, 2,  Let @ = B be an arbitrary integer. The relations ø − ( ) = (−1) ø ( ), n − ( ) = (−1) n  ( ) are valid. The function ø! ( ) is given by the first formula in (1) with @ = B , and n  ( ) can be obtained from the second formula in (1) by proceeding to the limit @\D B . For nonnegative B , the function n  ( ) can be represented in the form:

n  ( ) =



2



ø ( ) ln

 1 ½

−

2



−1



=0

(B − , − 1)! 2  O , ! 3P

where j (1) = −z , j (B ) = −z +

7

−1



=1

,

−1



1 ½

−





÷ (−1)  O =0





+2

2P

 j ( , + 1) + j (B + , + 1) , , ! (B + , )!

= 0.5772  is the Euler constant, j ( ) = [ln ( )]
 is

z

,

−2

w

the logarithmic derivative of the gamma function. S.2.5-4. Wronskians and similar formulas.

ú

( ø Á , ø −Á ) = −

2



ú

sin( @ ),





2 sin( @ )  ) + ø − Á ( ) ø Á −1 ( ) =  ,  ú

 Here, the notation (  , ) =   −   is used.

ø

Á ( ) ø − Á

+1 (

( ø Á , n Á ) =

ø

Á ( ) n Á

2





,

 )− ø Á

+1 (

+1 (

 ) n Á ( ) = −



2



.

S.2.5-5. Integral representations. The functions ø Á ( ) and n Á ( ) can be represented in the form of definite integrals (for  > 0):

 ø

Á ( ) = ù

n Á ( ) = ù



For | @ | < 12 ,  > 0,

z

0

9

z

9

0

)

− sin @ ù ÷ exp(− sinh a − @!a ) a ,

M

9





0

Á

w Á

Á˜g +  − ˜Á g cos @ )  − 

− ù ÷ (

21+  − 1 J 2 ( 1 − @ ) ùÈ÷ 1 2



Á

ù ÷ sin( cosh a ) M a , 0  For integer @ = B = 0, 1, 2,  ,

ø

ø

ø 

2



 )=

ù

0

2

 ( ) =

2 +1 (

z

1

( ) =

 

2

ù ù

0

0

z´J 2

2

n 0 ( ) = −

cos(B/a −  sin a ) a

z´J 2

g a. M

cos(a ) a M . (a 2 − 1) Á +1 J 2

w

2

sinh

sin(a ) a M , (a 2 − 1) Á +1 J 2

(Poisson’s formula).

w

 )=

M

0

Á z´J 2 2( 2) Á cos( cos a ) sin2 a a ù 1 1J 2 ( + @ ) M 0 2

 > 0,

0(

9

M

Á

Á ( ) =



Á ( ) =

ø

)

21+  − n Á ( ) = − 1 J 2 1 ù ÷ (2 − @ ) 1

For @ > − 21 ,

For @ = 0, 

9

sin( sin − @

ø

ø

9

cos( sin − @

M



ù ÷ cos( cosh a ) M a . 0

(Bessel’s formula),

cos( sin a ) cos(2B/a ) a ,

M

sin( sin a ) sin[(2B + 1)a ] a .

M

Page 749

750

SUPPLEMENTS

S.2.5-6. Asymptotic expansions. Asymptotic expansions as | | DGF :

ø

2

Á ( ) =

™





½ 4 − 2 @ − 4  P ¬ 6 +

cos O

2

n Á ( ) =



™



½ 4 − 2 @ − 4  P ¬ 6 +

sin O

)=

+

1

2

(4 @

−1

£ £

2

− 1)(4 @

  ø 2  (   ø 2 +1 (

Asymptotic behavior for large @

ø

Á ( ) ¬

Á (@ ) ¬

J

+

−1

+ [ (| |−2

+ 1)(2 )−2 +

−1

+ [ (| |−2

+ 1)(2 )−2

¡

6

−1

)®ƒ¢ ,

−1

)®ƒ¢ ,

6

¡

− 32 )  [4 @

−1

(−1) + ( @ , 2

=0 2

− (2

¡

) = (−1) 

+1

¡

¡

w

n Á ( ) ¬ −

n Á ( @ ) ¬ − 1 J 3

2

−2

@

O



6

( 21 + @ + ) ¡ . ! ( 21 + @ − )

w

¡

), −2

(cos  − sin  ) + [ (

( @\DGF ):

21 J 3 1 , (2 3) @ 1 J 3

w

− 1)2 ] =

) = (−1)  (cos  + sin  ) + [ (



32 3

+

(−1) ( @ , 2

=0

=0

1   Á O £ 2 @ 2 @lP ,

where  is fixed, and

ø

−1

(−1) + ( @ , 2 )(2 )−2 + + [ (| |−2 )®

¡ ! ¡ For nonnegative integer B and large  , 22

6

¡

=0

½ 4 − 2 @ − 4  P ¬ 6 +

+ cos O where ( @ ,

(−1) + ( @ , 2 )(2 )−2 + + [ (| |−2 )®

½ 4 − 2 @ − 4  P ¬ 6 +

− sin O

−1

 

2 @lP

).

−

Á

,

1 21 J 3 . (2 3) @ 1 J 3

6

w

S.2.5-7. Zeros and orthogonality properties of the Bessel functions. Each of the functions ø Á ( ) and n Á ( ) has infinitely many real zeros (for real @ ). All zeros are simple, except possibly for the point  = 0. The zeros = + of ø 0 ( ), i.e., the roots of the equation ø 0 (= + ) = 0, are approximately given by:

= + = 2.4 + 3.13 ( − 1) ¡

(

¡

= 1, 2,  ),

with a maximum error of 0.2%. Let “ = “ + be positive roots of the Bessel function ø Á (“ ), where @ > −1 and = 1, 2, 3,  ¡ Then the set of functions ø Á (“ + &  ) is orthogonal on the interval 0 ≤ & ≤  with weight & :

ù



0

ø

Á…O

“ + &  P

ø

Á…O

“  & & & = †P M

™

0

2  V ø Á
(“ + )W = 12 

1 2 2

ø

if ¡ Á +1 (“ + ) if

2 2

¡

≠, , =, .

S.2.5-8. Hankel functions (Bessel functions of the third kind). The Hankel functions of the first kind and the second kind are related to the Bessel functions by:

5 Á (1) ( H ) = ø Á ( H ) + ™Ön Á ( H ),

5 Á (2) ( H ) = ø Á ( H ) − ™Ön Á ( H ),

™ 2 = −1.

Page 750

751

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

Asymptotics for HCD

0: 2™

5

(1) 0 (

H )¬

5

(2) 0 (

H )¬ −

 2™

Asymptotics for | H | DGF :

5 Á (H ) ¬ (1)

5 Á (2) ( H ) ¬

ln H ,



2



Q

2

1 2

exp −™  H −

H



5 Á (2) ( H ) ¬

exp ™  H −

H

™

(@ ) w ( H 2) Á  (@ ) ™ w ( H 2) Á

5 Á (1) ( H ) ¬ −

ln H ,

Q



@ −

 1 2



1 4

@ −

S

 1 4



(Re @ > 0), (Re @ > 0).

(− < arg H < 2 ),



S



(−2 < arg H < ).





S.2.6. Modified Bessel Functions S.2.6-1. Definitions. Basic formulas.

The modified Bessel functions of the first kind, ®Á ( ), and the second kind, ; Á ( ) (also called the Macdonald function), of order @ are solutions of the modified Bessel equation

 and are defined by the formulas:

®Á (

½

)= ÷



=0



 +  
− (

2

2

Á ( 2)2  + , , ! ( @ + , + 1)

+ @ 2) = 0

; Á (

w

)=

(see below for ; Á ( ) with @ = 0, 1, 2,  ). The modified Bessel functions possess the properties:

; − Á (

M

) = ; Á ( ); 2 @¦®Á ( ) =  [ ®Á −1 ( ) − ®Á 1 M ®Á ( ) = [ ®Á −1 ( ) + ®Á  2

®

®

® 3 J 2 (

® 1 J 2 ( 2

1

)=



2



,

( ) = (−1)  ®  ( ), B = 0, 1, 2,  2 @ ; Á ( ) = − [ ; Á −1 ( ) − ; Á +1 ( )], +1 (  )], 1 M ; Á ( ) = − [ ; Á −1 ( ) + ; Á +1 ( )]. +1 (  )],  2

® −

M

S.2.6-2. Modified Bessel functions for @ = A"BçA

− Á − ®Á 2 ® sin  @

sinh  ,

1 2,

where B = 0, 1, 2, 

® −1 J 2 (

)=



2



2

cosh  , 1

” − sinh  + cosh  , ® −3 J 2 ( ) = ” − cosh  + sinh  ,  •   •     ½  1 (−1)  (B + , )! (B + , )!   − ½ ¬ − (−1) ® ,  +1 J 2 ( ) = £  , ! (B − , )! (2 ) , ! (B − , )! (2 )  2   =0  =0    ½  1 (−1)  (B + , )! (B + , )!   − ½ ¬ + (−1) ® , −  −1 J 2 (  ) = £  , ! (B − , )! (2 ) , ! (B − , )! (2 )  2   =0  =0 

 1 − ; 1 1 J 2 ( ) = 2   , ; 1 3 J 2 ( ) = 2  O 1 + 3P  − ,

 ½ (B + , )! ;  +1 J 2 ( ) = ; − −1 J 2( ) = 2   −  .  =0 , ! (B − , )! (2 ) )=



Page 751

752

SUPPLEMENTS

S.2.6-3. Modified Bessel functions for @ = B , where B = 0, 1, 2,  If @ = B is a nonnegative integer, then

;| (

) = (−1) 

+1

®



( ) ln

 1½ 2+

+

2

−1



(−1) + O

=0

2

2P

½   1 + (−1)  ÷ O 2 + =0 2 P

+ −  (B − +2

− 1)! !

¡

¡ + j (B + ¡

+ 1) + j ( + 1) ¡ ; ! (B + )!

¡

¡

B = 0, 1, 2,  ,

where j ( H ) is the logarithmic derivative of the gamma function; for B = 0, the first sum is dropped. S.2.6-4. Wronskians and similar formulas.

ú

®Á ( )® − Á +1 ( where

ú

( ®Á , ® − Á ) = −

) − ® − Á ( ) ®Á

−1 (

2

sin( @ ),





 )=−



2 sin( @ )



(  , ) =  
−  
.

 

,

ú

1 ( ®Á , ; Á ) = − ,

®Á

( ) ; Á



+1 (

 ) + ®Á

+1 (

 ) ; Á ( ) = 

1

,

S.2.6-5. Integral representations.

The functions ® Á ( ) and

®Á

( ) =



; Á (

) can be represented in terms of definite integrals:

 Á

J

1 22

ù

Á (@ + 1 ) 2

w

1 −1

exp(−a )(1 − a 2 )

; Á (

) = ùÈ÷ exp(− cosh a ) cosh( @!a ) a

; Á (

)=

0

; Á ( ) =

1 2



1 1 2

sin 

For integer @ = B ,

® ;

M

a



( > 0, @ > − 12 ), ( > 0),

ùÈ÷ cos( sinh a ) cosh( @!a ) M a @  0

( > 0, −1 < @ < 1),

ùÈ÷ sin( sinh a ) sinh( @!a ) M a @  0

( > 0, −1 < @ < 1).

1

( ) = 0(

J

−1 2

M

1

cos 

Á

z



ù

exp( cos a ) cos(B/a ) a

M

0

 ) = ùÈ÷ cos( sinh a ) a = ùÈ÷ M 0 0

(B = 0, 1, 2,  ),

cos(a ) £ a 2+1 M a

( > 0).

S.2.6-6. Asymptotic expansions as þDGF .

®Á





( ) = £ 2 

; Á (

)=



 2

½

™

(−1) +

1+ 6



+

−

The terms of the order of [  



™ −

2

− 1)(4 @

2

½

+

−1

− 32 )  [4 @ ! (8 ) +

2

− (2

¡

=1

(4 @

1+ 6

6

(4 @

=1

2

− 1)(4 @

2

− 32 )  [4 @ ! (8 ) +

¡

2

− (2

¡

¡

− 1)2 ] − 1)2 ]

¢

¢

,

.

 are omitted in the braces.

Page 752

753

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.7. Degenerate Hypergeometric Functions S.2.7-1. Definitions. The Kummer’s series. The degenerate hypergeometric functions ` (  ,  ;  ) and i (  ,  ;  ) are solutions of the degenerate hypergeometric equation:  

 + (  −  ) 
−  = 0. In the case  ≠ 0, −1, −2, −3,  , the function ` (  ,  ;  ) can be represented as Kummer’s series: ½ ( )    ` ( ,  ;  ) = 1 + ÷ , ( ) , !



=1

where (  )  =  (  + 1)  (  + , − 1), (  )0 = 1. Table 14 (see Subsection 2.1.2) presents some special cases where ` of simpler functions.

can be expressed in terms

S.2.7-2. Some transformations and linear relations. Kummer transformations:

 ` (  ,  ;  ) =  ` (  −  ,  ; − ),

i ( ,  ;  ) = 

1−

i

(1 +  −  , 2 −  ;  ).

Linear relations for ` : (  −  ) ` (  − 1,  ;  ) + (2  −  +  ) ` (  ,  ;  ) −  ` (  + 1,  ;  ) = 0,  (  − 1) ` (  ,  − 1;  ) −  (  − 1 +  ) ` (  ,  ;  ) + (  −  )/` (  ,  + 1;  ) = 0, (  −  + 1) ` (  ,  ;  ) −  ` (  + 1,  ;  ) + (  − 1) ` (  ,  − 1;  ) = 0, ` (  ,  ;  ) − ` ( − 1,  ;  ) − /` (  ,  + 1;  ) = 0,  (  +  ) ` (  ,  ;  ) − (  −  )/` (  ,  + 1;  ) − ` ( + 1,  ;  ) = 0, (  − 1 +  ) ` ( ,  ;  ) + (  −  ) ` (  − 1,  ;  ) − (  − 1) ` (  ,  − 1;  ) = 0. Linear relations for i :

i (  − 1,  ;  ) − (2 −  +  ) i (  ,  ;  ) +  ( −  + 1) i (  + 1,  ;  ) = 0, (  −  − 1) i ( ,  − 1;  ) − (  − 1 +  ) i (  ,  ;  ) + /i (  ,  + 1;  ) = 0, i (  ,  ;  ) −  i ( + 1,  ;  ) − i (  ,  − 1;  ) = 0, (  −  ) i (  ,  ;  ) − /i (  ,  + 1;  ) + i (  − 1,  ;  ) = 0, (  +  ) i (  ,  ;  ) +  (  −  − 1) i (  + 1,  ;  ) − /i (  ,  + 1;  ) = 0, (  − 1 +  ) i ( ,  ;  ) − i (  − 1,  ;  ) + ( − ( + 1) i (  ,  − 1;  ) = 0. S.2.7-3. Differentiation formulas and Wronskian. Differentiation formulas:

 ( ) ` (  + B ,  + B ;  ), M  ` ( ,  ;  ) =  ( ) M  M  i (  ,  ;  ) = (−1)  (  )  i (  + B ,  + B ;  ). 

M

 M ` (  ,  ;  ) = ` (  + 1,  + 1;  ),   M

M i (  ,  ;  ) = − i ( + 1,  + 1;  ), 

Wronskian:

ú

M

( ) ( ` , i ) = `‹i
 − `
 i = − w  ( )

w

−





.

Page 753

754

SUPPLEMENTS

S.2.7-4. Degenerate hypergeometric functions for B = 0, 1, 2,  (−1)  −1 ™ ` ( , B +1;  ) ln  B ! ( − B )

i (  , B + 1;  ) =

w

½

+ ÷

Ÿ =0

Ÿ

( )Ÿ V j (  + & ) − j (1 + & ) − j (1 + B + & )W  (B + 1) Ÿ &!

¢

+

 (B − 1)! ½ ( ) Ÿ

w

−1 =0

Ÿ ( − B )Ÿ  − , (1 − B ) Ÿ & !

where B = 0, 1, 2,  (the last sum is dropped for B = 0), j ( H ) = [ln ( H )]
‘ is the logarithmic w derivative of the gamma function,

j (1) = −z ,

j ( B ) = −z +

½ 

−1

−1

,

,

=1

where z = 0.5772  is the Euler constant. If  < 0, then the formula 1−

i ( ,  ;  ) = 

 i (

−  + 1, 2 −  ;  )

is valid for any  . For  ≠ 0, −1, −2, −3,  , the general solution of the degenerate hypergeometric equation can be represented in the form: = L 1 ` (  ,  ;  ) + L 2 i (  ,  ;  ), and for  = 0, −1, −2, −3,  , in the form: =

1−

 V L 1 ` (

−  + 1, 2 −  ;  ) + L

2

i (  −  + 1, 2 −  ;  )W .

S.2.7-5. Integral representations.

` ( ,  ;  ) = i ( ,  ;  ) = where

w

1  ( )  g a  −1 (1 − a )  −  ù ( ) (  −  ) 0 w  1  − g a  −1 (1 + a )  −  −1 a È ù ÷ ( ) 0 M

w

w w

−1

M

a

(for  >  > 0), (for  > 0,  > 0),

(  ) is the gamma function.

S.2.7-6. Asymptotic expansion as | | DGF .

` ( ,  ;  ) = w

w

` ( ,  ;  ) =

w

( )     ( )

/

−1

/ ½ 

=0

½ ( ) (− )−  ¬ ( −  )

½ i ( ,  ;  ) =   ¬ −

 ¬

w

/

−

where = [  

−



(−1)  =0



/

(  −  )  (1 −  )  −   + ® , B !

=0

 > 0,

(  )  (  −  + 1)  (− )−  + ® , B !

(  )  (  −  + 1)  −   + ® , B !

−F

 < 0,

<
.

Page 754

755

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.7-7. Whittaker functions. The Whittaker functions 8  equation:

,

 (

ú

) and

 ,  ( ) are linearly independent solutions of the Whittaker



 + V − 14 + 12 , + 

1 4

−“

2



−2

W

= 0.

The Whittaker functions are expressed in terms of degenerate hypergeometric functions as:

J   J 2`  +1 J 2 −  J 2 i  

8  ,  ( ) =   ú  ,  ( ) =  

+1 2 −

1 2 1 2

+ “ − , , 1 + 2“ ,   , + “ − , , 1 + 2“ ,   .

S.2.8. Hypergeometric Functions S.2.8-1. Definition. The hypergeometric series. The hypergeometric function  ( ƒ , ý , = ;  ) is a solution of the Gaussian hypergeometric equation:

 ( − 1) 

 + [( ƒ + ý + 1) − = ] 
+ ƒ…ý

= 0.

For = ≠ 0, −1, −2, −3,  , the function  ( ƒ , ý , = ;  ) can be expressed in terms of the hypergeometric series:

½ ( ƒ )  (ý )     (ƒ , ý , = ;  ) = 1 + ÷ , , !  =1 (= ) 

( ƒ )  = ƒ ( ƒ + 1)  ( ƒ + , − 1),

which, a fortiori, is convergent for | | < 1. Table 16 (see Subsection 2.1.2) presents some special cases where  of simpler functions.

can be expressed in terms

S.2.8-2. Basic properties. The function 

possesses the following properties:

 ( ƒ , ý , = ;  ) =  (ý , ƒ , = ;  ),  ( ƒ , ý , = ;  ) = (1 −  ) ; − 9 −:  (= − ƒ , = − ý , = ;  ),   ( ƒ , ý , = ;  ) = (1 −  )− 9 Oƒ , = − ý , = ; ,  −1P  ( ƒ )  (ý )   ( ƒ + B , ý + B , = + B ;  ). M   (ƒ , ý , = ;  ) =  (= )  M

If = is not an integer, then the general solution of the hypergeometric equation can be written in the form: = L 1  ( ƒ , ý , = ;  ) + L 2  1−;  ( ƒ − = + 1, ý − = + 1, 2 − = ;  ). S.2.8-3. Integral representations. For = > ý > 0, the hypergeometric function can be expressed in terms of a definite integral:

 (ƒ , ý , = ;  ) =

w

(= ) ù ( ý ) (= − ý )

w

w

1 0

a:

−1

(1 − a ); −:

−1

(1 − aÖ )− 9

M

a,

where (ý ) is the gamma function. w See M. Abramowitz and I. Stegun (1964) and H. Bateman and A. Erd´elyi (1953, Vol. 1) for more detailed information about hypergeometric functions.

Page 755

756

SUPPLEMENTS

S.2.9. Legendre Functions and Legendre Polynomials S.2.9-1. Definitions. Basic formulas.

.

The associated Legendre functions Á  ( H ) and Æ Á ( H ) of the first and the second kind are linearly independent solutions of the Legendre equation: (1 − H 2 )

‘'

‘ − 2 H ‘
+ [ @ ( @ + 1) − “ 2 (1 − H 2 )−1 ] = 0,

where the parameters @ and “ and the variable H can assume arbitrary real or complex values. For |1 − H | < 2, the formulas

.  Á Æ Á ( H

H −1 ) = #þO H +1P



(H ) = 2

w

1 H + 1 žJ 2 1−H O  O −@ , 1 + @ , 1 − “ ,  , (1 − “ ) H − 1 P 2 P

# =  © z w



1−H H +1 2 1−H + $ÊO O − @ , 1 + @ , 1 − “ , , 2 P H −1P 2 P (−“ ) (1 + @ + “ ) (“ ) © w , $ =  z w , ™ 2 = −1, 2 (1 + @ − “ ) 2

O − @ , 1 + @ , 1 + “ ,

w

are valid, where  (  ,  , ( ; H ) is the hypergeometric series (see Subsection S.2.8). For | H | > 1,

.  Á

2− (H ) = £

Á

−1

§w

(− 21 − @ ) − Á H (− @ − “ )

w



+ −1

(H

2

− 1)− žJ 2   O

1+@ −“ 2+@ −“ 2@ + 3 1 , , , 2 2 2 2 H •

Á (1 + @ ) 1 @ + “ 1 − @ − “ 1 − 2@ w 2 H Á +  ( H 2 − 1)− žJ 2 O − , , , 2 , 2 2 2 H • §w (1 + @ − “ ) £ ( @ + “ + 1) 2 + @ + “ 1 + @ + “ 2 @ + 3 1 Æ Á ( H ) =  © zf oÁ +1 w H − Á −  −1 ( H 2 − 1) žJ 2 O , , , 2 . 3 2 2 2 H • 2 (@ + 2 ) 2

+ £

w

. The functions

.

Á ( H ) ≡ Á 0 ( H ) and ÆIÁ ( H ) ≡ Æ 0Á ( H ) are called the Legendre functions. The modified associated Legendre functions, on the cut H =  , −1 <  < 1, of the real axis, are

defined by the formulas:

.  Á

V  © z . Á  ( + ™ 0) +  − 21 © z . Á  ( − ™ 0)W , 1 1 Æ Á ( ) =  © z·V  − 2 © z Æ Á ( + ™ 0) +  2 © z Æ Á ( − ™ 0)W . ( ) =

1 1 2 2 1 − 2

S.2.9-2. Trigonometric expansions. For −1 <  < 1, the modified associated Legendre functions can be represented in the form of trigonometric series as:

.  Á

9 2 (cos ) = £ 9

Æ Á (cos ) = £ where 0 <

9

9  ½ ( 12 + “ )  (1 + @ + “ )  9 ( @ + “ + 1) (sin ) ÷ sin[(2 , + @ + “ + 1) ], 3 3 (@ + 2 ) , ! (@ + 2 )  =0 w 1 9 ½ ( 2 + “ )  (1 + @ + “ )  9 ( @ + “ + 1) 2 w (sin )  ÷ cos[(2 , + @ + “ + 1) ], 3 3 (@ + 2 ) , ! (@ + 2 )  =0

+1





w

w

< .



Page 756

757

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.9-3. Some relations.

.  Á

(H ) =

.  Á

− −1 (

Æ Á ( H ) = For 0 <  < 1,

For −1 <  < 1,

.  Á

Æ Á

.

ú

.

 2 sin(“

)



.

 © zf



(− ) = − Æ Á (

.  Á

w

( @ + B + 1) . −  B = 0, 1, 2,  Á ( H ), ( @ − B + 1) . (1 + @ + “ ) . −  ¬ Á  (H ) − w Á ( H )® . (1 + @ − “ )

) cos[ ( @ + “



+1 (

 )=

, 2

2@ + 1  @ −“ +1

.

ú

Á  ( ) = (−1)  (1 −  2 ) ´J

.  Á

w

−1



 1 . Æ  Á ( )] − 2 Á ( 

) sin[ ( @ + “ )],



) sin[ ( @ + “ )].



@ +“ .  ( ), @ − “ + 1 Á −1 .  .  Á +1 ( ) = Á −1 ( ) − (2 @ + 1)(1 −  2 )1 J 2 Á  −1 ( ).

1 1− For B = 0, 1, 2,  , ( Á , ÆIÁ ) =

Á  (H ) = w

(− ) = Á  ( ) cos[ ( @ + “ )] − 2

. Wronskians:

.

H ),

( Á  , Æ Á ) = 2



M

M  

.

Á ( ),

( ) −

,

1−

, 2

, = 22  w

w



Á +  +1  Á  +2 +2  2 Á −  +1  w Á −  +2  .  2  2

Æ Á ( ) = (−1)  (1 −  2 ) ´J

2



M

w

M  ÆIÁ ( ). 

S.2.9-4. Integral representations. For B = 0, 1, 2,  , .  (H ) = w ( @ + B + 1) z H + cos a £ H 2 − 1  Á cos(B/a ) a , Re H > 0, Á ù M w ( @ +( @ 1)+ B +0 1) z Á Á Æ Á ( H ) = (−1)  w Á +1 ( H 2 − 1)− ´J 2 ù ( H + cos a )  − −1 (sin a )2 +1 a , 2 ( @ + 1) M 0 w Note that H ≠  , −1 <  < 1, in the latter formula.

Re @ > −1.

S.2.9-5. Legendre polynomials. . . The Legendre polynomials  =  ( ) and the Legendre functions Æ  ( ) are solutions of the linear differential equation:


(1 −  2 )  − 2  + B (B + 1) = 0. . The Legendre polynomials  ( ) and the Legendre functions Æ  ( ) are defined by the formulas: . ( ) = 1 M  ( 2 − 1) , Æ ( ) = 1 . ( ) ln 1 +  − ½  1 . + ( ). + ( ).    −1  − B ! 2   2 1− + =1 ¡ M .  =  ( ) can be calculated recursively using the relations: . ( ) = 1, . ( ) =  , . ( ) = 1 (3 2 − 1),  , . ( ) = 2B + 1  . ( ) − B . 0 1 2  +1 2 B +1  B +1  The first three functions Æ  = Æ  ( ) are given by: 1 1+  1+ 3 2 − 1 1 +  3 Æ 0 ( ) = ln , Æ 1 ( ) = ln − 1, Æ 2 ( ) = ln −  . 2 .1 −  2 1− 4 1− 2 The polynomials  ( ) have the explicit representation: . ( ) = 2− [½ ´J 2](−1)+ L + L   −2 + ,   2  −2 +  + =0 where [ # ] is the integer part of a number # .

The polynomials

.

−1 (

 ).

Page 757

758

SUPPLEMENTS

S.2.9-6. Zeros of the Legendre polynomials and the generating function.

.

All zeros of  ( ) are real and lie on the interval −1 <  < +1; the functions orthogonal system on the interval −1 ≤  ≤ +1, with

ù

+1

.

 (

−1

a

. )

+ ( )  = M

0

2 2B + 1

if B ≠ if B =

.

 ( ) form an

,

¡ ¡

.

The generating function is:

.

½

1

£ 1 − 2 °˜ + °

2

= ÷



=0

 ( ) ° 

(| ° | < 1).

S.2.9-7. Associated Legendre functions. The associated Legendre functions

.

+  ( ) of order ¡ are defined by the formulas:

+ . B = 1, 2, 3,  , = 0, 1, 2,  M +  ( ), ¡  . M . It is assumed by definition that  0 ( ) =  ( ). . + The functions  ( ) form an orthogonal system on the interval −1 ≤  ≤ +1, with +1 . + ( ). + ( )  =   0 2 (B + )! if B ≠ , , ù   ¡ if B = , . M −1 2B + 1 (B − )! ¡ . ≠ 0) are orthogonal on the interval −1 ≤  ≤ +1 with weight The functions  + ( ) (with ¡ (1 −  2 )−1 , that is, . if B ≠ , ,  0 +1 . + ( )  + ( )  B + )! (  = ù ¡ if B = , . (1 −  2 ) M −1 (B − )! ¡ ¡

.

+ 2 + J  ( ) = (1 −  )

2

S.2.10. Parabolic Cylinder Functions S.2.10-1. Definitions. Basic formulas. The Weber parabolic cylinder function c‚Á ( H ) is a solution of the linear differential equation:

‘'

‘ +  − 1 H 2 + @ + 1  4 2

= 0,

where the parameter @ and the variable H can assume arbitrary real or complex values. Another linearly independent solution of this equation is the function c − Á −1 (™ÖH ); if @ is noninteger, then c\Á (− H ) can also be taken as a linearly independent solution. The parabolic cylinder functions can be expressed in terms of degenerate hypergeometric functions as:

c\Á ( H ) = 21 J 2 exp  − 14 H 2  ¬

w

1



w 1  2 Á `  − 2Á , 12 , 12 H 2   2 − 2

For nonnegative integer @ = B , we have

+ 2−1 J

2

w

−1 w  2Á H`   −2

1 2

Á − 2 , 32 , 12 H 2  ® .

c  ( H ) = 2− ´J 2 exp  − 41 H 2  5   2−1 J 2 H  , B = 0, 1, 2,  ;  5  ( H ) = (−1)  exp  H 2  M  exp  − H 2  , H M where 5  ( H ) is the Hermite polynomial of order B .

Page 758

759

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.10-2. Integral representations.

c\Á ( H ) = c\Á ( H ) =

2

ü



exp 

H  ùÈ÷ a Á exp  − 21 a 2  cos  H´a −

1 2 4

1 Á exp  − 41 H 2  È ù ÷ a− (− @ ) 0

w

1 2

0

−1

exp  − H´a − 12 a 2 

M

@  a M



for Re @ > −1,

a

for Re @ < 0.

S.2.10-3. Asymptotic expansion as | H | DGF .

½ c\Á ( H ) = H Á exp  − 14 H 2  ¬ 

/

Á

=0

(−2)   − 2    B !

1 2

−

Á  2 

1

+ [þ | H |−2 H 2

/

−2

®

for | arg H | <

3  , 4

where (  )0 = 1, (  )  =  (  + 1)  (  + B − 1) for B = 1, 2, 3, 

S.2.11. Orthogonal Polynomials All zeros of each of the orthogonal polynomials ´  ( ) considered in this section are real and simple. The zeros of the polynomials ´  ( ) and ´  +1 ( ) are alternating. For Legendre polynomials see Subsections S.2.9-5, S.2.9-6, and S.2.9-7. S.2.11-1. Laguerre polynomials and generalized Laguerre polynomials. The Laguerre polynomials

0 

= 0  ( ) satisfy the linear differential equation

 

 + (1 −  ) 
+ B

=0

and are defined by the formulas:

0  (

 1   (−1)    − B 2   M     −  = B !  B ! Q

)=

−1

M

+

B 2 (B − 1)2   2!

−2

+ )))›S .

The first four polynomials are given by:

0

0

= 1,

To calculate

0

0  (

1

2

= 12 (

2

− 4 + 2),

0

3

= 16 (−

3

+ 9

2

− 18 + 6).

) for B ≥ 2, one can use the recurrence formulas:

0  The functions

0

= − + 1,

0  (

+1 (

1 V (2B + 1 −  ) 0  ( ) − B0  B +1

 )=

−1 (

 )W .

) form an orthonormal system on the interval 0 <  < F

ù ÷  0

−



0  ( ) 0 + ( )  M

=¦



=0

−



:

0 if B ≠ , ¡ 1 if B = .

¡

The generating function is:

½ 1 °˜ exp O − = ÷ 1−° 1 − °±P

with weight 

0  ( ) ° 

,

| ° | < 1.

The generalized Laguerre polynomials 0‹9 = 0·9 ( ) ( ƒ > −1) satisfy the linear differential equation  

 + ( ƒ + 1 −  ) 
+ B = 0

Page 759

760

SUPPLEMENTS

and are defined by the formulas:

0 9 ( 

½  1 −    9  M     +9  −  = B !  + M

)=



(− ) + . L   +− + 9 ! =0 ¡

The first two polynomials have the form:

0 09

0 19

= 1,

= ƒ +1− .

0 9 ( ) for B ≥ 2, one can use the recurrence formulas: To calculate l 0 9

+1 (

1

 )=

B +1

V (2B + ƒ + 1 −  )0 9 ( ) − (B + ƒ )0 9  

−1 (

 )W .

The functions l 0 9 ( ) form an orthogonal system on the interval 0 <  < F

ù ÷  9 

−



0

a

0 9 (  ) 0 +9 (  )  M

0

=

w

if B ≠

( ƒ + B + 1) B !

if B =

with weight h9 

−



:

,

¡

¡

.

The generating function is: (1 − ° )− 9

−1

exp O −

½

°˜

0 9 (  ) ° 

= ÷

1 − °±P



S.2.11-2. Chebyshev polynomials.

=0

{

1 b . The Chebyshev polynomials of the first kind

| ° | < 1.

,

{

 =  ( ) satisfy the linear differential equation (1 −  2 ) 

 −  
+ B 2 = 0

and are defined by the formulas:

{

 (−2)  B ! £ 1  ( ) = cos(B arccos  ) = (2B )! 1 −  2 M   V (1 −  2 )  − 2 W M [ ´J 2] B ½ (B − − 1)! + = (−1) ¡ (2 )  −2 + 2 + ! (B − 2 )! =0 ¡ ¡

where [ # ] stands for the integer part of a number # . The first five polynomials are:

{

0

= 1,

{

1

= ,

The recurrence formulas:

{

The functions

{

{

2

2

= 2

{

− 1,

{

3

= 4

3

− 3 ,

{

4

(B = 0, 1, 2,  ),

= 8

4

− 8

2

+ 1.

{

 ) = 2  ( ) −  −1 ( ), B ≥ 2; { { { { 2  ( ) + ( ) =  + + ( ) +  − + ( ), B ≥ . ¡ 

+1 (

 ( ) form an orthogonal system on the interval −1 <  < +1, with a 0 if B ≠ , { +1 {  ( ) + ( )  = 1 ¡ if B = ≠ 0, ù 2 £ 2 M ¡ 1 −  −1 if B = = 0.  ¡

The generating function is:

1 − °˜ 1 − 2 °˜ + °

{

½

2

= ÷



=0

 ( ) ° 

(| ° | < 1).

Page 760

761

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

2 b . The Chebyshev polynomials of the second kind ’  = ’  ( ) satisfy the linear differential equation (1 −  2 ) 

 − 3 
+ B (B + 2) = 0 and are defined by the formulas:

’  ( ) =

sin[(B + 1) arccos  ]

£ 1−

2

½ ´J

[

=

+

2]

(−1) +

¡

=0

(B − )! ¡ (2 )  ! (B − 2 )!

−2

+

(B = 0, 1, 2,  ).

¡

The recurrence formulas:

’ 

+1 (

 ) = 2/’  ( ) − ’ 

−1 (

 ),

B ≥ 2.

The generating function is:

½

1

1 − 2 °˜ + °

2

= ÷



=0

’  ( ) ° 

(| ° | < 1).

S.2.11-3. Hermite polynomials. The Hermite polynomial 5  = 5  ( ) satisfies the linear differential equation



 − 2 
+ 2B

=0

and is defined by the formulas: [ ´J 2]  ½ (−1) + B ! 5  ( ) = (−1)  exp   2  M  exp  − 2  = (2 )  ! ( B − 2 )!  + =0 ¡ ¡ M

−2

+

(B = 0, 1, 2,  ),

where [ # ] stands for the integer part of a number # . The first five polynomials are:

5

0

= 1,

5

1

= ,

5

2

= 4

2

− 2,

5

3

= 8

3

− 12 ,

5

4

= 16

4

− 48

2

+ 12.

The recurrence formulas:

5 

+1 (

 ) = 2"5  ( ) − 2B±5 

−1 (

 ),

B ≥ 2.

The functions 5  ( ) form an orthogonal system on the interval − F

<
with weight 

−



2

:

0 if B ≠ , 2 ù ÷ exp  −  5  ( ) 5 + ( ) M  = ™ £ 2  B ! if B = ¡ . −  ¡ ÷ The Hermite functions j  ( ) are introduced by the formula j  ( ) = exp  − 12  2  5  ( ), where B = 0, 1, 2,  The generating function is:

exp  − °

2

½

° 5  ( ) . B !  =0

+ 2 °˜  = ÷

Page 761

762

SUPPLEMENTS

S.2.11-4. Gegenbauer polynomials. The Gegenbauer polynomials L  (  ) ( ) satisfy the linear differential equation ( 2 − 1) 

 + (2  + 1) 
− B (2  + B ) = 0 and are defined by the formulas:

½  (2  + B ) (  + , ) (2  + B + , )( − 1)  . L  (  ) ( ) = w   2  + B , −B ,  + 12 ; 12 − 12   = w w (B + 1) (2  ) , ! (B − , )! 2  (  ) (2  + 2 , ) =0  w w w w The functions L  (  ) ( ) form an orthogonal system on the interval −1 <  < +1, with a 0 if B ≠ , +1 ¡ (2  + B ) (1 −  2 )  −1 J 2 L  (  ) ( ) L + (  ) ( )  = ù if B = . w  M  2 −1 2 −1 ¡ 2 (  + B )B ! (  )

w

The generating function is:

½

(1 − 2 °˜ + ° 2 )−  = ÷



=0

L  (  ) ( ) ° 

(| ° | < 1).

S.2.11-5. Jacobi polynomials.

.

. ,: ,:  9 =  9 ( ) satisfy the linear differential equation (1 −  2 ) 

 + V ý − ƒ − ( ƒ + ý + 2)W 
+ B (B + ƒ + ý + 1) = 0

The Jacobi polynomials

and are defined by the formulas:

 ½  (−1)  9 ,: =  L  + + L   +−: + ( − 1)  − + ( + 1) + , (1 −  )− 9 (1 +  )−: M  (1 −  ) 9 +  (1 +  ): +  S = 2−  9 2 B !  Q + =0 M where L  are binomial coefficients.

.



S.2.12. The Weierstrass Function S.2.12-1. Definitions. The Weierstrass function

<

= < ( H , 2, 3 ) is defined implicitly by the elliptic integral:

H =ù Ž

a M

4a − 2 a − 3 ü ÷ and satisfies the first-order nonlinear differential equation: ‘ ( <
)2 = 4 < 3 − 2 < − 3 . 3

S.2.12-2. Some properties. Below < ( H ) stands for Properties:

< (H

< ( H , 2,

) = < (− H ),

3 ).

< (H

1

+ H 2 ) = − < (H 1) − < (H 2) +

1 ¬ 4

<
(H < (H

2 − <
(H 2) ® . 1 ) − < (H 2) 1)

In the vicinity of the point H = 0, the Weierstrass function can be expanded into the series:

^_

< (H

)=

H

1 2

+



2

20

H 2+



3

28

H 4+



2 2

1200

H 6+

3 2 3 H 6160

8

+ ))) .

References for Section S.2: H. Bateman and A. Erd´elyi (1953, 1955), M. Abramowitz and I. A. Stegun (1964), F. W. J. Olver (1974), A. F. Nikiforov and V. B. Uvarov (1988), N. M. Temme (1996).

Page 762

S.3. TABLES OF INDEFINITE INTEGRALS

763

S.3. Tables of Indefinite Integrals ]

Throughout Section S.3 the integration constant L is omitted for brevity.

S.3.1. Integrals Containing Rational Functions S.3.1-1. Integrals containing  + ' . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

M

ù



 + '

=

1



ln |  + ' |.

(  + ' )  +1 ù (  + ' )  M  =  (B + 1) , B ≠ −1. 1   ù  +M ' =  2   + ' −  ln |  + ' |  .  2  1 1 ù  +M ' =  3 Q 2 (  + ' )2 − 2  (  + ' ) +  2 ln |  + ' | S .  1  + ' ù  (  M + ' ) = −  ln û  û. û û û  + '  1 û  ù  2 (  M + ' ) = −  +  2 ln û  û. û û û    û 1 ù (  +M ' )2 =  2 O ln |  + ' | +  + '3P .  2  1  2 ù (  + M ' )2 =  3 O + ' − 2  ln |  + ' | −  + '3P .  1 1  + ' ù  (  M + ' )2 =  (  + ' ) −  2 ln û  û. û û û  û   1 1 ù (  +M ' )3 =  2 Q −  + ' + 2(  + ' )2 S .

S.3.1-2. Integrals containing  +  11.

ù

12.

ù

13.

ù

14.

ù

15.

ù

16.

ù

17.

ù

18.

ù

19.

ù

and  +  .

 +  =  + (  −  ) ln |  +  |.  + wM  1  + = ln û M û ,  ≠  . For  =  , see integral 2 with B = −2. (  +  )(  +  )  −  û  + û û û   1 = M   ln |  +  | −  ln |  +  |  . (  +  )(  +  )  −  1  +  1 = + ln . M (  +  )(  +  )2 (  −  )(  +  ) (  −  )2 ûû  +  ûû û û      + = − ln û M û. 2 2 (  +  )(  +  ) (  −  )(  +  ) (  −  ) û  + û û û  2 − 2   2  2  2 = + ln |  +  | + ln |  +  |. M (  +  )(  +  )2 (  −  )(  +  ) (  −  )2 (  −  )2  1 1 1 2  + =− O + + ln û M2 û. 2 2 3 ( +  ) (  +  ) ( −  )  +   + 3P ( −  ) û  + û   1  û +  û   + = O + + ln û M û. 2 2 2 3 ( +  ) (  +  ) ( −  )  +   + 3P ( −  ) û  + û û  + û  2  1  2 2 2  = − O + + ln . M2 (  +  ) (  +  )2 (  −  )2  +   + 3P (  −  )3 ûû  +  ûû û û

Page 763

764

SUPPLEMENTS

S.3.1-3. Integrals containing 

ù

21.

ù

22.

ù

23.

ù

24.

ù

25.

ù

26.

ù

27.

ù

28.

ù

29.

ù

30.

ù

31.

ù

32.

ù

33.

ù

34.

ù

35.

ù

36.

ù

37.

ù

38.

ù

39.

ù

40.

ù

41.

ù

42.

ù

 M  2+

20.

=

1



arctan

2



2

M

+  2 )2

=

2 2(



 

+  2.

.

1  arctan . 2 3   3  3  + = 2 2 + arctan . M (  2 +  2 )3 4  (  +  2 )2 8  4 (  2 +  2 ) 8  5   2B − 1   = + M ù (  2 +M  2 )  ; B = 1, 2,   (  2 +  2 )  +1 2B± 2 (  2 +  2 )  2B± 2 1   = ln(  2 +  2 ). M  2+ 2 2   1 . =− M (  2 +  2 )2 2(  2 +  2 )   1 =− . M (  2 +  2 )3 4(  2 +  2 )2   1 M 2  +1 = − ; B = 1, 2,  2 2 ( +  ) 2B (  +  2 )  (



2

2

2

+  2)

+

  =  −  arctan . M  2+ 2  2   1   + =− arctan . M (  2 +  2 )2 2(  2 +  2 ) 2    2    1  M 2 3 = − 2 2 2 + 2 2 2 + 3 arctan . 2 ( +  ) 4(  +  ) 8 ( +  ) 8    2   1 M =− + ù (  2 +M  2 )  ; B = 1, 2,  (  2 +  2 )  +1 2B (  2 +  2 )  2B  3   2  2 − ln(  2 +  2 ). M 2 = 2  + 2 2  3   2 1 = + ln(  2 +  2 ). M (  2 +  2 )2 2(  2 +  2 ) 2  3  1  2 + ; B = 2, 3,  M 2  +1 = − 2 2 2  −1 2 ( +  ) 2(B − 1)(  +  ) 2B (  +  2 )   1  2 M2 2 = 2 ln 2 2 .  ( +  ) 2  +  1 1  2 M2 2 2 = 2 2 2 + 4 ln 2 2 .  ( +  ) 2 ( +  ) 2  +  1 1 1  2 M 2 3 = 2 2 2 2 + 4 2 2 + 6 ln 2 2 . 2  ( +  ) 4 ( +  ) 2 ( +  ) 2  +  1 1  = − 2 − 3 arctan . M  2 ( 2 +  2 )      1  3  =− 4 − 4 2 − arctan . M  2 (  2 +  2 )2   2 ( +  2 ) 2 5   1 1 1  2 =− 4 2 − 4 2 − 6 ln 2 . M 3 2 2 2 2  ( +  ) 2  2 ( +  )   + 2  1 7 15   =− 6 − 4 2 − − arctan . M  2 (  2 +  2 )3   4  (  +  2 )2 8  6 (  2 +  2 ) 8  7  1  1 1 3  2 − 4 2 =− 6 2 − 6 2 − 8 ln 2 . M 3 2 2 3 2 2 2  ( +  ) 2   ( +  ) 4 ( +  ) 2  + 2

Page 764

765

S.3. TABLES OF INDEFINITE INTEGRALS

S.3.1-4. Integrals containing  43.

ù

 M  2−

44.

ù

M 2

45.

ù

(

46.

ù

47.

ù

48.

ù

49.

ù

50.

ù

51.

ù

52.

ù

53.

ù

54.

ù

55.

ù

56.

ù

57.

ù

58.

ù

59.

ù

60.

ù

( (

 

−

M

−

2

=

−  2.

1  + ln û û. 2 û  − û

û  û 1  + + 3 ln û û. 2 )2 2 2 2 2 ( −  ) 4 û  − û û 3  +  3û  + = 2 2 + 4 2 ln 2 )3 2 2 2 4 ( −  ) 8  (  −  ) 16  5 ûû  −  2B − 1   û = + ; B M ù 2 )  +1 2B± 2 (  2 −  2 )  2B± 2 ( 2 −  2 ) =

−

M

2

2

2



û. û

û

= 1, 2, 

  1 = − ln |  2 −  2 |. M  2− 2 2   1 . = M (  2 −  2 )2 2(  2 −  2 )   1 = . M (  2 −  2 )3 4(  2 −  2 )2   1 = ; B = 1, 2,  M (  2 −  2 )  +1 2B (  2 −  2 )   2    + = − + ln û . M  2− 2 2 û  −  ûû û û  2   1  + = − ln û M û. 2 (  −  2 )2 2(  2 −  2 ) 4  û  − û û û  2   1  + M = − ln . − (  2 −  2 )3 4(  2 −  2 )2 8  2 (  2 −  2 ) 16  3 ûû  −  ûû û û  2    1 M M = − ; B = 1, 2,  ù (  2 −  2 )  +1 2B (  2 −  2 )  2B ( 2 −  2)  3   2  2 = − − ln |  2 −  2 |. M  2− 2 2 2  3   2 1 = + ln |  2 −  2 |. M (  2 −  2 )2 2(  2 −  2 ) 2  3  1  2 = − + ; B = 2, 3,  M (  2 −  2 )  +1 2(B − 1)(  2 −  2 )  −1 2B (  2 −  2 )   2  1 M = 2 ln û 2 û. 2 2  ( −  ) 2 û  − 2 û û û  1 1  2 M2 2 2 = 2 2 2 + 4 ln û 2 2 û .  ( −  ) 2 ( −  ) 2 û  − û û û 1  1 1  2 M2 2 3 = 2 2 2 2 + 4 2 2 + 6 ln û 2 2 û .  ( −  ) 4 ( −  ) 2 ( −  ) 2 û  − û û û

S.3.1-5. Integrals containing 

ù

62.

ù

63.

ù



+  3.

2 −  arctan £ . £  3  3   2  = 3 3 + M ù  3M +  3 . (  3 +  3 )2 3 ( +  3) 3 3   1 1 2 −   2 −  +  2 ln + £ arctan £ . M 3 = 3 2  + 6 ( +  )  3  3

M 3

61.

3

 +

3

=

1 (  +  )2 ln 6 2  2 −   + 

2

+

1

2

Page 765

766

SUPPLEMENTS

64.

ù

65.

ù

66.

ù 

67.

ù 

68.

ù 

69.

ù 

(







M

+  3 )2

3 2



=

3 3(



2 3

+  3)

+

1 ù 3 3

  M  3+

.

3

1 ln |  3 +  3 |. 3  1  3 = 3 ln û 3 M û. 3 3 ( +  ) 3 û  + 3 û û û  1 1  3 M3 3 2 = 3 3 3 + 6 ln û 3 3 ( +  ) 3 ( +  ) 3 û  +  1 1   û =− 3 − 3 ù . M M 2 ( 3 +  3 )     3+ 3  1  2 4 = − − − M ù 2 (  3 +  3 )2  6 3 6 ( 3 +  3 ) 3 6

M  3+

3

=

S.3.1-6. Integrals containing 



3

û. û û

  . M  3+ 3

−  3.

ù

71.

ù

72.

ù

73.

ù

74.

ù

75.

ù

76.

ù

77.

ù

78.

ù

1 2 +  arctan £ .  2£ 3  3    2 M 32 = 3 3 3 + 3ù M 3. 3 3 ( −  ) 3 ( −  ) 3  −   1  2 +  +  2 1 2 +  ln − £ arctan £ . M 3 = 3 2  − 6 ( −  )  3  3      2 1 = . + 3 ù M M 3 3 2 3 3 3 3 ( −  ) 3 ( −  ) 3  − 3  2  1 = − ln |  3 −  3 |. M  3− 3 3  1  3 M3 3 = 3 ln û 3 3 û .  ( −  ) 3 û  − û û û  1 1  3 = ln û 3 + M 32 û. 3 3 3 3 6  ( −  ) 3 ( −  ) 3 û  − 3 û û  1 1   û = − + . M M ù  2 ( 3 −  3 )  3  3  3− 3  1 4    2 = − − + 6 ù . M M 2 3 3 2 6 6 3 3 3  ( −  )   3 ( −  ) 3  − 3

70.

M

 − 3

3

=

1  2 +  +  ln 2 6 (  −  )2

S.3.1-7. Integrals containing 

ù

80.

ù

81.

ù

82.

ù

83.

ù

M 4

79.



 +   M 4

 +  2  M  4+  M  4−   M  4−

4

4

4

4

4

2

+

AC 4 . 1  2 +  £ 2 +  2 = ln + 4  3 £ 2  2 −  £ 2 +  2 2  1  2 = 2 arctan 2 . 2  1  2 +  £ 2 +  2 =− £ ln + 4 2  2 −  £ 2 +  2 2  1  + 1  = 3 ln û + arctan . 4 û −  ûû 2  3  û 2 û2 1  + = 2 ln û 2 û. 4 û  − 2 û û û 4

1 3

£ 2

1

£ 2

arctan

 £ 2 .  2− 2

arctan

 £ 2 .  2− 2

Page 766

767

S.3. TABLES OF INDEFINITE INTEGRALS

1  + 1  ln û − arctan . 4 û −  ûû 2   û  +  1û ln û = M + û.  (  + '  + )  û  + '  û



ù

2

85.

ù

M  4−

84.



4

=

¡

û

û

S.3.2. Integrals Containing Irrational Functions S.3.2-1. Integrals containing  1 J 2 .

ù

2.

ù

3.

ù

4.

ù

5.

ù

6.

ù

7.

ù

8.

ù

9.

ù

 1J

1.

10.

ù

11.

ù

12.

ù

13.

ù

14.

ù

 2 1J 2 2 ' 1 J 2 =  − arctan . M  2 +  2 2 3  2 3 J 2 2  2  1 J 2 2  3 ' 1 J 2  3J 2  = − + arctan . M  2 +  2 3 2 4 5   1J 2   1J 2 1 ' 1 J 2 = − arctan . + M (  2 +  2  )2  2 (  2 +  2  )  3   3J 2  2 3 J 2 3 2  1 J 2 3 ' 1 J 2 = + − arctan . M (  2 +  2  )2  2 ( 2 +  2  )  4( 2 +  2  )  5   2 ' 1 J 2 = arctan . M   (  2 +  2  ) 1 J 2  2 2 ' 1 J 2 . M 2 3 J 2 = − 2 1 J 2 − 3 arctan 2   (  +   )     1J 2 1 ' 1 J 2 M 2 2 1 J 2 = 2 2 2 + 3 arctan . 2  ( +   )    ( +   )   1J 2  2 2  + ' 1 J 2 M 2 = − 2  1 J 2 + 3 ln û û. 2  −    û  − ' 1 J 2 û û û  3J 2  2 3 J 2 2  2  1 J 2  3  + ' 1 J 2 = − − + ln M û û. 2 2 2 4 5  −  3   û  − ' 1 J 2 û û û  1J 2  1  + ' 1 J 2  1J 2 = − ln . M (  2 −  2  )2  2 (  2 −  2  ) 2  3 ûû  − ' 1 J 2 ûû û û  3J 2  3 2  1J 2 − 2  2  3 J 2 3  + ' 1 J 2 = − ln . M (  2 −  2  )2  4( 2 −  2  ) 2  5 ûû  − ' 1 J 2 ûû û û  1  + ' 1 J 2 = ln . M  ûû  − ' 1 J 2 ûû (  2 −  2  ) 1 J 2 û û  2   + ' 1 J 2 = − + ln . M (  2 −  2  ) 3 J 2  2  1 J 2  3 ûû  − ' 1 J 2 ûû û û   1J 2 1  + ' 1 J 2 = + ln . M  2 (  2 −  2  ) 2  3  ûû  − ' 1 J 2 ûû (  2 −  2  )2  1 J 2 û û 2

S.3.2-2. Integrals containing (  + ' )K J 2 . 2 ù (  + ' )K J 2 M  =  ( + 2) (  + ' )(K +2) J 2 . 2 (  + ' )(K +4) J 2  (  + ' )(K +2) J 2 ® . 16. ù  (  + ' )K J 2  = 2 ¬ M   +4 −  +2 2 (  + ' )(K +6) J 2 2  (  + ' )(K +4) J 2  2 (  + ' )(K 17. ù  2 (  + ' )K J 2  = 3 ¬ + M   +6 −  +4  +2 15.

J

+2) 2

® .

Page 767

768

SUPPLEMENTS

S.3.2-3. Integrals containing (

2

+  2 )1 J 2 .

1  (  2 +  2 )1 J 2 + M 2 1 ù  (  2 +  2 )1 J 2 M  = 3 (  2 +  2 )3 J 2 . 1 ù (  2 +  2 )3 J 2 M  = 4  (  2 +  2 )3 J 2 + 1 2 (  +  2 )1 J 2  = (  2 +  2 )1 J 2 −  ù

19. 20. 21.

+  2 )1 J

2

ù (

18.



ù

22.

£ 



M2

2

M

 +

2



2

 =

= ln V  + (

2

ln V  + (

2

2

+  2 )1 J 2 W .

3 2 2   (  +  2 )1 J 2 + 8  + (  2 +  2 )1 J 2 ln û

û

+ ) J W .

4

ln  + (

2

ln  + (

2

û û

+  2 )1 J

2

−  2 )1 J

2

.

û

.

û

û

û.



û

2 1 2

3  8

û û



2 2 1J 2 £  2M +  2 = ( +  ) . 24. ù ( 2 +  2 )−3 J 2  =  −2  ( 2 +  2 )−1 J 2 . M

ù

23.

S.3.2-4. Integrals containing (

2

−  2 )1 J 2 .

1  (  2 −  2 )1 J 2 − 2 M 1 ù  (  2 −  2 )1 J 2 M  = 3 (  2 −  2 )3 J 2 . 1 ù (  2 −  2 )3 J 2 M  = 4  (  2 −  2 )3 J 2 − 1 2 (  −  2 )1 J 2  = (  2 −  2 )1 J 2 −  ù 2

ù (

25. 26. 27. 28.



−  2 )1 J

2

 =

M



2

2

ù

S.3.2-5. Integrals containing ( 

2

û û

3 2   ( 8

2

2

−  2 )1 J

2

−  2 )1 J 2 +

û

û

 û. û û û û

.

3  8

4

3  8

4

û û

arccos û

 2 2 1J 2 £  M 2 −  2 = ln û  + ( −  ) û . û û   2 2 1J 2 30. ù £ = ( −  ) . M  2− 2 31. ù ( 2 −  2 )−3 J 2  = −  −2  ( 2 −  2 )−1 J 2 . M 29.

ln  + (

û

−  2 )1 J 2 .

1  2  ù (  2 −  2 )1 J 2 M  = 2  (  2 −  2 )1 J 2 + 2 arcsin  . 1 33. ù  (  2 −  2 )1 J 2  = − (  2 −  2 )3 J 2 . M 3 1 3 2 2 3J 2 34. ù (  −  )  =  (  2 −  2 )3 J 2 +  2  (  2 −  2 )1 J 2 + M 4 8 1 2  + (  2 −  2 )1 J 2 35. ù (  −  2 )1 J 2  = (  2 −  2 )1 J 2 −  ln û  M  û û   36. ù £ M = arcsin .   2− 2   = −(  2 −  2 )1 J 2 . 37. ù £ M  2− 2 38. ù (  2 −  2 )−3 J 2  =  −2  (  2 −  2 )−1 J 2 . M 32.

arcsin

 

.

û. û û

Page 768

S.3. TABLES OF INDEFINITE INTEGRALS

769

S.3.2-6. Reduction formulas. The parameters  ,  ,  , , and B below can assume arbitrary values, except for those at which ¡ denominators vanish in successive applications of a formula. Notation: ¨ =   +  . 1 + O  + +1 ¨ K + B¦  ù  + ¨ K −1  . 39. ù  (   +  )K  = M + B¦ + 1 M P ¡ 1 + + + − +1 ¨ K +1 + ( + B + B¦ + 1) ù  ¨ K +1  S . 40. ù  (   +  )K  = ¡ 'B ( + 1) Q M M 1 + + + ¨ ¨ 41. ù  (   +  )K  =  +1 K +1 −  ( + B + B¦ + 1) ù  +  K  S . ¡  ( + 1) Q M M ¡ 1 +  + −  +1 ¨ K +1 −  ( − B + 1) ù  + −  ¨ K  S . 42. ù  (   +  )K  = ¡  ( + B¦ + 1) Q M M

¡

S.3.3. Integrals Containing Exponential Functions    ù

1.



M

2.

ù



3.

ù

   

4.

ù

5.

ù B

6.

ù

7.

ù

8.

ù

9.

ù

M

1    .

 =  =







ln 

.

  1  =  O − 2 .   P    2 2 2  2   =   O − 2 + 3 . M    P   1  B B (B − 1)  −2 B ! B !      =   − 2   −1 +  − ))) + (−1)  −1   + (−1)   +1 S , 3 M     Q  = 1, 2,  . ( )    =    ½  (−1)  M  . ( ), where . ( ) is an arbitrary polynomial of degree B .    +1    M  =0  M  1   M  = −  +  K  D  ln | +   K |.

1   if  > 0,   £  arctan ”  K I• 
= 
  M    K +   −K 1  +  K £ −  ln ” if  < 0.  2 £ −   −  K £ −  •


£  +  K  −£  1 ln   £  £  +    + £  if  > 0,  K


= M £  +  K  £  +  K  2 if  < 0.  £ − arctan £ − M

S.3.4. Integrals Containing Hyperbolic Functions S.3.4-1. Integrals containing cosh  . 1

1.

ù cosh(  + ' ) M  =  sinh(  + ' ).

2.

ù

 cosh  M

 =  sinh  − cosh  .

Page 769

770

SUPPLEMENTS

 2 cosh  ù

3.

 = ( M

2

+ 2) sinh  − 2 cosh  .

½   2  −1  2  4. ù  sinh  − cosh  ® . cosh   = (2B )! ¬ (2 , − 1)! M  =1 (2 , )! ½   2  2  +1 sinh  − cosh  ® . 5. ù  2  +1 cosh   = (2B + 1)! ¬ (2 , )! M  =0 (2 , + 1)! 2

 K cosh 

 =  K sinh  −  K

6.

ù

7.

ù cosh2  M  = 12  + 14 sinh 2 .

M

−1

cosh  +  ( − 1) ù

 K

−2

cosh 

M

 .

ù cosh3  M  = sinh  + 13 sinh3  .  −1  1 ½ sinh[2(B − , ) ] L 2  9. ù cosh2    = L 2  2  + 2  −1 , B = 1, 2,  M 2 2 2(B − , )  =0   sinh[(2B − 2 , + 1) ] ½ sinh2  +1  1 ½ 10. ù cosh2  +1   = 2  L 2  +1 = L  , M 2 2B − 2 , + 1 2, + 1  =0  =0 8.

1



−1

11.

ù coshK  M  =

12.

ù cosh  cosh ' M  =  2 −  2 (  cosh ' sinh  −  cosh  sinh ' ).  2  = arctan    . M ù cosh   ½  −1 sinh   1 1 2  (B − 1)(B − 2)  (B − , ) = ¬ + ® , M ù 2 2  −1 2B − 1 cosh (2B − 3)(2B − 5)  (2B − 2 , − 1) cosh2  −2  −1  cosh    =1 B = 1, 2,  ½  −1 (2B − 1)(2B − 3)  (2B − 2 , + 1)  sinh  1 1 = + ¬ ® M ù 2  −2  2B 2  (B − 1)(B − 2)  (B − , ) cosh2  +1  cosh2   cosh   =1 (2B − 1)!! + arctan sinh  , B = 1, 2,  (2B )!!
sign   +  cosh  −£ arcsin if  2 <  2 ,  2 2  +  cosh    −  ù  +  M cosh  =

1  +  + £  2 −  2 tanh( 2) 2 2 £  2 −  2 ln  +  − £  2 −  2 tanh( 2) if  >  .

13. 14.

15.

16.



sinh  coshK

−1

 +

1



ù coshK

B = 1, 2, 

−2

 M

 .

S.3.4-2. Integrals containing sinh  . 1

17.

ù sinh(  + ' ) M  =  cosh(  + ' ).

18.

ù

 sinh 

19.

ù

 2 sinh 

20.

ù

 2  sinh 

 =  cosh  − sinh  . M

 = ( M M

2

+ 2) cosh  − 2 sinh  .

 = (2B )! ¬

½ 

=0

½  2  2  −1 cosh  − sinh  ® . (2 , )!  =1 (2 , − 1)!

Page 770

771

S.3. TABLES OF INDEFINITE INTEGRALS

½

ù

 2

22.

ù

 K sinh 

23.

2 ù sinh  M  = − 21  + 14 sinh 2 .

21.

+1

sinh 

M

M

 = (2B + 1)!



¬ =0

 =  K cosh  −  K

−1

 2  +1  2 cosh  − sinh  ® . (2 , + 1)! (2 , )! sinh  +  ( − 1) ù

 K

−2

sinh 

M

 .

3 3 ù sinh  M  = − cosh  + 13 cosh  .  −1 1 ½  sinh[2(B − , ) ] (−1)  L 2  25. ù sinh2    = (−1)  L 2  2  + 2  −1 , B = 1, 2,  M 2 2 2(B − , )  =0   1 ½ cosh[(2B − 2 , + 1) ] ½ cosh2  +1  26. ù sinh2  +1   = 2  (−1)  L 2  +1 = (−1)  +  L   , M 2 2B − 2 , + 1 2, + 1  =0  =0 B = 1, 2, 

24.

1

−1

ù sinhK  M  =

28.

ù sinh  sinh ' M  =  2 −  2   cosh  sinh ' −  cosh ' sinh   .  1  = ln û tanh . M ù sinh   2 ûû û û û  cosh  1 = ¬− M ù 2B − 1 sinh2  −1  sinh2   ½  −1 2  (B − 1)(B − 2)  (B − , ) 1 ® , B = 1, 2,  + (−1)  −1 (2B − 3)(2B − 5)  (2B − 2 , − 1) sinh2  −2  −1   =1 ½  −1  cosh  1 (2B −1)(2B −3)  (2B −2 , +1) 1 ® ¬ − 2  + (−1)  −1 M 2  +1 = ù 2  2B 2 (B −1)(B −2)  (B − , ) sinh  −2   sinh  sinh   =1 (2B − 1)!!  + (−1)  ln tanh , B = 1, 2,  (2B )!! 2  1  tanh( 2) −  + £  2 +  2 . ù  +  M sinh  = £ 2 2 ln  +  tanh( 2) −  − £  2 +  2 #* + $ sinh  $ #3 − $C  tanh( 2) −  + £  2 +  2  =  + £ 2 2 ln . ù  +  sinh  M    +  tanh( 2) −  − £  2 +  2

29. 30.

31.

32. 33.



sinhK



27.

−1

 cosh  − 1

S.3.4-3. Integrals containing tanh  34.

ù tanh  M  = ln cosh  .

35.

ù tanh2  M  =  − tanh  .



ù sinhK

−2

 M

 .

or coth  .

ù tanh3  M  = − 12 tanh2  + ln cosh  . ½  tanh2  −2  +1  , B = 1, 2,  37. ù tanh2    =  − M  =1 2B − 2 , + 1 ½ ½  tanh2  −2  +2  (−1)  L   , 38. ù tanh2  +1   = ln cosh  − = ln cosh  − 2 M  =1 2 , cosh   =1 2B − 2 , + 2 36.

B = 1, 2, 

Page 771

772

SUPPLEMENTS 1

39.

ù tanhK  M  = −  − 1 tanhK

40.

ù coth  M  = ln |sinh  |.

41.

2 ù coth  M  =  − coth  .

−1

 + ù tanhK

−2



 . M

ù coth3  M  = − 12 coth2  + ln |sinh  |. ½  coth2  −2  +1  43. ù coth2    =  − , B = 1, 2,  M  =1 2B − 2 , + 1 ½  ½  coth2  −2  +2  L  44. ù coth2  +1   = ln |sinh  |− = ln |sinh  |− , 2 M  =1 2 , sinh   =1 2B − 2 , + 2 42.

1

ù cothK  M  = −  − 1 cothK

45.

−1

 + ù cothK

−2

 M

B = 1, 2, 

 .

S.3.5. Integrals Containing Logarithmic Functions 1.

ù ln  M  =  ln  −  .

2.

ù

 ln  M

 = 12  2 ln  − 14  2 .

ù  K ln  M  =

3.

 

1 2

1  K +1 ln2 

+1

ln  −

1  K ( + 1)2

+1

if



≠ −1,

if



= −1.

2 2 ù (ln  ) M  =  (ln  ) − 2 ln  + 2 .

4.

 (ln  )2  = 12  2 (ln  )2 − 12  2 ln  + 14  2 . M 2K +1 2K +1 K +1  2  ) − ln  + if (ln  ( + 1)2 ( + 1)3 6. ù  K (ln  )2  =  + 1 M 3 1 if 3 ln   ½  7. ù (ln  )   = (−1)  (B + 1)Be (B − , + 1)(ln  )  −  , M B + 1  =0 5.

ù



≠ −1,



= −1.

B = 1, 2, 

ù (ln  ) & M  =  (ln  ) & −  ù (ln  ) & −1 M  ,  ≠ −1. +   +1 ½ (−1)  9. ù   (ln  ) +  = ( + 1)  ( − , + 1)(ln  ) + −  , B , = 1, 2,  ¡ ¡ ¡ M +1 (B + 1)  +1 ¡  =0 ¡ 1  +1 −1 10. ù  K (ln  ) &  = M  + 1  K (ln  ) & −  + 1 ù  K (ln  ) & M  ,  ,  ≠ −1. 8.

1

ù ln(  + ' ) M  =  (  +  ) ln( +  ) −  . 1 1  2 12. ù  ln(  + ' )  = ” 2 − 2 ln(  + ' ) − ” M 2  • 2 3 1  1 13. ù  2 ln(  + ' )  = ” 3 − 3 ln(  + ' ) − M 3  • 3

11.

”



2

−

2



3

3

−

 

 

2

.

• 2

+

 2 . 2 •

Page 772

773

S.3. TABLES OF INDEFINITE INTEGRALS

15.

ù

ln  1  + ln .  (  + ' )   + '  ln  1  1 =− + + 2 ln . 3 2 )


2  ( + ' ) 2 ( + '  ) 2£   + £' 2  + ' +  ¬ (ln  − 2) £  + ' + £  ln £ ®   ln    + ' − £ 
£  +M ' =

 2 £  + ' ¬ (ln  − 2) £  + ' + 2 £ −  arctan £ ®  − ln( 2 +  2 )  =  ln( 2 +  2 ) − 2 + 2  arctan(  ). M  ln( 2 +  2 )  = 12 V ( 2 +  2 ) ln( 2 +  2 ) −  2 W . M  2 ln( 2 +  2 )  = 13 V  3 ln( 2 +  2 ) − 23  3 + 2  2  − 2  3 arctan( M

ù

ln  M (  + ' ln  M (  + '

14.

ù

16.

17.

ù

18.

ù

19.

ù



)2

=−

if  > 0, if  < 0.

 )W .

S.3.6. Integrals Containing Trigonometric Functions S.3.6-1. Integrals containing cos 

(B = 1, 2,  ).

1

1.

ù cos(  + ' ) M  =  sin(  + ' ).

2.

ù

 cos 

3.

ù

 2 cos 

 = cos  +  sin  . M

 = 2 cos  + ( M

2

− 2) sin  .

½ ½  −1  2  −2     cos   = (2B )! ¬ (−1) (−1)  sin  + 4. ù  M (2B − 2 , )! (2B  =0  =0  2  −2  +1 ½   5. ù  2  +1 cos   = (2B + 1)! ¬ (−1)  sin  + M (2B − 2 , + 1)! (2B  =0 2

 K cos 

 =  K sin  +  K

6.

ù

7.

ù cos2  M  = 12  + 14 sin 2 .

8.

ù cos3  M  = sin  − 13 sin3  .

9.

ù

cos2  

10.

ù

11.

ù

12.

ù

13.

ù

M

M

 =

−1

cos  −  ( − 1) ù

 1  1 ½ L  + 22  2  22  −1

−1

 K

−2

cos 

M





2 −2 −1

− 2 , − 1)!



2 −2



− 2 , )!

cos  ® .

cos  ® .

 .

sin[(2B − 2 , ) ] L 2  . 2B − 2 ,

 =0  ½ 1 sin[(2B − 2 , + 1) ] cos2  +1   = 2  L 2  +1 . M 2 2B − 2 , + 1  =0   = ln û tan O +  . M cos  2 4 P ûû û û û  M 2 = tan  . cos   sin  1  = + ln û tan O +  . M cos3  2 cos2  2 û 2 4 P ûû û û

Page 773

774

SUPPLEMENTS

ù

15.

ù

M

14.





M2 



+

ù

16.

ù

17.

18.

ù

19.

ù ù

20.

ù

   cos2  ù

   cos  

23.

=0

 , B > 1. M B − 1 ù cos −2  (2B − 2)(2B − 4)  (2B − 2 , + 2) (2B − 2 , ) sin  − cos  (2B − 1)(2B − 3)  (2B − 2 , + 3) (2B − 2 , + 1)(2B − 2 , ) cos2  −2  −1

+



B −2

(B − 1)!   tan  + ln |cos  |  . (2B − 1)!! sin V (  −  ) W sin V (  +  ) W + , cos  cos '  = 2(  −  ) 2(  +  ) M


2 (  −  ) tan( 2) arctan  £  2− 2 £  2− 2  =


 M £  +  cos  1  2 −  2 + (  −  ) tan( £  2 −  2 ln ûû £  2 −  2 − (  −  ) tan( û   sin û  = 2 − 2 M ù  2 2 (  +  cos  ) (  −  )(  +  cos  )  −  2  1  tan  = £ arctan £ . M 2 2  2 +  2 cos2    +  2+ 2

1  tan    £  2 −  2 arctan £  2 −  2  =

 M £  2 −  2 −  tan   2 −  2 cos2  1 û ln 2  £  2 −  2 ûû £  2 −  2 +  tan 

   cos '

22.

−1



−1

ù

21.

½

=



cos 2

sin  (B − 1) cos 

=

cos 

M M

S.3.6-2. Integrals containing sin 



 ≠ A/ . if 

2

>  2,

2)

2 2

2) ûû if  >  . û û . M +  cos 

û û û

if 

2

>  2,

if 

2

>  2.

û û     =   ” 2 2 sin ' + 2 2 cos ' .  +  + •  2    = 2 ” cos2  + 2 sin  cos  + .  +4 r•  B (B − 1)   cos  −1   = (  cos  + B sin  ) + 2 2 2  +B  +B 2 ù

M

+1

   cos

−2

 M

 .

(B = 1, 2,  ).

1

24.

ù sin(  + ' ) M  = −  cos(  + ' ).

25.

ù

 sin 

26.

ù

 2 sin 

27.

ù

 3 sin 

28.

ù

 2  sin  ù

 2

30.

ù

 K sin 

31.

ù sin2  M  = 12  − 14 sin 2 .

29.

+1

 = sin  −  cos  . M

2

 = 2 sin  − ( M

 = (3 M M

sin 

M

2

3

− 6) sin  − (

 = (2B )! ¬ M

− 2) cos  .

½ 

(−1)  =0

 = (2B + 1)!

½ 

=0

 = − K cos  +  K

− 6 ) cos  .

½  −1  2  −2   2  −2  −1 cos  + (−1)  sin  ® . (2B − 2 , )! (2B − 2 , − 1)!  =0  2  −2  +1  2  −2  ¬ (−1)  +1 cos  + (−1)  sin  ® . (2B − 2 , + 1)! (2B − 2 , )! +1

−1

sin  −  ( − 1) ù

 K

−2

sin 

M

 .

Page 774

S.3. TABLES OF INDEFINITE INTEGRALS

 sin2 

 = 14 

2

− 14  sin 2 −

32.

ù

33.

3 3 ù sin  M  = − cos  + 13 cos  .

M

1 8

775

cos 2 .

 −1 1  (−1)  ½ sin[(2B − 2 , ) ]  34. ù sin   = 2  L 2   + 2  −1 (−1)  L 2  , M 2 2 2B − 2 ,  =0 ! ¡ are binomial coefficients (0! = 1). where L +  = , ! ( − , )! ¡  1 ½ cos[(2B − 2 , + 1) ] 2  +1   = 2 (−1)  +  +1 L 2  +1 . 35. ù sin M 2 2B − 2 , + 1  =0   36. ù = ln û tan û . M sin  2û û û û  37. ù M 2 = − cot  . sin    cos  1 38. ù M =− + ln û tan û . sin3  2 sin2  2 û 2û û B −û 2  cos   =− + 39. ù M  M  −2 , B > 1. ù  −1 sin  (B − 1) sin  B − 1 sin   −1 ½   sin  + (2B − 2 , ) cos  (2B − 2)(2B − 4)  (2B − 2 , + 2) 40. ù =− M sin2   (2B − 1)(2B − 3)  (2B − 2 , + 3) (2B − 2 , + 1)(2B − 2 , ) sin2  −2  +1   =0 2  −1 (B − 1)! +  ln |sin  | −  cot   . (2B − 1)!! sin[(  −  ) ] sin[(  +  ) ] − ,  ≠ A/ . 41. ù sin  sin '  = 2(  −  ) 2(  +  ) M


2  +  tan  2 arctan £ if  2 >  2 ,  £ 2− 2 2− 2    =


 42. ù M  − £  2 −  2 +  tan  2  +  sin  1 2 2 ln û £  2 −  2 û  + £  2 −  2 +  tan  2 ûû if  >  . û û û    cos û  43. ù = + . M M ù (  +  sin  )2 (  2 −  2 )(  +  sin  )  2 −  2  +  sin  £  2 +  2 tan   1 44. ù = arctan . M  2 +  2 sin2   
£  2 +  2

£  2 −  2 tan  1 arctan if  2 >  2 ,  £ 2 2   45. ù M =

   −  £  2 −  2 tan  +   2 −  2 sin2  1 ln û û if  2 >  2 . 2  £  2 −  2 ûû £  2 −  2 tan  −  ûû û sin   1 , cosû  . 46. ù £ M2 2 = − arcsin £ , 1 + , sin  1+, 2 sin   1 47. ù £ = − ln , cos  + £ 1 − , 2 sin2  . M û û , 1 − , 2 sin2  û û cos 1+, 2  , cos  £ £ 2 2 2 2 1 + , sin   = − 1 + , sin  − arcsin £ . 48. ù sin  M 2 2, 1+, 2 cos  £ 1−, 2 49. ù sin  £ 1 − , 2 sin2   = − 1 − , 2 sin2  − ln , cos  + £ 1 − , 2 sin2  . û û M 2 2, û û 2

Page 775

776

SUPPLEMENTS

    =   O 2 2 sin ' − 2 2 cos ' .  +  + P  2  O  sin2  − 2 sin  cos  + . sin2   = 2 M  +4  P   sin  −1  B (B − 1)  sin    = (  sin  − B cos  ) + 2 2 2 M  +B  +B 2 ù

50.

ù

   sin '

51.

ù

  

52.

ù

  

M

S.3.6-3. Integrals containing sin 

54. 55. 56. 57.

−2

 M

 .

and cos  .

cos[(  +  ) ] cos V (  −  ) W ù sin  cos ' M  = − 2(  +  ) − 2(  −  ) (  1 ù  2 cos2  M + ( 2 sin2  = ˜( arctan O  tan  P . ( tan  +   1 ù  2 cos2  M − ( 2 sin2  = 2 ˜( ln û ( tan  −  û . û û û û + −1  + 2  −2 + +1 ½  tan  L   + + −1 , ù cos2   M sin2 +  = 2, − 2 + 1  =0 ¡  ++  + ln |tan  | + ½ L  = L M ù cos2  +1  sin2 + +1   ++  ++  =0

53.

   sin 

 ≠ A/ .

,

B , ¡

= 1, 2, 

tan2  −2 +  , 2, − 2

¡

B , ¡

= 1, 2, 

S.3.6-4. Reduction formulas. The parameters  and  below can assume any values, except for those at which the denominators on the right-hand side vanish. sinK

−1

ù sinK  cos &  M  = −

59.

ù

60.

ù

61.

ù

62.

ù

63.

ù

64.

ù

S.3.6-5. Integrals containing tan  65.

 cos & +1 



−1 −2  +  +  ù sinK  cos&  M  . sinK +1  cos & −1   − 1 sinK  cos& −2   . sinK  cos &   = + M M  +  + ù −1 −1 sinK  cos &   −1 sinK  cos &   = O sin2  − M  +  + −2P ( − 1)(  − 1) + ù sinK −2  cos & −2  M  . ( +  )( +  − 2) sinK +1  cos & +1   +  + 2 sinK +2  cos &   . sinK  cos &   = + M  +1  +1 ù M +1 +1 sinK  cos &   +  + 2 sinK  cos & +2   . sinK  cos &   = − + M  +1  +1 ù M −1 +1 sinK  cos &   − 1 sinK −2  cos& +2   . sinK  cos &   = − + M  +1  +1 ù M +1 −1 sinK  cos &   −1 +2 −2 sinK  cos &   = + M  +1  + 1 ù sinK  cos &  M  .

58.

+

and cot  .

ù tan  M  = − ln |cos  |.

Page 776

777

S.3. TABLES OF INDEFINITE INTEGRALS

66.

ù tan2  M  = tan  −  .

67.

ù tan3  M  = 12 tan2  + ln |cos  |. ½  (−1)  (tan  )2  −2  +1 2  tan = (−1) − , B = 1, 2,     ù M 2B − 2 , + 1  =1 ½  (−1)  (tan  )2  −2  +2 2  +1  +1   = (−1) ln |cos  | − , B = 1, 2,  ù tan M 2B − 2 , + 2  =1  1 ù  + M  tan  =  2 +  2  +  ln |  cos  +  sin  |  .

tan    1 = £ arccos ” 1 − cos  ,  >  ,  > 0. M ù £  •  −  +  tan2 

68. 69. 70. 71. 72.

ù cot  M  = ln |sin  |.

73.

ù cot2  M  = − cot  −  .

ù cot3  M  = − 21 cot2  − ln |sin  |. ½  (−1)  (cot  )2  −2  +1 75. ù cot2    = (−1)   + , B = 1, 2,  M 2B − 2 , + 1  =1 ½  (−1)  (cot  )2  −2  +2 , B = 1, 2,   76. ù cot2  +1   = (−1)  ln |sin  | + 2B − 2 , + 2 M  =1  1 77. ù = 2 2  −  ln |  sin  +  cos  |  . M  +  cot   + 74.

S.3.7. Integrals Containing Inverse Trigonometric Functions 1. 2. 3. 4. 5. 6. 7. 8. 9.

  £ ù arcsin  M  =  arcsin  +  2 −  2 .  2  2  £ ù O arcsin ‹P M  = ‹O arcsin ‹P − 2 + 2  2 −  2 arcsin   1   £ ù  arcsin  M  = 4 (2 2 −  2 ) arcsin  + 4  2 −  2 .   3  1 £ ù  2 arcsin þM  = 3 arcsin  + 9 ( 2 + 2  2 )  2 −  2 .   £ ù arccos þM  =  arccos  −  2 −  2 .  2  2 £ ù O arccos ‹P M  = ‹O arccos *P − 2 − 2  2 −  2 arccos 1    £ ù  arccos  M  = 4 (2 2 −  2 ) arccos  − 4  2 −  2 .   3  1 £ ù  2 arccos  M  = 3 arccos  − 9 ( 2 + 2  2 )  2 −  2 .    ù arctan þM  =  arctan  − 2 ln(  2 +  2 ). 

.



.

Page 777

778

SUPPLEMENTS

ù

11.

ù

12.

ù

13.

ù

14.

ù

^_

1    = ( 2 +  2 ) arctan − .  M 2  2 3 2      3  2 arctan  = arctan − + ln(  2 +  2 ). þM 3  6 6    arccot  =  arccot + ln(  2 +  2 ).  M  2  1 2   2  arccot  = ( +  ) arccot + . þM 2  2 3 2      3  2 arccot  = arccot + − ln(  2 +  2 ). þM 3  6 6

 arctan

10.



References for Subsection S.3.3: H. B. Dwight (1961), I. S. Gradshteyn and I. M. Ryzhik (1980), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986, 1988).

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