Διαφορικές Εξισώσεις για Μηχανικούς- Σημειώσεις

Shmei¸seic Diaforik¸n Exis¸sewn ... gia MhqanikoÔc

Μανόλης Βάβαλης 1 Ιουνίου 2010

2

To keÐmeno autì morfopoi jhke se

c 2010 Copyright

LATEX.

Manìlhc Bˆbalhc

This work is licensed under the Creative Commons To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/us/ You can use, print, duplicate, share these notes as much as you want. You can base your own notes on these and reuse parts if you keep the license the same. If you plan to use these commercially (sell them for more than just duplicating cost) then you need to contact me and we will work something out. If you are printing a course pack for your students, then it is fine if the duplication service is charging a fee for printing and selling the printed copy. I consider that duplicating cost. 'Eqete to dikaÐwma na qrhsimopoi sete, ektup¸sete, antigrˆyete kai na dianèmete tic shmei¸seic autèc ìsec forèc jèlete. MporeÐte na basÐsete tic dikèc sac shmei¸seic se autèc  /kai na qrhsimopoi sete olìklhra mèrh touc, arkeÐ na diathr sete thn paroÔsa ˆdeia qr shc analloÐwth.

Eˆn skopeÔete na tup¸sete tic paroÔsec shmei¸seic gia

didaktikoÔc skopoÔc tìte mporeÐte na qre¸sete ìpou epijumeÐte to posì pou apaiteÐtai gia thn ektÔpwsh, thn diˆjesh kai thn dianom  tou ektupwmènou pakètou.

Perieqìmena Eisagwg 

1

2

3

4

4

0.1

ShmeÐwsh sqetikˆ me tic shmei¸seic autèc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

0.2

Eisagwg  stic diaforikèc exis¸seic

5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

SDE pr¸thc tˆxhc

10

1.1

LÔseic se morf  oloklhrwmˆtwn

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

PedÐa kateujÔnsewn

1.3

DiaqwrÐsimec Exis¸seic

1.4

Grammikèc exis¸seic kai oloklhrwtikoÐ parˆgontec

1.5

Antikatastˆseic

1.6

Autìnomec exis¸seic

10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Grammikèc SDE uyhlìterhc tˆxhc

35

2.1

Grammikèc SDE uyhlìterhc tˆxhc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Grammikèc SDE deÔterhc tˆxhc me stajeroÔc suntelestèc

. . . . . . . . . . . . . . . . . . . . . . . . .

35 38

2.3

Grammikèc SDE uyhlìterhc tˆxhc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.4

Talant¸seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.5

Mh-omogeneÐc exis¸seic

2.6

Exanagkasmènec talant¸seic kai suntonismìc

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 57

Sust mata SDE

63

3.1

Eisagwg  sta sust mata SDE

3.2

Grammikˆ sust mata SDE

3.3

Mèjodoc idiotim¸n

3.4

Sust mata dÔo diastˆsewn kai ta dianusmatikˆ pedÐa touc

. . . . . . . . . . . . . . . . . . . . . . . . .

74

3.5

Sust mata deÔterhc tˆxhc kai efarmogèc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.6

Idiotimèc me pollaplìthta

85

3.7

Ekjetikˆ Pinˆkwn

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

3.8

Mh-omogen  sust mata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Seirèc

F ourier

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

kai MDE

102

4.1

Probl mata sunoriak¸n tim¸n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.2

Trigwnometrikèc seirèc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

F ourier

4.3

Epiprìsjeta jèmata seir¸n

4.4

Seirèc hmitìnwn kai sunhmitìnwn

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.5

MDE, qwrismìc metablht¸n, kai h exÐswsh jermìthtac . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

BibliografÐa

123

Euret rio

125

3

Eisagwg  0.1

ShmeÐwsh sqetikˆ me tic shmei¸seic autèc.

Oi shmei¸seic autèc ˆrqisan na grˆfontai stic arqèc tou 2010 parˆllhla me tic dialèxeic mou sto eisagwgikì mˆjhma stic Diaforikèc Exis¸seic. Den filodoxoÔn oÔte na eÐnai pl reic oÔte prwtìtupec. Ousiastikˆ apoteloÔn perÐlhyh

Edwards and Penney, Differential Equations and Boundary Value Problems [EP], kai tou biblÐou Boyce and DiPrima, Elementary Differential Equations [BD], emploutismèno me dikèc mou parembˆseic me bˆsh

tou biblÐou twn twn

thn antÐdrash tou akroathrÐou mou.

Sac parakal¸ na epikoinwn sete mazÐ mou gia opoiad pote dikˆ mou lˆjh, kai

paral yeic   dikèc sac protˆseic.

EpiskefjeÐte thn istoselÐda

http://diaek.wordpress.com/

4

gia perissìterec plhroforÐec.

0.2. ΕΙΣΑΓΩΓŸΗ ΣΤΙΣ ΔΙΑΦΟΡΙʟΕΣ ΕΞΙӟΩΣΕΙΣ 0.2

5

Eisagwg  stic diaforikèc exis¸seic

0.2.1

Diaforikèc Exis¸seic

Oi nìmoi thc fusik c genikˆ diatup¸nontai san diaforikèc exis¸seic.

Gia ton lìgo autì ìloi oi epist monec kai oi

mhqanikoÐ qrhsimopoioÔn ˆlloi lÐgo ˆlloi polÔ diaforikèc exis¸seic. H katanìhsh twn diaforik¸n exis¸sewn epomènwc eÐnai aparaÐthth gia thn katanìhsh sqedìn otid pote ja melet sete sta maj matˆ sac. MporeÐte na jewr sete ta majhmatikˆ san thn gl¸ssa twn episthm¸n kai tic diaforikèc exis¸seic san èna apì ta shmantikìtera sustatikˆ thc. 'Eqete  dh sunant sei pollèc diaforikèc exis¸seic, endeqomènwc qwrÐc na to antilhfjeÐte.

'Eqete mˆlista  dh

lÔsei merikèc apì autèc sta maj mata tou LogismoÔ pou èqei pˆrei. Ac epikentrwjoÔme ìmwc t¸ra se mia diaforik  exÐswsh pou mˆllon den thn èqete dei xanˆ.

dx + x = 2 cos t, dt ìpou

x

eÐnai h exarthmènh metablht  kai

exÐswsh.

t

h anexˆrthth metablht .

(1) H exÐswsh (1) eÐnai mia klassik  diaforik 

Sthn pragmatikìthta eÐnai èna parˆdeigma miac diaforik c exÐswshc pr¸thc tˆxhc, mia kai emplèkei thn

pr¸th mìnon parˆgwgo thc exarthmènhc metablht c. H parapˆnw exÐswsh phgˆzei apì ton nìmo tou NeÔtwna gia thn yÔxh enìc s¸matoc ìtan h jermokrasÐa tou peribˆllontoc q¸rou talant¸netai ston qrìno.

0.2.2

EpÐlush diaforik¸n exis¸sewn

EpilÔw thn parapˆnw diaforik  exÐswsh shmaÐnei brÐskw to mia sunˆrthsh tou

t

thn opoÐa kaloÔme

x

x to opoÐo exartˆtai apì to t. Jèloume dhlad  na broÔme x, t, kai dx sthn exÐswsh (1) aut  na dt

tètoia ¸ste eˆn antikatast soume ta

isqÔei. H idèa eÐnai akrib¸c Ðdia me aut  pou sunantˆme stic algebrikèc exis¸seic, dhlad  se exis¸seic pou emplèkontai mìno to

x

kai to

t

kai ìqi oi parˆgwgoi. Sthn prokeÐmenh perÐptwsh to

x = x(t) = cos t + sin t eÐnai lÔsh. P¸c mporoÔme na epibebai¸soume kˆti tètoio? Aplˆ antikajist¸ntac to sthn exÐswsh! Pr¸ta prèpei na upologÐsoume thn

dx dx . BrÐskoume ìti dt dt

= − sin t + cos t.

Ac upologÐsoume t¸ra to dexiì mèloc thc exÐswshc (1)

dx + x = (− sin t + cos t) + (cos t + sin t) = 2 cos t. dt Prˆgmati!

Autì eÐnai Ðso me to dexiì thc mèloc.

cos t + sin t + e−t

Den telei¸same ìmwc!

MporoÔme na isqurisjoÔme oti h

x =

eÐnai epÐshc lÔsh. Ac dokimˆsoume,

dx = − sin t + cos t − e−t . dt 'Opwc kai prin ac thn antikatast soume sto aristerì mèloc thc exÐswshc (1)

dx + x = (− sin t + cos t − e−t ) + (cos t + sin t + e−t ) = 2 cos t. dt Kai prˆgmati eÐmaste bèbaioi ìti apoteleÐ lÔsh! Sunep¸c mporeÐ na upˆrqoun perissìterec apì mia lÔseic. Mˆlista, gia thn sugkekrimènh exÐswsh ìlec oi lÔseic pou upˆrqoun mporoÔn na sumptuqjoÔn kai na grafoÔn me thn ex c morf 

x = cos t + sin t + Ce−t ìpou

C

eÐnai mia kˆpoia stajerˆ. Sto sq ma 1 sthn epìmenh selÐda mporoÔme na broÔme tic grafikèc parastˆseic gia

merikèc apì autèc tic lÔseic.

Ja mˆjoume pwc mporoÔme na broÔme me susthmatikì trìpo ìlec autèc tic lÔseic se

lÐgec mèrec. Mhn nomÐsete ìti h epÐlush twn diaforik¸n exis¸sewn eÐnai eÔkolh upìjesh. TounantÐon, ìpwc ja doÔme sÔntoma, sun jwc eÐnai idiaÐtera dÔskolh. Den upˆrqoun genikèc mèjodoi ikanèc na lÔsoun opoiad pote diaforik  exÐswsh mac endiafèrei.

Ja prèpei na anafèroume ìti ja perioristoÔme sthn eÔresh analutik¸n ekfrˆsewn gia tic lÔseic kai

den endiaferìmaste gia arijmhtikèc proseggÐseic twn lÔsewn tic opoÐec endeqomènwc na mporoÔme na ektim soume

6

ΕΙΣΑΓΩΓΗ

upologistikˆ. Tètoiec proseggÐseic ja mac apasqol soun se maj mata episthmonik¸n upologism¸n kai arijmhtik c anˆlushc. 'Ena megˆlo mèroc tou maj matoc mac ja afierwjeÐ stic legìmenec Sun jeic Diaforikèc Exis¸seic (

ODEs

sta agglikˆ)   sunoptikˆ SDE (

sta agglikˆ),

ordinary dif f erential equations

ìpou me ton ìro autì dhl¸noume ìti èqoume mÐa mìnon

anexˆrthth metablht , en¸ ìlec oi emplekìmenec parˆgwgoi aforoÔn thn en lìgw metablht .

Sthn perÐptwsh pou

èqoume perissìterec apì mia eleÔjerec metablhtèc, odhgoÔmaste stic sun jwc legìmenec Merikèc Diaforikèc Ex-

partial dif f erential equations

is¸seic (

sta agglikˆ)

∗

  sunoptikˆ MDE (

P DEs

sta agglikˆ) me tic opoÐec ja

asqolhjoÔme sto deÔtero mèroc tou maj matoc. Akìma kai gia tic SDE, oi opoÐec ìson aforˆ thn basik  touc ènnoia, eÐnai eÔkola katanohtèc se bˆjoc, den eÐnai apl  upìjesh h eÔresh lÔsewn se ìla ta probl ma-

0

1

2

3

4

5

3

3

2

2

1

1

0

0

-1

-1

ta sta opoÐa emplèkontai. EÐnai shmantikì na gnwrÐzoume pìte mporoÔme na lÔsoume eÔkola kˆpoio tètoio prìblhma kai pwc sthn prˆxh mporoÔme na to kˆnoume autì. Akìma kai sthn perÐptwsh pou mporoÔme na broÔme thn lÔsh mac me qr sh upologistik¸n susthmˆtwn, kˆti pou prˆgmati sumbaÐnei sthn prˆxh, eÐnai aparaÐthto na katano soume ti paristˆ aut  kai pwc akrib¸c kai kˆtw apì poiec sunj kec proèkuye. Epiprìsjeta, eÐnai polÔ suqnˆ aparaÐthto na aplopoi soume tic exis¸seic mac   na tic metatrèyoume se kˆpoia ˆllh morf  thn opoÐa katanoeÐ to logismikì sÔsthma pou ja qrhsimopoi soume gia na tic lÔsoume. Endeqomènwc na qreiˆzetai na kˆnoume kˆpoiec shmantikèc upojèseic sto majhmatikì mac montèlo gia na mporèsoume

0

1

2

3

4

5

na pragmatopoi soume kˆpoiec apì tic aparaÐthtec autèc metatropèc. Gia na gÐnete ènac petuqhmènoc mhqanikìc   epist monac, ja anagkasteÐte suqnˆ na lÔsete probl mata pou potè prin den èqete antimetwpÐsei sto pareljìn.

Σχήμα 1: Γραφικές παραστάσεις μερικών από τις λύx σεις της εξίσωσης dx dt + 2 = cos t.

EÐnai

sunep¸c shmantikì na mˆjete diadikasÐec epÐlushc problhmˆtwn tic opoÐec ja mporèsete na efarmìsete gia thn epÐlush twn en lìgw problhmˆtwn. EÐnai sunhjismèno lˆjoc en gènei na perimènete ìti ja mˆjete suntagèc kai ètoimouc proc ˆmesh qr sh mhqanismoÔc gia na lÔsete ìla ta sqetikˆ probl mata pou ja sunant sete sthn karièra sac. Kˆti tètoio sÐgoura den prìkeitai na gÐnei sto mˆjhma autì.

0.2.3

Diaforikèc exis¸seic sthn prˆxh

Realistikì Prìblhma

Ac doÔme pwc mporoÔme na qrhsimopoi soume diaforikèc exis¸seic stic epist mec kai thn mhqanik . 'Estw ìti èqete kˆpoia realistikˆ probl mata ta opoÐa jèlete na katano sete. Gia to katafèrete autì prèpei na kˆnete kˆpoiec upo-

ex ghsh

afaÐresh

jèseic pou ja aplopoi soun ta prˆgmata kai ja sac epitrèyoun na dhmiourg sete èna majhmatikì montèlo. Me ˆlla lìgia, prèpei na metafrˆsete to realistikì sac prìblhma (thn pragmatikìthta eˆn protimˆte) se èna sÔnolo diaforik¸n exis¸sewn.

AfoÔ to katafèrete autì mporeÐte na efar-

Majhmatikì montèlo

επίλυση

mìsete majhmatikˆ ergaleÐa gia na dhmiourg sete kˆpoiac morf c majhmatik  lÔsh.

Majhmatik  lÔsh

'Omwc akìma den telei¸sate.

Prèpei na katano sete autì pou br kate. Prèpei dhlad  na diapist¸sete ti mporeÐ h majhmatik  lÔsh pou br kate na sac pei sqetikˆ to arqikì realistikì sac prìblhma. Pollˆ apì ta maj mata twn spoud¸n sac ja sac exoplÐsoun me katˆllhlec gn¸seic pou ja sac epitrèyoun na diatup¸nete majhmatikˆ montèla twn realistik¸n sac problhmˆtwn kai na katano sete ta apotelèsmatˆ sac. Sto mˆjhma autì ìmwc ja epikentrwjoÔme sthn majhmatik  anˆlush thc ìlhc diadikasÐac.

Arketèc forèc bèbaia

ja asqolhjoÔme me merikˆ realistikˆ probl mata gia na apokt soume kˆpoia bajÔterh katanìhsh kai kallÐterh diaÐsjhsh allˆ kai gia na apokt soume kˆpoio kÐnhtro gia autˆ pou ja kˆnoume parakˆtw allˆ kai gia ton trìpo pou ja ta kˆnoume.

∗ Pio katˆllhloc ìroc katˆ thn gn¸mh mou eÐnai Diaforikèc Exis¸seic me Merikèc Parag¸gouc, o opoÐoc dustuq¸c den eÐnai idiaÐtera diadedomènoc

0.2. ΕΙΣΑΓΩΓŸΗ ΣΤΙΣ ΔΙΑΦΟΡΙʟΕΣ ΕΞΙӟΩΣΕΙΣ

7

Ac doÔme èna parˆdeigma thc parapˆnw diadikasÐac.

Mia apì tic poio basikèc diaforikèc exis¸seic eÐnai to

exponential growth model).

prìtupo montèlo ekjetik c aÔxhshc ( bakthridÐwn se èna doqeÐo.

'Estw oti me

P

paristˆnoume ton plhjusmì kˆpoiwn

An upojèsoume ìti upˆrqei arket  trof  kai q¸roc tìte eÐnai eÔkolo na antilhfjoÔme

oti o rujmìc aÔxhshc tou plhjusmoÔ twn bakthridÐwn ja eÐnai anˆlogoc tou Ðdiou tou plhjusmoÔ. Dhlad , megˆloi plhjusmoÐ auxˆnontai taqÔtera. Eˆn me

t

sumbolÐzoume ton qrìno (ac poÔme se deuterìlepta) tìte katal goume sto

ex c majhmatikì montèlo

dP = kP dt ìpou

k>0

eÐnai kˆpoia jetik  stajerˆ.

Parˆdeigma 0.2.1:

'Estw ìti upˆrqoun 100 bakt ria thn qronik  stigm  0 kai 200 bakt ria metˆ apì 10c. Pìsa

bakt ria ja upˆrqoun 1 leptì metˆ thn arqik  stigm  0? (dhlad  se 60 deuterìlepta). Pr¸ta prèpei na lÔsoume thn exÐswsh. MporoÔme na isqurisjoÔme ìti h lÔsh dÐnetai apì thn sqèsh

P (t) = Cekt , ìpou

C

eÐnai kˆpoia stajerˆ. Ac to epibebai¸soume,

dP = Ckekt = kP. dt Prˆgmati eÐnai lÔsh. Ti katafèrame ìmwc? Den gnwrÐsoume to Xèroume ìti

P (0) = 100

C ìpwc den gnwrÐzoume oÔte to k. Xèroume ìmwc arketˆ ˆlla prˆgmata. P (10) = 200. Ac antikatast soume autˆ pou xèroume kai ac doÔme pou

ìpwc xèroume kai ìti

ja katal xoume.

100 = P (0) = Cek0 = C, 200 = P (10) = 100 ek10 . Sunep¸c,

2 = e10k

 

ln 2 10

= k ≈ 0.069.

'Ara gnwrÐzoume ìti

P (t) = 100 e(ln 2)t/10 ≈ 100 e0.069t . Dhlad  xèroume ìti metˆ apì 1 leptì,

t = 60,

o plhjusmìc ja eÐnai

P (60) = 6400.

Ac doÔme kai to Sq ma 2.

Entˆxei mèqri ed¸, allˆ ac prospaj soume na katano soume ta apotelèsmatˆ mac.

ShmaÐnei prˆgmati ìti metˆ

apì 60c ja èqoume akrib¸c 6400 bakt ria sto doqeÐo? Pro-

0

10

20

30

40

50

60

6000

6000

5000

5000

4000

4000

3000

3000

2000

2000

1000

1000

fan¸c ìqi! Mhn xeqnˆte ìti kˆname paradoqèc pou mporeÐ na mhn eÐnai swstèc. Eˆn ìmwc oi paradoqèc mac autèc eÐnai logikèc, tìte prˆgmati ja upˆrqoun perÐpou 6400 bakt ria. ShmeÐwste epÐshc ìti sthn pragmatikìthta to

P

eÐnai ènac

akèraioc arijmìc, kai ìqi kˆpoioc pragmatikìc arijmìc. Gia parˆdeigma to montèlo mac mporeÐ na mac diabebai¸sei ìti metˆ apì 61c,

P (61) ≈ 6859.35.

Sun jwc sthn prˆxh, to

k

sthn

P 0 = kP

eÐnai gnwstì,

kai prospajoÔme na lÔsoume thn exÐswsh gia diaforetikèc arqikèc sunj kec.

Ti ennooÔme me autì?

soume thn katˆstash jètontac dhlad  thn exÐswsh

P (0) = 1000

dP dt

=P

k = 1.

Ac aplopoi -

'Ac jewr soume

kˆtw apì thn proôpìjesh ìti

(h arqik  sunj kh).

P (t) = Cet

20

30

40

50

60

genik  lÔsh, mia kai h opoiad pote lÔsh thc exÐswshc mporeÐ na grafjeÐ sthn

morf  aut  gia kˆpoia katˆllhlh tim  thc stajerˆc

C

10

Σχήμα 2: Αύξηση του πληθυσμού των βακτηριδίων τα πρώτα 60 δευτερόλεπτα.

P (t) = 1000 et .

sugkekrimènh tim  thc

0 0

Tìte eÔkola mporoÔme

na broÔme thn ex c lÔsh

Ja onomˆsoume thn

0

C.

Qreiazìmaste mia arqik  sunj kh gia na prosdiorÐsoume thn

h opoÐa kajorÐzei thn sugkekrimènh lÔsh pou antistoiqeÐ sthn arqik  sunj kh. Suqnˆ me

8

ΕΙΣΑΓΩΓΗ

ton ìro sugkekrimènh lÔsh ja ennooÔme aplˆ kˆpoia lÔsh en¸ me ton ìro genik  lÔsh ja ennooÔme mia oikogèneia lÔsewn. Ac proqwr soume t¸ra se autèc pou ja onomˆsoume oi 4 jemeli¸deic exis¸seic oi opoÐec emfanÐzontai tìso suqnˆ pou kalì ja eÐnai na apomnhmoneÔsoume tic lÔseic touc.

MporoÔme sqetikˆ eÔkola na mantèyoume autèc

tic lÔseic enjumoÔmenoi idiìthtec twn ekjetik¸n, twn hmitonoeid¸n kai twn sunhmitonoeid¸n sunart sewn. epÐshc mporoÔme na epibebai¸soume to ìti eÐnai prˆgmati lÔseic.

EÔkola

Tètoiec epibebai¸seic apoteloÔn pˆnta mia kal 

genik  praktik  pou mporeÐ na mac prostateÔsei apotelesmatikˆ apì diˆfora probl mata (p.q.

na mhn jumìmaste

swstˆ thn lÔsh). Prèpei na mporeÐte na lÔnete me eukolÐa tic exis¸seic autèc. H pijanìthta na deÐte mia h dÔo apì autèc se 'OLA ta diagwnÐsmata tou maj matoc eÐnai idiaÐtera megˆlh. 'Iswc na mhn eÐnai kajìlou ˆsqhmh idèa na apomnhmoneÔsete tic lÔseic touc. Proswpikˆ eg¸ tic jumˆmai sun jwc. Den eÐmai ìmwc potè sÐgouroc gia thn mn mh mou me apotèlesma na elègqw to ti jumˆmai epibebai¸nontac to eˆn h sunˆrthsh pou jumˆmai eÐnai prˆgmati lÔsh. Fusikˆ, den eÐnai kai tìso dÔskolo na mantèyei kˆpoioc tic lÔseic autèc. Se kˆje perÐptwsh mhn mou peÐte ìti den sac proeidopoÐhsa. H pr¸th jemeli¸dhc exÐswsh eÐnai h ex c,

ìpou

k>0

kˆpoia stajerˆ.

Ed¸

y

eÐnai h

dy = ky, dx exarthmènh kai x h

anexˆrthth metablht .

H genik  lÔsh thc exÐswshc

aut  eÐnai

y(x) = Cekx . 'Eqoume  dh diapist¸sei ìti aut  eÐnai h lÔsh prohgoumènwc an kai tìte eÐqame qrhsimopoi sei ˆlla onìmata gia tic metablhtèc mac. Parìmoia, h exÐswsh

dy = −ky, dx ìpou

k>0

kˆpoia stajerˆ, èqei thn ex c genik  lÔsh

y(x) = Ce−kx . 0.2.1

Epibebai¸ste to gegonìc ìti h dojeÐsa

y

eÐnai pragmatikˆ h lÔsh thc exÐswshc.

T¸ra jewr ste thn ex c diaforik  exÐswsh deÔterhc tˆxhc

d2 y = −k2 y, dx2 gia kˆpoia stajerˆ

k > 0.

H genik  lÔsh thc exÐswshc aut c eÐnai

y(x) = C1 cos(kx) + C2 sin(kx). Shmei¸ste ìti epeid  èqoume diaforik  exÐswsh deÔterhc tˆxhc èqoume dÔo stajerèc sthn genik  thc lÔsh.

0.2.2

Epibebai¸ste to gegonìc ìti h dojeÐsa

y

eÐnai pragmatikˆ h lÔsh thc exÐswshc.

Tèloc, ac doÔme thn ex c diaforik  exÐswsh deÔterhc tˆxhc

d2 y = k2 y, dx2 gia kˆpoia stajerˆ

k > 0.

H genik  lÔsh thc exÐswshc aut c eÐnai

y(x) = C1 ekx + C2 e−kx ,   isodÔnama

y(x) = D1 cosh(kx) + D2 sinh(kx).

0.2. ΕΙΣΑΓΩΓŸΗ ΣΤΙΣ ΔΙΑΦΟΡΙʟΕΣ ΕΞΙӟΩΣΕΙΣ Gia ìsouc den jumoÔntai, oi

cosh

kai

sinh

9

orÐzontai wc ex c

ex + e−x , 2 x −x e −e sinh x = . 2

cosh x =

Merikèc forèc protimˆme na qrhsimopoioÔme tic sunart seic autèc parˆ autèc pou perièqoun ekjetikˆ kurÐwc lìgw kˆpoiwn polÔ elkustik¸n idiot twn touc ìpwc oi

d kˆpoio lˆjoc ed¸) kai dx

0.2.3

d dx

cosh x = sinh x

(ìqi, den upˆrqei

y

pou d¸same parapˆnw eÐnai prˆgmati lÔseic thc exÐswshc.

Ask seic

0.2.4

DeÐxte ìti h

x = e4t

0.2.5

DeÐxte ìti h

x = et

0.2.6

EÐnai h

0.2.7

EÐnai h sunˆrthsh

y = sin t

d¸ste ìlec tic dunatèc

0.2.8

kai

sinh x = cosh x.

Elèxte ìti oi morfèc thc

0.2.4

cosh 0 = 1, sinh 0 = 0,

eÐnai lÔsh thc

den eÐnai lÔsh thc

lÔsh thc


dy 2 dt

y = erx lÔsh timèc thc r .

Epibebai¸ste ìti h

x000 − 12x00 + 48x0 − 64x = 0.

x = Ce−2t

x000 − 12x00 + 48x0 − 64x = 0.

= 1 − y2 ?

Dikaiolog ste thn apˆnths  sac.

thc exÐswshc

y 00 + 2y 0 − 8y = 0

eÐnai lÔsh thc

x0 = −2x.

gia kˆpoia tim  thc paramètrou

BreÐte

C

r?

Eˆn nai,

pou antistoiqeÐ sthn arqik  sunj kh

x(0) = 100. 0.2.9

sthn arqik  sunj kh

0.2.10 2

x = C1 e−t + C2 e2t x(0) = 10.

Epibebai¸ste ìti h

x = 4.

eÐnai lÔsh thc

x00 − x0 − 2x = 0.

BreÐte

C1

kai

C2

pou na antistoiqoÔn

Qrhsimopoi¸ntac idiìthtec twn parag¸gwn gnwst¸n sunart sewn prospaj ste na breÐte mia lÔsh thc

(x0 )2 +

Kefˆlaio 1

SDE pr¸thc tˆxhc 1.1

LÔseic se morf  oloklhrwmˆtwn

SDE pr¸thc tˆxhc eÐnai mia exÐswsh thc morf c

dy = f (x, y), dx h thc morf c

y 0 = f (x, y). Genikˆ, den upˆrqei aplìc genikìc tÔpoc   diadikasÐa pou ja mporoÔse kˆpoioc na akolouj sei gia na brei tic lÔseic miac SDE ìpwc h parapˆnw. Stic epìmenec dialèxeic ja exetˆsoume sugkekrimènec peript¸seic gia tic opoÐec den eÐnai dÔskolo na broÔme tic lÔseic. Sthn parˆgrafo aut  ac upojèsoume ìti h

f

eÐnai mia sunˆrthsh mìnon miac metablht c, thc

x,

dhlad  èqei thn

ex c morf 

y 0 = f (x). MporoÔme na oloklhr¸soume wc proc

x

(1.1)

kai ta dÔo mèrh thc exÐswshc.

Z

y 0 (x) dx =

Z f (x) dx + C,

dhlad 

Z y(x) = Aut  h sunˆrthsh thc

f (x)

y(x) eÐnai ousiastikˆ h genik  lÔsh.

f (x) dx + C. Dhlad  gia na lÔsoume thn (1.1) br kame kˆpoia anti-parˆgwgo

kai katìpin kataskeuˆsame thn genik  lÔsh prosjètontac mia tuqaÐa stajerˆ.

Ed¸ eÐnai katˆllhlh stigm  na suzht soume èna jèma sqetikì me sto sumbolismì kai thn terminologÐa tou majhmatikoÔ logismoÔ.

Pollˆ biblÐa tou MajhmatikoÔ LogismoÔ dhmiourgoÔn sÔgqush ìtan me ton ìro olokl rwma

ennooÔn prwtarqikˆ to apokaloÔmeno orismèno olokl rwma.

To aìristo olokl rwma ìmwc eÐnai sthn ousÐa h an-

tiparˆgwgoc (sthn pragmatikìthta mia mono-parametrik  oikogèneia antiparag¸gwn). Sthn pragmatikìthta upˆrqei mìnon èna olokl rwma kai autì eÐnai to orismèno olokl rwma.

O mìnoc lìgoc pou qreiˆzetai kˆpoioc na qrhsi-

mopoi sei ton sumbolismì tou aìristou oloklhr¸matoc eÐnai epeid  mporoÔme pˆnta na grˆyoume thn antiparˆgwgo san èna (orismèno) olokl rwma. Prˆgmati me bˆsh to jemeli¸dec je¸rhma tou LogismoÔ mporoÔme na grˆyoume to

R

f (x) dx + C

wc ex c

Z

x

f (t) dt + C. x0 H olokl rwsh eÐnai aplˆ ènac trìpoc na upologÐsei kˆpoioc thn antiparˆgwgo (ènac trìpoc pou eÐnai pˆntote apotelesmatikìc, deÐte to parakˆtw parˆdeigma). H olokl rwsh orÐzetai bèbaia san to embadìn tou qwrÐou pou brÐsketai

10

1.1. ˟ΥΣΕΙΣ ΣΕ ΜΟΡΦŸΗ ΟΛΟΚΛΗΡΩ̟ΑΤΩΝ

11

kˆtw apì thn grafik  parˆstash thc sunˆrthshc, aplˆ tuqaÐnei na mac odhgeÐ kai sthn antiparˆgwgo.

Gia lìgouc

sumbatìthtac me diˆfora biblÐa allˆ kai ˆlla maj mata, ja suneqÐsoume na qrhsimopoioÔme ton sumbolismì tou aorÐstou oloklhr¸matoc gia na anaferjoÔme se antiparag¸gouc, pˆnta ìmwc ja prèpei na skeftìmaste to orismèno olokl rwma.

Parˆdeigma 1.1.1:

BreÐte thn genik  lÔsh thc exÐswshc

y 0 = 3x2 .

EÔkola parathroÔme ìti h genik  lÔsh prèpei na eÐnai thc morf c

y = x3 + C .

Ac to epibebai¸soume:

y 0 = 3x2 .

Odhghj kame akrib¸c pÐsw sthn exÐsws  mac. Sun jwc mia exÐswsh ìpwc h parapˆnw, sunodeÔetai apì mia arqik  sunj kh san thn arijmoÔc

x0

kai

y0

(to

x0

eÐnai sun jwc 0, allˆ ìqi pˆnta).

y(x0 ) = y0

gia kˆpoiouc

MporoÔme na grˆyoume thn lÔsh se mia polÔ ìmorfh

morf  san orismèno olokl rwma. 'Estw ìti to prìblhmˆ mac eÐnai

y 0 = f (x), y(x0 ) = y0 . Tìte h lÔsh Z x f (s) ds + y0 . y(x) =

eÐnai

(1.2)

x0 Ac to epibebai¸soume! H

y 0 = f (x)

(sÔmfwna me to jemeli¸dec je¸rhma tou logismoÔ gia parˆdeigma), eÐnai sÐgoura x0 0 0 0. x0 Parakal¸ shmei¸ste ìti to orismèno olokl rwma (anti-parˆgwgoc) eÐnai teleÐwc diaforetik  majhmatik  ontìthta

mia lÔsh. EÐnai ìmwc aut  pou ikanopoieÐ thn arqik  sunj kh? Prˆgmati eÐnai epeid 

y(x ) =

R

f (x) dx + y = y

apì to aìristo olokl rwma. To orismèno olokl rwma eÐnai ousiastikˆ ènac arijmìc. Sunep¸c h (1.2) eÐnai ènac tÔpoc ton opoÐo mporeÐ kaneÐc na qrhsimopoi sei endeqomènwc mèsw enìc upologist  gia na pˆrei sugkekrimènouc arijmoÔc. MporeÐ sunep¸c kˆpoioc na kˆnei thn grafik  parˆstash thc lÔshc kai na thn qrhsimopoi sei san na eÐnai mia opoiad pote sunˆrthsh. Den eÐnai dhlad  aparaÐthto na brei kˆpoioc thn analutik  morf  thc antiparag¸gou.

Parˆdeigma 1.1.2:

LÔste thn 2

y 0 = e−x ,

y(0) = 1.

Me bˆsh thn prohghjeÐsa suz thsh, h lÔsh prèpei na eÐnai h ex c

x

Z y(x) =

2

e−s ds + 1.

0 Den èqei kanèna nìhma na prospaj sete na breÐte thn lÔsh aut  se analutik  (merikoÐ thn lène kai kleist ) morf . Apì thn mia meriˆ den mporeÐte apì thn ˆllh den qreiˆzetai mia kai to en lìgw orismèno olokl rwma mporeÐ na d¸sei pollèc plhroforÐec. To olokl rwma autì parempiptìntwc eÐnai idiaÐtera shmantikì sthn Statistik  kai ìqi mìno. Qrhsimopoi¸ntac thn mèjodo aut  mporoÔme na lÔsoume kai exis¸seic thc morf c

y 0 = f (y). Ac grˆyoume thn exÐswsh se mia ˆllh morf  qrhsimopoi¸ntac ton sumbolismì

Leibniz

dy = f (y). dx MporoÔme t¸ra na qrhsimopoi soume to je¸rhma thc antÐstrofhc sunˆrthshc tou logismoÔ gia na antistrèyoume touc rìlouc twn metablht¸n

x

kai

y

katal gontac sthn exÐswsh

dx 1 = . dy f (y) H parapˆnw prospˆjeiˆ mac dÐnei thn entÔpwsh ìti kˆnoume aplèc algebrikèc prˆxeic me ta elkustikì na kˆnei kaneÐc algebrikèc prˆxeic me ta

dx kai dy jewr¸ntac ta aploÔc arijmoÔc.

dx

kai

dy .

EÐnai sÐgoura

Sthn parapˆnw perÐptwsh

ìla doÔleyan mia qarˆ. Prosoq  ìmwc, tètoiec praktikèc aplopoihmènwn prˆxewn mporeÐ na mac dhmiourg soun sobaroÔc mpelˆdec, idiaÐtera ìtan emplèkontai perissìterec apì mia anexˆrthtec metablhtèc. Sto shmeÐo autì mporoÔme aplˆ na oloklhr¸soume

Z x(y) = Telikˆ, ac prospaj soume na lÔsoume wc proc

y.

1 dy + C. f (y)

12

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ

Parˆdeigma 1.1.3:

Mantèyame ìti h

y 0 = ky

èqei thn ex c lÔsh

upologÐzontac thn lÔsh aut  ex' arq c. Shmei¸ste pr¸ta ìti h ìti

y 6= 0.

y=0

Cekx .

Ac epikur¸soume thn manteyiˆ mac aut 

eÐnai profan¸c mia lÔsh. Ac upojèsoume ìmwc

'Eqoume loipìn

dx 1 = . dy ky Oloklhr¸nontac paÐrnoume

x(y) = x = T¸ra ac lÔsoume wc proc

1 ln |y| + C 0 . k

y 0

ekC ekx = |y|. Eˆn antikatast soume ton ìro

ekC

0

me kˆpoia tuqaÐa stajerˆ

'Etsi èqoume enswmat¸sei kai thn lÔsh

y = 0,

C

mporoÔme na apallaqtoÔme apì thn apìluth tim .

katal gontac ousiastikˆ sthn Ðdia genik  lÔsh pou arqikˆ mantèyame,

y = Cekx . Parˆdeigma 1.1.4: Xèroume ìti h

BreÐte thn genik  lÔsh thc

y=0

y0 = y2 .

eÐnai mia lÔsh. Ac upojèsoume proc stigm  ìti

y 6= 0

kai ac grˆyoume

dx 1 = 2. dy y Oloklhr¸nontac èqoume

x= LÔnoume wc proc

−1 + C. y

1 kai katal goume sthn ex c genik  lÔsh C−x

y=

y= Parathr ste tic idiaiterìthtec thc lÔshc.

1 C −x

 

y = 0.

An gia parˆdeigma jèsoume

C = 1,

tìte ìso plhsiˆzoume to

x = 1

parathroÔme miac suneq¸c kai me gorgoÔc rujmoÔc, auxanìmenh tim  thc lÔshc katal gontac argˆ h gr gora se #ekrhxh. lÔsh thc.

EÐnai en gènei dÔskolo aplˆ parathr¸ntac thn Ðdia thn exÐswsh na katalˆboume to pwc sumperifèretai h H exÐswsh

y 0 = y 2 faÐnetai mia qarˆ kai eÐnai orismènh (−∞, C)   thc morf c (C, ∞).

pantoÔ, h lÔsh thc ìmwc eÐnai orismènh mìnon se

kˆpoio diˆsthma thc morf c

Klassikˆ probl mata pou mac odhgoÔn se diaforikèc exis¸seic tic opoÐec mporoÔme na epilÔsoume me apl  olokl rwsh eÐnai probl mata pou aforoÔn taqÔthta, epitˆqunsh kai apìstash.

Tètoia probl mata sÐgoura èqete  dh

sunant sei sta maj mata tou LogismoÔ.

Parˆdeigma 1.1.5:

Jewr ste èna ìqhma pou kineÐtai me taqÔthta

et/2 mètra to deuterìlepto, ìpou me t paristˆnoume

to qrìno se deuterìlepta. Pìso makriˆ ja èqei fjˆsei se 2 deuterìlepta kai pìso se 10 deuterìlepta?. 'Estw

x

h apìstash pou dianÔei to ìqhma. IsqÔei h exÐswsh

x0 = et/2 . MporoÔme aplˆ na oloklhr¸soume kai ta dÔo mèlh thc exÐswshc kai na katal xoume sthn ex c exÐswsh

x(t) = 2et/2 + C. Shmei¸ste ìti akìma den èqoume upologÐsei to

C.

GnwrÐzoume ìmwc ìti ìtan to

t=0

tìte

x = 0,

dhlad :

x(0) = 0

opìte

0 = x(0) = 2e0/2 + C = 2 + C. 'Ara

C = −2

kai sunep¸c

x(t) = et/2 − 2. MporoÔme t¸ra na antikatast soume sthn exÐswsh gia na upologÐsoume thn apìstash pou ja dianÔsei to ìqhma metˆ apì 2 kai 10 deuterìlepta

x(2) = 2e2/2 − 2 ≈ 3.44

,

mètra

x(10) = 2e10/2 − 2 ≈ 294

.

mètra

1.1. ˟ΥΣΕΙΣ ΣΕ ΜΟΡΦŸΗ ΟΛΟΚΛΗΡΩ̟ΑΤΩΝ Parˆdeigma 1.1.6:

13

Upojèste ìti to ìqhma epitaqÔnei me rujmì

t2 m/s2 .

Thn qronik  stigm 

t = 0

to ìqhma

brÐsketai se apìstash 1 mètrou apì thn arqik  tou jèsh kai kin te me taqÔthta 10 m/c. PoÔ ja brÐsketai to ìqhma thn qronik  stigm 

t = 10?

Autì eÐnai èna prìblhma deÔterhc tˆxhc. Eˆn parast soume me h taqÔthtˆ tou kai

x00

x00 = t2 , Fusikˆ, eˆn jèsoume

x0 = v

x(0) = 1,

to opoÐo mporoÔme na lÔsoume wc proc LÔste wc proc

1.1.1

v

v,

kai katìpin lÔste wc proc

x.

Ask seic

LÔste thn exÐswsh

dy dx

= x2 + x

1.1.3

LÔste thn exÐswsh

dy dx

= sin 5x

1.1.4

LÔste thn exÐswsh

dy dx

=

1.1.5

LÔste thn exÐswsh

y0 = y3

1.1.6

LÔste thn exÐswsh

y 0 = (y − 1)(y + 1)

dy 1.1.7 LÔste thn exÐswsh dx LÔste thn exÐswsh

x0 (0) = 10.

=

gia

gia

1 gia x2 −1 gia

y 00 = sin x

y(1) = 3.

y(0) = 2. y(0) = 0.

y(0) = 1.

1 gia y 2 +1 gia

v(0) = 10,

kai katìpin na oloklhr¸soume gia na broÔme to

1.1.2

1.1.8

thn apìstash pou dianÔei to ìqhma, tìte

eÔkola katal goume sto ex c prìblhma pr¸thc tˆxhc

v 0 = t2 ,

1.1.1

x

h epitˆquns  tou. To prìblhma (h exÐswsh kai oi arqikèc sunj kec) eÐnai tìte to ex c

gia

y(0) = 3.

y(0) = 0. y(0) = 0.

x.

x0

eÐnai

14

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ

1.2

PedÐa kateujÔnsewn M AT LAB http://www.math.uiuc.edu/iode/ .

Sto shmeÐo autì den eÐnai ˆsqhmh idèa na egkatast sete sthn mporeÐte na breÐte sthn istoselÐda

sac to logismikì sÔsthma

IODE

to opoÐo

'Opwc eÐdame, oi exis¸seic pr¸thc tˆxhc me tic opoÐec ja asqolhjoÔme èqoun thn ex c genik  morf 

y 0 = f (x, y). En gènei, den eÐnai dunatìn na lÔsoume ìlec tic exis¸seic autoÔ tou tÔpou analutikˆ. Ja epijumoÔsame na gnwrÐzame toulˆqiston kˆpoia stoiqeÐa twn lÔse¸n touc, ìpwc h genik  morf  touc kai h en gènei sumperiforˆ touc,   èstw na mporoÔsame na upologÐsoume proseggÐseic twn lÔsewn twn exis¸sewn aut¸n.

1.2.1

PedÐa kateujÔnsewn

MporoÔme eÔkola na diapist¸soume ìti h exÐswsh shmeÐo tou epipèdou

(x, y).

y 0 = f (x, y)

ousiastikˆ mac dÐnei thn tim  thc klÐshc thc

y

se kˆje

MporoÔme na kˆnoume thn grafik  parˆstash thc en lìgw klÐshc qrhsimopoi¸ntac èna

mikrì eujÔgrammo tm ma se diˆfora shmeÐa, arketˆ ¸ste na sqhmatÐsoume mia kajar  eikìna tou ti akrib¸c paristˆ ìpwc autì faÐnetai sto Sq ma 1.1.

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

3

3

3

3

2

2

2

2

1

1

1

1

0

0

0

0

-1

-1

-1

-1

-2

-2

-2

-2

-3

-3

-3 -3

-2

-1

0

1

2

3

Σχήμα 1.1: Πεδίο κατευθύνσεων της εξίσωσης y 0 = xy.

-3 -3

-2

y(x0 ) = y0 ,

0

1

2

3

Σχήμα 1.2: Το πεδίο κατευθύνσεων της εξίσωσης y 0 = xy και η γραφική παράσταση της λύσης που ικανοποιεί τις συνθήκες y(0) = 0.2, y(0) = 0, και y(0) = −0.2.

Thn parapˆnw eikìna onomˆzoume pedÐo kateujÔnsewn thc exÐswshc. sunj kh

-1

mporoÔme na xekin soume apì to shmeÐo

(x0 , y0 )

Eˆn mac dojeÐ mia sugkekrimènh arqik 

sto epÐpedo kai na proqwr soume sthn grafik 

parˆstash thc sugkekrimènhc lÔshc aplˆ akolouj¸ntac katˆllhla tic klÐseic tou pedÐou. Parathr ste sqetikˆ to Sq ma 1.2. Parathr¸ntac to pedÐo kateujÔnsewn loipìn mporoÔme na pˆroume pollèc plhroforÐec sqetikˆ me thn sumperiforˆ twn lÔsewn. Gia parˆdeigma, sto Sq ma 1.2 mporoÔme eÔkola na doÔme pwc pˆne oi lÔseic ìtan h arqik  sunj kh eÐnai

y(0) > 0, y(0) = 0

kai

y(0) < 0.

Shmei¸ste ìti mia mikr  allag  stic arqikèc sunj kec mporeÐ na epifèrei drastikèc

allagèc sthn sumperiforˆ thc lÔshc. Apì thn ˆllh meriˆ, eˆn kˆnoume thn grafik  parˆstash merik¸n lÔsewn thc exÐswshc

y 0 = −y ,

parathroÔme ìti apì ìpou kai na xekin soume, ìlec oi lÔseic teÐnoun sto mhdèn kaj¸c to

sto ˆpeiro. DeÐte to Sq ma 1.3 sthn paroÔsa selÐda.

x

teÐnei

1.2. ΠΕğΙΑ ΚΑΤΕΥȟΥΝΣΕΩΝ

15

-3

-2

-1

0

1

2

3

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3 -3

-2

-1

0

1

2

3

Σχήμα 1.3: Πεδίο κατευθύνσεων της εξίσωσης y 0 = −y μαζί με την γραφική παράσταση μερικών συγκεκριμένων λύσεων.

1.2.2

'Uparxh kai monadikìthta

Duo eÐnai ta jemeli¸dh erwt mata pou mac apasqoloÔn ìson aforˆ to ex c prìblhma

y 0 = f (x, y),

y(x0 ) = y0 .

i

( ) Upˆrqei lÔsh?

ii

( ) EÐnai h lÔsh monadik  (eˆn upˆrqei)? Mhn biasteÐte na jewr sete ta parapˆnw erwt mata rhtorikˆ me profan  apˆnthsh kai sta dÔo to nai. Upˆrqoun peript¸seic pou (anˆloga me to

f (x, y))

h apˆnthsh se kˆpoio apì autˆ mporeÐ kˆllista na eÐnai ìqi.

Epeid  genikˆ oi exis¸seic mac prokÔptoun apì realistikèc katastˆseic kai fusikˆ probl mata, faÐnetai logikì na upojèsoume ìti pˆnta upˆrqei lÔsh. H en lìgw lÔsh epÐshc eÐnai logikì na upojèsoume ìti eÐnai monadik  mia kai ìloi mac pisteÔoume ìti o kìsmoc mac eÐnai nteterministikìc. Eˆn den upˆrqei lÔsh,   h lÔsh den eÐnai monadik  tìte pijanìn na prospajoÔme na lÔsoume kˆpoio majhmatikì montèlo pou den antistoiqeÐ swstˆ sto arqikì realistikì mac prìblhma. Se kˆje perÐptwsh, eÐnai pˆnta shmantikì na gnwrÐzoume pìte èqoume prìblhma kai an eÐnai dunatìn giatÐ.

Parˆdeigma 1.2.1:

Prospaj ste na lÔsete thn exÐswsh:

y0 = Oloklhr¸ste gia na breÐte thn lÔsh

1 , x

y(0) = 0.

y = ln |x| + C .

Shmei¸ste ìti h lÔsh aut  den upˆrqei gia

x = 0.

DeÐte

to 1.4 sthn epìmenh selÐda.

Parˆdeigma 1.2.2:

LÔste thn exÐswsh:

y0 = 2 Shmei¸ste ìti h sunˆrthsh

y = x

2

p |y|,

y(0) = 0.

eÐnai mia lÔsh kai ìti h

apoteleÐ lÔsh mìno sthn perÐptwsh pou èqoume

x > 0).

y = 0

eÐnai epÐshc lÔsh (shmeÐwste ìmwc ìti h

x2

DeÐte to Sq ma 1.5 sthn epìmenh selÐda.

EÐnai sthn pragmatikìthta dÔskolo na sumperˆnei kˆpoioc to eˆn mia diaforik  exÐswsh èqei monadik  lÔsh   ìqi parathr¸ntac aplˆ to pedÐo twn dieujÔnse¸n thc. je¸rhma (gnwstì kai san je¸rhma tou qwrÐc apìdeixh.

∗ Charles

´ Emile Picard (1856 – 1941)

Upˆrqei ìmwc kˆpoia elpÐda.

Ac doÔme pr¸ta to parakˆtw

P icard∗ ) to opoÐo parìlo pou den eÐnai dÔskolo na apodeiqjeÐ ja to deqjoÔme

16

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

3

3

3

3

2

2

2

2

1

1

1

1

0

0

0

0

-1

-1

-1

-1

-2

-2

-2

-2

-3

-3

-3 -3

-2

-1

0

1

2

3

Σχήμα 1.4: Πεδίο διευθύνσεων της εξίσωσης y 0 = 1/x.

Je¸rhma 1.2.1

-3 -3

-2

(x0 , y0 ),

0

1

2

3

Σχήμα 1.5: Πεδίο διευθύνσεων της εξίσωσης y 0 = p 2 |y| με δύο λύσεις που ικανοποιούν την συνθήκη y(0) = 0.

P icard).

(Je¸rhma Ôparxhc kai monadikìthtac tou

san sunˆrthsh dÔo bebaÐwc metablht¸n) kai eˆn h parˆgwgoc apì to

-1

Eˆn h

f (x, y)

eÐnai suneq c (jewr¸ntac thn

∂f upˆrqei kai eÐnai suneq c se kˆpoia perioq  gÔro ∂y

tìte h lÔsh tou probl matoc

y 0 = f (x, y), upˆrqei (toulˆqiston gia kˆpoia

x)

Shmei¸ste ìti ta probl mata

y(x0 ) = y0 ,

kai eÐnai monadik .

y 0 = 1/x, y(0) = 0

kai

p y 0 = 2 |y|, y(0) = 0

den ikanopoioÔn tic proôpojèseic tou

jewr matoc. Akìma kai sthn perÐptwsh pou mporoÔme na efarmìsoume to parapˆnw je¸rhma ofeÐloume na eÐmaste prosektikoÐ pˆnw sto jèma thc Ôparxhc. EÐnai arketˆ pijanìn h lÔsh na upˆrqei mìnon proswrinˆ. RÐxte mia matiˆ sto ex c prìblhma

Parˆdeigma 1.2.3:

y0 = y2 , ìpou

A

y(0) = A,

kˆpoia stajerˆ.

A 6= 0, ìpote h y den eÐnai Ðsh me mhdèn toulˆqiston 1 . Eˆn y(0) = A, tìte C = 1/A ˆra y = C−x

Xèroume pwc na broÔme thn lÔsh tou. Pr¸ta ac upojèsoume ìti gia kˆpoio

x

kontˆ sto 0. Opìte

x0 = 1/y2 ,

ˆra

x = −1/y + C , y=

Eˆn

A = 0,

tìte h

y = 0 eÐnai profan¸c mia lÔsh. A = 1 èqoume èkrhxh thc lÔshc

Eˆn sugkekrimèna thc

x

sunep¸c

1 . −x

1/A

gia

x = 1.

parìlo pou h exÐswsh fainìtan mia qarˆ gia kˆje tim  tou

Sunep¸c, den èqoume Ôparxh lÔshc gia kˆpoia tim 

x.

H exÐswsh

y0 = y2

den prodÐdei kˆpoio prìblhma

kai fusikˆ eÐnai orismènh pantoÔ.

Sto mˆjhma autì ja epikentrwjoÔme se probl mata diaforik¸n exis¸sewn ta opoÐa èqoun monadik  lÔsh, sun jwc gia kˆje tim  tou

x.

EÐnai ìmwc aparaÐthto na suneidhtopoi soume ìti to jèma thc Ôparxhc kai thn monadikìthtac thc

lÔshc mporeÐ kˆllista na prokÔyei, akìma kai apì exis¸seic ìpwc h

y0 = y2

oi opoÐec den prodiajètoun gia kˆti tètoio.

Prèpei toulˆqiston na eÐmaste se jèsh na diapist¸soume thn Ôparxh enìc tètoiou probl mata kai na prospaj soume na katano soume ta aÐtia.

1.2. ΠΕğΙΑ ΚΑΤΕΥȟΥΝΣΕΩΝ 1.2.3

1.2.1 to

x?

17

Ask seic

Sqediˆste to pedÐo dieujÔnsewn thc exÐswshc

y 0 = ex−y .

Pwc sumperifèrontai oi lÔseic thc ìtan auxˆnoume

MporeÐte na mantèyete mia sugkekrimènh lÔsh parathr¸ntac aplˆ to pedÐo twn dieujÔnse¸n thc?

1.2.2

Sqediˆste to pedÐo dieujÔnsewn thc exÐswshc

y 0 = x2 .

1.2.3

Sqediˆste to pedÐo dieujÔnsewn thc exÐswshc

y0 = y2 .

1.2.4

EÐnai dunatìn na lÔsete thn exÐswsh

1.2.5

'Estw

k, A, B, C

y0 =

xy gia cos x

y(0) = 1?

Dikaiolog ste thn apˆnths  sac.

eÐnai kˆpoiec stajerèc. AntistoiqÐste tic ex c diaforikèc exis¸seic me ta pedÐa kateujÔnsewn

pou dÐnontai parakˆtw.

1.

dy dx

= ky ,

ìpou

k>0

2.

dy dx

= ky ,

ìpou

k<0

3.

dy dx

= 2x(A − x)

4.

dy dx

= y(B − y)

5.

dy dx

= x2 + exp(x2 ) cos2 (3y)

6.

dy dx

= −ey − C(1 − cos(x))

2

−1

2

−1.2

1.5

1.5

−1.4 1

1 −1.6 0.5

−1.8 −2

y

0

y

y

0.5

−2.2

−0.5

0

−0.5

−2.4 −1

−1 −2.6

−1.5

−2 −3

−1.5

−2.8

−2

−1

0 x

1

2

3

−3

−8

−6

−4

−2

0 x

2

4

6

8

−2 −3

−2

−1

0 x

1

2

3

18

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ

4

2

3

3.5 1.5

2

1

1

0.5

0

3 2.5

y

y

y

2 1.5 1 0

−1

−0.5

−2

0.5 0 −0.5 −1 −1

−0.5

0

0.5

1 x

1.5

2

2.5

3

−1 −1

−3 −0.5

0

0.5 x

1

1.5

2

−1.5

−1

−0.5

0 x

0.5

1

1.5

1.3. ΔΙΑΧΩџΙΣΙΜΕΣ ΕΞΙӟΩΣΕΙΣ 1.3

19

DiaqwrÐsimec Exis¸seic

y 0 = f (x), tìte mporoÔme na thn lÔsoume aplˆ oloklhr¸nontac: R y = f (x) dx + C . Dustuq¸c o trìpoc autìc den mporeÐ na epektajeÐ eÔkola kai gia exis¸seic thc genik c morf c y 0 = f (x, y). Oloklhr¸nontac kai ta duo mèlh èqoume Z y = f (x, y) dx + C. 'Opwc eÐdame an h exÐsws  mac eÐnai sthn morf 

Prosèxte ìti to olokl rwma exartˆtai apì to

1.3.1

y.

DiaqwrÐsimec Exis¸seic

Ac exetˆsoume thn perÐptwsh pou h exÐswsh eÐnai diaqwrÐsimh, dhlad  an èqei thn morf 

y 0 = f (x)g(y), ìpou

f (x)

kai

g(y)

kˆpoiec sunart seic. Ac grˆyoume thn exÐswsh aut  sthn legìmenh morf 

Leibniz

dy = f (x)g(y). dx T¸ra ac thn xanagrˆyoume wc ex c

dy = f (x) dx. g(y) 'Eqoume sunep¸c diatup¸sei thn exÐswsh me trìpo pou na mporoÔme na oloklhr¸soume kai ta dÔo mèlh.

Dhlad 

èqoume

Z

dy = g(y)

Z f (x) dx + C.

Sunep¸c eˆn mporèsoume na upologÐsoume analutikˆ to olokl rwma èqoume elpÐdec na lÔsoume thn prokÔptousa algebrik  exÐswsh wc proc

Parˆdeigma 1.3.1:

y.

Jewr ste thn exÐswsh

y 0 = xy. Pr¸ta suneidhtopoi ste ìti h Ac grˆyoume thn exÐswsh san

y = 0 apoteleÐ lÔsh, opìte ac asqolhjoÔme dy = xy , opìte dx Z Z dy = x dx + C. y

apì ed¸ kai pèra me thn perÐptwsh

y 6= 0.

UpologÐzontac thn antiparˆgwgo katal goume sthn

ln |y| =

x2 +C 2

  isodÔnama sthn

|y| = e ìpou

D>0

kˆpoia stajerˆ. Mia kai h

y=0

x2 2

+C

=e

x2 2

eC = De

x2 2

,

eÐnai lÔsh kai lìgw thc apìluthc tim c mporoÔme na katal xoume sthn

lÔsh

y = De gia kˆje arijmì

D

x2 2

,

(sumperilambanomènou tou mhdèn kai ton arnhtik¸n arijm¸n).

Ac to elègxoume:

y 0 = Dxe Swstˆ!

x2 2

= x(De

x2 2

) = xy.

20

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ OfeÐloume ìmwc na eÐmaste lÐgo pio prosektikoÐ me thn mèjodo aut .

Oloklhr¸noume ousiastikˆ wc proc dÔo

diaforetikèc metablhtèc kai autì den faÐnetai na eÐnai tìso swstì. FaÐnetai ìti den kˆnoume thn Ðdia prˆxh kai sta dÔo mèrh thc exÐswshc. Ac doulèyoume me perissìterh majhmatik  austhrìthta.

dy = f (x)g(y) dx Ac xanagrˆyoume thn exÐswsh, lambˆnontac up ìyin to ìti h

y = y(x)

eÐnai sunˆrthsh tou

x

ìpwc eÐnai kai h

dy ! dx

1 dy = f (x) g(y) dx x. Z 1 dy dx = f (x) dx + C. g(y) dx

Ac oloklhr¸soume kai ta dÔo mèrh thc exÐswshc wc proc

Z

Me allag  metablht¸n t¸ra katal goume sthn

Z

1 dy = g(y)

Z f (x) dx + C.

Oloklhr¸same.

1.3.2

'Emmesec LÔseic

EÐnai fanerì ìti kˆpoiec forèc mporeÐ na mhn mporoÔme na proqwr soume sthn diatÔpwsh thc lÔshc se analutik  (kleist ) morf  akìma kai an mporèsoume na upologÐsoume ìla ta emplekìmena oloklhr¸mata. jewr soume thn ex c diaqwrÐsimh exÐswsh

y0 =

Gia parˆdeigma, ac

xy . y2 + 1

DiaqwrÐzoume tic metablhtèc

y2 + 1 dy = y

  1 y+ dy = x dx y

kai oloklhr¸noume gia na pˆroume

y2 x2 + ln |y| = +C 2 2   kallÐtera

y 2 + 2 ln |y| = x2 + C. Den eÐnai eÔkolo na broÔme thn lÔsh thc diaforik c exÐswshc se analutik  (kleist ) morf  epeid  den eÐnai eÔkolo na lÔsoume thn parapˆnw algebrik  exÐswsh aut  wc

y.

Ja kaloÔme loipìn kˆje sunˆrthsh

y

pou ikanopoieÐ thn

parapˆnw algebrik  exÐswsh èmmesh lÔsh parìlo pou den mporoÔme na thn diatup¸soume se analutik  morf . EÐnai eÔkolo na elègxoume an prˆgmati mia algebrik  exÐswsh emperièqei prˆgmati èmmesh lÔsh h opoÐa ikanopoieÐ thn diaforik  exÐswsh. Gi autì oloklhr¸noume kai èqoume

  2 = 2x. y 0 2y + y EÐnai profanèc ìti h diaforik  exÐswsh epalhjeÔetai. Ta prˆgmata mporeÐ na gÐnoun lÐgo pio dÔskola sthn perÐptwsh pou prèpei na upologÐsoume merikèc timèc thc

y

kai na prèpei na qrhsimopoi soume kˆpoio tèqnasma. Gia parˆdeigma,

mporoÔme na kˆnoume thn grafik  parˆstash thc

x

san sunˆrthsh thc

y,

kai metˆ na anapodogurÐsoume to qartÐ. Oi

upologistèc mporoÔn na mac bohj soun ousiastikˆ me tètoia teqnˆsmata, prèpei ìmwc na eÐmaste prosektikoÐ. Shmei¸ste ìti h parapˆnw exÐswsh èqei epiprìsjeta san lÔsh thn eÐnai h

y 2 + 2 ln |y| = x2 + C

idiˆzousec lÔseic.

mazÐ me thn

y = 0.

y = 0.

Sthn perÐptwsh aut  h genik  lÔsh

Autèc tic apìkentrec lÔseic ìpwc h

y = 0 pollèc forèc tic onomˆzoume

1.3. ΔΙΑΧΩџΙΣΙΜΕΣ ΕΞΙӟΩΣΕΙΣ 1.3.3

21

ParadeÐgmata

Parˆdeigma 1.3.2:

LÔste thn exÐswsh

x2 y 0 = 1 − x2 + y 2 − x2 y 2 , y(1) = 0.

ParagontopoioÔme pr¸ta to dexiì mèroc gia na pˆroume thn exÐswsh

x2 y 0 = (1 − x2 )(1 + y 2 ). DiaqwrÐzoume tic metablhtèc, oloklhr¸noume kai lÔnoume wc proc

0

y

2

y 1−x = , 1 + y2 x2 0 y 1 = 2 − 1, 1 + y2 x −1 − x + C, arctan(y) = x   −1 y = tan −x+C . x T¸ra lÔnoume thn exÐswsh gia thn arqik  sunj kh,

0 = tan(−2 + C)

kai paÐrnoume ìti

C = 2

( 

2 + π,

etc. . . ).

Sunep¸c, h lÔsh pou yˆqnoume eÐnai h

 y = tan Parˆdeigma 1.3.3:

 −1 −x+2 . x

Upojèste ìti sto kulikeÐo ftiˆqnoun ton kafè sac qrhsimopoi¸ntac brastì nerì (100bajm¸n

KelsÐou) kai èstw ìti protimˆtai na pÐnete ton kafè sac stouc 70 bajmoÔc.

Eˆn h jermokrasÐa tou peribˆllontoc

eÐnai 26 bajmoÔc kai èqete diapist¸sei ìti 1 leptì metˆ thn paraskeu  tou o kafèc sac èqei jermokrasÐa 95 bajmoÔ pìte prèpei na pieÐte ton kafè sac? 'Estw

T

h jermokrasÐa tou kafè kai

A

h jermokrasÐa tou peribˆllontoc (kulikeÐou),

tìte gia kˆpoio

k

h

jermokrasÐa tou kafè ikanopoieÐ thn exÐswsh:

dT = k(A − T ). dt Gia to prìblhmˆ mac

A = 26, T (0) = 100, T (1) = 95.

C

DiaqwrÐzoume tic metablhtèc kai oloklhr¸noume (

kai

D

paristoÔn tuqaÐec stajerèc) gia na pˆroume ìti

dT 1 = k, A − T dt ln(A − T ) = −kt + C, A − T = D e−kt , T = A − D e−kt . T = 26 − D e−kt . QrhsimopoioÔme thn pr¸th sunj kh 100 = T (0) = 26 − D kai katal goume sto D = −74. −kt −k Opìte èqoume T = 26 + 74 e . QrhsimopoioÔme thn 95 = T (1) = 26 + 74 e . LÔnoume wc proc k kai paÐrnoume 95−26 k = − ln 74 ≈ 0.07. T¸ra lÔnoume wc proc to t gia to opoÐo h jermokrasÐa tou kafè eÐnai 74 bajmoÔc. LÔnoume

Dhlad 

dhlad  thn exÐswsh

70 = 26 + 74e−0.07t

Parˆdeigma 1.3.4:

gia na pˆroume

LÔste thn exÐswsh

Shmei¸noume pr¸ta ìti h

y=0

y0 =

t=−

ln 70−26 74 0.07

≈ 7.43

leptˆ.

−xy 2 . 3

eÐnai mia lÔsh (mia idiˆzousa lÔsh). Upojètontac ìti

−3 0 y = x, y2 3 x2 = + C, y 2 3 6 y= 2 = 2 . x /2 + C x + 2C

y 6= 0

èqoume

22

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ

1.3.4

Ask seic

1.3.1

LÔste thn exÐswsh

y 0 = x/y.

1.3.2

LÔste thn exÐswsh

y 0 = x2 y .

1.3.3

LÔste thn exÐswsh

dx dt

= (x2 − 1) t,

1.3.4

LÔste thn exÐswsh

dx dt

= x sin(t),

1.3.5

LÔste thn exÐswsh

dy dx

= xy + x + y + 1.

1.3.6

BreÐte mia èmmesh lÔsh thc exÐswshc

dy 1.3.7 LÔste thn exÐswsh x dx

2

gia

gia

− y = 2x y ,

x(0) = 0.

x(0) = 1. Upìdeixh: Paragontopoi ste to dexiì mèroc.

xy 0 = y + 2x2 y ,

gia

y(0) = 10.

ìpou

y(1) = 1.

1.4. ΓΡΑΜΜΙʟΕΣ ΕΞΙӟΩΣΕΙΣ ΚΑΙ ΟΛΟΚΛΗΡΩΤΙΚΟŸΙ ΠΑџΑΓΟΝΤΕΣ 1.4

23

Grammikèc exis¸seic kai oloklhrwtikoÐ parˆgontec

'Enac apì touc shmantikìterouc tÔpouc diaforik¸n exis¸sewn pou ja melet soume eÐnai oi grammikèc exis¸seic. Sthn pragmatikìthta mìnon se lÐgec peript¸seic ja asqolhjoÔme me mh-grammikèc exis¸seic. Ac epikentrwjoÔme proc to parìn stic grammikèc exis¸seic pr¸thc tˆxhc. Mia pr¸thc tˆxhc diaforik  exÐswsh eÐnai grammik  eˆn mporoÔme na thn grˆyoume sthn ex c morf :

y 0 + p(x)y = f (x). H lèxh grammik  en prokeimènw shmaÐnei grammik  wc proc

y.

(1.3)

H exˆrthsh wc proc

x

mporeÐ fusikˆ na eÐnai pio

perÐplokh. Oi lÔseic twn grammik¸n exis¸sewn èqoun endiafèrousec idiìthtec. mporoÔn na orÐzontai oi sunart seic thc lÔshc eÐnai h Ðdia me aut  twn

Gia parˆdeigma, h lÔsh upˆrqei arkeÐ na

p(x) kai f (x) sto diˆsthma pou mac endiafèrei. Sthn perÐptwsh aut  o omalìthta p(x) kai f (x), eÐnai dhlad  ìso omal  ìso kai oi dÔo autèc sunart seic. To

shmantikìtero ìmwc gegonìc, toulˆqiston gia t¸ra, eÐnai to ìti mporoÔme na diatup¸soume mia saf  mèjodo gia thn epÐlush grammik¸n diaforik¸n exis¸sewn pr¸thc tˆxhc. Ac pollaplasiˆsoume kai ta dÔo mèrh thc exÐswshc (1.3) me kˆpoia sunˆrthsh

r(x)

tètoia ¸ste

i d h r(x)y 0 + r(x)p(x)y = r(x)y . dx MporoÔme bèbaia na oloklhr¸soume kai ta dÔo mèlh

i d h r(x)y = r(x)f (x). dx y en¸ to aristerì mèroc eÐnai h antiparˆgwgoc miac sunˆrthshc. y . Fusikˆ ìla ta parapˆnw kˆtw apì thn proôpìjesh ìti h sunˆrthsh r(x) eÐnai r(x) onomˆzete oloklhrwtikìc parˆgontac kai katˆ sunèpeia h prokÔptousa mèjodoc

Parathr ste ìti to dexiì mèroc den exartˆtai apì to MporoÔme na lÔsoume wc proc gnwst .

H sunˆrthsh aut 

onomˆzete mèjodoc oloklhrwtikoÔ parˆgonta. Yˆqnoume loipìn gia mia sunˆrthsh pollaplasiasmènh me

p(x).

r(x)

tètoia ¸ste eˆn thn paragwgÐsoume, ja pˆroume thn Ðdia thn sunˆrthsh

Problèpetai ekjetik  sunˆrthsh!

R

r(x) = e

p(x)dx

Ac kˆnoume ìmwc tic prˆxeic.

y 0 + p(x)y = f (x), e

R

p(x)dx 0

R

Gia na pˆroume bèbaia thn

y

R

p(x)y = e p(x)dx f (x), R d h R p(x)dx i e y = e p(x)dx f (x), dx Z R R e p(x)dx y = e p(x)dx f (x) dx + C, Z R  R y = e− p(x)dx e p(x)dx f (x) dx + C .

y +e

p(x)dx

se kleist  morf  prèpei na mporoÔme na upologÐsoume kai ta dÔo emplekìmena

oloklhr¸mata se kleist  morf .

Parˆdeigma 1.4.1:

LÔste thn exÐswsh

y 0 + 2xy = ex−x Parathr ste pr¸ta ìti

p(x) = 2x

Pollaplasiˆzoume kai ta dÔo mèrh me

kai

r(x)

f (x) = ex−x

2

y(0) = −1.

2

R

. O oloklhrwtikìc parˆgontac eÐnai

gia na pˆroume 2

2

2

2

ex y 0 + 2xex y = ex−x ex , d h x2 i e y = ex . dx

r(x) = e

p(x) dx

2

= ex

.

24

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ

Oloklhr¸noume 2

ex y = ex + C, 2

2

y = ex−x + Cex . LÔnoume tic arqikèc sunj kec

−1 = y(0) = 1 + C ,

gia na pˆroume ìti 2

C = −2.

H lÔsh loipìn eÐnai

2

y = ex−x − 2ex . R

Parathr ste ìti den endiaferìmaste poia antiparˆgwgo ja pˆroume ìtan upologÐzoume to

e

p(x)dx

.

MporoÔme

dhlad  na prosjèsoume mia stajerˆ olokl rwshc allˆ aut  den prìkeitai na paÐxei kˆpoio rìlo mia kai en tèlei ja enswmatwjeÐ sthn stajerˆ olokl rwshc thc epìmenhc antiparag¸gou pou ja upologÐsoume.

1.4.1

Dokimˆste to.

Prosjèste mia stajerˆ olokl rwshc ìtan ja upologÐsete ton oloklhrwtikì parˆgonta kai

deÐxte ìti h lÔsh pou ja katal xete tautÐzetai me aut  pou br kame parapˆnw. Mia sumboul :

Mhn prospaj sete na apomnhmoneÔsete ton telikì tÔpo.

EÐnai dÔskolo   kallÐtera, eÐnai eu-

kolìtero na jumˆste thn diadikasÐa kai na thn epanalˆbete. Epeid  den mporoÔme pˆntote na upologÐzoume ta emplekìmena oloklhr¸mata se kleist  morf  eÐnai qr simo na xèroume pwc mporoÔme na grˆyoume thn lÔsh sthn morf  orismènou oloklhr¸matoc.

Mhn xeqnˆte ìti to orismèno

olokl rwma eÐnai kˆti to opoÐo mporeÐc na upologÐsoume qrhsimopoi¸ntac upologist    kompouterˆki. 'Estw h exÐswsh

y 0 + p(x)y = f (x),

y(x0 ) = y0 .

MporoÔme na d¸soume thn lÔsh grˆfontac ta oloklhr¸mata san orismèna oloklhr¸mata wc ex c

y(x) = e

R − xx p(s) ds

x

Z

Rt

e

0

x0

p(s) ds

 f (t) dt + y0 .

(1.4)

x0 Prèpei na eÐmaste prosektikoÐ sthn qr sh twn metablht¸n olokl rwshc bebaÐwc. Profan¸c mporoÔme na ulopoi soume thn parapˆnw lÔsh ston upologist  ètsi ¸ste autìc na mac d¸sei ˆmesa ìpoia arijmhtik  tim  thc lÔshc tou zht soume.

1.4.2

Elègxete an h lÔsh (1.4) ikanopoieÐ thn sunj kh

1.4.3

Grˆyte thn lÔsh tou ex c probl matoc

y 0 + y = ex san orismèno olokl rwma.

2

y(x0 ) = y0 .

−x

,

y(0) = 10.

prospaj ste na aplopoi sete thn lÔsh sac ìso eÐnai dunatìn.

Shmei¸ste ìti den ja

mporèsete na grˆyete thn lÔsh se kleist  morf .

Parˆdeigma 1.4.2:

Ac doÔme t¸ra mia apl  efarmog  twn grammik¸n exis¸sewn tupik  gia mia kathgorÐa prob-

lhmˆtwn pou suqnˆ sunantˆme sthn prˆxh. Gia parˆdeigma, qrhsimopoioÔme suqnˆ grammikèc exis¸seic gia na melet soume thn sugkèntrwsh miac ousÐac (qhmik c ousÐac, rÔpou,

. . .)

se èna mèso (nerì).

Sugkekrimèna, mia dexamen  100 lÐtrwn perièqei 10 kilˆ alatioÔ dialumèna se 60 lÐtra neroÔ. Diˆluma neroÔ kai alatioÔ (ˆlmh) puknìthtac 0.1 kilˆ anˆ lÐtro eisˆgetai sto doqeÐo me rujmì 5 lÐtra to leptì. To diˆluma anakateÔetai kalˆ kai exˆgetai apì to doqeÐo me rujmì 3 lÐtra to leptì. Pìso alˆti ja perièqei h dexamen  ìtan ja èqei gemÐsei? Prèpei na kataskeuˆsoume thn exÐswsh. Ac sumbolÐsoume me

t ∆x)

h dexamen , kai me

to qrìno (se leptˆ).

sumbolÐzoume me

eÐnai perÐpou

∆x ≈ (rujmìc

eisagwg c

PaÐrnoume to ìrio ìtan to

dx = (rujmìc dt

× sugkèntrwsh ∆t → 0

eisagwg c

x

thn posìthta tou alatioÔ (se kilˆ) pou perièqei

Tìte gia mia mikr  allag 

)∆t − (rujmìc

eisagwg c

∆t

ston qrìno h allag  sto

x

(pou thn

exagwg c

× sugkèntrwsh

exagwg c

exagwg c

× sugkèntrwsh

exagwg c

)∆t.

kai parathroÔme ìti

× sugkèntrwsh

) − (rujmìc

eisagwg c

).

1.4. ΓΡΑΜΜΙʟΕΣ ΕΞΙӟΩΣΕΙΣ ΚΑΙ ΟΛΟΚΛΗΡΩΤΙΚΟŸΙ ΠΑџΑΓΟΝΤΕΣ

25

'Eqoume loipìn

rujmìc eisagwg c sugkèntrwsh eisagwg c rujmìc exagwg c sugkèntrwsh exagwg c

= 5, = 0.1, = 3, =

x ìgkoc

=

x . 60 + (5 − 3)t

Katal goume sunep¸c sthn exÐswsh

  dx x = (5 × 0.1) − 3 . dt 60 + 2t Thn opoÐa bebaÐwc mporoÔme na grˆyoume sthn morf  (1.3) wc ex c

3 dx + x = 0.5. dt 60 + 2t Ac prospaj soume na thn lÔsoume t¸ra. O oloklhrwtikìc parˆgontˆc thc eÐnai

Z r(t) = exp

3 dt 60 + 2t



 = exp

 3 ln(60 + 2t) = (60 + 2t)3/2 . 2

Pollaplasiˆzontac kai ta duo mèlh èqoume

(60 + 2t)3/2

3 dx + (60 + 2t)3/2 x = 0.5(60 + 2t)3/2 , dt 60 + 2t i d h (60 + 2t)3/2 x = 0.5(60 + 2t)3/2 , dt Z (60 + 2t)3/2 x =

C.

GnwrÐzoume ìti

Z

x = (60 + 2t)−3/2

1 (60 + 2t)5/2 + C(60 + 2t)−3/2 , 10

60 + 2t + C(60 + 2t)−3/2 . 10

t = 0, x = 10.

10 = x(0) =

(60 + 2t)3/2 dt + C(60 + 2t)−3/2 , 2

x = (60 + 2t)−3/2

x= Ac broÔme t¸ra to

0.5(60 + 2t)3/2 dt + C,

'Ara

60 + C(60)−3/2 = 6 + C(60)−3/2 , 10

 

C = 4(603/2 ) ≈ 1859.03. Jèloume na broÔme to dhlad  ìtan

t = 20.

x

ìtan gemÐsei h dexamen . ParathroÔme ìti h dexamen  eÐnai gemˆth ìtan

60 + 2t = 100,

'Ara

x(20) =

60 + 40 + C(60 + 40)−3/2 ≈ 10 + 1859.03(100)−3/2 ≈ 11.86. 10

H sugkèntrwsh loipìn ìtan gemÐsei h dexamen  ja eÐnai perÐpou 0.1186 kg/liter en¸ h arqik  tim  thc  tan

0.167 kg/liter.

1/6

 

26

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ

1.4.1

Ask seic

Stic parakˆtw ask seic prospaj ste na d¸sete thn lÔsh se kleist  morf .

Eˆn kˆti tètoio den eÐnai dunatìn,

mporeÐte na thn af sete sthn morf  orismènou oloklhr¸matoc.

1.4.4

LÔste thn exÐswsh

y 0 + xy = x.

1.4.5

LÔste thn exÐswsh

y 0 + 6y = ex .

1.4.6

LÔste thn exÐswsh

y 0 + 3x2 y = sin(x) e−x

1.4.7

LÔste thn exÐswsh

y 0 + cos(x)y = cos(x).

1.4.8

LÔste thn exÐswsh

1 x2 +1

1.4.9

'Eqoume dÔo lÐmnec kai apì thn mia rèei nerì sthn ˆllh.

y 0 + xy = 3,

3

ìtan

apì thn kˆje lÐmnh eÐnai 500 lÐtra thn ¸ra.

, ìtan

y(0) = 1.

y(0) = 0. H rujmìc thc ro  eisagwg c kai exagwg c neroÔ

H pr¸th lÐmnh perièqei 100 qiliˆdec lÐtra neroÔ kai h deÔterh 200

qiliˆdec lÐtra. 'Ena butÐo pou metafèrei 500 kilˆ miac toxik c ousÐac anatrèpetai mèsa sthn sthn pr¸th lÐmnh ìpou kai qÔnetai to perieqìmenì tou. Upojètontac ìti h toxik  ousÐa dialÔetai amèswc kai omoiìmorfa breÐte 1. Thn sugkèntrwsh thc toxik c ousÐac san sunˆrthsh tou qrìnou (se deuterìlepta) stic dÔo lÐmnec. 2. Pìte h sugkèntrwsh sthn pr¸th lÐmnh ja eÐnai mikrìterh apì 0.01 kilˆ anˆ lÐtro. 3. Pìte ja eÐnai h sugkèntrwsh sthn deÔterh lÐmnh ìso to dunatìn pio uyhl .

dx dt

= −k(x − A) ìpou x eÐnai h A h jermokrasÐa tou peribˆllontoc, kai k > 0 mia stajerˆ. Upojèste ìti A = A0 cos ω t gia kˆpoiec stajerèc A0 kai ω . Upojèste dhlad  ìti h jermokrasÐa tou peribˆllontoc metabˆlletai periodikˆ (pq lìgw metabol c thc jermokrasÐac metaxÔ mèrac kai nÔqtac) kai breÐte thn genik  lÔsh x tou probl 1.4.10

O Nìmoc diˆdoshc thc jermìthtac tou NeÔtwna mac diabebai¸nei ìti

jermokrasÐa enìc s¸matoc,

t

eÐnai o qrìnoc,

matoc. Exetˆste eˆn oi arqikèc sunj kec ephreˆzoun ousiastikˆ thn lÔsh sto makrinì mèllon. DikaiologeÐste ta sumperˆsmatˆ sac.

1.5. ΑΝΤΙΚΑΤΑΣԟΑΣΕΙΣ 1.5

27

Antikatastˆseic

'Opwc akrib¸c kai ìtan prospajoÔme na upologÐsoume oloklhr¸mata, ètsi kai gia thn epÐlush diaforik¸n exis¸sewn mporoÔme na prospaj soume na kˆnoume allagèc metablht¸n gia na katal xoume se aploÔsterec exis¸seic tic opoÐec eÐnai eukolìtero na lÔsoume.

1.5.1

Antikatˆstash

Mia kai h exÐswsh

y 0 = (x − y + 1)2 . den eÐnai oÔte diaqwrÐsimh oÔte grammik  ac prospaj soume na allˆxoume tic metablhtèc thc ètsi ¸ste h nèa morf  thc (wc proc tic nèec metablhtèc) na eÐnai aploÔsterh. Ja qrhsimopoi soume mia nèa metablht  thn jewr soume san sunˆrthsh tou

v,

thn opoÐa ja

x. v = x − y + 1.

Prèpei na broÔme thn

y

0

sunart sei twn

0

v,v

kai

x.

ParagwgÐzontac (wc proc

x)

èqoume

v0 = 1 − y0 .

'Ara

y0 = 1 − v0 .

AntikajistoÔme sthn arqik  diaforik  exÐswsh gia na pˆroume

1 − v0 = v2 . Dhlad ,

v0 = 1 − v2 .

Thn exÐswsh aut  xèroume na thn lÔsoume.

1 dv = dx. 1 − v2 'Ara

v + 1 1 = x + C, ln 2 v − 1 v + 1 2x+2C , v − 1 = e  

v+1 v−1

= De2x

gia kˆpoia stajerˆ

D.

Shmei¸ste ìti h

v=1

kai h

v = −1

eÐnai epÐshc lÔseic.

T¸ra ac antikatast soume proc ta pÐsw gia na pˆroume

x−y+2 = De2x , x−y ìpwc kai tic ˆllec dÔo lÔseic exÐswsh wc proc

x−y+1 = 1

 

y = x,

kai

x − y + 1 = −1

 

y = x + 2.

Ac lÔsoume kai thn pr¸th

y. x − y + 2 = (x − y)De2x , x − y + 2 = Dxe2x − yDe2x , −y + yDe2x = Dxe2x − x − 2, y (−1 + De2x ) = Dxe2x − x − 2, y=

Shmei¸ste ìti tim 

D=0

mac odhgeÐ sthn lÔsh

y = x + 2,

Dxe2x − x − 2 . De2x − 1 en¸ kamiˆ tim  tou

D

den mac odhgeÐ sthn

y = x.

H parapˆnw mèjodoc allag c metablht¸n gia thn epÐlush diaforik¸n exis¸sewn efarmìzetai ìpwc kai h antÐstoiqh mejodologÐa ston Logismì. BasÐzetai dhlad  se manteyièc. Gia mia exÐswsh mporeÐ na upˆrqoun polloÐ trìpoi (  na mhn upˆrqei kanènac trìpoc) na efarmosjeÐ mia tètoia mèjodoc, mporeÐ ìmwc na eÐnai dÔskolo na broÔme kˆpoia apì autèc. Prèpei en gènei na basistoÔme sthn majhmatik  mac diaÐsjhsh kai koultoÔra. Upˆrqoun ìmwc kˆpoiec genikèc arqèc pou anˆloga me thn exÐswsh ja mporoÔsan na mac bohj soun. Merikèc apì autèc eÐnai oi parakˆtw.

28

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ 'Otan deic

dokÐmase

yy 0 y2 y0 (cos y)y 0 (sin y)y 0 y 0 ey

y2 y3 sin y cos y ey

Sun jwc prospajoÔme na apallagoÔme apì to pio perÐploko mèroc thc exÐswshc elpÐzontac ìti ètsi ja thn aplopoi soume.

O parapˆnw pÐnakac eÐnai apl¸c ènac praktikìc genikìc kanìnac.

Upˆrqei bebaÐwc endeqìmeno na

mhn sac bohj sei kai na qreiasjeÐ na metatrèyete thn manteyiˆ sac. Dhlad , eˆn den doulèyei (den san odhg sei se mia aploÔsterh exÐswsh) kˆpoia antikatˆstash ofeÐloume na dokimˆsoume kˆpoia ˆllh.

1.5.2

Exis¸seic

Bernoulli

Upˆrqoun merikoÐ tÔpoi exis¸sewn gia touc opoÐouc upˆrqoun metasqhmatismoÐ pou efarmìzontai se kˆje perÐptwsh me epituqÐa. 'Enac tètoioc tÔpoc exis¸sewn eÐnai h exÐswsh

Bernoulli† .

y 0 + p(x)y = q(x)y n . H exÐswsh aut  moiˆzei polÔ me grammik  exÐswsh, mia kai eˆn den up rqe o ìroc

n=0

 

n=1

h exÐswsh eÐnai grammik  kai mporoÔme na thn lÔsoume.

v = y 1−n

qrhsimopoi soume thn ex c allag  metablht¸n exÐswsh. Shmei¸ste ìti den eÐnai aparaÐthto o

Parˆdeigma 1.5.1:

n

yn

prˆgmati ja  tan grammik . Gia

Se ìlec tic ˆllec peript¸seic mporoÔme na

gia na metatrèyoume thn exÐswsh

Bernoulli

se grammik 

na eÐnai akèraioc.

LÔste thn exÐswsh

xy 0 + y(x + 1) + xy 5 = 0, Pr¸ta prèpei na suneidhtopoi soume ìti èqoume mia exÐswsh

y(1) = 1.

Bernoulli (p(x) = (x + 1)/x

and

q(x) = −1).

Antika-

jistoÔme

v = y 1−5 = y −4 , Dhlad  èqoume,

−y 5 0 v 4

= y0 .

v 0 = −4y −5 y 0 .

'Ara

xy 0 + y(x + 1) + xy 5 = 0, −xy 5 0 v + y(x + 1) + xy 5 = 0, 4 −x 0 v + y −4 (x + 1) + x = 0, 4 −x 0 v + v(x + 1) + x = 0, 4 kai telikˆ

v0 −

4(x + 1) v = 4. x

Katal xame loipìn se mia grammik  exÐswsh thn opoÐa ja lÔsoume me oloklhrwtikì parˆgonta. Ac upojèsoume ìti

x>0

opìte

|x| = x.

H upìjes  mac aut  den eÐnai perioristik  mia kai h arqik  mac sunj kh eÐnai sto shmeÐo

Z r(x) = exp

x = 1.

 −4(x + 1) e−4x dx = e−4x−4 ln(x) = e−4x x−4 = . x x4

† Upˆrqoun pollèc exis¸seic pou lègontai exis¸seic Bernoulli. Oi Bernoullis  tan mia olìklhrh oikogèneia shmantik¸n Elbet¸n majhmatik¸n. Oi exis¸seic me tic opoÐec ja asqolhjoÔme sto mˆjhma sqetÐzontai me ton Jacob Bernoulli (1654 1705).

1.5. ΑΝΤΙΚΑΤΑΣԟΑΣΕΙΣ

29

T¸ra

  d e−4x e−4x v = 4 , dx x4 x4 Z x e−4x e−4s v= 4 4 ds + 1, 4 x s 1  Z x −4s  e 4x 4 v=e x 4 ds + 1 . s4 1 Prosèxte ìti den mporoÔme na upologÐsoume to olokl rwma se kleist  morf . 'Opwc ìmwc èqoume  dh tonÐsei, eÐnai pl rwc apodektì na diatup¸soume thn lÔsh me èna orismèno olokl rwma. Ac antikatast soume proc ta pÐsw.'

y

−4

x

 Z =e x 4 4x 4

1

y=

1.5.3

 e−4s ds + 1 , s4

e−x  R x x 4 1

e−4s s4

ds + 1

1/4 .

OmogeneÐc exis¸seic

'Enac ˆlloc tÔpoc exis¸sewn pou mporoÔme na lÔsoume me allagèc metablht¸n eÐnai oi legìmenec omogeneÐc exis¸seic. Upojèste ìti mporoÔme na grˆyoume thn diaforik  mac exÐswsh wc ex c

y0 = F

y x

.

Ed¸ prospajoÔme ton metasqhmatismì

v=

y x

kai

y 0 = v + xv 0 .

sunep¸c

Shmei¸ste ìti h exÐsws  mac èqei metasqhmatisjeÐ wc ex c

v + xv 0 = F (v)

xv 0 = F (v) − v

 

 

v0 1 = . F (v) − v x

'Ara mia èmmesh lÔsh eÐnai h ex c

Z

Parˆdeigma 1.5.2:

1 dv = ln |x| + C. F (v) − v

LÔste thn exÐswsh

x2 y 0 = y 2 + xy, Ac thn fèroume pr¸ta sthn ex c morf 

v = y/x

y 0 = (y/x)2 + y/x.

y(1) = 1. T¸ra mporoÔme na qrhsimopoi soume ton metasqhmatismì

gia na pˆroume mia diaqwrÐsimh exÐswsh

xv 0 = v 2 + v − v = v 2 , h opoÐa èqei thn ex c lÔsh

Z

1 dv = ln |x| + C, v2 −1 = ln |x| + C, v −1 v= . ln |x| + C

30

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ

Antikajist¸ntac proc ta pÐsw èqoume

−1 , ln |x| + C −x y= . ln |x| + C

y/x =

Jèloume

y(1) = 1,

opìte

1 = y(1) = 'Ara

C = −1

kai h lÔsh mac eÐnai h ex c

y=

1.5.4

−1 −1 = . ln |1| + C C −x . ln |x| − 1

Ask seic

1.5.1

LÔste thn exÐswsh

xy 0 + y(x + 1) + xy 5 = 0,

1.5.2

LÔste thn exÐswsh

2yy 0 + 1 = y 2 + x,

1.5.3

LÔste thn exÐswsh

1.5.4

LÔste thn exÐswsh

y 0 + xy = y 4 , me y(0) = 1. p yy 0 + x = x2 + y 2 .

1.5.5

LÔste thn exÐswsh

y 0 = (x + y − 1)2 .

1.5.6

LÔste thn exÐswsh

x+y y0 = √ 2 y

2

y +1

, me

me

me

y(1) = 1.

y(0) = 1.

y(0) = 1.

1.6. ΑΥԟΟΝΟΜΕΣ ΕΞΙӟΩΣΕΙΣ 1.6

31

Autìnomec exis¸seic

Ac epikentrwjoÔme t¸ra se exis¸seic tou tÔpou

dx = f (x), dt ìpou h parˆgwgoc twn lÔsewn exartˆtai mìnon apì thn (exarthmènh metablht )

x.

Oi exis¸seic tou tÔpou autoÔ

lègontai autìnomec exis¸seic. To epÐjeto autìnomh prokÔptei apì to gegonìc ìti eˆn jewr soume thn perÐptwsh pou h metablht 

t

paristˆ qrìno, tìte oi exis¸seic autèc eÐnai anexˆrthtec apì ton qrìno.

Ac epistrèyoume sto prìblhma tou kafè. O nìmoc thc diˆdoshc tou NeÔtwna mac diabebai¸nei ìti

dx = −k(x − A), dt ìpou

x

eÐnai h jermokrasÐa,

t

eÐnai eÐnai o qrìnoc,

k

eÐnai kˆpoia stajerˆ kai

A

eÐnai h jermokrasÐa tou peribˆllontoc

q¸rou. DeÐte to Sq ma 1.6 gia èna parˆdeigma. Shmei¸ste ìti h Ta shmeÐa tou

x

krÐsimo shmeÐo.

x=A

eÐnai lÔsh (sto parˆdeigma

ˆxona sta opoÐa èqoume

A = 5).

f (x) = 0

x = A

eÐnai eustaj s.

odhgoÔn se ousiastikˆ diaforetikèc lÔseic gia arketˆ megˆlo

t→∞

To shmeÐo dhlad 

Sthn pragmatikìthta, kˆje krÐsimo shmeÐo antistoiqeÐ se mia lÔsh isorropÐac.

parathr¸ntac thn grafik  parˆstash, ìti h lÔsh

ìtan

AutoÔ tou eÐdouc tic lÔseic tic lème lÔseic isorropÐac.

ta lème krÐsima shmeÐa.

èqoume

t.

x = A

eÐnai èna

Shmei¸ste epÐshc,

Dhlad  mikrèc diataraqèc sto

x

den

An loipìn allˆxoume lÐgo thn arqik  sunj kh, tìte

x → A. Tètoia krÐsima shmeÐa ta lème eustaj . Sto tetrimmèno parˆdeigmˆ mac faÐnetai ìti ìlec A ìtan to t → ∞. An èna krÐsimo shmeÐo den eÐnai eustajèc tìte lème ìti autì eÐnai astajèc.

oi lÔseic teÐnoun sto

0

5

10

15

0

20

10

10

5

5

0

5

10

15

20

10

10

5

5

0

0

0

-5

-5

-10

-5

-10 0

5

10

15

-5 0

20

Σχήμα 1.6: Πεδίο κατευθύνσεων και η γραφική παράσταση μερικών λύσεων της εξίσωσης x0 = −0.3(x − 5).

5

10

15

20

Σχήμα 1.7: Το πεδίο κατευθύνσεων και η γραφική παράσταση μερικών λύσεων της εξίσωσης x0 = −0.1x(5 − x).

Ac jewr soume t¸ra thn logistik  exÐswsh

dx = kx(M − x), dt gia kˆpoiouc jetikoÔc arijmoÔc

k

kai

M.

H exÐswsh aut  qrhsimopoieÐtai suqnˆ san plhjusmiakì montèlo eˆn

gnwrÐsoume ìti o o plhjusmìc enìc eÐdouc den mporeÐ na uperbeÐ ton arijmì katastrofikèc problèyeic gia ton pagkìsmio plhjusmì.

M.

To montèlo autì odhgeÐ se ligìterec

Shmei¸ste ìti sthn pragmatikìthta den upˆrqei arnhtikìc

x gia majhmatikoÔc lìgouc. x = 0 kai x = 5. To krÐsimo shmeÐo

plhjusmìc, ja diathr soume ìmwc thn dunatìthta na èqoume arnhtikèc timèc gia to Ac doÔme to Sq ma 1.7 gia parˆdeigma. Parathr ste ta dÔo krÐsima shmeÐa, sto

x=5

eÐnai eustajèc en¸ autì sto

x=0

eÐnai astajèc.

32

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ Den eÐnai aparaÐthto na èqoume tic akribeÐc lÔseic miac exÐswshc gia na apofanjoÔme sqetikˆ me thn sumperiforˆ

twn lÔsewn thc gia megˆlec timèc thc eleÔjerhc metablht c twn. Gia parˆdeigma apì ta parapˆnw mporoÔme eÔkola na doÔme ìti

  5 lim x(t) = 0 t→∞  

DU  

an an

−∞

an

x(0) > 0, x(0) = 0, x(0) < 0.

'Opou DU shmaÐnei den upˆrqei. EÐnai dÔskolo, aplˆ parathr¸ntac to pedÐo twn kateujÔnsewn, na apofanjoÔme gia to ti sumbaÐnei ìtan

x(0) < 0.

MporeÐ h lÔsh na mhn upˆrqei ìtan to

t

teÐnei sto

∞.

SkefjeÐte ìti h exÐswsh

y0 = y2 ,

ìpwc èqoume  dh dei ufÐstatai mìnon gia kˆpoio peperasmèno qronikì diˆsthma. To Ðdio mporeÐ na sumbaÐnei kai me thn twrin  exÐswsh. 'Opwc ja diapist¸soume den upˆrqei lÔsh thc exÐswshc tou parapˆnw paradeÐgmatoc gia opoiod pote qronik  stigm . Gia na to doÔme ìmwc autì ja prèpei na prospaj soume na thn lÔsoume. Se kˆje perÐptwsh, h lÔsei teÐnei sto

−∞,

endeqomènwc polÔ gr gora.

Arketèc forèc autì pou ousiastikˆ mac apasqoleÐ eÐnai h sumperiforˆ thc lÔshc se bˆjoc qrìnou kai sunep¸c se mia tètoia perÐptwsh Ðswc na spatalˆme ˆskopa enèrgeia prospaj¸ntac na broÔme akrib¸c thn lÔsh. EÐnai eukolìtero aplˆ na parathr soume to diˆgramma fˆshc   eikìna fˆshc, h opoÐa eÐnai ènac aplìc trìpoc gia na optikopoi soume thn sumperiforˆ twn autìnomwn exis¸sewn. Sunep¸c sqediˆzoume ton

x

Sthn prokeÐmenh perÐptwsh upˆrqei mia anexˆrthth metablht  h

x.

ˆxona, shmei¸noume ìla ta krÐsima shmeÐa kai metˆ sqediˆzoume bèlh metaxÔ touc. Proc

ta epˆnw bèlh paristoÔn jetikìthta kai proc ta kˆtw arnhtikèc timèc.

y=5 y=0

Exoplismènoi me to diˆgramma fˆshc, mporoÔme eÔkola na sqediˆsoume prìqeira pwc perÐpou ja eÐnai oi lÔseic.

1.6.1

Prospaj ste na sqediˆsete merikèc lÔseic. Epibebai¸ste to apotèlesma sac qrhsimopoi¸ntac thn parapˆnw

grafik  parˆstash. Apì thn stigm  pou èqoume sthn diˆjesh mac to diˆgramma fˆshc, eÔkola mporoÔme na apofanjoÔme poia apì ta krÐsima shmeÐa eÐnai eustaj  kai poia astaj .

eustaj c

astaj c

Epeid  kˆje majhmatikì montèlo pou mac apasqoleÐ apoteleÐ mia en dunˆmei prosèggish kˆpoiac pragmatik c katˆstashc, ta astaj  shmeÐa shmatodotoÔn sun jwc sobarˆ probl mata. Ac exetˆsoume t¸ra thn logistik  exÐswsh me katanˆlwsh. gia thn montelopoÐhsh plhjusmiak¸n allag¸n.

Oi logistikèc exis¸seic qrhsimopoioÔntai eurÔtata

Upojèste ìti mia omˆda anjr¸pwn trèfetai me èna eÐdoc z¸ou sto

opoÐo basÐzesai gia thn epibÐws  thc. Ektrèfei loipìn mia agèlh tètoiwn z¸wn kai ta katanal¸nei me rujmì z¸a ton qrìno. 'Estw ìti to se èth). 'Estw epÐshc ìti

M

x

paristˆ to pl joc twn z¸wn (ac poÔme se qiliˆdec) kai to

t

h

tètoia

paristˆ qrìno (ac poÔme

eÐnai eÐnai o elˆqistoc plhjusmìc kˆtw apì ton opoÐo den epitrèpetai h katanˆlwsh twn

1.6. ΑΥԟΟΝΟΜΕΣ ΕΞΙӟΩΣΕΙΣ z¸wn.

k>0

33

eÐnai mia stajerˆ pou exartˆtai apì to pìso gr gora anaparˆgontai ta z¸a.

H exÐsws  mac èqei thn

ex c morf 

dx = kx(M − x) − h. dt 'Eqoume ìti

dx = −kx2 + kM x − h. dt 'Ara ta krÐsima shmeÐa

A

B

kai

eÐnai

A= 1.6.2

kM +

p (kM )2 − 4hk 2k

B=

kM −

Sqediˆste to diˆgramma fˆshc gia diaforetikèc katastˆseic.

A > B,

 

A = B,

 

A

kai

B

p

(kM )2 − 4hk . 2k

Shmei¸ste ìti autèc oi katastˆseic eÐnai oi

kai oi dÔo migadikèc (dhlad  den upˆrqei pragmatik  lÔsh).

EÔkola diapist¸noume ìti eˆn

h = 1,

tìte ta

A

B eÐnai jetikˆ kai ˆnisa. To sqetikì grˆfhma dÐnetai sto B to opoÐo eÐnai perÐpou 1,55 qiliˆdec, tìte o plhjusmìc B tìte o plhjusmìc twn z¸wn ja exaleifjeÐ, me apotèlesma

kai

Sq ma 1.8. 'Oso o plhjusmìc paramènei megalÔteroc tou twn z¸wn paramènei en¸ eˆn pèsei kˆtw apì to ìrio tou

na exaleifjeÐ kai o anjr¸pinoc plhjusmìc lìgw èlleiyhc trof c.

0

5

10

15

20

0

5

10

15

20

10

10

10

10

8

8

8

8

5

5

5

5

2

2

2

2

0

0

0

0

0

5

10

15

20

Σχήμα 1.8: Το πεδίο κατευθύνσεων και μερικές λύσεις της εξίσωσης x0 = −0.1x(8 − x) − 1.

'Otan

h = 1.6,

èqoume ìti

A = B.

0

5

10

15

20

Σχήμα 1.9: Το πεδίο κατευθύνσεων και μερικές λύσεις της εξίσωσης x0 = −0.1x(8 − x) − 1.6.

Upˆrqei mìnon èna krÐsimo shmeÐo to opoÐo eÐnai astajèc. 'Otan o plhjusmìc

twn z¸wn pèsei kˆtw apì tic 1,6 qiliˆdec autìc ja teÐnei sto mhdèn en¸ eˆn eÐnai megalÔteroc apì 1,6 qiliˆdec tìte ja teÐnei na gÐnei 1,6 qiliˆdec. 'Eqoume loipìn mia idiaÐtera epikÐndunh katˆstash ìpou eˆn lìgw kˆpoiou endeqìmenou mikroÔ lˆjouc apomakrunjoÔme lÐgo apì to shmeÐo isorropÐac ja epèljei katastrof .

H katˆstash loipìn den

epidèqetai to paramikrì lˆjoc. DeÐte to Sq ma 1.9 Tèloc eˆn katanal¸noun 2 qiliˆdec z¸a ton qrìno, o plhjusmìc twn z¸wn ja exaleifjeÐ opwsd pote kai anexˆrthta apì ton arqikì pl joc twn z¸wn. DeÐte to Sq ma 1.10 sthn epìmenh selÐda.

1.6.1

1.6.3

Ask seic

'Estw

x0 = x2 .

a) Sqediˆste to diˆgramma fˆshc, breÐte ta krÐsima shmeÐa kai shmei¸ste poia apì autˆ eÐnai

eustaj  kai poia astaj . b) Sqediˆste merikèc qarakthristikèc lÔseic thc exÐswshc. c) BreÐte to lÔshc pou antistoiqeÐ sthn arqik  sunj kh

x(0) = −1.

limt→∞ x(t)

thc

34

ΚΕ֟ΑΛΑΙΟ 1. ΣΔΕ ΠџΩΤΗΣ ΤŸΑΞΗΣ 0

5

10

15

20

10

10

8

8

5

5

2

2

0

0

0

5

10

15

20

Σχήμα 1.10: Πεδίο κατευθύνσεων και μερικές λύσεις της εξίσωσης x0 = −0.1x(8 − x) − 2.

1.6.4

'Estw

x0 = sin x.

a) Sqediˆste to diˆgramma fˆshc gia

−4π ≤ x ≤ 4π .

Sto diˆshma autì breÐte ta krÐsima

shmeÐa kai shmei¸ste poia apì autˆ eÐnai eustaj  kai poia astaj . b) Sqediˆste merikèc qarakthristikèc lÔseic thc exÐswshc. c) BreÐte to

1.6.5

'Estw ìti h

x0 = f (x),

limt→∞ x(t)

f (x)

thc lÔshc pou antistoiqeÐ sthn arqik  sunj kh

eÐnai jetik  gia

0 < x < 1

kai arnhtik  eidˆllwc.

x(0) = 1.

a) Sqediˆste to diˆgramma fˆshc gia

breÐte ta krÐsima shmeÐa kai shmei¸ste poia apì autˆ eÐnai eustaj  kai poia astaj . b) Sqediˆste merikèc

qarakthristikèc lÔseic thc exÐswshc.

c) BreÐte to

limt→∞ x(t)

thc lÔshc pou antistoiqeÐ sthn arqik  sunj kh

x(0) = 0.5. 1.6.6

Xekin ste me thn logistik  exÐswsh

dx dt

= kx(M − x).

Ac allˆxoume lÐgo ton trìpo pou katanal¸noume. Sug-

kekrimèna, ac upojèsoume ìti to posì pou katanal¸noume eÐnai anˆlogo tou upˆrqontoc plhjusmoÔ. Ja katanal¸noume dhlad 

hx

gia kˆpoio

h > 0.

a) Kataskeuˆste thn diaforik  exÐswsh pou analogeÐ b) ApodeÐxte ìti eˆn

h exÐswsh paramènei logistik . c) Ti sumbaÐnei ìtan

kM < h?

kM > h,

tìte

Kefˆlaio 2

Grammikèc SDE uyhlìterhc tˆxhc 2.1

Grammikèc SDE uyhlìterhc tˆxhc

Ac jewr soume thn ex c genik  morf  miac grammik c diaforik c exÐswshc deÔterhc tˆxhc

A(x)y 00 + B(x)y 0 + C(x)y = F (x). Suqnˆ diairoÔme kai ta dÔo mèrh me

A

gia na thn pˆroume sthn ex c morf 

y 00 + p(x)y 0 + q(x)y = f (x),

(2.1)

p = B/A, q = C/A, kai f = F/A. H lèxh grammik  shmaÐnei ìti h exÐswsh y , y , kai y 00 oÔte autèc emfanÐzontai san orÐsmata ˆllwn sunart sewn. Sthn eidik  perÐptwsh f (x) = 0 èqoume mia omogen  equation

ìpou

den perièqei dunˆmeic twn sunart sewn

0

y 00 + p(x)y 0 + q(x)y = 0.

(2.2)

'Eqoume  dh sunant sei kˆpoiec omogeneÐc grammikèc diaforikèc exis¸seic deÔterhc tˆxhc.

y 00 + k2 y = 0 00

Duo lÔseic eÐnai:

2

y −k y =0

Duo lÔseic eÐnai:

y1 = cos kx, kx

y1 = e ,

y2 = sin kx.

y2 = e−kx .

Eˆn mporèsoume kai broÔme dÔo lÔseic miac grammik c omogenoÔc exÐswshc, tìte èqoume apokt sei kai shmantik  epiprìsjeth plhroforÐa.

Je¸rhma 2.1.1

(Upèrjesh). An

y1

kai

y2

eÐnai dÔo lÔseic thc omogenoÔc exÐswshc (2.2) tìte h

y(x) = C1 y1 (x) + C2 y2 (x), eÐnai epÐshc lÔsh thc (2.2) gia kˆpoiec stajerèc

C1

kai

C2 .

Dhlad , mporoÔme na prosjèsoume lÔseic (  na pollaplasiˆsoume lÔseic me kˆpoion arijmì) kai to apotèlesma na eÐnai epÐshc lÔsh. Ac apodeÐxoume t¸ra to pr¸to mac je¸rhma.

H apìdeix  tou ja mac bohj sei ousiastikˆ na katano soume tic

ènnoiec, ta qarakthristikˆ kai touc mhqanismoÔc pou aforoÔn tic grammikèc diaforikèc exis¸seic. Apìdeixh: 'Estw

y = C1 y1 + C2 y2 . 00

Tìte

0

y + py + qy = (C1 y1 + C2 y2 )00 + p(C1 y1 + C2 y2 )0 + q(C1 y1 + C2 y2 ) = C1 y100 + C2 y200 + C1 py10 + C2 py20 + C1 qy1 + C2 qy2 = C1 (y100 + py10 + qy1 ) + C2 (y200 + py20 + qy2 ) = C1 · 0 + C2 · 0 = 0

35

36

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ H apìdeixh mporeÐ na diatupwjeÐ akìma pio aplˆ an qrhsimopoi soume ton sumbolismì twn telest¸n. Me ìso pio

aplˆ lìgia gÐnetai, ac jewr soume telest  kˆti pou tou dÐnoume kˆpoiec sunart seic kai autìc mac epistrèfei kˆpoiec sunart seic (ìpwc akrib¸c dÐnoume arijmoÔc se mia sunˆrthsh kai mac epistrèfei arijmoÔc). Ac orÐsoume ton diaforikì telest 

L

wc ex c

Ly = y 00 + py 0 + qy. To ìti o telest c

L

eÐnai grammikìc shmaÐnei ìti

L(C1 y1 + C2 y2 ) = C1 Ly1 + C2 Ly2

kai h apìdeixh oloklhr¸netai wc

ex c.

Ly = L(C1 y1 + C2 y2 ) = C1 Ly1 + C2 Ly2 = C1 · 0 + C2 · 0 = 0. y 00 − k2 y = 0 eÐnai oi y1 = cosh kx kai y2 = sinh kx. JumhjeÐte ìti sÔmfwna me x −x sinh x = e −e . 'Ara, me bˆsh to je¸rhma thc upèrjeshc, oi parapˆnw eÐnai 2

DÔo ˆllec lÔseic thc exÐswshc ex +e−x gnwstì orismì, kai

cosh x =

2

lÔseic mia kai eÐnai grammikoÐ sunduasmoÐ twn dÔo gnwst¸n ekjetik¸n lÔsewn. Arketèc forèc eÐnai polÔ bolikìtero na qrhsimopoi soume tic sunart seic

sinh

kai

cosh

kai ìqi tic ekjetikèc. Ac

jumhjoÔme merikèc apì tic idiìthtèc touc.

2.1.1

cosh 0 = 1

sinh 0 = 0

d cosh x = sinh x dx cosh2 x − sinh2 x = 1

d sinh x = cosh x dx

ApodeÐxte tic parapˆnw idiìthtec qrhsimopoi¸ntac tic ekfrˆseic pou sundèoun tic

sinh kai cosh me thn ekjetik 

sunˆrthsh. Ta erwt mata thc Ôparxhc kai tic monadikìthtac thc lÔshc eÐnai, sthn perÐptwsh twn grammik¸n exis¸sewn, eÔkolo na apanthjoÔn.

Je¸rhma 2.1.2

('Uparxh kai monadikìthta). 'Estw ìti oi

p, q, f

eÐnai suneqeÐc sunart seic kai ìti oi

a, b0 , b1

eÐnai

stajerèc. H exÐswsh

y 00 + p(x)y 0 + q(x)y = f (x), èqei akrib¸c mia lÔsh

y(x)

h opoÐa ikanopoieÐ tic ex c arqikèc sunj kec

y(a) = b0 Gia parˆdeigma, h exÐswsh

y 00 + y = 0

me

y(0) = b0

kai

y 0 (a) = b1 . y 0 (0) = b1

èqei thn ex c lÔsh

y(x) = b0 cos x + b1 sin x. H exÐswsh

y 00 − y = 0

me

y(0) = b0

kai

y 0 (0) = b1

èqei thn ex c lÔsh

y(x) = b0 cosh x + b1 sinh x. Shmei¸ste parakal¸ to gegonìc ìti h qr sh twn

cosh kai sinh mac epitrèpei na lÔsoume wc proc tic arqikèc sunj kec

me polÔ pio xekˆjaro trìpo sugkritikˆ me to an qrhsimopoioÔsame ekjetikèc sunart seic. 'Hdh ja parathr sate ìti kˆje SDE deÔterhc tˆxhc sundèetai me dÔo arqikèc sunj kec. Autì eÐnai anamenìmeno mia kai gia na lÔsoume mia tètoia exÐswsh prèpei ousiastikˆ na oloklhr¸soume dÔo forèc me apotèlesma na emplakoÔn dÔo stajerèc olokl rwshc tic timèc twn opoÐwn bebaÐwc prèpei kˆpote na prosdiorÐsoume. Autì mporeÐ na gÐnei mìnon eˆn èqoume dÔo epiprìsjetec exis¸seic tic opoÐec mac prosfèroun oi arqikèc sunj kec. Er¸thsh:

Upojèste ìti oi

y1

kai

y2

eÐnai dÔo diaforetikèc metaxÔ touc lÔseic thc omogenoÔc exÐswshc (2.2).

MporeÐ kˆje ˆllh lÔsh na dojeÐ (qrhsimopoi¸ntac upèrjesh) sthn morf  H apˆnthsh eÐnai profan¸c nai! Upojètontac ìmwc ìti oi ex c ènnoia. Ja lème ìti oi

y1

kai

y2

y1

kai

y2

y = C1 y1 + C2 y2 ?

eÐnai arketˆ diaforetikèc metaxÔ touc me thn

eÐnai grammikˆ anexˆrthtec eˆn den eÐnai mia apì autèc pollaplˆsia (me stajerˆ)

thc ˆllhc. Eˆn breÐte dÔo grammikˆ anexˆrthtec lÔseic, tìte kˆje ˆllh lÔsh mporeÐ na grafjeÐ sthn morf 

y = C1 y1 + C2 y2 .

2.1. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ y = C1 y1 + C2 y2

Sthn perÐptwsh aut  h

37

lègetai genik  lÔsh.

y1 = sin x kai y2 = cos x eÐnai lÔseic thc y 00 + y = 0. EÐnai profanèc ìti oi sin kai cos den mporeÐ na eÐnai pollaplˆsia h mia thc ˆllhc. Eˆn sin x = A cos x gia kˆpoia stajerˆ A, tìte jètontac x = 0 èqoume A = 0 = sin x, prˆgma ˆtopo. 'Ara oi y1 kai y2 eÐnai grammikˆ anexˆrthtec. Sunep¸c h Gia parˆdeigma, eÔkola brÐskoume ìti oi

y = C1 cos x + C2 sin x eÐnai h genik  lÔsh thc

2.1.1

00

y + y = 0.

Ask seic

y = ex

2.1.2

DeÐxte ìti oi

2.1.3

Mantèyte mia lÔsh thc

2.1.4 kai

y2

y = e2x

eÐnai grammikˆ anexˆrthtec.

y 00 + 5y = 10x + 5.

ApodeÐxte thn arq  thc upèrjeshc gia mh-omogeneÐc exis¸seic. 'Estw ìti h eÐnai mia lÔsh thc

2.1.5

y=x

kai

r

Ly2 = g(x) (kai oi dÔo èqoun to Ðdio telest  L). x2 y 00 − xy 0 = 0

BreÐte dÔo grammikˆ anexˆrthtec lÔshc thc

ax2 y 00 + bxy 0 + cy = 0 Euler. Gia na tic lÔsoume qrhsimopoioÔmai thn manteyiˆ y = xr wc proc r (gia eukolÐa mac ac upojèsoume ìti x ≥ 0). y=x

r

Upojèste ìti

(b − a)2 − 4ac > 0.

kai d¸ste thn genik  lÔsh thc. Upìdeixh: Jèste

kai breÐte thn katˆllhlh tim  tou

'Estw ìti

(b − a)2 − 4ac = 0.

Euler

onomˆzontai Exis¸seic

  Exis¸seic

Cauchy −

thn opoÐa antikajistoÔme sthn exÐswsh kai lÔnoume

ax2 y 00 + bxy 0 + cy = 0. Upìdeixh: Jèste (b − a) − 4ac = 0   ìtan (b − a)2 − 4ac < 0?

a) D¸ste thn genik  lÔsh thc

r.

b) Ti sumbaÐnei ìtan

Ja epanèljoume argìtera sthn perÐptwsh pou

2.1.7

y1 eÐnai mia lÔsh thc Ly1 = f (x) y eÐnai lÔsh thc Ly = f (x)+g(x).

.

Shmei¸ste ìti oi exis¸seic thc morf c

2.1.6

DeÐxte ìti h

2

(b − a)2 − 4ac < 0.

D¸ste thn genik  lÔsh thc

ax2 y 00 + bxy 0 + cy = 0.

Upìdeixh: Jèste

y = xr ln x

gia na breÐte thn deÔterh lÔsh H diadikasÐa eÔreshc miac deÔterhc lÔshc kˆpoiac grammik c omogenoÔc exÐswshc ìtan  dh gnwrÐzeic mia lÔsh lègetai mèjodoc elˆttwshc thc tˆxhc.

2.1.8

'Estw ìti h

y1

eÐnai lÔsh thc exÐswshc

y 00 + p(x)y 0 + q(x)y = 0. Z

y2 (x) = y1 (x)

DeÐxte ìti kai h

R

e− p(x) dx dx (y1 (x))2

eÐnai lÔsh. Ac lÔsoume t¸ra merikèc fhmismènec exis¸seic.

2.1.9(exÐswsh

Chebychev

1hc tˆxhc) 'Estw

(1 − x2 )y 00 − xy 0 + y = 0.

a) DeÐxte ìti h

y = x

eÐnai lÔsh.

Qrhsimopoi ste thn mèjodo elˆttwshc thc tˆxhc gia na breÐte mia deÔterh grammikˆ anexˆrthth lÔsh.

b)

c) D¸ste

thn genik  lÔsh.

2.1.10(ExÐswsh

Hermite

2hc tˆxhc) 'Estw

y 00 − 2xy 0 + 4y = 0.

a) DeÐxte ìti h

y = 1 − 2x2

eÐnai lÔsh.

Qrhsimopoi ste thn mèjodo elˆttwshc thc tˆxhc gia na breÐte mia deÔterh grammikˆ anexˆrthth lÔsh. thn genik  lÔsh.

b)

c) D¸ste

38

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ

2.2

Grammikèc SDE deÔterhc tˆxhc me stajeroÔc suntelestèc

Jewr ste to prìblhma

y 00 − 6y 0 + 8y = 0,

y 0 (0) = 6.

y(0) = −2,

H parapˆnw exÐswsh eÐnai mia grammik  kai omogen c exÐswsh deÔterhc tˆxhc me stajeroÔc suntelestèc. Me ton ìro stajeroÐ suntelestèc ennooÔme ìti oi sunart seic pou pollaplasiˆzontai me touc ìrouc den exartiìntai dhlad  apì to

y 00 , y 0 ,

kai

y

eÐnai stajerèc,

x.

Mac boleÔei idiaÐtera (ja doÔme se lÐgo giatÐ) na jewr soume thn lÔsh thc parapˆnw exÐswshc san mia sunˆrthsh   opoÐa den allˆzei ousiastikˆ eˆn thn paragwgÐsoume me apotèlesma na elpÐzoume ìti ènac katˆllhloc grammikìc sunduasmìc thc en lìgw sunˆrthshc kai twn parag¸gwn thc ja mac d¸sei mhdèn. Me autì to skeptikì eÐnai logikì na exetˆsoume san lÔsh thn ex c sunˆrthsh kai

00

2 rx

y =r e

y = erx .

BebaÐwc èqoume

y 0 = rerx

kai antikajist¸ntac paÐrnoume

y 00 − 6y 0 + 8y = 0, r2 erx − 6rerx + 8erx = 0, r2 − 6r + 8 = 0

(diairèste kai ta dÔo mèlh me

erx ),

(r − 2)(r − 4) = 0. Sunep¸c gia

2.2.1

r=2

  gia

r = 4,

Exetˆste katˆ pìso h

Oi sunart seic

e2x

h

y1

erx

eÐnai mia lÔsh. Katal goume loipìn stic

kai h

y2

y1 = e2x

kai

y2 = e4x .

eÐnai lÔseic.

e4x eÐnai grammikˆ anexˆrthtec. Eˆn den  tan ja mporoÔsame na grˆyoume e4x = Ce2x , e = C , prˆgma adÔnaton na sumbaÐnei. Sunep¸c mporoÔme na grˆyoume thn genik  lÔsh

kai

apì to opoÐo èqoume ìti

2x

sthn ex c morf 

y = C1 e2x + C2 e4x . C1

Prèpei na lÔsoume wc proc in

x=0

kai

C2 .

To apply the initial conditionc we first find

y 0 = 2C1 e2x + 4C2 e4x .

We plug

kai solve.

−2 = y(0) = C1 + C2 , 6 = y 0 (0) = 2C1 + 4C2 . Apì to parapˆnw sÔsthma algebrik¸n exis¸sewn eÔkola paÐrnoune ìti katal xoume ìti

C1 = −7.

3 = C1 + 2C2 ,

kai

5 = C2

me apotèlesma na

H genik  loipìn lÔsh eÐnai h ex c

y = −7e2x + 5e4x . Ac genikeÔsoume thn parapˆnw mejodologÐa gia genikèc peript¸seic. 'Eqoume loipìn thn exÐswsh

ay 00 + by 0 + cy = 0, ìpou

a, b, c

eÐnai kˆpoiec stajerèc. Dokimˆzontac thn manteyiˆ

(2.3)

rx

y=e

paÐrnoume

ar2 erx + brerx + cerx = 0, ar2 + br + c = 0. H exÐswsh

ar2 + br + c = 0

onomˆzetai qarakthristik  exÐswsh thc SDE. LÔnontac thn exÐswsh aut  wc proc

r

kai

qrhsimopoi¸ntac ton tÔpo tou triwnÔmou.

r1 , r 2 = Opìte katal goume ìti oi

er1 x

kai

er2 x

eÐnai lÔseic.

−b ±

√

b2 − 4ac . 2a

Upˆrqei bèbaia kai h perÐptwsh pou

eÔkolo na antimetwpisjeÐ..

Je¸rhma 2.2.1.

'Estw ìti

r1

kai

r2

eÐnai oi rÐzec tic qarakthristik c exÐswshc.

r1 = r2 ,

allˆ aut  eÐnai

2.2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΔşΥΤΕΡΗΣ ΤŸΑΞΗΣ ΜΕ ΣΤΑΘΕΡϟΥΣ ΣΥΝΤΕΛΕΣԟΕΣ i

( ) An

r1

kai

r2

b2 − 4ac > 0),

eÐnai pragmatikèc kai diaforetikèc metaxÔ touc (

39

tìte h genik  lÔsh thc (2.3) eÐnai h

ex c

y = C1 er1 x + C2 er2 x . ii)

(

An

r1 = r2 (b2 − 4ac = 0),

tìte h genik  lÔsh thc (2.3) eÐnai h ex c

y = (C1 + C2 x) er1 x . y 00 − k2 y = 0 thc opoÐac h −kx kx sunep¸c e kai e eÐnai oi dÔo

Ac doÔme èna akìma parˆdeigma thc pr¸thc perÐptwshc jewr¸ntac thn exÐswsh qarakthristik  exÐswsh eÐnai h ex c

2

2

r −k = 0

dhlad 

(r − k)(r + k) = 0

kai

grammikˆ anexˆrthtec lÔseic.

Parˆdeigma 2.2.1:

BreÐte thn genik  lÔsh thc exÐswshc

y 00 − 8y 0 + 16y = 0. H qarakthristik  exÐswsh eÐnai h ex c

r2 − 8r + 16 = (r − 4)2 = 0

h opoÐa èqei dipl  rÐza

r1 = r2 = 4 .

'Ara h

genik  lÔsh eÐnai

y = (C1 + C2 x) e4x = C1 e4x + C2 xe4x . 2.2.2 H

EÐnai oi

e4x

e4x

kai

xe4x

grammikèc anexˆrthtec lÔseic?

y = xe4x

eÐnai profan¸c lÔsh. Gia thn

èqoume

y 0 = e4x + 4xe4x

kai

y 00 = 8e4x + 16xe4x .

Antikajist¸ntac

paÐrnoume

y 00 − 8y 0 + 16y = 8e4x + 16xe4x − 8(e4x + 4xe4x ) + 16xe4x = 0. ˆra eÐnai lÔsh kai mˆlista epeid  h

xe4x = Ce4x

sunepˆgetai to ˆtopo ìti

x = C

oi dÔo lÔseic eÐnai grammikˆ

anexˆrthtec.

Prèpei na shmei¸soume ìti eÐnai sthn prˆxh exairetikˆ spˆnio na èqoume dipl  rÐza. Gia na katal xoume se dipl  rÐza prèpei ousiastikˆ na èqoume epilèxei ek twn protèrw katˆllhla tou suntelestèc thc diaforik c exÐswshc. Ac d¸soume tèloc mia sÔntomh apìdeixh tou giatÐ mac boleÔei h lÔsh

xerx

ìtan èqoume dipl  rÐza. H en lìgw

perÐptwsh loipìn eÐnai h oriak  katˆstash sthn opoÐa èqoume dÔo rÐzec nai men diafèroun metaxÔ touc allˆ elˆqista. er2 x −er1 x Parathr ste ìti h eÐnai mia lÔsh ìtan oi rÐzec eÐnai diaforetikèc. 'Otan to 1 teÐnei sto 2 h en lìgw lÔsh r2 −r1 rx rx teÐnei sthn parˆgwgo thc wc proc dhlad  sthn , kai sunep¸c kai aut  eÐnai epÐshc lÔsh sthn perÐptwsh

r

e

r

r

xe

dipl c rÐzac.

2.2.1

MigadikoÐ arijmoÐ kai o tÔpoc tou

Euler

Profan¸c oi rÐzec enìc poluwnÔmou mporeÐ na eÐnai migadikèc. Gia parˆdeigma h exÐswsh

r2 + 1 = 0

èqei dÔo migadikèc

rÐzec kai pragmatik . Ac kˆnoume mia sÔntomh anaskìphsh twn migadik¸n arijm¸n . èna zeugˆri pragmatik¸n arijm¸n, èna shmeÐo tou epipèdou.

(a, b)

ìpou

i2 = −1.

Ac jumhjoÔme ìti migadikìc arijmìc

(a, b) × (c, d)

orismìc

=

(ac − bd, ad + bc).

BebaiwjeÐte ìti katanoeÐte (kai mporeÐte na dikaiolog sete) tic parakˆtw tautìthtec:

2

• i = −1, i3 = −i, i4 = 1, 1 • = −i, i • (3 − 7i)(−2 − 9i) = · · · = −69 − 13i, • (3 − 2i)(3 + 2i) = 32 − (2i)2 = 32 + 22 = 13, •

1 3−2i

=

1 3+2i 3−2i 3+2i

=

eÐnai

Prosjètoume dÔo migadikoÔc arijmoÔc me ton profan  trìpo en¸ touc pollaplasiˆzoume

wc ex c

2.2.3

a + ib

BoleÔei kˆpoiec forèc na jewroÔme ènan migadikì arijmì san

3+2i 13

=

3 13

+

2 i. 13

40

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ Parakˆtw ja qrhsimopoi soume thn ekjetik  sunˆrthsh

eÔkola eˆn antikatast soume sto anˆptugma

ex = 1 + to

x

me

a + ib

mporoÔme eÔkola na upologÐsoume kai thn

Je¸rhma 2.2.2

(TÔpoc tou

a+ib

e

e

a+ib

a ib

=e e

ex+y = ex ey

isqÔoun kai gia

x

kai

y

migadikoÔc.

kai sunep¸c eˆn mporèsoume na upologÐsoume thn

. Gia autì ja qrhsimopoi soume ton gnwstì tÔpo tou

eib

Euler.

Euler). eiθ = cos θ + i sin θ

kai

e−iθ = cos θ − i sin θ.

Elègxte tic parakˆtw tautìthtec qrhsimopoi¸ntac ton tÔpo tou

cos θ = 2.2.5

enìc migadikoÔ arijmoÔ h opoÐa mporeÐ na orisjeÐ

x x2 x3 + + + ··· 1! 2! 3!

opìte kai eÔkola blèpoume ìti idiìthtec ìpwc h

H idiìthta aut  mac odhgeÐ sthn ex c idiìthta

2.2.4

ea+ib

T aylor

eiθ + e−iθ 2

kai

sin θ =

Tautìthtec diplˆsiac gwnÐac: Qrhsimopoi ste ton tÔpo tou

Euler: eiθ − e−iθ . 2i

Euler

kai ta dÔo mèrh thc

ei(2θ) = eiθ


2

gia na

sumperˆnete ìti:

cos 2θ = cos2 θ − sin2 θ

kai

sin 2θ = 2 sin θ cos θ.

Qreiˆzetai na xekajarÐsoume ton sumbolismì mac kai thn orologÐa mac. Se èna migadikì arijmì pragmatikì mèroc kai o

b

Re(a + bi) = a 2.2.2

a+ib o a onomˆzetai

fantastikì mèroc tou arijmoÔ autoÔ. PolÔ diadedomènoc eÐnai o ex c sumbolismìc kai

Im(a + bi) = b.

Migadikèc rÐzec

Ac upojèsoume t¸ra ìti h qarakthristik  exÐswsh migadikèc rÐzec. Dhlad  èqoume

2

b − 4ac < 0

'Opwc mporoÔme na doÔme, oi lÔseic ja eÐnai

ar2 + br + c = 0

thc diaforik c exÐswshc

ay 00 + by 0 + cy = 0

èqei

kai katˆ sunèpeia oi rÐzec eÐnai oi ex c.

√ −b b2 − 4ac r1 , r 2 = ±i . 2a 2a pˆnta se zeÔgh α ± iβ . Kai

sthn perÐptwsh aut  mporoÔme na grˆyoume

thn lÔsh me ton Ðdio trìpo ìpwc kai prohgoumènwc

y = C1 e(α+iβ)x + C2 e(α−iβ)x . 'Omwc h ekjetik  sunˆrthsh èqei migadikˆ orÐsmata kai ja qreiasjeÐ na epilèxoume katˆllhlouc migadikoÔc arijmoÔc gia tic stajerèc

C1

kai

C2

ètsi ¸ste h lÔsh na mhn emplèkei migadikoÔc (kˆti pou profan¸c epijumoÔme). Kˆti tètoio

eÐnai men efiktì allˆ endeqomènwc na apaiteÐ pollèc (kai Ðswc dÔskolec) prˆxeic. Se kˆje perÐptwsh ìmwc, ìpwc ja doÔme, den eÐnai aparaÐthto. MporoÔme na qrhsimopoi soume ton tÔpo tou

Euler.

(α+iβ)x

y1 = e

Pr¸ta ac jèsoume

kai

y2 = e(α−iβ)x .

Ac shmei¸soume ìti

y1 = eαx cos βx + ieαx sin βx, y2 = eαx cos βx − ieαx sin βx. Den xeqnˆme ìti kˆje grammikìc sunduasmìc lÔsewn eÐnai kai autìc lÔsh. Sunep¸c oi

y1 + y2 = eαx cos βx, 2 y1 − y2 y4 = = eαx sin βx, 2i y3 =

eÐnai epÐshc lÔseic kai mˆlista me pragmatikì pedÐo orismoÔ kai tim¸n. Den eÐnai idiaÐtera dÔskolo na apodeÐxoume ìti eÐnai kai grammikˆ anexˆrthtec (den eÐnai h mia pollaplˆsio thc ˆllhc). 'Etsi katal goume sto parakˆtw je¸rhma.

2.2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΔşΥΤΕΡΗΣ ΤŸΑΞΗΣ ΜΕ ΣΤΑΘΕΡϟΥΣ ΣΥΝΤΕΛΕΣԟΕΣ Je¸rhma 2.2.3.

41

Eˆn oi rÐzec thc qarakthristik c exÐswshc thc diaforik c exÐswshc

ay 00 + by 0 + cy = 0. α ± iβ ,

eÐnai oi

tìte h genik  thc lÔsh eÐnai

y = C1 eαx cos βx + C2 eαx sin βx. Parˆdeigma 2.2.2:

y 00 + k2 y = 0, gia kˆpoia stajerˆ k > 0. rÐzec thc eÐnai r = ±ik kai me bˆsh to parapˆnw

BreÐte thn genik  lÔsh thc exÐswshc

H qarakthristik  thc exÐswshc eÐnai

2

2

r + k = 0.

Oi

je¸rhma h

genik  lÔsh eÐnai

y = C1 cos kx + C2 sin kx. Parˆdeigma 2.2.3:

y 00 − 6y 0 + 13y = 0, y(0) = 0, y 0 (0) = 10. r − 6r + 13 = 0 oi rÐzec thc opoÐac eÐnai r = 3 ± 2i.

BreÐte thn lÔsh tou probl matoc

2

H qarakthristik  thc diaforik c exÐswshc eÐnai

Me bˆsh to

parapˆnw je¸rhma h genik  lÔsh eÐnai

y = C1 e3x cos 2x + C2 e3x sin 2x. Gia na broÔme thn lÔsh tou sugkekrimènou probl matoc qrhsimopoioÔme thn pr¸th arqik  sunj kh gia na pˆroume

0 = y(0) = C1 e0 cos 0 + C2 e0 sin 0 = C1 . 'Ara

C1 = 0

kai sunep¸c

y = C2 e3x sin 2x.

ParagwgÐzontac èqoume

y 0 = 3C2 e3x sin 2x + 2C2 e3x cos 2x. Qrhsimopoi¸ntac thn ˆllh arqik  sunj kh èqoume

10 = y 0 (0) = 2C2 ,

 

C2 = 5 .

'Ara h lÔsh pou yˆqnoume eÐnai

y = 5e3x sin 2x. 2.2.3

Ask seic

2.2.6

BreÐte thn genik  lÔsh thc exÐswshc

2y 00 + 2y 0 − 4y = 0.

2.2.7

BreÐte thn genik  lÔsh thc exÐswshc

y 00 + 9y 0 − 10y = 0.

2.2.8

LÔste to prìblhma

y 00 − 8y 0 + 16y = 0

2.2.9

LÔste to prìblhma

y 00 + 9y 0 = 0

gia

gia

y(0) = 2, y 0 (0) = 0.

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

2.2.10

BreÐte thn genik  lÔsh thc exÐswshc

2y 00 + 50y = 0.

2.2.11

BreÐte thn genik  lÔsh thc exÐswshc

y 00 + 6y 0 + 13y = 0.

2.2.12

BreÐte thn genik  lÔsh thc exÐswshc

y 00 = 0

2.2.13

H mèjodo thc paragrˆfou aut c mporeÐ na efarmosjeÐ kai se exis¸seic tˆxhc megalÔterhc tou dÔo.

qrhsimopoi¸ntac thn mèjodo thc paragrˆfou aut c. Ja

asqolhjoÔme me tètoiec exis¸seic an¸terhc tˆxhc argìtera. Prospaj ste ìmwc na lÔsete thn ex c exÐswsh pr¸thc tˆxhc

2y 0 + 3y = 0

2.2.14 èqoume

qrhsimopoi¸ntac thn mèjodo thc paragrˆfou aut c.

Ac epistrèyoume sthn exÐswsh tou

(b − a)2 − 4ac < 0.

xr = er ln x .

Euler

pou sunant same sthn ˆskhsh 2.1.6 sth selÐda 37.

'Estw ìti

ax2 y 00 + bxy 0 + cy = 0.

Upìdeixh:

BreÐte ènan tÔpo gia thn genik  lÔsh thc exÐswshc

42

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ

2.3

Grammikèc SDE uyhlìterhc tˆxhc

Genikˆ, oi pleioyhfÐa twn diaforik¸n exis¸sewn pou emfanÐzontai sthn prˆxh se efarmogèc eÐnai deÔterhc tˆxhc. Exis¸seic megalÔterhc tˆxhc den emfanÐzontai suqnˆ kai en gènei èqei epikrat sei h ˆpoyh ìti o fusikìc mac kìsmoc eÐnai deÔterhc tˆxhs. H antimet¸pish SDE megalÔterhc tˆxhc eÐnai parìmoia me aut n twn SDE deÔterhc tˆxhc.

Prèpei ìmwc na

diasafhjeÐ h ènnoia thc grammik c anexarthsÐac. Exˆllou h en lìgw ènnoia qrhsimopoieÐtai se pollèc ˆllec perioqèc twn Majhmatik¸n kai se pollˆ ˆlla shmeÐa akìma kai aut¸n twn shmei¸sewn. Sunep¸c axÐzei na thn katano soume pl rwc. Ac arqÐsoume me thn ex c genik  omogen  grammik  exÐswsh

y (n) + pn−1 (x)y (n−1) + · · · + p1 (x)y 0 + p0 (x)y = 0. Je¸rhma 2.3.1

(Upèrjesh). Eˆn

y1 , y2 ,

yn

...,

(2.4)

eÐnai lÔseic thc omogenoÔc exÐswshc (2.4), tìte h

y(x) = C1 y1 (x) + C2 y2 (x) + · · · + Cn yn (x), eÐnai epÐshc lÔsh sthc (2.4) gia opoiesd pote stajerèc

C1 ,

...,

Cn .

IsqÔei to parakˆtw je¸rhma Ôparxhc kai monadikìthtac gia mh-omogeneÐc grammikèc exis¸seic.

Je¸rhma 2.3.2 sunart seic kai oi

('Uparxhc kai monadikìthtac).

a, b0 , b1 , . . . , bn−1

'Estw ìti oi sunart seic

p0 , p1 , . . . , pn−1 ,

kai

f

eÐnai suneqeÐc

eÐnai stajerèc. H exÐswsh

y (n) + pn−1 (x)y (n−1) + · · · + p1 (x)y 0 + p0 (x)y = f (x), . èqei akrib¸c mia lÔsh

y(x)

oi opoÐa ikanopoieÐ tic parakˆtw arqikèc sunj kec

y 0 (a) = b1 ,

y(a) = b0 , 2.3.1

'Eqoume  dh anafèrei ìti dÔo sunart seic

...,

y (n−1 )(a) = bn−1 .

Grammik  anexarthsÐa

laplˆsio thc ˆllhc.

y2 ,

...,

yn

'Otan èqoume

n

y1

kai

y2

eÐnai grammikˆ anexˆrthtec an h miˆmÐa den mporeÐ na eÐnai pol-

sunart seic mporoÔme na doulèyoume parapl sia wc ex c.

Oi sunart seic

y1 ,

eÐnai grammikˆ anexˆrthtec an h exÐswsh

c1 y1 + c2 y2 + · · · + cn yn = 0, èqei mìnon thn tetrimmènh lÔsh

c1 = c2 = · · · = cn = 0. Eˆn mia apì tic stajerèc thc exÐswshc (èstw, qwrÐc periorismì c1 6= 0, tìte mporoÔme na ekfrˆsoume thn y1 san grammikì sunduasmì

thc genikìthtac, h pr¸th) eÐnai mh-mhdenik 

twn upoloÐpwn. Eˆn oi sunart seic den eÐnai grammikˆ anexˆrthtec tìte lème ìti eÐnai grammikˆ exarthmènec.

Parˆdeigma 2.3.1:

DeÐxte ìti oi

ex , e2x , e3x

eÐnai grammikˆ anexˆrthtec.

Ac to kˆnoume me diˆforec mejìdouc. Ta perissìtera didaktikˆ biblÐa (sumperilambanomènwn twn [EP] kai [F]) eisˆgoun thn Jètontac

W ronskian. Kˆti tètoio z = ex sthn exÐswsh

den eÐnai aparaÐthto sthn perÐptws  mac.

c1 ex + c2 e2x + c3 e3x = 0. kai qrhsimopoi¸ntac jemeli¸deic idiìthtec twn ekjetik¸n sunart sewn èqoume

c1 z + c2 z 2 + c3 z 3 = 0. To aristerì mèroc eÐnai èna polu¸numo bajmoÔ trÐa wc proc

z.

MporeÐ eÐte na eÐnai tautotikˆ Ðso me to mhdèn eÐte na

èqei to polÔ 3 rÐzec. H parapˆnw exÐswsh profan¸c isqÔei gia ìla ta

c1 = c2 = c3 = 0

z,

kai oi dojeÐsec sunart seic eÐnai grammikˆ anexˆrthtec.

Ac dokimˆsoume ènan ˆllo trìpo xekin¸ntac apì thn sqèsh

c1 ex + c2 e2x + c3 e3x = 0.

ˆra eÐnai tautotikˆ Ðsh me to mhdèn, dhlad 

2.3. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ h opoÐa bebaÐwc isqÔei gia kˆje

x.

43

Ac diairèsoume kai ta dÔo mèlh me

e3x

èqoume thn sqèsh

c1 e−2x + c2 e−x + c3 = 0. h opoÐa epÐshc isqÔei gia kˆje

x,

ˆra, mporoÔme na pˆroume to ìrio ìtan

x→∞

kai na katal xoume ìti

c3 = 0 .

opìte

h exÐsws  mac paÐrnei thn ex c morf 

c1 ex + c2 e2x = 0. Aplˆ mènei na epanalˆboume thn parapˆnw diadikasÐa dÔo akìma forèc gia na doÔme ìti

c2 = 0

kai

c1 = 0.

Ac dokimˆsoume ènan trÐto trìpo xekin¸ntac apì thn sqèsh

c1 ex + c2 e2x + c3 e3x = 0. MporoÔme kataskeuˆsoume ìsec exis¸seic, me agn¸stouc ta

c1 , c2

kai

c3 ,

epijumoÔme tic opoÐec katìpin mporoÔme

na qrhsimopoi soume gia na broÔme touc en lìgw agn¸stouc. Kˆti tètoio bèbaia apaiteÐ arketèc prˆxeic. MporoÔme epÐshc pr¸ta na paragwgÐsoume kai ta dÔo mèlh kai metˆ na parˆgoume me ton parapˆnw trìpo kai ˆllec exis¸seic. Gia na aplopoi soume thn diadikasÐa ac diairèsoume me

ex .

c1 + c2 ex + c3 e2x = 0. Jètoume

x=0

kai paÐrnoume thn exÐswsh

c1 + c2 + c3 = 0.

ParagwgÐzontac kai ta dÔo mèrh èqoume

c2 ex + 2c3 e2x = 0, x = 0 èqoume ìti c2 + 2c3 = 0. Tèloc diairoÔme xanˆ me ex kai paragwgÐzoume gia na pˆroume 4c3 e2x = 0. èqoume c3 = 0. 'Ara kai c2 = 0 mia kai c2 = −2c3 kai c1 = 0 epeid  c1 + c2 + c3 = 0.

kai jètontac Opìte

Parˆdeigma 2.3.2:

Oi sunart seic

ex , e−x ,

kai

cosh x

eÐnai grammikˆ exarthmènec.

Aplˆ efarmìste ton orismì

tou uperbolikoÔ sunhmitìnou:

cosh x =

2.3.2

ex + e−x . 2

SDE uyhlìterhc tˆxhc me stajeroÔc suntelestèc

H antimet¸pish omogen¸n grammik¸n exis¸sewn an¸terhc tˆxhc me stajeroÔc suntelestèc eÐnai se megˆlo bajmì parìmoia me thn mejodologÐa pou anaptÔxame parapˆnw. Apl¸c prèpei na broÔme perissìterec grammikˆ anexˆrthtec lÔseic. Eˆn loipìn h exÐswsh eÐnai

nth

tˆxhc qreiˆzetai na broÔme

n

grammikˆ anexˆrthtec lÔseic. Ac xekajarÐsoume

ta prˆgmata me èna parˆdeigma.

Parˆdeigma 2.3.3:

UpologÐste thn genik  lÔsh thc exÐswshc

y 000 − 3y 00 − y 0 + 3y = 0. Antikajist¸ntac thn manteyiˆ

y = erx

(2.5)

èqoume

r3 erx − 3r2 erx − rerx + 3erx = 0. DiairoÔme me

erx

kai paÐrnoume.

r3 − 3r2 − r + 3 = 0. Den eÐnai tetrimmènh upìjesh na broÔme tic rÐzec enìc poluwnÔmou. Upˆrqoun tÔpoi gia tic rÐzec enìc poluwnÔmou bajmoÔ 3 kai 4 an kai eÐnai idiaÐtera polÔplokoi. autì den shmaÐnei ìti den upˆrqoun oi rÐzec.

Gia polu¸numa megalÔterou bajmoÔ den upˆrqoun tÔpoi.

EÐnai gnwstì ìti èna polu¸numo

nstou

bajmoÔ èqei

apì autèc mporeÐ na eÐnai ìpwc mporeÐ merikèc apì autèc tic rÐzec na eÐnai migadikèc.

n

rÐzec.

Fusikˆ Merikèc

Fusikˆ upˆrqoun pollˆ kai

exairetikˆ logismikˆ sust mata ta opoÐa mporoÔn na upologÐsoun proseggÐseic twn riz¸n enìc poluwnÔmou kˆpoiou logikoÔ bajmoÔ. Epiprìsjeta, upˆrqoun kai majhmatikˆ apotelèsmata thc jewrÐac arijm¸n ta opoÐa mac dÐnoun thn dunatìthta na upologÐsoume akrib¸c tic rÐzec.

Gia parˆdeigma gnwrÐzoume ìti o stajerìc ìroc kˆje poluwnÔmou

44

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ

isoÔtai me to ginìmeno ìlwn twn riz¸n tou. Gia parˆdeigma èstw èqoume

r3 − 3r2 − r + 3 = (r − r1 )(r − r2 )(r − r3 )

opìte kai

∗

3 = (−r1 )(−r2 )(−r3 ) = (1)(−1)(−r3 ) = r3 . EÐnai eÔkolo na epibebai¸soume ìti h thc (2.5).

r3 = 3

eÐnai mia rÐza.

Sunep¸c gnwrÐzoume ìti oi

e−x , ex

kai

e3x

eÐnai lÔseic

EpÐshc eÔkola mporoÔme na doÔme ìti eÐnai grammikˆ anexˆrthtec, kai eÐnai treic, dhlad  akrib¸c ìsec

qreiazìmaste. 'Ara h genik  lÔsh eÐnai

y = C1 e−x + C2 ex + C3 e3x . Gia na ikanopoi soume tic arqikèc sunj kec

y(0) = 1, y 0 (0) = 2,

kai

y 00 (0) = 3

èqoume

1 = y(0) = C1 + C2 + C3 , 2 = y 0 (0) = −C1 + C2 + 3C3 , 3 = y 00 (0) = C1 + C2 + 9C3 . H lÔsh tou parapˆnw algebrikoÔ grammikoÔ sust matoc eÐnai

C1 = −1/4, C2 = 1

kai

C3 = 1/4

kai sunep¸c h lÔsh

thc diaforik c exÐswshc pou ikanopoieÐ tic dojeÐsec arqikèc sunj kec eÐnai h

y=

−1 −x 1 e + ex + e3x . 4 4

Ac upojèsoume t¸ra ìti ìlec oi rÐzec eÐnai pragmatikèc allˆ me kˆpoia pollaplìthta (dhlad  merikèc lÔseic epanalambˆnontai).

Ac perioristoÔme sthn perÐptwsh pou èqoume mia rÐza

r

me pollaplìthta

k.

Sthn perÐptwsh

aut , kai sto pneÔma thc mejodologÐac pou anaptÔxame gia thn anˆlogh perÐptwsh gia tic exis¸seic deÔterhc tˆxhc, anagnwrÐzoume tic lÔseic

erx , xerx , x2 erx , . . . , xk−1 erx kai h genik  lÔsh prokÔptei san ènac grammikìc touc sunduasmìc.

Parˆdeigma 2.3.4:

LÔste thn exÐswsh

y (4) − 3y 000 + 3y 00 − y 0 = 0. H qarakthristik  exÐswsh eÐnai

r4 − 3r3 + 3r2 − r = 0. Shmei¸ste ìti

r4 − 3r3 + 3r2 − r = r(r − 1)3 .

'Ara oi rÐzec, me pollaplìthta, eÐnai oi

r = 0, 1, 1, 1

opìte prokÔptei h

parakˆtw genik  lÔsh

(c0 + c1 x + c2 x2 ) ex | {z }

y=

ìroi proerqìmenoi apì thn r = 1

+

c4 |{z}

.

apì thn r = 0

Entel¸c parìmoia me thn perÐptwsh thc deÔterhc tˆxhc mporoÔme na antimetwpÐsoume to endeqìmeno na prokÔyoun migadikèc rÐzec. Oi migadikèc rÐzec ìpwc gnwrÐzoume èrqontai se zeÔgh

r = α ± iβ .

Oi antÐstoiqec lÔseic eÐnai oi

(c0 + c1 x + · · · + ck−1 xk ) eαx cos βx + (d0 + d1 x + · · · + dk−1 xk ) eαx sin βx. ìpou

c0 ,

...,

ck−1 , d0 ,

Parˆdeigma 2.3.5:

...,

dk−1

eÐnai tuqaÐec stajerèc.

LÔste thn exÐswsh

y (4) − 4y 000 + 8y 00 − 8y 0 + 4y = 0. H qarakthristik  exÐswsh eÐnai

r4 − 4r3 + 8r2 − 8r + 4 = 0, (r2 − 2 + 2)2 = 0,
2 (r − 1)2 + 2 = 0. ∗ Sto mˆjhma tou EpisthmonikoÔ logismoÔ ja anaptÔxete kai ja analÔsete diˆforec arijmhtikèc mejìdouc gia ton upologismì proseggÐsewn twn riz¸n akìma kai sthn genik  perÐptwsh poluwnÔmwn opoioud pote bajmoÔ.

2.3. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ 'Ara oi rÐzec eÐnai oi

1±i

45

me pollaplìthta 2. Sunep¸c h genik  lÔsh eÐnai h ex c

y = (c0 + c1 x) ex cos x + (d0 + d1 x) ex sin x. O trìpoc pou lÔsame thn qarakthristik  exÐswsh eÐnai manteÔontac ousiastikˆ thn lÔsh kai dokimˆzontac tic manteyièc mac. Kˆti tètoio den eÐnai pˆnta eÔkolo. Ac mhn xeqnˆme bèbaia ìti mporoÔme na prospaj soume na qrhsimopoi soume kˆpoio apì ta upˆrqonta logismikˆ sust mata eÔreshc riz¸n poluwnÔmwn.

2.3.3

Ask seic

2.3.1

BreÐte thn genik  lÔsh thc exÐswshc

y 000 − y 00 + y 0 − y = 0.

2.3.2

BreÐte thn genik  lÔsh thc exÐswshc

y (4) − 5y 000 + 6y 00 = 0.

2.3.3

BreÐte thn genik  lÔsh thc exÐswshc

y 000 + 2y 00 + 2y 0 = 0.

2.3.4

Upojèste ìti h

(r − 1)2 (r − 2)2 = 0 eÐnai h qarakthristik  b) BreÐte thn genik  lÔsh thc.

exÐswsh miac diaforik c exÐswshc.

a)

BreÐte mia

tètoia diaforik  exÐswsh.

2.3.5

Upojèste ìti mia exÐswsh tètarthc tˆxhc èqei thn parakˆtw lÔsh

diaforik  exÐswsh.

b)

y = 2e4x x cos x. a)

BreÐte mia tètoia

BreÐte tic arqikèc sunj kec tic opoÐec ikanopoieÐ h dojeÐsa lÔsh.

2.3.6

BreÐte thn genik  lÔsh thc exÐswshc thc ˆskhshc 2.3.5.

2.3.7

'Estw ìti

f (x) = ex − cos x, g(x) = ex + cos x,

kai

h(x) = cos x.

EÐnai oi

f (x), g(x),

kai

h(x)

grammikˆ

anexˆrthtec? Eˆn nai, apodeÐxte to, eˆn ìqi, breÐte ènan grammikì sunduasmì pou mac boleÔei.

2.3.8

'Estw ìti

f (x) = 0, g(x) = cos x,

kai

h(x) = sin x.

EÐnai oi

f (x), g(x),

kai

h(x)

grammikˆ anexˆrthtec? Eˆn

nai, apodeÐxte to, eˆn ìqi, breÐte ènan grammikì sunduasmì pou mac boleÔei.

2.3.9

EÐnai oi

x, x2 ,

kai

x4

grammikˆ anexˆrthtec? Eˆn nai, apodeÐxte to, eˆn ìqi, breÐte ènan grammikì sunduasmì

pou mac boleÔei.

2.3.10

EÐnai oi

ex , xex ,

kai

sunduasmì pou mac boleÔei.

x2 ex

grammikˆ anexˆrthtec?

Eˆn nai, apodeÐxte to, eˆn ìqi, breÐte ènan grammikì

46

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ

2.4

Talant¸seic

Ac rÐxoume mia matiˆ se kˆpoiec efarmogèc twn grammik¸n exis¸sewn deÔterhc tˆxhc me stajeroÔc suntelestèc.

2.4.1

Merikˆ ParadeÐgmata

k

F (t) m

To pr¸to mac parˆdeigma eÐnai èna sÔsthma mˆzas-elathrÐou. soume loipìn èna swmatÐdio mˆzac

m > 0

eÐnai sundedemèno me èna elat rio me stajerˆ elathrÐou (  alli¸c stajerˆ tou Qouk)

k>0

(ac upojèsoume se

N ewtons

eÐnai sundedemènh se ènan toÐqo.

apìsbesh c

anˆ mètro) h ˆllh ˆkrh tou opoÐou

Epiprìsjeta, upojètoume ìti efarmìzoume

sto swmatÐdio mia exwterik  dÔnamh

F (t).

Tèloc upojètoume ìti to sÔsthma

upìkeitai se apìsbesh h opoÐa kajorÐzetai apì mia stajerˆ 'Estw

x

Ac jewr -

(ac upojèsoume se kilˆ) to opoÐo

h metatìpish tou swmatidÐou upojètontac ìti thn

x = 0

c ≥ 0. x auxˆnei

eÐnai h jèsh isorropÐac kai h

ìso

metakineÐte to swmatÐdio proc ta dexiˆ (apomakrunìmeno apì ton toÐqo). H dÔnamh pou askeÐ to elat rio sto swmatÐdio eÐnai sÔmfwna me ton nìmo tou Qouknìmoc tou Qouk anˆlogo thc metatìpishc. EÐnai dhlad  Ðsh me

kx

sthn antÐjeth

kateÔjunsh. Parìmoia, h dÔnamh pou ofeÐletai sthn apìsbesh eÐnai anˆlogh me thn taqÔthta tou swmatidÐou. SÔmfwna me ton deÔtero nìmo tou NeÔtwnadeÔteroc nìmoc tou NeÔtwna h sunolik  dÔnamh isoÔtai me mˆza epÐ epitˆqunsh. Dhlad  èqoume thn ex c

mx00 + cx0 + kx = F (t) grammik  SDE deÔterhc tˆxhc me stajeroÔc suntelestèc. Ac kajorÐsoume thn orologÐa pou aforˆ thn exÐswsh aut . Lème ìti mia kÐnhsh eÐnai

i

( ) exanagkasmènh, an

F 6≡ 0 (F

ii

den eÐnai tautotikˆ mhdèn),

( ) mh-exanagkasmènh   eleÔjerh, an

iii)

me apìsbesh, an

iv)

qwrÐc apìsbesh, an

(

(

c > 0,

F ≡ 0,

kai

c = 0.

Parìlh thn aplìthtˆ tou, to parapˆnw sÔsthma emfanÐzetai ston pur na poll¸n kai shmantik¸n pragmatik¸n

†

efarmog¸n. Pollèc ˆllec efarmogèc mporoÔn an melethjoÔn san aplopoi seic enìc sust matoc mˆzac elathrÐou . Ac doÔme dÔo paradeÐgmata.

RLC pou faÐnetai sthn R ohms, èna phnÐo me suntelest  autepagwg c L henries, kai ènac puknwt c qwrhtikìthtac C f arads. Upˆrqei epÐshc kai mia hlektrik  phg  (p.q. mia mpatarÐa) pou mac dÐnei hlektrik  tˆsh E(t) volts thn qronik  stigm  t (ac upojèsoume deuterìlepta). Ac upojèsoume tèloc ìti Q(t) eÐnai to fortÐo tou puknwt  se columbs kai ìti I(t) to reÔma pou diarrèei to kÔklwma. EÐnai gnwstì ìti h sqèsh twn dÔo 0 ex c Q = I . Epiprìsjeta, qrhsimopoi¸ntac jemeli¸deic nìmouc katal goume sthn exÐswsh OrÐste èna parˆdeigma hlektrologÐac. Jewr ste to kÔklwma

parˆpleurh eikìna. Upˆrqei mia antÐstash

C

E

L

R

aut¸n posot twn eÐnai h

LI 0 + RI + Q/C = E .

ParagwgÐzontac thn èqoume

LI 00 (t) + RI 0 (t) +

1 I(t) = E 0 (t). C

Aut  eÐnai mia mh-omogen c grammik  exÐswsh deÔterhc tˆxhc me stajeroÔc suntelestèc.

L, R,

kai

C

Epiprìsjeta, mia kai ta

eÐnai ìla jetikˆ, to sÔsthma autì sumperifèretai akrib¸c san èna sÔsthma mˆzac elathrÐou. H jèsh tou

swmatidÐou antikatastˆjhke me to reÔma, h mˆza me ???, h apìsbesh me thn antÐstash kai h stajerˆ tou elathrÐou me thn stajerˆ qwrhtikìthtac. H diaforˆ tˆshc apoteleÐ thn exwterik  dÔnamh. 'Ara gia stajer  tˆsh èqoume eleÔjerh kÐnhsh. To epìmeno parˆdeigmˆ mac sumperifèretai proseggistikˆ mìnon san èna sÔsthma mˆzac elathrÐou. 'Estw loipìn ìti èqoume mia mˆza

g

m sthn ˆkrh enìc ekkremoÔc m kouc L.

Jèloume na broÔme mia exÐswsh gia thn gwnÐa

h dÔnamh thc barÔthtac. Aplˆ stoiqeÐa fusik c mac dÐnoun thn ex c exÐswsh

θ00 + † Perissìterec

g sin θ = 0. L

plhroforÐec sthn ex c selÐda http : //el.wikipedia.org/wiki/

θ(t).

'Estw

2.4. ΤΑΛΑΝԟΩΣΕΙΣ

47

Prˆgmati h parapˆnw exÐswsh prokÔptei apì ton deÔtero nìmo tou NeÔtwna, ìpou h dÔnamh isoÔtai me thn mˆza epÐ thn epitˆqunsh.

Profan¸c h epitˆqunsh eÐnai

Lθ00

efaptìmenh sunist¸sa thc dÔnamhc thc barÔthtac. lìgo exafanÐzetai.

kai h mˆza eÐnai

m.

To ginìmeno touc ofeÐlei na eÐnai Ðso me

Se autì ofeÐletai o ìroc

Ac proqwr soume t¸ra sthn ex c prosèggish.

sin θ ≈ θ. Autì gÐnetai ˆmesa apodektì apì to sq ma 2.1 ìpou mporoÔme na (se radians) oi grafikèc parastˆseic twn sin θ kai θ ousiastikˆ tautÐzontai. -1.0

-0.5

0.0

mg sin θ.

To

m

gia kˆpoion parˆxeno

θ èqoume ìti −0.5 < θ < 0.5

Gia mikrèc se apìluth tim  gwnÐec doÔme ìti gia perÐpou

0.5

1.0

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.0

-0.5

0.0

0.5

1.0

Σχήμα 2.1: Οι γραφικές παραστάσεις των sin θ και θ (σε radians).

Sunep¸c ìtan oi aiwr seic eÐnai mikrèc, mporoÔme na ekmetalleutoÔme to gegonìc ìti oi gwnÐec

θ

eÐnai pˆnta mikrèc kai na montelopoi soume thn sumperiforˆ tou ekremoÔc me

thn ex c aploÔsterh grammik  exÐswsh

L

θ00 +

θ

g θ = 0. L

Shmei¸ste ìti lìgw thc parapˆnw aploÔsteushc ta sfˆlmata pou endeqomènwc na prokÔyoun eÐnai dunatìn na megejÔnontai suneq¸c ìso parèrqetai o qrìnoc thc ai¸rhshc me apotèlesma to montèlo na mac d¸sei mia ousiastikˆ diaforetik  sumperiforˆ apì thn pragmatik  sumperiforˆ tou sust matoc. 'Opwc ja doÔme, to montèlo mac odhgeÐ sto sumpèrasma ìti to plˆtoc thc talˆntwshc eÐnai anexˆrthto thc periìdou kˆti pou bebaÐwc den isqÔei sthn prˆxh se èna ekkremèc. periìdouc kai mikrèc aiwr seic (p.q.

Parìla autˆ, gia arketˆ mikrèc qronikèc

gia ekkrem  megˆlou m kouc) to parapˆnw montèlo proseggÐzei ikanopoihtikˆ

to fusikì fainìmeno. 'Otan antimetwpÐzoume realistikˆ probl mata, polÔ suqnˆ anagkazìmaste na kˆnoume tètoiou eÐdouc paradoqèc kai aplopoi seic. Gia na diapist¸soume oi aplopoi seic mac autèc apodektèc, sto plaÐsio bebaÐwc thc ekˆstote melèthc mac, eÐnai profan¸c aparaÐthto na katanooÔme to fusikì prìblhma tìso apì thn meriˆ thc fusik c ìso kai apì thn meriˆ twn majhmatik¸n.

2.4.2

EleÔjerh kÐnhsh qwrÐc apìsbesh

Epeid  den mporoÔme akìmh na lÔsoume mh-omogeneÐc exis¸seic sthn parˆgrafo aut  ja asqolhjoÔme me eleÔjerh (mh-exanagkasmènh) kÐnhsh. Ac xekin soume me thn perÐptwsh pou den upˆrqei apìsbesh, dhlad  ìtan

c = 0,

opìte

kai èqoume

mx00 + kx = 0. Eˆn diairèsoume me

m

kai upojèsoume ìti

ω0

eÐnai ènac arijmìc tètoioc ¸ste

morf 

x00 + ω02 x = 0. 'Opwc xèroume h genik  lÔsh thc exÐswshc aut c eÐnai

x(t) = A cos ω0 t + B sin ω0 t.

ω02 = k/m

tìte h exÐsws  mac paÐrnei thn

48

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ

Parathr ste kat' arq n ìti, qrhsimopoi¸ntac gnwst  trigwnometrik  tautìthta, èqoume thn ex c sqèsh me tic ˆllec dÔo stajerèc

C

kai

γ A cos ω0 t + B sin ω0 t = C cos(ω0 t − γ). √

C = A2 + B 2 kai tan γ = B/A. x(t) = C cos(ω0 t − γ), ìpou C kai γ tuqaÐec stajerèc.

Den eÐnai dÔskolo na doÔme ìti ex c

2.4.1

sunep¸c mporoÔme na grˆyoume thn genik  lÔsh wc

Dikaiolog ste thn tautìthta kai epibebai¸ste tic exis¸seic gia ta

C

kai

γ.

Kajìson eÐnai genikˆ eukolìtero na qrhsimopoi soume thn pr¸th morf  thc lÔshc gia na broÔme tic timèc twn stajer¸n

A kai B pou ikanopoioÔn tic arqikèc sunj kec h deÔterh morf  eÐnai stenìtera sundedemènh me to pragmatikì C kai γ èqoun poll  ìmorfh diermhneÐa. Eˆn parathr soume thn lÔsh sthn ex c morf 

prìblhma. Oi stajerèc

x(t) = C cos(ω0 t − γ) blèpoume ìti to plˆtoc thc talˆntwshc eÐnai san metatìpish fˆshc.

C , ω0

eÐnai h (gwniak ) suqnìthta, kai

γ

eÐnai mia posìthta gnwst 

MetatopÐzei to grˆfhma thc sunˆrthshc proc ta dexiˆ   proc ta aristerˆ.

H posìthta

ω0

onomˆzetai fusik  (gwniak ) suqnìthta. H kÐnhsh pou mìlic perigrˆyame eÐnai gnwst  san apl  armonik  kÐnhsh. Ac kˆnoume mia parat rhsh pou aforˆ thn lèxh gwniak  tou ìrou . H

ω0

dÐdetai se

kai ìqi se kÔklouc anˆ monˆda qrìnou ìpwc sun jwc metrˆme thn suqnìthta. perifèreia enìc kÔklou eÐnai topojethjeÐ h stajerˆ

2π ,

2π ,

h sun jhc suqnìthta dÐdetai apì thn sqèsh

radians

anˆ monˆda qrìnou,

Epeid  ìmwc ìpwc gnwrÐzoume ìti h

ω0 . EÐnai dhlad  apl¸c jèma to pou ja 2π

kˆti to opoÐo apoteleÐ eleÔjerh epilog  tou kajenìc.

H perÐodoc miac kÐnhshc isoÔtai me to antÐstrofo thc suqnìthtac (se kÔklouc anˆ monˆda qrìnou) kai sunep¸c èqoume

2π . EÐnai dhlad  o qrìnoc pou apaiteÐtai gia na oloklhrwjeÐ mia pl rhc talˆntwsh. ω0

Parˆdeigma 2.4.1:

'Estw ìti

m = 2 kg

kai

k = 8 N/m.

To sÔsthma mˆzac elathrÐou brÐsketai se èna ìqhma to

opoÐo kineÐtai me taqÔthta 1 m/s. To ìqhma sugkroÔetai kai stamatˆ. To swmatÐdio to opoÐo mèqri tìte  tan se jèsh 0.5 mètra makriˆ apì thn jèsh isorropÐac (ekteÐnontac to elat rio) af netai eleÔjero kai arqÐzei na talant¸netai. Poia eÐnai h suqnìthta kai poio to plˆtoc thc en lìgw talˆntwshc? 'Etsi katal goume sto ex c prìblhma

2x00 + 8x = 0,

x(0) = 0.5,

p √ ω0 = k/m = 4 = 2. 2/2π = 1/π ≈ 0.318 Hertz .

MporoÔme ˆmesa na upologÐsoume to (peristrofèc anˆ deuterìlepto) eÐnai

x0 (0) = 1.

Dhlad  h gwniak  suqnìthta eÐnai 2.

H suqnìthta

H genik  lÔsh eÐnai

x(t) = A cos 2t + B sin 2t. Jètontac

x(0) = 0

0

0 A = 0.5. √ Tìte x (t) = −0.5 sin 2t + B cos 2t. Jètontac x (0) = 1 √ 2 2 eÐnai C = A + B = 1.25 ≈ 1.118. H lÔsh eÐnai

shmaÐnei

to plˆtoc thc talˆntwshc

èqoume

B = 1.

Sunep¸c,

x(t) = 0.5 cos 2t + sin 2t. H grafik  parˆstash thc lÔshc

x(t)

dÐnetai sto Sq ma 2.2 sthn paroÔsa selÐda.

Gia thn eleÔjerh kÐnhsh qwrÐc apìsbesh, h lÔsh thc morf c

x(t) = A cos ω0 t + B sin ω0 t, antistoiqeÐ stic arqikèc sunj kec

x(0) = A

kai

x0 (0) = B .

EÐnai fanerì ìti h parapˆnw morf  eÐnai polÔ bolikìterh eˆn jèloume na upologÐsoume tic timèc twn sÔgkrish me to eˆn jèlame na upologÐsoume to plˆtoc kai thn metatìpish thc fˆshc.

A

kai

B,

se

2.4. ΤΑΛΑΝԟΩΣΕΙΣ

49 0.0

2.5

5.0

7.5

10.0

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0

0.0

2.5

5.0

7.5

10.0

Σχήμα 2.2: Απλή ταλάντωση χωρίς απόσβεση.

2.4.3

EleÔjerh kÐnhsh

Ac estiˆsoume t¸ra sthn exanagkasmènh kÐnhsh kai ac xanagrˆyoume thn exÐswsh wc ex c

mx00 + cx0 + kx = 0,  

x00 + 2px0 + ω02 x = 0, ìpou

r ω0 =

k , m

p=

c . 2m

H qarakthristik  exÐswsh eÐnai

r2 + 2pr + ω02 = 0. Oi rÐzec eÐnai

r = −p ±

q

p2 − ω02 .

H morf  thc lÔshc thc diaforik c exÐswshc exartˆtai apì to eˆn oi rÐzec eÐnai pragmatikèc   migadikèc. Pragmatikèc rÐzec èqoume mìnon ìtan o parakˆtw arijmìc eÐnai mh-arnhtikìc.

p2 − ω02 = To prìshmo tou

p2 − ω02

 c 2 k c2 − 4km − = . 2m m 4m2

eÐnai to Ðdio me to prìshma

c2 − 4km.

Sunep¸c èqoume pragmatikèc rÐzec an h

c2 − 4km

eÐnai

mh-arnhtik .

Ισχυρά φθίνουσα ταλάντωση 'Otan

c2 − 4km > 0,

lème ìti to sÔsthma eÐnai isqurˆ fjÐnwn.

metaxÔ touc pragmatikèc rÐzec, mikrìterh apì to

p

r1

r2 . Shmei¸ste p −p ± p2 − ω02

kai

opìte h posìthta

Sthn perÐptwsh aut , upˆrqoun dÔo diaforetikèc

ìti kai oi dÔo eÐnai arnhtikèc, mia kai h

p

p2 − ω02

eÐnai pˆnta

eÐnai pˆnta arnhtik .

'Ara h lÔsh eÐnai

x(t) = C1 er1 t + C2 er2 t . AfoÔ oi

r1 , r 2

eÐnai arnhtikèc,

x(t) → 0

ìtan

t → ∞.

Autì shmaÐnei ìti to swmatÐdio ja teÐnei na akinhtopoihjeÐ ìso

pernˆei o qrìnoc. To Sq ma 2.3 sthn epìmenh selÐda perièqei merikèc grafikèc parastˆseic lÔsewn gia diaforetikèc arqikèc timèc.

50

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ Parathr ste ìti ousiastikˆ den èqoume talˆntwsh. Mˆlista faÐnetai ìti h grafik  parˆstash thc lÔshc tèmnei

x

ton

ˆxona mìnon mia forˆ gegonìc to opoÐo mporoÔme eÔkola na epibebai¸soume wc ex c.

lÔsoume thn

0 = C1 er1 t + C2 er2 t .

'Ara

C1 er1 t = −C2 er2 t ,

Ac prospaj soume na

kai sunep¸c

−C1 = e(r2 −r1 )t . C2 Upˆrqei loipìn mia to polÔ lÔsh gia

Parˆdeigma 2.4.2:

x0 (0) = 0.

t ≥ 0.

Upojèste ìti af noume to swmatÐdio apì thn jèsh isorropÐac. Dhlad  èqoume

Tìte èqoume

x(t) =

x(0) = x0

kai


x0 r1 er2 t − r2 er1 t . r1 − r2

EÐnai eÔkolo na dei kaneÐc ìti h parapˆnw lÔsh ikanopoieÐ tic arqikèc sunj kec.

Κρίσιμα φθίνουσα κίνηση 0

c2 − 4km = 0, lème ìti to sÔsthma eÐnai rÐza, thn −p pollaplìthtac 2. 'Ara h lÔsh

25

50

75

100

'Otan

krÐsima fjÐnonkrÐsima fjÐnon sÔsthma. Sthn perÐptwsh aut  èqoume

mia

mac eÐnai

1.5

1.5

x(t) = C1 e−pt + C2 te−pt . 1.0

1.0

H sumperiforˆ enìc krÐsima fjÐnontoc sust matoc eÐnai polÔ parìmoia me aut  enìc isqurˆ fjÐnontoc sust matoc. Mˆlista sthn ousÐa èna krÐsima fjÐnon sÔsthma eÐnai me kˆpoia ènnoia to ìrio enìc isqurˆ fjÐnontoc sust matoc. Epeid  oi diaforikèc exis¸seic apoteloÔn proseggÐseic pragmatik¸n susthmˆtwn, eÐnai exairetikˆ spˆnio na sunant 0.5 soume sthn prˆxh mia krÐsima fjÐnousa kÐnhsh. EÐnai logikì 0.5 mia kÐnhsh na eÐnai eÐte lÐgo isqurˆ fjÐnousa eÐte lÐgo

asjen¸c fjÐnousa. Ac mhn proqwr soume loipìn se leptomèreiec pou aforoÔn krÐsima fjÐnousec kin seic.

Ασθενώς φθίνουσα κίνηση 0

5

10

15

20

0.0

25

c −4km < 0, lème ìti èqoume èna asjen¸c fjÐnon

'Otan

30

1.0

0.0

2

1.0

sÔsthma. Sthn perÐptwsh aut  rÐzec 0 25 50 èqoume migadikèc 75 100

q

0.5

Σχήμα 2.3: Ισχυρά για διάφορες αρr = −pφθίνουσα ± p2 −κίνηση ω02 q χικές τιμές. √

0.5

= −p ±

0.0

ω02 − p2

= −p ± iω1 ,

0.0

ìpou

-0.5

−1

ω1 =

p

ω02 − p2 .

H lÔsh mac eÐnai

x(t) = e−pt (A cos ω1 t + B sin ω1 t) ,

-0.5

 

x(t) = Ce−pt cos(ω1 t − γ). -1.0

-1.0 0

5

10

15

20

25

30

Σχήμα 2.4: Ισχυρά φθίνουσα κίνηση και οι περικλείουσες καμπύλες της.

H grafik  parˆstash miac tètoiac lÔshc dÐnetai sto Sq ma 2.4.

Shmei¸ste kai sthn perÐptwsh aut  èqoume ìti

x(t) → 0

ìtan

kampÔlec twn dÔo aut¸n kampÔlwn. stigm .

t → ∞.

To en lìgw sq ma perilambˆnei kai tic perikleÐousec

Ce−pt

kai

−Ce−pt .

H lÔsh talant¸netai metaxÔ

Oi perikleÐousec kampÔlec dÐnoun to mègisto plˆtoc thc talˆntwshc se kˆje qronik 

H metatìpish fˆshc

γ

aplˆ metatopÐzei to grˆfhma eÐte proc ta dexiˆ   proc ta aristerˆ allˆ pˆnta mèsa

sto q¸ro pou kajorÐzoun oi perikleÐousec kampÔlec (oi opoÐec bebaÐwc den metabˆllontai eˆn metablhjeÐ to

γ ).

Shmei¸ste tèloc ìti h gwniak  yeudo-suqnìthta (den kaloÔme apl¸c suqnìthta epeid  h lÔsh den eÐnai sthn ousÐa periodik  sunˆrthsh)

ω1

elatt¸netai ìso o suntelest c

mia kai eˆn suneqÐsoume na metabˆloume to

c

c

(kai sunep¸c kai o

p)

auxˆnei.

Kˆti tètoio eÐnai logikì

se kˆpoio shmeÐo h lÔsh mac ja arqÐsei na moiˆzei san thn lÔsh pou

antistoiqeÐ sthn isqurˆ fjÐnousa   thn krÐsima fjÐnousa perÐptwsh h opoÐa bebaÐwc den talant¸netai kajìlou.

2.4. ΤΑΛΑΝԟΩΣΕΙΣ

51

Apì thn ˆllh meriˆ ìso elatt¸noume to

c

to

ω1

teÐnei sto

ω0

(paramènontac pˆnta mikrìtero) kai h lÔsh ìlo kai

perissìtera moiˆzei me thn sun jh periodik  kÐnhsh qwrÐc apìsbesh. Sthn perÐptwsh aut , ìso to perikleÐousec kampÔlec teÐnoun na ekfulistoÔn se eujeÐec parˆllhlec me ton ˆxona tou

x

p

teÐnei sto 0, oi

.

Leptomèreiec sqetikˆ me thn optik  je¸rhsh thc fusik c sto parapˆnw jèma allˆ kai gia pollˆ apì autˆ pou ja akolouj soun sto parìn kefˆlaio mporeÐ kaneÐc na brei stic shmei¸seic tou maj matoc Fusik c tou PanepisthmÐou

‡

Ajhn¸n .

2.4.4

Ask seic

2.4.2

Jewr ste èna sÔsthma mˆzac elathrÐou me mˆza

c = 1.

a) Diatup¸ste thn diaforik  exÐswsh pou analogeÐ sto parapˆnw sÔsthma kai breÐte thn lÔsh tou. b) EÐnai

m = 2,

stajerˆ elathrÐou

k = 3,

kai stajerˆ apìsbeshc

to en lìgw sÔsthma asjen¸c, krÐsima   oriakˆ fjÐnon? c) Eˆn to sÔsthma eÐnai krÐsima fjÐnon, breÐte mia tim  tou

c

h opoÐa to kˆnei krÐsima fjÐnon.

2.4.3

LÔste thn 'Askhsh 2.4.2 gia

2.4.4

Ac upojèsoume ìti agnooÔme thn apìsbesh kai ìti qrhsimopoioÔme tic diejneÐc monˆdec mètrhshc (mètra-kilˆ-

m = 3, k = 12,

kai

c = 12.

deuterìlepta) 'Estw ìti jèloume na qrhsimopoi soume èna elat rio me stajerˆ 4

N/m

gia na zugÐzoume antikeÐmena.

'Estw epÐshc ìti topojetoÔme èna swmatÐdio sto elat rio kai to af noume na talantwjeÐ. metr¸ntac thn suqnìthta talˆntwshc kai diapist¸soume ìti aut  eÐnai 0.8 to swmatÐdio? b) BreÐte ènan tÔpo gia thn mˆza tou swmatidÐou

2.4.5

m

Hz

a) Eˆn diapist¸soume

(kÔkloi to deuterìlepto) ti mˆza èqei

wc proc thn suqnìthta

ω

se

Hz .

'Estw ìti prosjètoume to endeqìmeno Ôparxhc apìsbeshc sto sÔsthma thc 'Askhshc 2.4.4.

Epiprìsjeta,

upojètoume ìti den gnwrÐzoume thn tim  thc stajerˆc tou elathrÐou, èqoume ìmwc dÔo swmatÐdia anaforˆc bˆrouc 1 kai 2 kil¸n me ta opoÐa ja diametr soume to sÔsthma wc ex c.

TopojetoÔme to kajèna apì autˆ sto sÔsthma kai

Hz , en¸ gia to 2 k kai thc stajerˆc apìsbeshc c. a) Eˆn diapist¸soume metr¸ntac thn suqnìthta talˆntwshc kai diapist¸soume ìti aut  eÐnai 0.2 Hz (kÔkloi to deuterìlepto) ti mˆza èqei to swmatÐdio? b) BreÐte ènan tÔpo gia thn mˆza tou swmatidÐou m wc proc thn suqnìthta ω se Hz . metrˆme thn suqnìthta talˆntwshc. 'Estw ìti gia to swmatÐdio tou 1 kiloÔ h suqnìthta  tan 0.8

kil¸n  tan 0.39

Hz .

a) BreÐte thn tim  thc stajerˆc elathrÐou

‡ http://web.cc.uoa.gr/

~ctrikali/aplets_web/fysiki_i/pendulum/lroom.htm

52

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ

2.5

Mh-omogeneÐc exis¸seic

2.5.1

EpÐlush mh-omogen¸n exis¸sewn

MporoÔme loipìn pwc na lÔsoume (h toulˆqiston na prospaj soume na lÔsoume) grammikèc omogeneÐc exis¸seic me stajeroÔc suntelestèc.

Ac apallagoÔme t¸ra apì ton periorismì tou na eÐnai h exÐswsh omogen c.

shmaÐnei ìti sto sÔsthmˆ to opoÐo prospajoÔme na montelopoi soume dra mia exwterik  dÔnamh.

Kˆti tètoio

Mia tètoia, mh-

omogen c exÐswsh eÐnai h ex c

y 00 + 5y 0 + 6y = 2x + 1.

(2.6)

Shmei¸ste ìti h exÐswsh paramènei me stajeroÔc suntelestèc. Dhlad  oi suntelestèc twn

y 00 , y 0 ,

kai

y

paramènoun

stajerèc. Genikˆ ja grˆfoume thn exÐswsh sthn morf 

L

Ly = 2x + 1

den ephreˆzei ton sullogismì kai tic enèrgeièc mac.

BrÐskoume thn genik  lÔsh

yc

ìtan h sugkekrimènh morf  tou diaforikoÔ telest 

O trìpoc me ton opoÐo ja lÔsoume thn (2.6) eÐnai o ex c.

thc analogoÔshc omogenoÔc exÐswshsanalogoÔsa omogen c exÐswsh

y 00 + 5y 0 + 6y = 0. EpÐshc brÐskoume, me kˆpoion trìpo, mia sugkekrimènh lÔsh

yp

(2.7)

thc (2.6) opìte eÔkola diapist¸noume ìti h

y = yc + yp eÐnai h genik  lÔsh thc exÐswshc (2.6). H Shmei¸ste ìti h

yp

yc

suqnˆ onomˆzetai sumplhrwmatik  lÔsh.

mporeÐ na eÐnai opoiad pote lÔsh.

w = yp − y˜p

thn sunˆrthsh thc diaforˆc touc

'Estw ìti br kate mia diaforetik  lÔsh

kai ac antikatast soume thn

w

y˜p .

Ac orÐsoume

kai sta dÔo mèrh thc exÐswshc gia na

pˆroume

w00 + 5w0 + 6w = (yp00 + 5yp0 + 6yp ) − (˜ yp00 + 5˜ yp0 + 6˜ yp ) = (2x + 1) − (2x + 1) = 0. H parapˆnw diadikasÐa gÐnetai aploÔsterh eˆn qrhsimopoi soume sumbolismì telest¸n. Shmei¸ste ìti

L

eÐnai ènac

grammikìc telest c opìte èqoume

Lw = L(yp − y˜p ) = Lyp − L˜ yp = (2x + 1) − (2x + 1) = 0. 'Ara h

w = yp − y˜p

eÐnai mia lÔsh thc (2.7) kai sunep¸c oi opoiesd pote dÔo lÔseic thc (2.6) diafèroun metaxÔ touc

katˆ mia lÔsh thc omogenoÔc exÐswshc (2.7). Epeid  h

yc

eÐnai h genik  thc omogenoÔc exÐswshc, h lÔsh

y = yc + yp

sumperilambˆnei ìlec tic lÔseic thc (2.6). Ta parapˆnw mac odhgoÔn sto ex c shmantikì sumpèrasma.

Eˆn breÐte mia opoiad pote sugkekrimènh lÔsh me

kˆpoio trìpo kai mia opoiad pote ˆllh diaforetik  lÔsh me opoiond pote trìpo (p.q.

manteyiˆ) tìte mporeÐte na

breÐte thn genikeumènh lÔsh tou exÐswshc. H genik  aut  lÔsh mporeÐ na diatupwjeÐ me entel¸c diaforetikì trìpo eˆn epilèxoume ˆllec sugkekrimènec lÔseic. Se kˆje perÐptwsh ìmwc h genik  lÔsh paristˆ thn Ðdia klˆsh lÔsewn, dhlad  sunart sewn pou ikanopoioÔn thn diaforik  exÐswsh.

Apì thn en lìgw klˆsh ja epilèxoume thn (monadik 

eˆn upˆrqei) katˆllhlh lÔsh tou probl matoc pou ikanopoieÐ kai thn exÐswsh kai tic arqikèc sunj kec.

2.5.2

Mèjodoc aprosdiìristwn suntelest¸n

EÐnai profan¸c epijumhtì na mporoÔme na manteÔoume thn lÔsh thc (2.6). Shmei¸ste ìti to kai to aristerì mèroc thc exÐswshc ja eÐnai epÐshc polu¸numo eˆn epilèxoume thn

y

2x+1 eÐnai èna polu¸numo,

na eÐnai èna polu¸numo tou Ðdiou

bajmoÔ. Ac prospaj soume loipìn jètontac

y = Ax + B. Antikajist¸ntac èqoume

y 00 + 5y 0 + 6y = (Ax + B)00 + 5(Ax + B)0 + 6(Ax + B) = 0 + 5A + 6Ax + 6B = 6Ax + (5A + 6B). 'Ara

6Ax + (5A + 6B) = 2x + 1.

Opìte

A = 1/3

kai

B=

−1/9. Autì shmaÐnei ìti

sumplhrwmatikì prìblhma èqoume. ('Askhsh!)

yc = C1 e−2x + C2 e−3x .

yp =

1 3

x−

1 9

=

3x−1 . LÔnontac to 9

2.5. ΜΗ-ΟΜΟΓΕΝşΙΣ ΕΞΙӟΩΣΕΙΣ

53

'Ara h genik  lÔsh thc (2.6) eÐnai

3x − 1 . 9 sunj kec y(0) = 0

y = C1 e−2x + C2 e−3x + Ac upojèsoume ìti epiprìsjeta mac dÐnontai oi arqikèc

y 0 = −2C1 e−2x − 3C2 e−3x + 1/3.

0 = y(0) = C1 + C2 − LÔnontac èqoume

C1 = 1/3

kai

C2 = −2/9.

y(x) = 2.5.1

Elègxte ìti h

y

kai

y 0 (0) = 1/3.

Pr¸ta èqoume ìti

Tìte

1 9

1 1 = y 0 (0) = −2C1 − 3C2 + . 3 3

'Ara h lÔsh eÐnai

1 −2x 2 −3x 3x − 1 3e−2x − 2e−3x + 3x − 1 e − e + = . 3 9 9 9

apoteleÐ lÔsh thc exÐswshc.

ShmeÐwsh: 'Ena sÔnhjec lˆjoc eÐnai na upologÐsoume tic stajerèc qrhsimopoi¸ntac tic arqikèc sunj kec kai thn

yc

yp .

kai katìpin na prosjèsoume thn sugkekrimènh lÔsh

y = yc + yp

pr¸ta na upologÐsoume thn

Autì den ja mac odhg sei sthn lÔsh pou zhtˆme. Prèpei

kai katìpin na lÔsoume wc proc tic stajerèc qrhsimopoi¸ntac tic arqikèc

sunj kec. Me parìmoio trìpo mporoÔme na antimetwpÐsoume peript¸seic pou sto dexiì mèloc tic exÐswshc èqoume ekjetik  sunˆrthsh   hmÐtona kai sunhmÐtona. Gia parˆdeigma

y 00 + 2y 0 + 2y = cos 2x Ac upologÐsoume thn

yp •qrhsimopoi¸ntac

thn manteyiˆ

y = A cos 2x + B sin 2x. Antikajist¸ntac thn

y

sthn exÐswsh èqoume

−4A cos 2x − 4B sin 2x − 4A sin 2x + 4B cos 2x + 2A cos 2x + 2B sin 2x = cos 2x. −4A + 4B + 2A = 1 B = 1/5. 'Ara

Gia na eÐnai to dexiì meroc Ðso me to aristerì prèpei na isqÔei

−2A + 4B = 1

kai

2A + B = 0

kai sunep¸c

A = −1/10

kai

yp = A cos 2x + B sin 2x =

dhlad 

−4B − 4A + 2B = 0.

'Ara

− cos 2x + 2 sin 2x . 10

Me parìmoio trìpo mporoÔme na antimetwpÐsoume thn perÐptwsh pou to dexiì mèroc perièqei ekjetikèc sunart seic, opìte kai h mantexiˆ mac ja eÐnai epÐshc ekjetik  sunˆrthsh. Gia parˆdeigma, an

Ly = e3x , L eÐnai ènac opoiosd pote diaforikìc telest c y = Ae3x . EÐnai shmantikì na suneidhtopoi soume ìti (ìpou

me stajeroÔc suntelestèc) tìte h manteyiˆ mac ja eÐnai h basizìmenoi ston kanìna tou ginomènou gia thn parag¸gish

mporoÔme na sunduˆsoume katˆllhla manteyièc. Gia parˆdeigma, gia thn exÐswsh

Ly = (1 + 3x2 ) e−x cos πx, ac jewr soume thn manteyiˆ

y = (A + Bx + Cx2 ) e−x cos πx + (D + Ex + F x2 ) e−x sin πx thn opoÐa ja antikatast soume sthn exÐswsh elpÐzontac na pˆroume to sÔsthma twn algebrik¸n exis¸sewn pou eÐnai aparaÐthto gia na upologÐsoume tic timèc twn

A, B, C, D, E, F .

There ic one hiccup in all thic. It could be that our guesc actually solvec the associated homogeneouc equation. That ic, suppose we have Ac xedialÔnoume ton basikì mhqanismì thc mejìdou aut c jewr¸ntac thn exÐswsh

y 00 − 9y = e3x .

54

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ

Mia profan c manteyiˆ eÐnai h

y = Ae3x ,

thn opoÐa ìmwc an antikatast soume sto aristerì mèroc ja pˆroume

y 00 − 9y = 9Ae3x − 9Ae3x = 0 6= e3x . A ètsi ¸ste to dexiì mèroc na eÐnai Ðso me e3x .

Den upˆrqei trìpoc epilog c tou

Gia na apofÔgoume thn pollaplìthta

thc sumplhrwmatik c lÔshc (profan¸c se aut  ofeÐletai to prìblhma), mporoÔme na thn pollaplasiˆsoume me UpologÐzoume dhlad  pr¸ta thn

yc

(thn lÔsh thc

x.

Ly = 0)

yc = C1 e−3x + C2 e3x kai parathroÔme ìti upˆrqei se aut n o ìroc

e3x

o opoÐoc sumpÐptei me thn epijumht  manteyiˆ mac.

y = Axe3x . y 00 = 4Ae3x + 9Axe3x . 'Ara

apofÔgoume kˆti tètoio an metatrèyoume thn manteyiˆ mac se sun jwc paÐrnontac

y 0 = Ae3x + 3Axe3x

kai

MporoÔme na

T¸ra mporoÔme na proqwr soume wc

y 00 − 9y = 4Ae3x + 9Axe3x − 9Axe3x = 4Ae3x . Gia na eÐnai to dexiì mèloc Ðso me

e3x

èqoume ìti

4A = 1

kai ˆra

A = 1/4.

MporoÔme loipìn na grˆyoume thn genik 

lÔsh wc ex c

y = yc + yp = C1 e−3x + C2 e3x + Upˆrqei perÐptwsh o pollaplasiasmìc thc manteyiˆc me

x

1 3x xe . 4

na mhn mac epitrèyei na apofÔgoume sÔmptwsh lÔsewn.

Gia parˆdeigma,

y 00 − 6y 0 + 9 = e3x . yc = C1 e3x + C2 xe3x .

Shmei¸ste ìti

Sunep¸c h manteyiˆ

perÐptwsh ac exetˆsoume thn ex c epilog  me

x

2 3x

y = Ax e

.

y = Axe3x

den ja mac odhg sei poujenˆ.

Se aut  thn

En gènei mporoÔme na pollaplasiˆsoume thn manteyiˆ mac

ìsec forèc qreiˆzetai gia na apofÔgoume thn sÔmptwsh lÔsewn.

until all duplication ic gone.

'Oqi ìmwc

perissìterec forèc! Eˆn pollaplasiˆsoume perissìterec forèc apo ìsec prèpei ja odhghjoÔme kai pˆli se adièxodo. Tèloc ac exetˆsoume thn perÐptwsh pou to dexiì mèroc eÐnai ˆjroisma kˆpoiwn ìrwn, ìpwc gia parˆdeigma to

Ly = e2x + cos x. Sthn en lìgw perÐptwsh brÐskoume mia sunˆrthsh

v

u

h opoÐa apoteleÐ lÔsh thc exÐswshc

Lu = e2x

kai mia sunˆrthsh

Lv = cos x (antimetwpÐzoume kˆje ìro xeqwristˆ). Katìpin shmei¸ste ìti an 2x jèsoume y = u+v , èqoume Ly = e +cos x. Autì ofeÐletai sto gegonìc ìti o telest c L eÐnai grammikìc kai mporoÔme 2x na qrhsimopoi soume upèrjesh twn lÔsewn. Me ton trìpo autì èqoume ìti Ly = L(u + v) = Lu + Lv = e + cos x. h opoÐa apoteleÐ lÔsh thc exÐswshc

Analutikèc epiprìsjetec analutikèc teqnikèc leptomèreiec thc parapˆnw mejìdou mporeÐte na breÐte sto biblÐo twn

Edwards

2.5.3

kai

P enney

[EP] allˆ kai se kˆje ˆllo biblÐo diaforik¸n exis¸sewn.

Mèjodoc diakÔmanshc twn paramètrwn

H mèjodoc twn aprosdiìristwn suntelest¸n parìlo pou mac bohjˆ na lÔsoume pollˆ basikˆ probl mata diaforik¸n exis¸sewn dustuq¸c den mporeÐ na antimetwpÐsei pollˆ ˆlla. sthn perÐptwsh pou to dexiì mèroc thc exÐswshc

Ly = f (x)

Sthn pragmatikìthta eÐnai apotelesmatik  mìnon

èqei mìnon peperasmèno arijmì grammikˆ anexˆrthtwn

parag¸gwn, opìte kai eÐnai dunatìn na d¸sei kˆpoioc mia manteyiˆ pou tic perilambˆnei ìlec touc. Kˆti tètoio eÐnai idiaÐtera dÔskolo kˆpoiec forèc. Gia parˆdeigma, ac jewr soume thn exÐswsh

y 00 + y = tan x. Shmei¸ste ìti oi parˆgwgoi thc

tan x

deÐqnoun teleÐwc diaforetikèc metaxÔ touc kai eÐnai adÔnaton na grafoÔn san

grammikìc sunduasmìc parag¸gwn kat¸terhc tˆxhc. Prˆgmati èqoume

sec2 x, 2 sec2 x tan x,

klp . . . .

Gia thn epÐlush thc exÐswshc aut c qreiˆzetai mia nèa mèjodoc. Ac anaptÔxoume loipìn thn mèjodo diakÔmanshc twn paramètrwndiakÔmansh twn paramètrwn, h opoÐamporeÐ na mac bohj sei na antimetwpÐsoume opoiad pote exÐswsh thc morf c

Ly = f (x),

arkeÐ bebaÐwc na eÐnai efiktì na upologÐsoume ta emplekìmena oloklhr¸mata.

Gia na

aplousteÔsoume thn suz thsh, kai fusikˆ qwrÐc periorismì thc genikìthtac, ac perioristoÔme se exis¸seic deÔterhc

2.5. ΜΗ-ΟΜΟΓΕΝşΙΣ ΕΞΙӟΩΣΕΙΣ

55

tˆxhc, èqontac bebaÐwc upìyin ìti h en lìgw mèjodoc mporeÐ me thn Ðdia eukolÐa na antimetwpÐsei exis¸seic uyhlìterhc tˆxhc (endeqomènwc mèsw eÔkolwn allˆ baret¸n upologism¸n). Ac prospaj soume na lÔsoume thn ex c exÐswsh.

Ly = y 00 + y = tan x. Pr¸ta brÐskoume thn sumplhrwmatik  lÔsh thc

y1 = cos x

kai

y2 = sin x.

Ly = 0

brÐskontac sqetikˆ eÔkola ìti

yc = C1 y1 + C2 y2

ìpou

Gia na prospaj soume t¸ra na broÔme mia lÔsh thc mh-omogenoÔc exÐswshc ac jewr soume

thn ex c sunˆrthsh

yp = y = u1 y1 + u2 y2 , ìpou oi

u1

kai

u2

eÐnai sunart seic kai ìqi stajerèc. An prospaj soume na ikanopoi soume thn exÐswsh

ja katal xoume se mia sunj kh pou sundèei tic

u1

kai

u2 .

Ly = tan x

Pr¸ta bebaÐwc prèpei na upologÐsoume (prosoq , me ton

kanìna tou ginomènou!)

y 0 = (u01 y1 + u02 y2 ) + (u1 y10 + u2 y20 ). Mia kai lìgw thc poluplokìthtac thc eÐnai sugkritikˆ dÔskolo na diaqeiristoÔme thn parapˆnw sqèsh, ac thn aplopoi soume jètontac thn ex c perioristik  sqèsh

(u01 y1 +u02 y2 ) = 0.

Kˆti tètoio ìqi mìnon eÐnai epitreptì allˆ kai

ousiastikˆ anagkaÐo mia kai prèpei na broÔme dÔo ˆgnwstec sunart seic gegonìc pou apaiteÐ kai mia deÔterh exÐswsh pèra apì thn arqik  diaforik  exÐswsh. 'Etsi kai o upologismìc thc deÔterhc parag¸gou eÐnai polÔ eukolìteroc.

y 0 = u1 y10 + u2 y20 , y 00 = (u01 y10 + u02 y20 ) + (u1 y100 + u2 y200 ). T¸ra epeid  mac  tan thc

y2 eÐnai lÔseic thc y 00 + y = 0, gnwrÐzoume ìti y100 = −y1 0 00 0 00 morf c y + ay + by = 0 ja eÐqame yi = −ayi − byi .)

y1

kai

kai

y200 = −y2 .

(ShmeÐwsh: An h exÐswsh

'Ara

y 00 = (u01 y10 + u02 y20 ) − (u1 y1 + u2 y2 ). Shmei¸ste ìti

y 00 = (u01 y10 + u02 y20 ) − y, kai sunep¸c

y 00 + y = Ly = u01 y10 + u02 y20 . Gia na ikanopoieÐ h

y

thn

Ly = f (x)

f (x) = u01 y10 + u02 y20 . pou jèsame gia tic u1

prèpei na èqoume

Prèpei na lÔsoume tic dÔo exis¸seic (sunj kec)

kai

u2

u01 y1 + u02 y2 = 0, u01 y10 + u02 y20 = f (x). MporoÔme t¸ra na lÔsoume wc proc tic

u01

kai

u02

opoiasd pote sugkekrimènh exÐswsh thc morf c

sunart sei twn

Ly = f (x).

f (x), y1

kai

y2 .

Oi parapˆnw exis¸seic eÐnai Ðdiec gia

Upˆrqei sunep¸c genikìc tÔpoc upologismoÔ thc lÔshc

ston opoÐo arkeÐ na kˆnei kˆpoioc tic aparaÐthtec antikatastˆseic, eÐnai ìmwc kallÐtero na epanalˆbei thn parakˆtw diadikasÐa. Sthn perÐptws  mac oi dÔi exis¸seic paÐrnoun thn morf 

u01 cos x + u02 sin x = 0, −u01 sin x + u02 cos x = tan x. 'Ara

u01 cos x sin x + u02 sin2 x = 0, −u01 sin x cos x + u02 cos2 x = tan x cos x = sin x. kai sunep¸c

u02 (sin2 x + cos2 x) = sin x, u02 = sin x, u01 =

− sin2 x x = − tan x sin x. cos

56

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ u02 gia na upologÐsoume tic u1 kai u2 . Z Z 1 (sin x) − 1 0 u1 = u1 dx = − tan x sin x dx = ln + sin x, 2 (sin x) + 1 Z Z u2 = u02 dx = sin x dx = − cos x.

T¸ra prèpei na oloklhr¸soume tic

u01

kai

'Ara h sugkekrimènh lÔsh mac eÐnai

(sin x) − 1 1 + cos x sin x − cos x sin x = yp = u1 y1 + u2 y2 = cos x ln 2 (sin x) + 1 = H genik  lÔsh thc

y 00 + y = tan x

eÐnai loipìn

y = C1 cos x + C2 sin x +

2.5.4

(sin x) − 1 1 . cos x ln 2 (sin x) + 1

(sin x) − 1 1 . cos x ln 2 (sin x) + 1

Ask seic

2.5.2

BreÐte mia sugkekrimènh lÔsh thc exÐswshc

y 00 − y 0 − 6y = e2x .

2.5.3

BreÐte mia sugkekrimènh lÔsh thc exÐswshc

y 00 − 4y 0 + 4y = e2x .

2.5.4

LÔste to ex c prìblhma

2.5.5

Diatup¸ste thn genik  morf  miac sugkekrimènhc lÔshc thc

y 00 + 9y = cos 3x + sin 3x

gia

y(0) = 2, y 0 (0) = 1. y (4) − 2y 000 + y 00 = ex

qwrÐc na lÔsete wc proc

touc suntelestèc.

2.5.6

Diatup¸ste thn genik  morf  kai breÐte mia sugkekrimènh lÔsh thc exÐswshc

y (4) − 2y 000 + y 00 = ex + x + sin x

qwrÐc na lÔsete wc proc touc suntelestèc.

2.5.7

a) Qrhsimopoi ste thn mèjodo thc diakÔmanshc twn paramètrwn gia na breÐte mia sugkekrimènh lÔsh thc

exÐswshc

y 00 − 2y 0 + y = ex .

b) BreÐte mia sugkekrimènh lÔsh thc parapˆnw exÐswshc qrhsimopoi¸ntac thn mèjodo

twn aprosdiìristwn suntelest¸n. c) EÐnai oi dÔo lÔseic pou br kate parapˆnw Ðdiec? Ti sumbaÐnei?

2.5.8

BreÐte mia sugkekrimènh lÔsh thc exÐswshc

plekìmena orismèna oloklhr¸mata.

y 00 − 2y 0 + y = sin x2 .

Den qreiˆzetai na upologÐzetai ta em-

2.6. ΕΞΑΝΑΓΚΑΣ̟ΕΝΕΣ ΤΑΛΑΝԟΩΣΕΙΣ ΚΑΙ ΣΥΝΤΟΝΙΣ̟ΟΣ 2.6

57

Exanagkasmènec talant¸seic kai suntonismìc

Ac epistrèyoume t¸ra sto sÔsthma mˆzac elathrÐou kai ac exetˆsoume

k

thn perÐptwsh twn exanagkasmènwn talant¸sewn. Ac exetˆsoume dhlad  thn

F (t) m

exÐswsh

mx00 + cx0 + kx = F (t) gia kˆpoia mh-mhdenik  exÐswsh elathrÐou,

m

F (t).

Ac jumhjoÔme ìti sto sÔsthma mˆzac

eÐnai h mˆza tou swmatidÐou,

c

eÐnai h apìsbesh,

k

apìsbeshFc(t) eÐnai kˆpoia

eÐnai h stajerˆ tou elathrÐou kai

exwterik  dÔnamh pou dra pˆnw sto swmatÐdio. Sun jwc h exwterik  dÔnamh eÐnai periodik , ìpwc pq swmatÐdia peristrefìmena ektìc kèntrou bˆrouc,   periodikoÔc  qouc klp. 'Otan ja anaptÔxoume teqnikèc seir¸n

F ourier

ja doÔme ìti sthn ousÐa mporoÔme na antimetw-

pÐsoume kˆje periodik  sunˆrthsh, opoioud pote tÔpou, jewr¸ntac sunart seic thc morf c

F (t) = F0 cos ω t

( 

sin),

me ousiastikˆ akrib¸c ton Ðdio trìpo.

2.6.1

Exanagkasmènh kÐnhsh qwrÐc apìsbesh kai suntonismìc

Qˆrin aplìthtac ac exetˆsoume pr¸ta thn perÐptwsh qwrÐc apìsbesh (

c = 0).

'Eqoume loipìn thn exÐswsh

mx00 + kx = F0 cos ω t. h opoÐa èqei thn ex c sumplhrwmatik  lÔsh (lÔsh thc antÐstoiqhc omogenoÔc exÐswshc)

xc = C1 cos ω0 t + C2 sin ω0 t, ìpou

p

ω0 =

k/m. H

ω0

lègetai (gwniak ) fusik  suqnìthta.

EÐnai h suqnìthta sthn opoÐa to sÔsthma jèlei na

talantwjeÐ qwrÐc thn ep reia exwterik¸n paragìntwn. Ac upojèsoume ìti

ω0 6= ω ,

ac dokimˆsoume thn manteyiˆ

xp = A cos ω t

kai ac lÔsoume wc proc

A.

Shmei¸ste ìti

den eÐnai aparaÐthto na èqoume ìro hmitìnou sthn manteyiˆ mac mia kai sto aristerì mèroc ja èqoume oÔtwc   ˆllwc sunhmÐtona.

Fusikˆ mporeÐ kˆpoioc na sumperilˆbei ìrouc hmitìnou sthn manteyiˆ, oi suntelestèc ìmwc twn ìrwn

aut¸n ja apodeiqjeÐ ìti eÐnai mhdèn. LÔnontac parìmoia (kˆnte to san ˆskhsh) me thn mèjodo twn aprosdiìristwn suntelest¸n kai qrhsimopoi¸ntac thn parapˆnw manteyiˆ brÐskoume ìti

xp =

F0 cos ω t. m(ω02 − ω 2 )

H genik  lÔsh eÐnai

x = C1 cos ω0 t + C2 sin ω0 t +

F0 cos ω t. m(ω02 − ω 2 )

h opoÐa mporeÐ na dojeÐ sthn ex c morf 

x = C cos(ω0 t − γ) +

F0 cos ω t. m(ω02 − ω 2 )

EÐnai dhlad  upèrjesh dÔo sunhmitonoeid¸n kumˆtwn diaforetik¸n suqnot twn.

Parˆdeigma 2.6.1:

kai ìti

x(0) = 0

kai

'Estw ìti

0.5x00 + 8x = 10 cos πt

0

x (0) = 0.

Ac anagnwrÐsoume tic paramètrouc pr¸ta:

ω = π , ω0 =

p

8/0.5

x = C1 cos 4t + C2 sin 4t + Ac lÔsoume t¸ra wc proc

C2 = 0

kai

C1 =

C1

kai

C2

= 4, F0 = 10, m = 1.

'Ara h genik  lÔsh eÐnai

20 cos πt. 16 − π 2

qrhsimopoi¸ntac tic arqikèc sunj kec. EÐnai eÔkolo na diapist¸soume ìti

−20 . 'Ara 16−π 2

x=

20 (cos πt − cos 4t). 16 − π 2

58

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ •

Parathr ste thn sumperiforˆ thc lÔshc sto Sq ma 2.5 wc proc to mègisto plˆtoc thc.

Prˆgmati eˆn qrhsi-

mopoi soume thn trigwnometrik  tautìthta

 2 sin

A−B 2



 sin

A+B 2

 = cos B − cos A

ja doÔme ìti

x= Shmei¸ste ìti to

x

20 16 − π 2

     4+π 4−π t sin t . 2 sin 2 2

eÐnai èna uyhl c suqnìthtac kÔma to eÔroc tou opoÐou diamorf¸netai apì èna kÔma qamhl c

suqnìthtac.

ω0 = ω . Profan¸c, den eÐnai dunatìn h A cos ω t se sunduasmì me thn mèjodo twn apros-

'Estw t¸ra manteyiˆ

diìristwn suntelest¸n na mac odhg sei sthn lÔsh. n prokeÐmenh perÐptwsh blèpoume ìti h sunˆrthsh

0

5

10

15

20

10

10

5

5

0

0

-5

-5

Sth-

cos ω t

apoteleÐ lÔsh thc antÐstoiqhc omogenoÔc exÐswshc.

'Ara,

xp = At cos ω t+ Bt sin ω t mia kai deÔterh parˆgwgoc thc t cos ω t emperièqei qreiˆzetai na dokimˆsoume thn sunˆrthsh

hmÐtona. Grˆfoume loipìn thn exÐswsh

F0 x +ω x= cos ω t. m 00

2

Antikajist¸ntac sto aristerì mèroc èqoume

2Bω cos ω t − 2Aω sin ω t =

F0 cos ω t. m -10

'Ara

F0 2mω

F0 A = 0 kai B = 2mω . H sugkekrimènh t sin ω t kai h genik  lÔsh eÐnai

x = C1 cos ω t + C2 sin ω t +

-10 0

5

10

15

20

lÔsh eÐnai

Σχήμα 2.5: Η γραφική παράσταση της cos 4t).

F0 t sin ω t. 2mω

20 16−π 2 (cos πt−

O shmantikìc ìroc eÐnai o teleutaÐoc (h sugkekrimènh 0

5

10

15

lÔsh pou br kame).

20

5.0

5.0

EÔkola blèpoume ìti o ìroc autìc

auxˆnetai qwrÐc ìrio ìtan

t → ∞.

talant¸netai metaxÔ tou

−F0 t F0 t kai tou . 2mω 2mωp

Sthn pragmatikìthta,

dÔo talant¸netai mìnon metaxÔ twn tim¸n 2.5

2.5

opoÐec gÐnontai, ìso auxˆnei to

t,

±

Oi pr¸toi

C12 + C22 ,

oi

ìlo kai mikrìterec se

sÔgkrish me tic me tic antÐstoiqec timèc talˆntwshc tou teleutaÐou ìrou. 0.0

0.0

Sto Sq ma 2.6 blèpoume thn grafik 

parˆstash thc lÔshc ìtan

C1 = C2 = 0 , F0 = 2 , m = 1 ,

ω = π. -2.5

-2.5

O exanagkasmìc enìc sust matoc se talˆntwsh se kˆpoia sugkekrimènh suqnìthta mac odhgeÐ se polÔ parˆxenec talant¸seic. AutoÔ tou eÐdouc h sumperiforˆ onomˆze-

-5.0

-5.0

taisuntonismìc   fusikìc suntonismìc.

Kˆpoiec forèc o

suntonismìc bèbaia endeqomènwc na eÐnai epijumhtìc. 0

5

10

15

Σχήμα 2.6: Η γραφική παράσταση της

20

1 π t sin πt.

Gi-

a parˆdeigma, JumhjeÐte ìti ta paidiˆ xekinˆn thn koÔnia thc paidik c qarˆc me talant¸seic mikroÔc plˆtouc gia na katal xoun, auxˆnontˆc to, na suntonistoÔn se talant¸seic me to epijumhtì plˆtoc kai suqnìthta.

'Allec forèc o suntonismìc mporeÐ na eÐnai katastrofikìc. Metˆ apì èna seismì kˆpoia ktÐria paramènoun sqedìn anèpafa en¸ ˆlla katarrèoun. Autì ofeÐletai sto gegonìc ìti diaforetikˆ ktÐria èqoun kai diaforetikèc suqnìthtec

2.6. ΕΞΑΝΑΓΚΑΣ̟ΕΝΕΣ ΤΑΛΑΝԟΩΣΕΙΣ ΚΑΙ ΣΥΝΤΟΝΙΣ̟ΟΣ

59

suntonismoÔ (jumhjeÐte bebaÐwc kai ton anˆlogo jrÔlo twn teiq¸n thc IeriqoÔ). Sunep¸c h melèth thc suqnìthtac suntonismoÔc enìc sust matoc mporeÐ na eÐnai zwtik c shmasÐac. Suqnˆ san parˆdeigma katastrofikoÔ suntonismoÔ suqnìthtac anafèretai kai h katˆrreush thc gèfurac

T acomaN arrows

stic arqèc tou 20ou ai¸na stic H.P.A.. Kˆti tètoio eÐnai ìmwc lˆjoc mia kai kˆpoio ˆllo fainìmeno eÐnai to kurÐwc

§

upeÔjuno gia thn en lìgw katastrof  .

2.6.2

Exanagkasmènh talˆntwsh me apìsbesh kai suntonismìc sthn prˆxh

Profan¸c sthn pragmatikìthta ta prˆgmata den eÐnai tìso aplˆ ìso ta perigrˆyame parapˆnw.

Gia parˆdeigma,

upˆrqei apìsbesh opìte kai h exÐsws  mac paÐrnei thn ex c morf 

mx00 + cx0 + kx = F0 cos ω t, gia kˆpoio

c > 0.

(2.8)

'Eqoume  dh lÔsei to antÐstoiqo omogenèc prìblhma. Opìte jètoume

c p= 2m

r ω0 =

k . m

AntikajistoÔme thn exÐswsh (2.8) me thn

x00 + 2px0 + ω02 x =

F0 cos ω t. m

EÔkola brÐskoume ìti oi rÐzec tou qarakthristikoÔ poluwnÔmou tou antÐstoiqou omogenoÔc probl matoc eÐnai oi ex c

p r1 , r2 = −p ± p2 − ω02 . H morf  thc genik c lÔshc tou omogenoÔc probl matoc exartˆtai, 2 2 2 prìshmo tou p − ω0 , h isodÔnama apì to prìshmo tou c − 4km. Dhlad   r1 t r2 t 2  if c > 4km, C1 e + C2 e −pt −pt 2 xc = C1 e + C2 te if c = 4km,   −pt e (C1 cos ω1 t + C2 sin ω1 t) if c2 < 4km , ìpou

ω1 =

p

ω02 − p2 .

Se kˆje perÐptwsh parathroÔme ìti

xc (t) → 0

ìtan

t → ∞.

ìpwc  dh eÐdame, apì to

Epiprìsjeta, tÐpote den

mac empodÐzei na prospaj soume na oloklhr¸soume thn epÐlush tou probl matoc qrhsimopoi¸ntac thn mèjodo twn aprosdiìristwn suntelest¸n se sunduasmì me thn manteyiˆ

xp = A cos ω t+B sin ω t.

Shmei¸ste ed¸ ìti ta parapˆnw

anaferjènta katastrofikˆ senˆria den eÐnai pijanìn na sumboÔn sthn pragmatikìthta ìpou upˆrqei kˆpoia, èstw kai polÔ mikr , apìsbesh pou apotrèpei ton suntonismì. Ant' autoÔ èqoume fainìmena suntonismìc kˆpoiou ˆllou tÔpou me parapl sia ennoiologÐa. Ac antikatast soume loipìn kai ac lÔsoume wc proc

A

kai

B.

Metˆ apì pollèc kai aniarèc prˆxeic èqoume



F0 cos ω t. (ω02 − ω 2 )B − 2ωpA sin ω t + (ω02 − ω 2 )A + 2ωpB cos ω t = m Telikˆ paÐrnoume

(ω02 − ω 2 )F0 m(2ωp)2 + m(ω02 − ω 2 )2 2ωpF0 B= . m(2ωp)2 + m(ω02 − ω 2 )2 A=

EÔkola brÐskoume ìti mia kai

C=

√

A2 + B 2

èqoume

C=

F0 p . m (2ωp)2 + (ω02 − ω 2 )2

§ K. Billah kai R. Scanlan, Resonance, Tacoma Narrows Bridge Failure, kai Undergraduate Physics Textbooks, American Journal of Physics, 59(2), 1991, 118–124, http://www.ketchum.org/billah/Billah-Scanlan.pdf

60

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ

Sunep¸c h sugkekrimènh mac lÔsh eÐnai

xp =

(ω02 − ω 2 )F0 2ωpF0 cos ω t + sin ω t m(2ωp)2 + m(ω02 − ω 2 )2 m(2ωp)2 + m(ω02 − ω 2 )2

Enallaktikˆ, qrhsimopoi¸ntac ton deÔtero sumbolismì èqoume plˆtoc talˆntwshc (ìtan

C

kai metakÐnhshc fˆshc

γ

ìpou

ω 6= ω0 ) tan γ =

B 2ωp = 2 . A ω0 − ω 2

'Ara èqoume

xp = 'Otan

ω = ω0

blèpoume ìti

F0 p cos(ω t − γ). 2 m (2ωp) + (ω02 − ω 2 )2

A = 0, B = C =

F0 kai 2mωp

γ = π/2.

O akrib c tÔpoc thc lÔshc den èqei ìsh spoudaiìthta èqei h basik  idèa apì thn opoÐa proèkuye. Den qreiˆzetai loipìn na apomnhmoneÔsete ton sugkekrimèno tÔpo en¸ eÐnai aparaÐthto na enjumoÔmaste thn emplekìmenec idèec. Shmei¸ste ìti akìma kai sthn perÐptwsh pou allˆxei èstw kai lÐgo to dexiì mèroc tìso h morf  ìso kai h sumperiforˆ

•

thc lÔshc mporeÐ na allˆxei drastikˆ. Sunep¸c den upˆrqei ousiastikìc lìgoc apomnhmìneushc tou sugkekrimènou autoÔ tÔpou kai eÐnai saf¸c protimìtero na ton anaparˆgoume kˆje forˆ pou ja ton qreiastoÔme. Gia lìgouc pou ja exhg soume se lÐgo, ac onomˆsoume thn lÔsh

xtr .

Ac onomˆsoume epÐshc thn

xp

xc

metabatik  lÔsh kai ac thn sumbolÐsoume me

pou br kame parapˆnw eustaj  periodik  lÔsh kai ac thn sumbolÐsoume me

xsp .

H

genik  lÔsh tou probl matoc mac eÐnai

x = xc + xp = xtr + xsp . Shmei¸ste ìti h 0

5

10

15

t→∞

20

5.0

5.0

xc = xtr

teÐnei sto mhdèn ìtan to

san ìroc pou perilambˆnei thn ekjetik  sunˆrthsh

me arnhtikì ìrisma.

'Ara gia megˆla

t,

o ìroc

amelhtèoc kai ousiastikˆ epikrateÐ mìno h

xsp

xtr

eÐnai

thn ep reia

thc opoÐac eÐnai kai to mìno pou aisjanìmaste. Shmei¸ste 2.5

2.5

epÐshc ìti h

xsp

den emplèkei tuqaÐec stajerèc kai oi ar-

qikèc sunj kec ephreˆzoun mìnon thn

xtr .

Autì shmaÐnei

ìti h epÐdrash twn arqik¸n sunjhk¸n ja eÐnai amelhtèa 0.0

0.0

sthn arq  (gia mikrì

t).

Autìc eÐnai o lìgoc pou onomˆsame

parapˆnw thn en lìgw lÔsh metabatik .

Ex aitÐac thc

sumperiforˆc aut c, mporoÔme na epikentrwjoÔme sthn eu-2.5

-2.5

staj  periodik  lÔsh kai na agno soume thn metabatik . 'Eqontac katˆ nou ta parapˆnw, parathr ste sto Sq -

-5.0

-5.0

ma 2.7 tic grafikèc parastˆseic twn lÔsewn gia diaforetikèc arqikèc sunj kec.

0

5

10

15

xtr teÐnei c). 'Oso to c), tìso

Parathr ste ìti h taqÔthta me thn opoÐa h

20

sto mhdèn exartˆtai apì to

Σχήμα 2.7: Γραφικές παραστάσεις των λύσεων με διαφορετικές αρχικές συνθήκες και τις εξής παραμέτρους k = 1, m = 1, F0 = 1, c = 0.7, και ω = 1.1.

p

p (  ìso pio megˆlo eÐnai xtr gÐnetai amelhtèa. 'Ara ìso

pio megˆlo eÐnai to poio gr gora h

(kai ˆra apì to

pio mikr 

eÐnai h apìsbesh, tìso megalÔterh eÐnai h the metabatik  perÐodos. To sumpèrasma autì eÐnai sÔmfwno me thn parat rhsh ìti ìtan h

c = 0,

oi arqikèc sunj kec ephreˆzoun thn

sumperiforˆ thc lÔshc pˆnta (dhlad  h metabatik  perÐodos eÐnai gia pˆnta). Ac perigrˆyoume t¸ra ti shmaÐnei suntonismìc ìtan upˆrqei apìsbesh. Mia kai den upˆrqoun antifˆseic katˆ thn epÐlush me thn mèjodo twn aprosdiìristwn suntelest¸n, den upˆrqei ìroc o opoÐoc na teÐnei sto ˆpeiro. Ac doÔme ìmwc poia eÐnai h mègisth tim  tou plˆtouc thc eustajoÔc periodik c lÔshc. thc

xsp .

Eˆn kˆnoume thn grafik  parˆstash tou

stajerèc) mporoÔme na broÔme to mègistì tou. suqnìthta suntonismoÔ.

To mègisto plˆtoc

C

san sunˆrthsh tou

H tim  tou

C(ω)

ω

ω

'Estw

C

to plˆtoc thc talˆntwshc

(jewr¸ntac ìlec tic ˆllec paramètrouc

pou antistoiqeÐ sto mègisto autì onomˆzetai fusik 

eÐnai gnwstì me to ìnoma fusikì plˆtoc suntonismoÔ.

Sunep¸c

ìtan upˆrqei apìsbesh milˆme gia fusikìc suntonismìc antÐ gia aplì suntonismì. Endeiktikèc grafikèc parastˆseic

2.6. ΕΞΑΝΑΓΚΑΣ̟ΕΝΕΣ ΤΑΛΑΝԟΩΣΕΙΣ ΚΑΙ ΣΥΝΤΟΝΙΣ̟ΟΣ c

gia treic diaforetikèc timèc tou

61

dÐdontai sto Sq ma 2.8. 'Opwc eÐnai fanerì to fusikì plˆtoc suntonismoÔ auxˆnei

ìtan h apìsbesh elatt¸netai kai o fusikìc suntonismìc exafanÐzetai ìtan h apìsbesh eÐnai megˆlh.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

2.5

2.5

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Σχήμα 2.8: Η γραφική παράσταση του C(ω) που αναδεικνύει συντονισμό στην πράξη με παραμέτρους k = 1, m = 1, F0 = 1. Η επάνω καμπύλη είναι για c = 0.4, η μεσαία για c = 0.8, και η κάτω για c = 1.6.

Gia na broÔme to mègisto prèpei pr¸ta na upologÐsoume thn parˆgwgo

C 0 (ω) =

EÐnai bebaÐwc mhdèn ìtan

ω=0

h ìtan

Den eÐnai dÔskolo na apodeÐxoume ìti h posìthta

timèc tou

ω ).

An to

ω = 0

h opoÐa eÔkola brÐskoume ìti eÐnai

−4ω(2p2 + ω 2 − ω02 )F0
3/2 . m (2ωp)2 + (ω02 − ω 2 )2

2p2 + ω 2 − ω02 = 0. Me ˆlla lìgia q ω = ω02 − 2p2 or 0

suntonismoÔ (eÐnai dhlad  to shmeÐo ìpou h

C 0 (ω)

ω02 − 2p2

C(ω) èqei

ìtan

eÐnai jetik  ìtan h

p ω02 − 2p2

eÐnai h fusik  suqnìthta

to mègistì thc, ìpou sthn perÐptwsh aut 

C 0 (ω) > 0 gia

mikrèc

eÐnai to mègisto tìte den upˆrqei ousiastikˆ fusikìc suntonismìc mia kai upojètoume

ìti gia to sÔsthmˆ mac èqoumc

ω > 0.

Sthn perÐptwsh aut  to plˆtoc auxˆnei ìtan h exanagkasmènh(??) suqnìthta

elatt¸netai.

ω0 . p) elatt¸nontai, tìso h fusik  suqnìthta suntonismoÔ plhsiˆzei thn ω0 . 'Ara h apìsbesh eÐnai polÔ mikr , h ω0 apoteleÐ mia kal  prosèggish thc suqnìthtac suntonismoÔ. H sumperiforˆ sumfwneÐ me thn parat rhsh ìti ìtan to c = 0, tìte h ω0 eÐnai h suqnìthta suntonismoÔ.

Parìmoia mporoÔme na doÔme ìti eˆn den sumbaÐnei fusikìc suntonismìc, h suqnìthta eÐnai mikrìterh apì to 'Oso h apìsbesh ìtan aut 

c

(kai sunep¸c kai h

H parapˆnw sumperiforˆ eÐnai poÔ pio perÐplokh eˆn h exwterik  dÔnamh den eÐnai kÔma sunhmitìnou, allˆ gia parˆdeigma eÐnai èna tetrˆgwno kÔma. seir¸n

Ja exetˆsoume thn perÐptwsh aut  mìlic exoplistoÔme me to ergaleÐo twn

F ourier.

2.6.3

Ask seic

2.6.1

BreÐte ènan tÔpo gia thn

xsp

2.6.2

BreÐte ènan tÔpo gia thn

xsp

eˆn h exÐswsh eÐnai eˆn h exÐswsh eÐnai

mx00 + cx0 + kx = F0 sin ω t.

Upojèste ìti

mx00 + cx0 + kx = F0 cos ω t + F1 cos 3ω t.

c > 0.

Upojèste ìti

c > 0.

mx00 + cx0 + kx = F0 cos ω t kai èstw m > 0 kai k > 0. Jewr ste thn sunˆrthsh C(ω). Gia poiec timèc tou c (lÔste wc proc m, k , kai F0 ) den ja upˆrqei fusikìc suntonismìc (gia poiec timèc c den upˆrqei mègisto tou C(ω) ìtan ω > 0).

2.6.3

'Estw h exÐswsh

62 2.6.4

ΚΕ֟ΑΛΑΙΟ 2. ΓΡΑΜΜΙʟΕΣ ΣΔΕ ΥΨΗ˟ΟΤΕΡΗΣ ΤŸΑΞΗΣ mx00 + cx0 + kx = F0 cos ω t kai èstw c > 0 kai k > 0. Jewr ste thn sunˆrthsh C(ω). Gia c, k, kai F0 ) den ja upˆrqei fusikìc suntonismìc (gia poiec timèc m den upˆrqei ω > 0).

'Estw h exÐswsh

m C(ω)

poiec timèc tou

(lÔste wc proc

mègisto tou

for

Kefˆlaio 3

Sust mata SDE 3.1

Eisagwg  sta sust mata SDE

Sun jwc den èqoume mìnon mia exarthmènh metablht  kai mìnon mÐa diaforik  exÐswsh allˆ èna sÔsthma allhloexart¸menwn diaforik¸n exis¸sewn ìpou emplèkontai pollèc exarthmènec metablhtèc wc proc tic opoÐec prèpei na lÔsoume. Sthn perÐptwsh pou to prìblhma pou meletoÔme emplèkei perissìterec apì mia exarthmènec metablhtèc, èstw tic

y1 , y2 ,

...,

yn

mporeÐ na èqoume diaforikèc exis¸seic pou na emplèkoun tìso ìlec autèc tic metablhtèc ìso kai tic

parag¸gouc twn.

Gia parˆdeigma,

y100 = f (y10 , y20 , y1 , y2 , x).

Sun jwc ìtan emplèkontai dÔo exarthmènec metablhtèc

èqoume kai dÔo exis¸seic wc ex c

y100 = f1 (y10 , y20 , y1 , y2 , x), y200 = f2 (y10 , y20 , y1 , y2 , x), gia kˆpoiec sunart seic

f1

kai

f2 .

H parapˆnw duˆda exis¸sewn onomˆzetai sÔsthma diaforik¸n exis¸sewn. Gia thn

akrÐbeia, eÐnai èna sÔsthma deÔterhc tˆxhc. Kˆpoiec forèc èna sÔsthma eÐnai eÔkolo na epilujeÐ arkeÐ na lÔsoume thn mia exÐswsh wc proc thn mÐa exarthmènh metablht  kai metˆ na lÔsoume wc proc thn deÔterh.

Parˆdeigma 3.1.1:

Jewr ste to parakˆtw sÔsthma pr¸thc tˆxhc

y10 = y1 , y20 = y1 − y2 , me arqikèc sunj kec thc morf c

y1 (0) = 1, y2 (0) = 2.

Shmei¸ste ìti h genik  lÔsh thc pr¸thc exÐswshc eÐnai

0 exÐswsh, aut  paÐrnei thn ex c morf  y2 eÔkola mporoÔme na lÔsoume wc proc

y2 .

x

= C1 e − y2 ,

y2 =

C1 x e 2

+ C2 e−x .

Antikajist¸ntac aut  thn

y1

sthn deÔterh

h opoÐa eÐnai grammik  exÐswsh pr¸thc tˆxhc kai sunep¸c

Prˆgmati me thn mèjodo twn oloklhrwtik¸n paragìntwn èqoume

ex y2 =  

y1 = C1 ex .

C1 2x e + C2 , 2

Sunep¸c h genik  lÔsh tou sust matoc eÐnai h ex c,

y1 = C1 ex , y2 = MporoÔme t¸ra na prosdiorÐsoume ta brÐskoume ìti

C1 = 1

kai

C1

kai

C2

C1 x e + C2 e−x . 2

qrhsimopoi¸ntac tic arqikèc sunj kec. Antikajist¸ntac to

x=0

C2 = 3/2.

Genikˆ, den ja eÐmaste pˆnta tìso tuqeroÐ na mporoÔme na lÔsoume èna sÔsthma brÐskontac ènan ènan tou agn¸stouc ìpwc kˆname parapˆnw, allˆ ja qreiasteÐ na touc broÔme ìlouc mazÐ.

63

64

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ San parˆdeigma efarmog c, ac jewr soume kai pˆli sust mata mˆzac elathrÐou.

elat rio stajerˆc

k

kai topojetoÔme sta ˆkra tou dÔo swmatÐdia mˆzac

swmatÐdia san bagonèta ta opoÐa kinoÔntai ousiastikˆ qwrÐc trib . 'Estw

'Estw loipìn ìti èqoume èna

m1 kai m2 . MporoÔme na jewr soume ta x1 h metatìpish tou enìc bagonètou kai x2

h metatìpish tou deÔterou apì thn jèsh isorropÐac. Me ton ìro jèsh isorropÐac ennooÔme thn jèsh twn swmatidÐwn ìtan den upˆrqei tˆsh sto elat rio. Dhlad , topojetoÔme ta dÔo bagonèta kˆpou ètsi ¸ste na mhn upˆrqei kajìlou tˆsh sta elat ria kai shmei¸noume tic jèseic touc san jèseic mhdèn. Dhlad , q¸ro apì thn jèsh pou dhl¸nei to

x2 = 0.

to elat rio ja epimhkunjeÐ (h ja sumpiesteÐ) katˆ

k(x2 − x1 )

x1 = 0

dhl¸nei diaforetik  jèsh sto

Eˆn metakin soume ta bagonèta apì thn jèsh isorropÐac tìte, epeid 

x2 − x1 ,

h dÔnamh pou ja askhjeÐ sto pr¸to bagonèto ja eÐnai

en¸ h dÔnamh pou ja askhjeÐ sto deÔtero bagonèto ja eÐnai Ðsh kai antÐjethc kateÔjunshc. SÔmfwna me

ton deÔtero nìmo tou NeÔtwna èqoume

m1 x001 = k(x2 − x1 ), m2 x002 = −k(x2 − x1 ). Den eÐnai efiktì na lÔsoume to parapˆnw sÔsthma wc proc dhlad  na lÔsoume tautìqrona wc proc to

x1

kai to

x2

x1 pr¸ta.

Prèpei

k

ìpwc exˆllou kai eÐnai

anamenìmeno mia kai h kÐnhsh tou enìc bagonètou exartˆtai apì thn kÐnhsh tou

m2

m2

ˆllou. Prin suzht soume to pwc ja antimetwpÐsoume to parapˆnw sÔsthma, ac shmei¸soume ìti, katˆ kˆpoia ènnoia, arkeÐ na asqolhjoÔme mìnon me sust mata pr¸thc tˆxhc. Ac jewr soume mia diaforik  exÐswsh

nsthc

tˆxhc

y (n) = F (y (n−1) , . . . , y 0 , y, x). Ac orÐsoume t¸ra tic ex c metablhtèc

u1 , . . . , u n

kai ac diatup¸soume to sÔsthma wc ex c

u01 = u2 u02 = u3 . . .

u0n−1 = un u0n = F (un , un−1 , . . . , u2 , u1 , x). MporoÔme t¸ra na prospaj soume na lÔsoume wc proc

u3

...

un

kai na jèsoume

y = u1 .

Parathr ste ìti h

y

u1 , u2 ,

...,

un

kai mìlic tic broÔme na ˆgno soume' tic

u2 ,

aut  eÐnai lÔsh thc arqik c exÐswshc.

Parìmoia eÐnai h diadikasÐa gia thn epÐlush enìc sust matoc diaforik¸n exis¸sewn uyhlìterhc tˆxhc. MporoÔme gia parˆdeigma, na metatrèyoume èna sÔsthma

n×k

exis¸sewn pr¸thc tˆxhc me

Parˆdeigma 3.1.2:

n×k

k

diaforik¸n exis¸sewn me

k

agn¸stouc, ìlec tˆxhc

n,

se èna sÔsthma

agn¸stouc.

Merikèc forèc mporoÔme na qrhsimopoi soume thn parapˆnw idèa me antÐstrofh forˆ. Jew-

r ste to sÔsthma

x0 = 2y − x, ìpou h anexˆrthth metablht  eÐnai h

t.

y 0 = x,

Jèloume na broÔme thn lÔsh pou ikanopoieÐ tic ex c arqikèc sunj kec

x(0) = 1,

y(0) = 0. ParagwgÐzontac thn pr¸th exÐswsh èqoume

y 00 = x0

00

kai sunep¸c gnwrÐzoume thn

0

x0

san sunˆrthsh twn

x

kai

y.

0

y = x = 2y − x = 2y − y . 'Ara èqoume t¸ra thn exÐswsh 'Eqontac thn

y

y 00 + y 0 − 2y = 0

thc opoÐac thn lÔsh

y = C1 e−2t + C2 et x wc ex c

mporoÔme na broÔme eÔkola.

mporoÔme na antikatast soume kai na upologÐsoume thn

x = y 0 = −2C1 e−2t + C2 et . Ac qrhsimopoi soume t¸ra tic arqikèc sunj kec kai

1 = 3C2 .

'Ara

C1 = −1/3

kai

C2 = 1/3

1 = x(0) = −2C1 + C2

kai

0 = y(0) = C1 + C2 .

kai h lÔsh eÐnai h ex c

x=

2e−2t + et , 3

y=

−e−2t + et . 3

Sunep¸c,

C1 = −C2

3.1. ΕΙΣΑΓΩΓŸΗ ΣΤΑ ΣΥΣԟΗΜΑΤΑ ΣΔΕ 3.1.1

65

Epibebai¸ste ìti prˆgmati aut  eÐnai lÔsh.

To parapˆnw parˆdeigma eÐnai èna grammikì sÔsthma pr¸thc tˆxhc. Tautìqrona eÐnai kai èna mia kai oi exis¸seic den exart¸ntai apì thn anexˆrthth metablht 

t.

MporoÔme eÔkola kˆnoume thn grafik  parˆstash tou pedÐou dieujÔnsewn   dianusmatikì pedÐo kˆje autìnomou sust matoc.

EÐnai h Ðdia grafik  parˆstash tou pedÐou kateujÔnsewn pou sunant same prohgoumènwc, mìnon pou

t¸ra antÐ na zwgrafÐsoume se kˆje shmeÐo thn klÐsh thc lÔshc ja zwgrafÐsoume th kateÔjunsh (kai to mègejoc). To prohgoÔmeno parˆdeigma

x0 = 2y − x, y 0 = x

mac dhl¸nei ìti sto shmeÐo

(x, y)

h kateÔjunsh sthn opoÐa ofeÐloume

na badÐsoume gia na ikanopoi soume tic exis¸seic tou sust matoc eÐnai h kateÔjunsh pou orÐzei to ex c diˆnusma

(2y − x, x)•.

H taqÔthta me thn opoÐa prèpei na badÐsoume prèpei na eÐnai Ðsh me to mètro tou en lìgw dianÔsmatoc.

ZwgrafÐzoume loipìn to diˆnusma

(2y − x, x)

sto shmeÐo

(x, y)

kai kˆnoume to Ðdio gia ìpoio shmeÐo tou

xy -epipèdou

epijumoÔme. Endeqomènwc bebaÐwc na qreiasjeÐ na allˆxoume klÐmata sto grˆfhmˆ mac eˆn jèloume na prosjèsoume pollˆ shmeÐo sto Ðdio pedÐo kateujÔnsewn. DeÐte sto Sq ma 3.1. MporoÔme t¸ra na zwgrafÐsoume thn diadrom  thc lÔshc sto epÐpedo. Sugkekrimèna, upojèste ìti h lÔsh dÐdetai

x = f (t), y = g(t), opìte kai mporoÔme na epilèxoume èna diˆsthma tou t (ac epilèxoume gia parˆdeigma 0 ≤ t ≤ 2) kai na zwgrafÐsoume ìla ta shmeÐa (f (t), g(t)) gia t mèsa sto diˆsthma pou epilèxoume. H eikìna pou ja

apì tic

pˆroume me ton trìpo autì sun jwc onomˆzetai parˆstash fˆsewn (  parˆstash fˆshc epipèdou). Thn sugkekrimènh kampÔlh pou paÐrnoume onomˆzoume troqiˆ   kampÔlh epÐlushc.

MporeÐte na deÐte èna qarakthristikì parˆdeigma

(1, 0) kai kineÐtai pˆnw sto pedÐo dieujÔnsewn gia t. Mia kai èqoume  dh lÔsei to en lìgw sÔsthma xèroume ta x(2) kai y(2). 'Eqoume loipìn ìti x(2) ≈ 2.475 kai y(2) ≈ 2.457. To shmeÐo autì antistoiqeÐ sto ˆnw dexiˆ ˆkro thc grafik c parˆstashc thc lÔshc sto Sq ma 3.2. Sto Sq ma autì h kampÔlh xekinˆ apo to shmeÐo

apìstash 2 monˆdec tou

sto Sq ma. -1

0

1

2

3

-1

0

1

2

3

3

3

3

3

2

2

2

2

1

1

1

1

0

0

0

0

-1

-1

-1 -1

0

1

2

3

Σχήμα 3.1: Το πεδίο κατευθύνσεων του συστήματος x0 = 2y − x, y 0 = x.

-1 -1

0

1

2

3

Σχήμα 3.2: Το πεδίο κατευθύνσεων του συστήματος x0 = 2y − x, y 0 = x με την τροχιά της λύσης που ξεκινά από το σημείο ωιτη (1, 0) για 0 ≤ t ≤ 2.

Parathr ste thn omoiìthta twn parapˆnw sqhmˆtwn me autˆ pou sunant same gia ta autìnoma sust mata miac diˆstashc. FantasteÐte t¸ra pìso pio perÐploka ja gÐnoun ta prˆgmata eˆn prosjèsoume akìma mia diˆstash. Shmei¸ste epÐshc ìti mporoÔme na zwgrafÐsoume parastˆseic fˆsewn kai troqi¸n sto

xy -epÐpedo akìma kai gia mh

autìnoma sust mata. Sthn perÐptwsh aut  ìmwc den mporoÔme na zwgrafÐsoume pedÐa dieujÔnsewn, mia kai ta en lìgw pedÐa allˆzoun ìtan to

3.1.1

t

allˆzei. MporoÔme bebaÐwc na èqoume gia kˆje

t

kai èna diaforetikì pedÐo dieujÔnsewn.

Ask seic

3.1.2

BreÐte thn genik  lÔsh tou sust matoc

x01 = x2 − x1 + t, x02 = x2 .

3.1.3

BreÐte thn genik  lÔsh tou sust matoc

x01 = 3x1 − x2 + et , x02 = x1 .

66

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

3.1.4

Grˆyte thn exÐswsh

3.1.5

Grˆyte tic exis¸seic

ay 00 + by 0 + cy = f (x)

san èna sÔsthma SDE pr¸thc tˆxhc.

x00 + y 2 y 0 − x3 = sin(t), y 00 + (x0 + y 0 )2 − x = 0

san èna sÔsthma SDE pr¸thc tˆxhc.

3.2. ΓΡΑΜΜΙʟΑ ΣΥΣԟΗΜΑΤΑ ΣΔΕ 3.2

67

Grammikˆ sust mata SDE

Prin xekin sete thn melèth thc paragrˆfou aut c eÐnai anagkaÐo na kˆnete mia sÔntomh epanˆlhyh thc Grammik c 'Algebrac. Ja qreiasjeÐte kˆpoia basikˆ stoiqeÐa ìpwc pollaplasiasmìc pinˆkwn, orÐzousec kai idiotimèc. Ac asqolhjoÔme pr¸ta me sunart seic dianusmˆtwn kai pinˆkwn. Autèc den eÐnai tÐpote parapˆnw parˆ dianÔsmata kai pÐnakec ta stoiqeÐa twn opoÐwn exart¸ntai apì kˆpoia metablht  h opoÐa ac upojèsoume ìti eÐnai h diˆnusma sunart sewn

~ x(t)

t.

Tìte mia

eÐnai kˆti pou èqei thn ex c morf 

 x1 (t)  x2 (t)    ~ x(t) =  .  .  ..  

xn (t) Parìmoia ènac pÐnakac sunart sewn eÐnai kˆti pou èqei thn ex c morf 



H parˆgwgoc

A0 (t)

 

a11 (t)  a21 (t)  A(t) =  .  ..

a12 (t) a22 (t)

an1 (t)

an2 (t)

. . .

··· ··· .

.

.

···

 a1n (t) a2n (t)   . . .  . ann (t)

dA eÐnai kai aut  ènac pÐnakac sunart sewn o opoÐoc sthn dt

ij sth

jèsh èqei thn sunˆrthsh

a0ij (t).

Gia tic pÐnakec sunart sewn isqÔoun oi Ðdioi kanìnec parag¸gishc me autoÔc twn sumbatik¸n sunart sewn. Dhlad  gia kˆje pÐnakec sunart sewn stajerì pÐnaka

C

A

kai

B,

gia kˆje stajerì pragmatikì arijmì

c

kai gia kˆje sumbatikì

isqÔoun (upojètoume ìti oi diastˆseic twn pinˆkwn eÐnai sunepeÐc me tic prˆxeic kai tic isìthtec

bebaÐwc) oi tautìthtec

(A + B)0 = A0 + B 0 (AB)0 = A0 B + AB 0 (cA)0 = cA0 (CA)0 = CA0 (AC)0 = A0 C 'Ena grammikì sÔsthma SDE pr¸thc tˆxhc eÐnai èna sÔsthma to opoÐo mporeÐ na grafjeÐ wc ex c

~ x 0 (t) = P (t)~ x(t) + f~(t). P

~ x ~ x 0 = P~ x + f~.

eÐnai ènac pÐnakac sunart sewn, kai

apì to

t

kai ja grˆfoume

kai

f~

eÐnai dianÔsmata sunart sewn. Suqnˆ ja apokrÔptoume thn exˆrthsh

H lÔsh eÐnai profan¸c èna diˆnusma sunart sewn

~ x

to opoÐo ikanopoieÐ to

sÔsthma. Gia parˆdeigma oi exis¸seic

x01 = 2tx1 + et x2 + t2 , x1 − x2 + et , x02 = t mporoÔn na grafoÔn sthn morf 

~ x0 =



2t 1/t

  2 et t ~ x+ t . −1 e

Ja epikentrwjoÔme kurÐwc se exis¸seic oi opoÐec den eÐnai mìnon grammikèc, allˆ kai me stajeroÔc suntelestèc. Dhlad , o pÐnakac 'Otan

f~ = ~0

P

eÐnai ènac sumbatikìc pÐnakac kai stoiqeÐa pragmatikoÔc arijmoÔc kai den exartˆtai apì to

(to mhdenikì diˆnusma), tìte lème ìti to sÔsthma eÐnai omogenèc.

t.

H arq  thc upèrjeshc isqÔei kai

gia omogen  grammikˆ sust mata ìpwc akrib¸c isqÔei kai gia mia mìno omogen  exÐswsh.

68

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

Je¸rhma 3.2.1

(Upèrjesh). 'Estw

~ x 0 = P~ x

èna grammikì omogenèc sÔsthma SDE. 'Estw ìti

~ x1 , . . . , ~ xn

eÐnai

n

lÔseic thc exÐswsh, tìte h

~ x = c1 ~ x1 + c2 ~ x2 + · · · + cn ~ xn , eÐnai epÐshc lÔsh. An epiprìsjeta, eÐnai èna sÔsthma

n

exis¸sewn (

P

(3.1)

eÐnai

n × n),

kai ta

~ x1 , . . . , ~xn

eÐnai grammikˆ

anexˆrthta, tìte kˆje lÔsh mporeÐ na grafjeÐ sthn morf  thc exÐswshc (3.1).

H ènnoia thc grammik  anexarthsÐa sunart sewn me dianÔsmata eÐnai ousiastikˆ h Ðdia me aut  twn sumbatik¸n dianusmˆtwn me stoiqeÐa pragmatikoÔc arijmoÔc. Dhlad , ta dianÔsmata

~ x1 , . . . , ~ xn

eÐnai grammikˆ anexˆrthta an kai

mìnon an h

c1 ~ x1 + c2 ~ x2 + · · · + cn ~ xn = ~0 èqei mìnon mia lÔsh, thn

c1 = c2 = · · · = cn = 0. c1 ~ x1 + c2 ~ x2 + · · · + cn ~ xn

O grammikìc sunduasmìc

mporeÐ pˆnta na grafjeÐ sthn morf 

X(t) ~c, X(t) eÐnai ènac pÐnakac c1 , . . . , cn . O X(t) onomˆzetai

ìpou

me st lec ta dianÔsmata

~ x1 , . . . , ~ xn ,

kai ìpou

~c

eÐnai to sthlo-diˆnusma me stoiqeÐa

jemeli¸dhc pÐnakac,   jemeli¸dhc pÐnakac epÐlushc.

MporoÔme na lÔsoume èna sÔsthma omogen¸n grammik¸n exis¸sewn pr¸thc tˆxhc me thn Ðdia praktik  pou qrhsimopoi same gia mia mình exÐswsh tou Ðdiou tÔpou.

Je¸rhma 3.2.2.

Eˆn

~ x 0 = P~ x + f~

eÐnai èna grammikì sÔsthma SDE kai eˆn

~ xp

mia opoiad pote sugkekrimènh

lÔsh tou, tìte kˆje lÔsh tou mporeÐ na grafjeÐ sthn morf 

~ x=~ xc + ~ xp , ìpou

~xc

~ x 0 = P~ x).

eÐnai mia lÔsh thc antÐstoiqhc omogenoÔc exÐswshc (

Sunep¸c h diadikasÐa eÐnai Ðdia me prohgoumènwc. BrÐskoume mia sugkekrimènh lÔsh thc mh-omogenoÔc exÐswshc, metˆ brÐskoume mia genik  lÔsh thc antÐstoiqhc omogenoÔc exÐswshc kai tèloc prosjètoume tic dÔo autèc lÔseic. 'Estw ìti èqoume brei thn ex c genik  lÔsh arqikèc sunj kec thc morf c

~ x(t0 ) = ~b

ìpou ~ b

~ x 0 = P~ x + f~

kai jèloume na broÔme thn lÔsh pou ikanopoieÐ kˆpoiec

kˆpoio stajerì diˆnusma. 'Estw epÐshc ìti to

pÐnakac epÐlushc thc antÐstoiqhc omogenoÔc exÐswshc (dhlad  oi st lec tou

X

X(t)

thn ex c morf 

~ x(t) = X(t)~c + xp (t). AkoloÔjwc prèpei na broÔme èna diˆnusma

~c

tètoio ¸ste

~b = ~ x(t0 ) = X(t0 )~c + xp (t0 ). Me ˆlla lìgia prèpei na lÔsoume to ex c mh-omogenèc sÔsthma grammik¸n algebrik¸n exis¸sewn

X(t0 )~c = ~b − xp (t0 ) wc proc

~c.

Parˆdeigma 3.2.1:

Sto

§3.1

lÔsame to ex c sÔsthma

x01 = x1 , x02 = x1 − x2 . me arqikèc sunj kec

x1 (0) = 1, x2 (0) = 2.

Mia kai prìkeitai gia èna omogenèc sÔsthma, èqoume

 1 ~ x0 = 1

f~ = ~0.

Grˆfoume to sÔsthmˆ mac wc ex c



0 ~ x, −1

~ x(0) =

  1 . 2

eÐnai o jemeli¸dhc

eÐnai lÔseic). Tìte h genik  lÔsh èqei

3.2. ΓΡΑΜΜΙʟΑ ΣΥΣԟΗΜΑΤΑ ΣΔΕ Br kame ìti h genik  lÔsh eÐnai h

x1 = c1 et

69 kai

x2 =

c1 t e 2

+ c2 e−t .

'Ara se morf  pinˆkwn, o jemeli¸dhc pÐnakac

epÐlushc eÐnai o ex c

 X(t) =

et 1 t e 2

0 e−t

 .

Den eÐnai dÔskolo na doÔme ìti oi st lec tou pÐnaka autoÔ eÐnai grammikˆ anexˆrthtec. Sunep¸c gia na lÔsoume to prìblhma arqik¸n tim¸n prèpei na lÔsoume thn exÐswsh

X(0)~c = ~b,   me ˆlla lìgia,

   1 0 ~c = . 1 2  1  twn gramm¸n brÐskoume ìti ~ c = 3/2 . 'Ara h lÔsh      t 1 et e 0 ~ x(t) = X(t)~c = 1 t −t 3 = 1 t 3 −t . e e e + 2e 2 2 2 

1 1 2

Metˆ apì mia apl  prˆxh metaxÔ

mac eÐnai

EÐnai ìpwc blèpoume Ðdia me thn lÔsh pou  dh br kame parapˆnw.

3.2.1

3.2.1

Ask seic

Grˆyte to sÔsthma

x01 = 2x1 − 3tx2 + sin t, x02 = et x1 + 3x2 + cos t

sthn morf 

~ x 0 = P (t)~ x + f~(t).

 1  −2t ~ x 0 = [ 13 31 ] ~ x èqei dÔo lÔseic [ 11 ] e4t kai −1 e . b) D¸ste thn genik  lÔsh. c) D¸ste thn genik  lÔsh sthn morf  x1 =?, x2 =? (dhlad  d¸ste thn morf  kˆje stoiqeÐou thc lÔshc).  1  t t 3.2.3 Epibebai¸ste ìti ta [ 1 1 ] e kai −1 e eÐnai grammikˆ anexˆrthta. Upìdeixh: Antikatast ste to t = 0. h1i h 1 i h 1 i t t 2t 3.2.4 Epibebai¸ste ìti ta 1 e , −1 e kai −1 e eÐnai grammikˆ anexˆrthta. Upìdeixh: Prèpei profan¸c na

3.2.2

a) epibebai¸ste ìti to sÔsthma

0

1

1

drˆsete poio èxupna apì ìti sthn prohgoÔmenh ˆskhsh.

3.2.5

Epibebai¸ste ìti ta



t t2



kai

h

t3 t4

i

eÐnai grammikˆ anexˆrthta.

70

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

3.3

Mèjodoc idiotim¸n

Sthn parˆgrafo aut  ja mˆjoume p¸c ja mporoÔme na lÔnoume grammikˆ omogen  sust mata SDE me stajeroÔc suntelestèc qrhsimopoi¸ntac thn mèjodo twn idiotim¸n. 'Estw to ex c grammikì omogenèc sÔsthma

~ x 0 = P~ x. Ac prospaj soume na epekteÐnoume thn mèjodo pou  dh qrhsimopoi same gia mia exÐswsh stajer¸n suntelest¸n qrhsimopoi¸ntac thn manteyiˆ

~v eλt ,

ìpou

~v

eλt .

Epeid  ìmwc to

~ x

eÐnai diˆnusma, eÐnai logikì na protim soume thn ex c manteyiˆ

eÐnai èna opoiod pote stajerì diˆnusma. Thn antikajistoÔme loipìn sthn exÐswsh kai èqoume

λ~v eλt = P ~v eλt . eλt

DiairoÔme me

kai parathroÔme ìti prèpei na broÔme èna arijmì

λ

kai èna diˆnusma

~v

tètoia ¸ste na ikanopoioÔn

thn exÐswsh

λ~v = P ~v . Gia na lÔsoume to parapˆnw prìblhma ja qreiasjoÔme merikˆ basikˆ stoiqeÐa grammik c ˆlgebra.

3.3.1

Idiotimèc kai idiodianÔsmata pinˆkwn

'Estw o tetragwnikìc kai stajerìc pÐnakac

~v

A.

'Estw epÐshc ìti upˆrqei ènac arijmìc

λ

kai èna mh mhdenikì diˆnusma

tètoia ¸ste

A~v = λ~v . Onomˆzoume to en lìgw

Parˆdeigma 3.3.1:

λ

A

idiotim  tou pÐnaka

O pÐnakac

[ 20 11 ]

kai to

~v

λ=2

èqei mia idiotim 



2 0

1 1

to idiodiˆnusma pou antistoiqeÐ sto

λ.

to opoÐo antistoiqeÐ sto ex c idiodiˆnusma

[ 10 ]

epeid 

      1 2 1 = =2 . 0 0 0

Ac xanagrˆyoume thn parapˆnw exÐswsh sthn ex c morf 

(A − λI)~v = ~0. ~v mìnon ìtan o pÐnakac A − λI eÐnai mh-antistrèyimoc. Eˆn  tan antistrèy(A − λI)−1 (A − λI)~v = (A − λI)−1~0 to opoÐo sunepagetai ìti ~v = ~0. 'Ara o A èqei to

ParathroÔme ìti upˆrqei mh-mhdenik  lÔsh imoc ja mporoÔsame na èqoume

λ

idiotim  an kai mìnon an to

λ

eÐnai lÔsh thc exÐswshc

det(A − λI) = 0. Shmei¸ste ìti h parapˆnw exÐswsh shmaÐnei ìti mporoÔme na broÔme mia idiotim  qwrÐc na prèpei aparaÐthta na upologÐsoume to antÐstoiqo idiodiˆnusma to opoÐo mporoÔme na upologÐsoume argìtera, afoÔ èqoume  dh upologÐsei to

λ

pr¸ta.

Parˆdeigma 3.3.2:

BreÐte ìlec tic idiotimèc tou pÐnaka

h2 1 1i

1 2 0 . 0 0 2

'Eqoume



2 det 1 0

1 2 0

  1 1 0 − λ 0 2 0

0 1 0

  0 2−λ 0 = det  1 1 0

1 2−λ 0

 1 0  = 2−λ = (2 − λ)2 ((2 − λ)2 − 1) = −(λ − 1)(λ − 2)(λ − 3).

kai sunep¸c oi idiotimèc eÐnai oi

λ = 1, λ = 2,

kai

λ = 3.

3.3. ̟ΕΘΟΔΟΣ ΙΔΙΟΤΙ̟ΩΝ Shmei¸ste ìti gia ènan

n×n

71 det(A − λI)

pÐnaka, to polu¸numo

eÐnai bajmoÔ

n,

kai sunep¸c ja èqoume genikˆ

n

idiotimèc. Gia na broÔme to idiodiˆnusma pou antistoiqeÐ sto

λ,

qrhsimopoioÔme thn exÐswsh

(A − λI)~v = ~0, kai kai lÔnoume gia èna mh-mhdenikì diˆnusma

Parˆdeigma 3.3.3:

~v .

An to

λ

eÐnai prˆgmati idiotim  tìte kˆti tètoio eÐnai pˆnta efiktì.

BreÐte to idiodiˆnusma pou antistoiqeÐ sthn idiotim 

λ=3

tou pÐnaka

h2 1 1i 1 2 0 0 0 2

.

'Eqoume

 2 (A − λI)~v = 1 0

  1 1 0  − 3 0 0 2

1 2 0

LÔnontac to parapˆnw sÔsthma brÐskoume ìti na epilèxoume mia opoiad pote tim  gia to

v2

0 1 0

    −1 v1 0 0 v2  =  1 0 v3 1

1 −1 0

  v1 1 0  v2  = ~0. v3 −1

v1 − v2 = 0, v3 = 0, ìpou h v2 eÐnai mia eleÔjerh metablht . MporoÔme h1i v1 = v2 kai fusikˆ v3 = 0. Gia parˆdeigma, ~v = 1 . Ac

kai na jèsoume

0

dokimˆsoume:



2 1 0

      1 1 3 1 0 1 = 3 = 3 1 . 2 0 0 0

1 2 0

EÐnai loipìn swstì.

3.3.1(eÔkolh)

EÐnai ta idiodianÔsmata monadikˆ? MporeÐte na breÐte èna diaforetikì apì autì pou br kate parapˆnw

idiodiˆnusma gia thn idiotim 

3.3.2

λ = 3?

Upˆrqei kˆpoia sqèsh metaxÔ twn dÔo aut¸n dianusmˆtwn?

H mèjodoc twn idiotim¸n me pragmatikèc idiotimèc qwrÐc pollaplìthta

Jewr ste thn exÐswsh

~ x 0 = P~ x. BrÐskoume tic idiotimèc t¸ra ìti oi sunart seic

· · · + cn~vn eλn t λ1 ,

tou pÐnaka ...,

P,

~vn eλn t

kai ta antÐstoiqa idiodianÔasmata

~v1 , ~v2 , . . . , ~vn . Parathr ste ~ x = c1~v1 eλ1 t + c2~v2 eλ2 t +

eÐnai lÔseic thc exÐswshc kai sunep¸c h

eÐnai mia lÔsh.

Je¸rhma 3.3.1. idiotimèc,

λ1 , λ2 , . . . , λn ~v1 eλ1 t , ~v2 eλ2 t ,

...,

'Estw

λn

~ x 0 = P~ x.

tìte upˆrqoun

An

n

P

eÐnai ènac

n×n pÐnakac o opoÐoc èqei n diaforetikèc metaxÔ touc pragmatikèc ~v1 , . . . , ~vn , kai h genik  lÔsh

grammikˆ anexˆrthta antÐstoiqa idiodianÔsmata

thc SDE mporeÐ na grafjeÐ wc ex c

~ x = c1~v1 eλ1 t + c2~v2 eλ2 t + · · · + cn~vn eλn t . Parˆdeigma 3.3.4:

BreÐte thn genik  lÔsh tou sust matoc



2 ~ x 0 = 1 0

to

h'Eqoume i 1

 dh brei ìti oi idiotimèc eÐnai oi

1, 2, 3.

1 2 0

 1 0 ~ x. 2

Br kame ìti to idiodiˆnusma pou antistoiqeÐ sthn idiotim  3 eÐnai

1 . Parìmoia mporoÔme na broÔme ìti to idiodiˆnusma pou antistoiqeÐ sthn idiotim  1 eÐnai to 0 h 0 i 1 antistoiqeÐ sthn idiotim  2 eÐnai to (elègxte to san ˆskhsh). 'Ara h genik  lÔsh mac eÐnai −1

       1 0 1 et + e3t ~ x = −1 et +  1  e2t + 1 e3t = −et + e2t + e3t  . 0 −1 0 −e2t 

h

1 −1 0

i

en¸ autì pou

72

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

3.3.2

Epibebai¸ste ìti h parapˆnw prˆgmati apoteleÐ lÔsh tou sust matoc.

ShmeÐwsh: Eˆn grˆyoume mia grammik  omogen  exÐswsh exis¸sewn pr¸thc tˆxhc (ìpwc  dh kˆname sthn parˆgrafo

nsthc tˆxhc me stajeroÔc suntelestèc §3.1) tìte h exÐswsh idiotim¸n

san èna sÔsthma

det(P − λI) = 0. §2.2

eÐnai ousiastikˆ to Ðdio me thn qarakthristik  exÐswsh pou sunant same stic paragrˆfouc

3.3.3

kai

§2.3.

Migadikèc idiotimèc

'Enac pÐnakac mporeÐ na èqei migadikèc idiotimèc akìma kai sthn perÐptwsh pou ìla ta stoiqeÐa tou eÐnai pragmatikoÐ arijmoÐ. Gia parˆdeigma ac jewr soume to sÔsthma



1 ~ x = −1  1 0

 1 ~ x. 1

 1 P = −1 1 .   1−λ 1 det(P − λI) = det = (1 − λ)2 + 1 = λ2 − 2λ + 2 = 0. −1 1−λ

Ac upologÐsoume tic idiotimèc tou pÐnaka

SumperaÐnoume loipìn ìti

λ = 1 ± i.

Ta antÐstoiqa idiodianÔsmata eÐnai bebaÐwc kai autˆ migadikˆ

(P − (1 − i)I)~v = ~0,   i 1 ~v = ~0. −1 i iv1 + v2 = 0

Profan¸c apì tic dÔo exis¸seic

kai

−v1 + iv2 = 0

ja epilèxoume na lÔsoume mìnon thn mÐa (h ˆllh

eÐnai pollaplˆsiˆ thc). Eˆn epiprìsjeta epilèxoume gia parˆdeigma, Parìmoia brÐskoume ìti sthn idiotim 

1+i

v2 = 1ja

antistoiqeÐ to idiodiˆnusma

katal xoume sto idiodiˆnusma

−i 1 .

~v = [ 1i ].

MporoÔme loipìn na grˆyoume thn lÔsh wc ex c

~ x = c1

      i (1−i)t −i (1+i)t c1 ie(1−i)t − c2 ie(1+i)t . e + c2 e = 1 1 c1 e(1−i)t + c2 e(1+i)t 1

T¸ra ìmwc prèpei na yˆxoume gia poÐec migadikèc timèc twn

c1

kai

c2

ikanopoioÔntai oi arqikèc sunj kec.

poroÔsame na qrhsimopoi soume thn praktik  pou basÐzetai ston tÔpo tou

Ja m-

Euler kai thn opoÐa sunant same parapˆnw

allˆ ac ergastoÔme kˆpwc pio èxupna pr¸ta. MporoÔme na isqurisjoÔme ìti den qreiˆzetai na asqolhjoÔme me to deÔtero idiodiˆnusma (oÔte me thn deÔterh idiotim ) mia kai ìlec oi migadikèc idiotimèc (enìc pragmatikoÔ pÐnaka) sunantiìntai se zeÔgh suzug¸n. MporeÐte eˆn Jèlete na apodeÐxete thn endiafèrousa aut  idiìthta san ˆskhsh. Pr¸ta ac jumhjoÔme ìti mporoÔme na upologÐsoume to pragmatikì mèroc enìc migadikoÔ arijmoÔ ìpou h gramm  pˆnw apì ton

z

pinˆkwn. Eˆn o

P

a + ib = a − ib. 'Etsi paÐrnoume ton suzug  migadikì a ¯ = a. Parìmoia mporoÔme na pˆroume ton x=P~ x = P~ x  tìte P = P . Shmei¸ste epÐshc ìti P ~

shmaÐnei

ìti gia kˆje pragmatikì arijmì

a

èqoume ìti

eÐnai pragmatikìc

z

arijmì tou

wc ex c

z.

z+¯ z , 2

Shmei¸ste

suzug  dianusmˆtwn kai

¯ v. (P − λI)~v = (P − λI)~ Sunep¸c an to

~v

eÐnai èna idiodiˆnusma pou antistoiqeÐ sthn idiotim 

a − ib. t¸ra ìti a + ib

a + ib,

tìte to

~v

eÐnai èna idiodiˆnusma pou

antistoiqeÐ sthn idiotim  Ac upojèsoume

eÐnai mia migadik  idiotim  tou

P

h opoÐa antistoiqeÐ sto idiodiˆnusma

~v

opìte kai

èqoume ìti h

~ x1 = ~v e(a+ib)t eÐnai mia lÔsh (me migadikèc timèc) tou

0

~ x = P~ x.

Tìte parathr ste ìti kai h

~ x2 = ~ x1 = ~v e(a−ib)t

ea+ib = ea−ib

kai sunep¸c kai h

3.3. ̟ΕΘΟΔΟΣ ΙΔΙΟΤΙ̟ΩΝ

73

eÐnai epÐshc lÔsh. H sunˆrthsh

~ x3 = Re ~ x1 = Re ~v e(a+ib)t =

~ x1 + ~ x1 ~ x1 + ~ x2 = 2 2

eÐnai kai aut  lÔsh kai mˆlista me pragmatikèc timèc. Parìmoia mia kai to

~ x4 = Im ~ x1 =

z−¯ z eÐnai to fantastikì mèroc h 2i

Im z =

~ x1 − ~ x2 . 2i

eÐnai mia lÔsh pragmatik¸n tim¸n epÐshc. EÔkola mporoÔme na diapist¸soume ìti oi

~ x3

kai

~ x4

eÐnai grammikˆ anexˆrtht-

ec. Epistrèfontac t¸ra sto prìblhmˆ mac èqoume

     t 
i (1−i)t i ie cos t − et sin t e = et cos t + iet sin t = t . 1 1 e cos t + iet sin t

~ x1 =

EÐnai eÔkolo na diapist¸soume ìti oi

 −et sin t , t e cos t  t  e cos t Im ~ x1 = t , e sin t 

Re ~ x1 =

eÐnai oi lÔseic pou yˆqnoume. H genik  lÔsh eÐnai

 ~ x = c1 'Otan ta

c1

kai

c2

  t    −et sin t e cos t −c1 et sin t + c2 et cos t + c = . 2 et cos t et sin t c1 et cos t + c2 et sin t

eÐnai pragmatikoÐ arijmoÐ tìte h lÔsh aut  èqei pragmatikèc timèc. MporoÔme t¸ra na lÔsoume wc

proc opoiesd pote arqikèc sunj kec mac dojoÔn wc ex c. 'Otan èqoume migadikèc idiotimèc tìte autèc èrqontai se suzug  zeÔgh.

λ = a + ib

kai brÐskoume to antÐstoiqo idiodiˆnusma

~v .

Shmei¸ste ìti ta

Epilègoume mia apì autèc, èstw thn

Re ~v e(a+ib)t

kai

Im ~v e(a+ib)t

eÐnai epÐshc

lÔseic thc exÐswshc kai mˆlista grammikˆ anexˆrthta metaxÔ touc kai me pedÐo tim¸n touc pragmatikoÔc. SuneqÐzoume thn diadikasÐa epilègontac to èna apì to epìmeno zeÔgoc twn migadik¸n idiotim¸n   thn epìmenh pragmatik  idiotim  kai upologÐzw ta antÐstoiqa idiodianÔsmata. Me ton trìpo autì oi idiotimèc ja mac odhg soun se

n

n

diaforetikèc metaxÔ touc (pragmatikèc   migadikèc)

grammikˆ anexˆrthtec lÔseic.

MporoÔme loipìn t¸ra na broÔme genikèc lÔseic me pedÐo tim¸n touc pragmatikoÔc kˆje omogenoÔc sust matoc tou opoÐou o pÐnakac èqei diaforetikèc metaxÔ touc idiotimèc. H perÐptwsh pou upˆrqoun idiotimèc kˆpoiac pollaplìthtac eÐnai pio perÐplokh kai me aut n ja asqolhjoÔme sthn parˆgrafo

3.3.4

§3.6.

Ask seic

3×3

3.3.3

'Estw o

3.3.4

a) BreÐte thn genik  lÔsh tou sust matoc

pÐnakac

A

me mia idiotim  3 kai antÐstoiqo idiodiˆnusma

(grˆyte pr¸ta to sÔsthma sthn morf 

0

~ x = A~ x).

x01 = 2x1 , x02 = 3x2

1 −1 . BreÐte to 3

h

~v =

i

A~v .

qrhsimopoi¸ntac thn mèjodo twn idiotim¸n

b) LÔste to sÔsthma lÔnontac thn kˆje exÐswsh mình thc kai

sugkrÐnatè thn genik  lÔsh pou ja breÐte me ton trìpo autì me aut  pou br kate parapˆnw.

3.3.5

BreÐte thn genik  lÔsh tou sust matoc

x01 = 3x1 + x2 , x02 = 2x1 + 4x2

qrhsimopoi¸ntac thn mèjodo twn

x01 = x1 − 2x2 , x02 = 2x1 + x2

qrhsimopoi¸ntac thn mèjodo twn

idiotim¸n.

3.3.6

BreÐte thn genik  lÔsh tou sust matoc

idiotim¸n. Prospaj ste ètsi ¸ste h lÔsh sac na mhn sumperilambˆnei migadikˆ ekjetikˆ.

3.3.7

a) BreÐte tic idiotimèc kai ta idiodianÔsmata tou pÐnaka

sust matoc

3.3.8

h

A =

0

~x = A~ x.

BreÐte tic idiotimèc kai ta idiodianÔsmata tou pÐnaka

h −2 −1 −1 i 3 2 1 −3 −1 0

.

9 −2 −6 −8 3 6 . 10 −2 −6

i

b) BreÐte thn genik  lÔsh tou

74

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

3.4

Sust mata dÔo diastˆsewn kai ta dianusmatikˆ pedÐa touc

Ac epikentrwjoÔme gia lÐgo se omogen  sust mata sto epÐpedo. Sugkekrimèna ac exetˆsoume thn morf  twn dianusmatik¸n pedÐwn kai thn exˆrths  touc apì tic idiotimèc. 'Eqoume loipìn ènan

2×2

pÐnaka

P

kai to ex c sÔsthma

   0 x x =P . y y

(3.2)

Ac doÔme pwc mporoÔme, me bˆsh tic idiotimèc kai ta idiodianÔsmata tou pÐnaka, na anakalÔyoume optikˆ pwc moiˆzei to dianusmatikì pedÐo. PerÐptwsh 1. -3

-2

-1

0

1

2

3

3

3

Upojètoume ìti oi idiotimèc eÐnai prag-

matikèc kai jetikèc.

Ac broÔme tic dÔo idiotimèc kai ac

kˆnoume thn grafik  touc parˆstash. Gia parˆdeigma, ac jewr soume ton ex c pÐnaka 2

2

[ 10 12 ].

Oi idiotimèc eÐnai 1 kai 2

kai ta antÐstoiqa idiodianÔsmata eÐnai ta

[ 10 ]

kai

[ 11 ].

DeÐte

to Sq ma 3.3. 1

1

x kai y eÐnai suneujeiakˆ me ~v kˆpoiac idiotim c [ xy ] = a~v gia kˆpoio arijmì a. Tìte

Ac upojèsoume t¸ra ìti ta

thn gramm  pou orÐzei èna idiodiˆnusma 0

0

λ.

Dhlad  èqoume

èqoume -1

-1

-2

-2

-3

-3

 0   x x =P = P (a~v ) = a(P ~v ) = aλ~v . y y H parˆgwgoc loipìn eÐnai pollaplˆsio tou

-3

-2

-1

0

1

2

3

kajorÐzetai se megˆlo bajmì apì to

~v .

~v

kai sunep¸c

'Otan

λ > 0,

h

dieÔjunsh thc parag¸gou tautÐzetai me thn dieÔjunsh tou

~v

Σχήμα 3.3: Τα ιδιοδιανύσματα του P .

ìtan to

a

eÐnai jetikì, en¸ ìtan eÐnai arnhtikì h dieÔ-

junsh eÐnai akrib¸c antÐjeth.

Ac qrhsimopoi soume bèl-

h gia na parast soume tic kateujÔnseic.

DeÐte to Sq -

ma 3.4 sthn paroÔsa selÐda. Sumplhr¸noume to sq ma prosjètontac merikˆ akìma bèlh kai prosjètontac thn grafik  parˆstash merik¸n lÔsewn.

DeÐte to Sq ma 3.5 sthn epìmenh selÐda.

Parathr ste ìti h eikìna faÐnetai san na upˆrqei mia phg  apì

thn opoÐa xephdˆn bèlh. Gia ton lìgo autì onomˆzoume to tÔpo autì thc grafik c parˆstashc phg    merikèc forèc astaj  kìmboastaj c kìmboc. PerÐptwsh 2.

'Estw t¸ra ìti kai oi dÔo idiotimèc eÐnai arnhtikèc.

thc perÐptwshc 1 me allagmèna ìla tou ta prìshma, idiodianÔsmata eÐnai ta Ðdia, dhlad 

[ 10 ]

kai

[ 11 ].

 −1 −1 

0 −2 .

Gia parˆdeigma, ac jewr soume ton pÐnaka

Oi idiotimèc tou eÐnai

−1

kai

−2

kai ta antÐstoiqa

Tìso oi upologismoÐ mac ìso kai oi eikìnec eÐnai parìmoiec me thn

perÐptwsh 1. H mình diaforˆ eÐnai to ìti oi idiotimèc eÐnai arnhtikèc kai sunep¸c h forˆ twn bel¸n èqei antistrafeÐ. PaÐrnoume loipìn thn eikìna tou Sq matoc 3.6 sthn paroÔsa selÐda.

Kˆje tètoiou eÐdouc eikìna thn onomˆzoume

katabìjra h merikèc forèc eustaj  kìmboeustaj c kìmboc. PerÐptwsh 3.

'Estw ìti mia idiotim  eÐnai jetik  kai h ˆllh arnhtik , ìpwc gia parˆdeigma o pÐnakac

opoÐoc èqei idiotimèc 1 kai

−2

kai antÐstoiqa idiodianÔsmata ta Ðdia me parapˆnw, dhlad 

[ 10 ]

kai



1 −3 .



1

1 0 −2



Antistrè-

foume loipìn ta bèlh se mia gramm  (aut  pou antistoiqeÐ sthn arnhtik  idiotim ) kai paÐrnoume thn eikìna tou Sq matoc 3.7 sthn epìmenh selÐda. Eikìnec autoÔ tou eÐdouc onomˆzontai sagmatikˆ shmeÐasagmatikì shmeÐo. Stic epìmenec treic peript¸seic ja upojèsoume ìti oi idiotimèc eÐnai migadikèc. BebaÐwc stic peript¸seic autèc kai ta idiodianÔsmata eÐnai migadikˆ kai sunep¸c den ja mporèsoume na kˆnoume thn grafik  touc parˆstash sto epÐpedo.

PerÐptwsh 4. Upojètoume ìti oi idiotimèc eÐnai fantastikèc, dhlad  èqoun thn morf 

P =



0 1 −4 0



oi idiotimèc tou opoÐou eÐnai

±2i

kai ta idiodianÔsmata

1 [ 2i ]

kai



±ib.

'Estw loipìn o pÐnakac

1 −2i . Ac jewr soume thn idiotim 



2i

kai

3.4. ΣΥΣԟΗΜΑΤΑ ğΥΟ ΔΙΑΣԟΑΣΕΩΝ ΚΑΙ ΤΑ ΔΙΑΝΥΣΜΑΤΙʟΑ ΠΕğΙΑ ΤΟΥΣ -3

-2

-1

0

1

2

3

-3

-2

-1

0

1

75 2

3

3

3

3

3

2

2

2

2

1

1

1

1

0

0

0

0

-1

-1

-1

-1

-2

-2

-2

-2

-3

-3

-3 -3

-2

-1

0

1

2

3

Σχήμα 3.4: Τα ιδιοδιανύσματα του P με τις κατευθύνσεις. -3

-2

-1

0

1

2

-3 -3

-2

-1

0

1

2

3

Σχήμα 3.5: Παράδειγμα διανυσματικού πεδίου πηγής μαζί με τα ιδιοδιανύσματα και τις λύσεις.

3

-3

-2

-1

0

1

2

3

3

3

3

3

2

2

2

2

1

1

1

1

0

0

0

0

-1

-1

-1

-1

-2

-2

-2

-2

-3

-3

-3 -3

-2

-1

0

1

2

3

Σχήμα 3.6: Παράδειγμα διανυσματικού πεδίου καταβόθρας μαζί με τα ιδιοδιανύσματα και τις λύσεις.

to idiodiˆnusmˆ thc

1 [ 2i ]

-3 -3

-2

-1

0

1

2

3

Σχήμα 3.7: Παράδειγμα διανυσματικού πεδίου σαγματικού σημείου μαζί με τα ιδιοδιανύσματα και τις λύσεις.

kai shmei¸ste ìti to pragmatikì kai fantastikì mèroc thc

~v ei2t

eÐnai

    1 i2t cos 2t Re e = , 2i −2 sin 2t     1 i2t sin 2t Im e = . 2i 2 cos 2t MporoÔme bebaÐwc na pˆroume ìpoio grammikì sunduasmì touc jèloume. epilèxoume ja mac to kajorÐsoun oi arqikèc sunj kec.

Ton poiìn sugkekrimèna apì autoÔc ja

Gia parˆdeigma, to pragmatikì mèroc eÐnai mia parametrik 

exÐswsh èlleiyhc ìpwc kai to fantastikì mèroc kai sunep¸c kˆje grammikìc sunduasmìc touc.

Den eÐnai dÔskolo

na suneidhtopoi soume ìti ta parapˆnw isqÔoun genikˆ gia thn perÐptwsh pou èqoume kajarˆ fantastikèc idiotimèc opìte kai lème ìti èqoume lÔseic elleiptik¸n (dianusmatik¸n pedÐwn).

O tÔpoc autìc thc eikìnac suqnˆ lègete

76

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

kèntro. DeÐte to Sq ma 3.8. -3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

3

3

3

3

2

2

2

2

1

1

1

1

0

0

0

0

-1

-1

-1

-1

-2

-2

-2

-2

-3

-3

-3 -3

-2

-1

0

1

2

3

Σχήμα 3.8: Παράδειγμα διανυσματικού πεδίου κέντρου.

-3 -3

-2

-1

0

1

2

3

Σχήμα 3.9: Παράδειγμα διανυσματικού πεδίου σπειροειδούς πηγής.

PerÐptwsh 5. 'Estw t¸ra ìti oi idiotimèc èqoun jetikì pragmatikì mèroc. 'Estw loipìn ìti oi idiotimèc eÐnai gia kˆpoio eÐnai

1 [ 2i ]

a > 0. Gia  1

kai

−2i .

fantastikˆ mèrh thc

 1 1 P = −4 1 oi idiotimèc tou 1 1 + 2i kai to idiodiˆnusma [ 2i ]

1 ± 2i

a ± ib

parˆdeigma, èstw o pÐnakac

opoÐou eÐnai

PaÐrnoume thn idiotim 

kai shmei¸noume ìti ta pragmatikˆ kai

~v e(1+2i)t

kai ta idiodianÔsmata

eÐnai

    1 (1+2i)t cos 2t e = et , 2i −2 sin 2t     1 (1+2i)t sin 2t Im e = et . 2i 2 cos 2t Re

Parathr ste to

et

sthn arq  thc lÔshc.

Autì shmaÐnei ìti oi lÔseic auxˆnoun se mègejoc speiroeid¸c ìso apo-

makrÔnontai apì thn arq  twn axìnwn. Sunep¸c èqoume mia speiroeid  phg speiroeid c phg . DeÐte to Sq ma 3.9. PerÐptwsh 6. Tèloc ac jewr soume thn perÐptwsh twn migadik¸n idiotim¸n me arnhtikì akèraio mèroc. Dhlad 

 −a ± ib 1

ja èqoume idiotimèc thc morf c

−1 ± 2i kai ta idiodianÔsmata −2i kai fantastikì mèroc thc ~ v e(1+2i)t

kai

gia kˆpoio

1 [ 2i ].

a > 0.

Gia parˆdeigma èstw o pÐnakac

PaÐrnoume thn idiotim 

P =

 −1 −1  4 −1

oi idiotimèc tou

1 −1 − 2i kai to idiodiˆnusma [ 2i ] kai to pragmatikì

ta opoÐa eÐnai

    1 (−1−2i)t cos 2t −t Re e =e , 2i 2 sin 2t     1 (−1−2i)t − sin 2t Im e = e−t . 2i 2 cos 2t Parathr ste to

e−t

sthn arq  thc lÔshc. Autì shmaÐnei ìti oi lÔseic elatt¸noun to mègejìc touc ìso apomakrÔnon-

tai speiroeid¸c apì thn arq  twn axìnwn.

'Ara èqoume mia speiroeid  katabìjraspeiroeid c katabìjra.

DeÐte to

Sq ma 3.10 sthn paroÔsa selÐda. Ac sunoyÐsoume thn sumperiforˆ twn grammik¸n omogen¸n disdiˆstatwn susthmˆtwn ston ex c PÐnaka 3.1.

3.4.1

3.4.1

k > 0.

Ask seic

jewr ste thn ex c exÐswsh enìc sust matoc mˆzac elathrÐou

mx00 + cx0 + kx = 0,

a) Metatrèyte thn exÐswsh aut  se sÔsthma exis¸sewn pr¸thc tˆxhc.

me

m > 0, c ≥ 0

kai

b) KajorÐste thn sumperiforˆ tou

3.4. ΣΥΣԟΗΜΑΤΑ ğΥΟ ΔΙΑΣԟΑΣΕΩΝ ΚΑΙ ΤΑ ΔΙΑΝΥΣΜΑΤΙʟΑ ΠΕğΙΑ ΤΟΥΣ -3

-2

-1

0

1

2

77

3

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3 -3

-2

-1

0

1

2

3

Σχήμα 3.10: Παράδειγμα διανυσματικού πεδίου σπειροειδούς καταβόθρας.

Ιδιοτιμές Πραγματικές και αμφότερες θετικές Πραγματικές και αμφότερες αρνητικές Πραγματικές και με αντίθετα πρόσημα Καθαρά φανταστικές Μιγαδικές με θετικό ακέραιο μέρος Μιγαδικές με αρνητικό ακέραιο μέρος

Συμπεριφορά πηγή / ασταθής κόμβος καταβόθρα / ευσταθής κόμβος σαγματικό ςεντερ ποιντ / έλλειψη σπειροειδής πηγή σπειροειδής καταβόθρα

Πίνακας 3.1: Σύνοψη της συμπεριφοράς γραμμικών ομογενών δισδιάστατων συστημάτων.

sust matoc gia diaforetikèc timèc twn

m, c, k.

c) MporeÐte na exhg sete, me bˆsh thn fusik  sac antÐlhyh, giatÐ

emfanÐzontai ìlec ta diaforetikˆ eÐdh twn sumperifor¸n thc lÔshc sto en lìgw sÔsthma?

3.4.2

MporeÐte na anakalÔyete ti sumbaÐnei sthn perÐptwsh pou

P = [ 10 11 ]

ìpou èqoume mia dipl  idiotim  kai mìnon

èna idiodiˆnusma? Pwc ja moiˆzei h antÐstoiqh eikìna?

3.4.3 toiqeÐ?

MporeÐte na anakalÔyete ti sumbaÐnei sthn perÐptwsh pou

P = [ 11 11 ].

Se poia apì tic gnwstèc eikìnec antis-

78

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

3.5

Sust mata deÔterhc tˆxhc kai efarmogèc

3.5.1

Sust mata mˆzac elathrÐou qwrÐc ustèrhsh

Parìlo pou ìpwc èqoume anafèrei, sun jwc exetˆzoume sust mata pr¸thc tˆxhc mìnon, merikèc forèc eÐnai idiaÐtera apotelesmatikì na meletoÔme fainìmena ìpwc akrib¸c emfanÐzontai sthn prˆxh kai ta opoÐa endeqomènwc na eÐnai deÔterhc tˆxhc. Gia parˆdeigma, èstw ìti èqoume 3 swmatÐdia sundedemèna me elat ria metaxÔ dÔo toÐqwn. Profan¸c to pl joc twn swmatidÐwn (kai twn elathrÐwn) na eÐnai polÔ megalÔtero allˆ qˆrin aplìthtac ac perioristoÔme sta 3.

Ac upojèsoume epÐshc ìti h trib  eÐnai amelhtèa, dhlad  èqoume èna sÔsthma qwrÐc ustèrhsh.

ta swmatÐdia èqoun mˆza

m1 , m2 ,

kai

m3

h metatìpish tou pr¸tou swmatidÐou apì thn jèsh isorropÐac en¸ ta kai trÐtou swmatidÐou. metatìpishc (ìso to

x1

'Estw epÐshc ìti

k1 , k2 , k3 , kai k4 . 'Estw tèloc ìti x1 eÐnai x2 kai x3 eÐnai oi metatìpishc tou deÔterou

kai ta elat ria èqoun stajerèc

Ac jewr soume ìpwc sunhjÐzoume, ìti h kÐnhsh proc ta dexiˆ antistoiqeÐ se jetikèc timèc auxˆnei tìso to pr¸to swmatÐdio metakineÐtai dexiˆ). DeÐte to Sq ma 3.11.

k1

k2

k3

m1

k4

m2

m3

Σχήμα 3.11: Σύστημα σωματιδίων με ελατήρια. To aplì autì sÔsthma emfanÐzetai se aprosdìkhta Ðswc pollèc efarmogèc me parˆxeno trìpo.

'Iswc autì na

ofeÐletai sto ìti o kìsmoc mac apoteleÐtai ousiastikˆ apì ˆpeira to pl joc mikrˆ allhlepidr¸nta swmatÐdia   tm mata Ôlhc. Ac epikentrwjoÔme se exis¸seic pou aforoÔn sust mata tri¸n swmatidÐwn. Me bˆsh ton nìmo tou

Hooke

èqoume

ìti h dÔnamh pou efarmìzei èna elat rio se kˆpoio swmatÐdio isoÔtai me to ginìmeno thc stajerˆc tou elathrÐou epÐ thn metatìpis  tou. Me bˆsh ton deÔtero nìmo tou NeÔtwna deÔtero nìmo tou NeÔtwna h dÔnamh isoÔtai me thn mˆza epÐ thn epitˆqunsh. Sunep¸c, eˆn ajroÐsoume tic dunˆmeic pou askoÔntai se kˆje swmatÐdio, prosèqontac na bˆloume to swstì prìshmo se kˆje ìro anˆloga me thn kateÔjunsh sthn opoÐa dra h dÔnamh, katal goume sto ex c sÔsthma exis¸sewn.

m1 x001 = −k1 x1 + k2 (x2 − x1 )

= −(k1 + k2 )x1 + k2 x2 ,

m2 x002 = −k2 (x2 − x1 ) + k3 (x3 − x2 )

= k2 x1 − (k2 + k3 )x2 + k3 x3 ,

m3 x003

= k3 x2 − (k3 + k4 )x3 .

= −k3 (x3 − x2 ) − k4 x3

Ac orÐsoume touc pÐnakec



m1 M = 0 0

0 m2 0

 0 0  m3



kai

−(k1 + k2 ) k2 K= 0

k2 −(k2 + k3 ) k3

 0  k3 −(k3 + k4 )

kai ac grˆyoume tic parapˆnw exis¸seic sthn ex c morf 

M~ x 00 = K~ x. Sto shmeÐo autì ja mporoÔsame na eisagˆgoume 3 nèec metablhtèc kai na katal xoume se èna sÔsthma 6 exis¸sewn pr¸thc tˆxhc. MporoÔme na isqurisjoÔme ìti to nèo sÔsthma eÐnai pio aplì na epilujeÐ sugkrinìmeno me to arqikì sÔsthma deÔterhc tˆxhc. Ac onomˆsoume to

3.5.1

~ x

diˆnusma metatìpishc, to

Epanalˆbate thn parapˆnw diadikasÐa (breÐte touc pÐnakec

perigrˆyte thn perÐptwsh twn O antÐstrofoc tou pÐnaka

n M

M

kai

swmatidÐwn? eÐnai o

 M

−1

1 m1

= 0 0

0 1 m2

0

 0 0  1 m3

M

pÐnaka mˆzac, kai to

K)

K

pÐnaka akamyÐac.

gia 4 swmatÐdia kai gia 5 swmatÐdia kai

3.5. ΣΥΣԟΗΜΑΤΑ ΔşΥΤΕΡΗΣ ΤŸΑΞΗΣ ΚΑΙ ΕΦΑΡΜΟßΕΣ opìte jètontac

A = M −1 K

èqoume to sÔsthma sthn ex c morf 

79

~ x 00 = M −1 K~ x,

  sthn ex c

~ x 00 = A~ x. Pollˆ sust mata tou pragmatikoÔ kìsmou mac mporoÔn na montelopoihjoÔn me aut  thn exÐswsh. Gia lìgouc aploÔsteushc, ja asqolhjoÔme mìno me probl mata maz¸n-elathrÐwn. Ac prospajoÔme mia lÔsh thc morf c

~ x = ~v eαt . Gia thn manteyiˆ mac aut n èqoume,

~ x 00 = α2~v eαt .

Antikajist¸ntac èqoume ìti

α2~v eαt = A~v eαt . Diair¸ntac me

eαt

èqoume

α2~v = A~v .

Sunep¸c mia idiotim  tou

A

eÐnai

α2

kai to antÐstoiqo idiodiˆnusma eÐnai

~v

kai

ˆra br kame mia lÔsh. Sto parˆdeigmˆ mac, ìpwc kai se pollèc ˆllec peript¸seic, o prokÔpton pÐnakac

A

èqei arnhtikèc pragmatikèc

idiotimèc (mazÐ endeqomènwc me mia mhdenik  idiotim ). Ac melet soume loipìn mìnon thn perÐptwsh aut  ed¸. 'Otan mia idiotim 

λ

tètoioc ¸ste

eÐnai arnhtik , shmaÐnei ìti to

2

−ω = λ.

Tìte

α = ±iω .

α2 = λ

eÐnai arnhtikì. Sunep¸c upˆrqei kˆpoioc pragmatikìc arijmìc

ω

kai h lÔsh pou mantèyame eÐnai

~ x = ~v (cos ω t + i sin ω t). PaÐrnontac pragmatikˆ kai fantastikˆ mèrh (shmei¸ste ìti to

~v sin ω t

Eˆn mia idiotim  eÐnai mhdèn, tìte ta

3.5.2

~v

eÐnai pragmatikì), brÐskoume ìti ta

~v cos ω t

kai

eÐnai grammikˆ anexˆrthtec lÔseic.

DeÐxte ìti an o pÐnakac

lÔsh tou

~x 00 = A~ x

ìpou

a

kai

~v

kai

~v t

eÐnai epÐshc lÔseic eˆn to

A èqei mia mhdenik  tim  b eÐnai tuqaÐec stajerèc.

me

~v

~v

eÐnai to antÐstoiqo idiodiˆnusma.

to antÐstoiqo idiodiˆnusma, tìte to

~ x = ~v (a + bt)

eÐnai

'Estw o n × n pÐnakac A o opoÐoc èqei n diaforetikèc metaxÔ touc arnhtikèc idiotimèc tic opoÐec −ω12 > −ω22 > · · · > −ωn2 , en¸ ta antÐstoiqa idiodianÔmatˆ touc me ~v1 , ~v2 , . . . , ~vn . An o A eÐnai (opìte kai èqoume ìti ω1 > 0), tìte

Je¸rhma 3.5.1. sumbolÐzoume me antistrèyimoc

~ x(t) =

n X

~vi (ai cos ωi t + bi sin ωi t),

i=1

eÐnai h genik  lÔsh tou

~ x 00 = A~ x, gia kˆpoiec stajerèc

ai

kai

bi .

An o

A

èqei mia mhdenik  idiotim , dhlad 

ω1 = 0 ,

en¸ ìlec oi ˆllec idiotimèc eÐnai

arnhtikèc kai diaforetikèc metaxÔ touc tìte h genik  lÔsh èqei thn ex c morf 

~ x(t) = ~v1 (a1 + b1 t) +

n X

~vi (ai cos ωi t + bi sin ωi t).

i=2

Shmei¸ste ìti mporoÔme na qrhsimopoi soume to parapˆnw je¸rhma gia na broÔme thn genik  lÔsh problhmˆtwn ìpwc autˆ pou perigrˆyame sthn eisagwg  aut c thc paragrˆfou, akìma kai sthn perÐptwsh pou kˆpoia apì ta swmatÐdia  /kai kˆpoia apì ta elat ria den upˆrqoun.

Gia parˆdeigma, ìtan mac dojeÐ ìti èqoume 2 swmatÐdia me 2

elat ria, aplˆ paÐrnoume mìnon tic exis¸seic pou analogoÔn sta swmatÐdia kai jètoume Ðsec me mhdèn ìlec tic stajerèc twn elathrÐwn pou den upˆrqoun.

80

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

k1

k2 m1

m2

Σχήμα 3.12: Σύστημα μάζας ελατηρίου.

3.5.2

ParadeÐgmata

Parˆdeigma 3.5.1: kai ìpou

'Estw ìti èqoume to sÔsthma pou perigrˆfetai sto Sq ma 3.12, ìpou

m1 = 2, m2 = 1, k1 = 4,

k2 = 2 .

Oi exis¸seic pou dièpoun to sÔsthma eÐnai oi ex c

 2 0

  0 00 −(4 + 2) ~ x = 1 2

 2 ~ x, −2

 

 −3 2

~ x 00 = Oi idiotimèc tou

A

eÐnai

λ = −1, −4

 1 ~ x. −2

(ˆskhsh) kai ta idiodianÔsmata

kai



1 −1



antÐstoiqa (ˆllh ˆskhsh).

ω2 = 2. Sunep¸c h genik  lÔsh     1 1 ~ x= (a1 cos t + b1 sin t) + (a2 cos 2t + b2 sin 2t) . 2 −1

Me bˆsh to parapˆnw je¸rhma èqoume ìti

ω1 = 1

[ 12 ]

kai

eÐnai

Oi dÔo ìroi thc lÔshc paristoÔn tic dÔo fusikoÔc trìpouc talˆntwshc kai oi dÔo (gwniakèc) suqnìthtec eÐnai oi . H grafik  parˆstash twn dÔo aut¸n ìrwn dÐdetai sto Sq ma 3.13. 0.0

2.5

5.0

7.5

10.0

0.0

2.5

5.0

7.5

10.0

2

2

1.0

1.0

1

1

0.5

0.5

0

0

0.0

0.0

-1

-1

-0.5

-0.5

-2

-1.0

-2 0.0

2.5

5.0

7.5

10.0

-1.0 0.0

2.5

5.0

7.5

10.0

Σχήμα 3.13: Οι δύο τρόποι ταλάντωσης ενός συστήματος μάζας ελατηρίου. Στα αριστερά τα σωματίδια κινούνται στην ίδια κατεύθυνση ενώ στα δεξιά σε αντίθετη κατεύθυνση. Ac diatup¸soume thn exÐswsh wc ex c

~ x=

    1 1 c1 cos(t − α2 ) + c cos(2t − α1 ). 2 −1 2

O pr¸toc ìroc,

    1 c1 cos(t − α1 ) c1 cos(t − α1 ) = , 2 2c1 cos(t − α1 )

3.5. ΣΥΣԟΗΜΑΤΑ ΔşΥΤΕΡΗΣ ΤŸΑΞΗΣ ΚΑΙ ΕΦΑΡΜΟßΕΣ

81

analogeÐ sthn perÐptwsh pou ta swmatÐdia kinoÔntai sugqronismèna sthn Ðdia kateÔjunsh. O deÔteroc ìroc,



   1 c2 cos(2t − α2 ) c2 cos(2t − α2 ) = , −1 −c2 cos(2t − α2 )

analogeÐ sthn perÐptwsh pou ta swmatÐdia kinoÔntai sugqronismèna allˆ se antÐjetec kateujÔnseic. H genik  lÔsh eÐnai ènac sunduasmìc twn dÔo aut¸n ìrwn.

Dhlad , h arqikèc sunj kec kajorÐzoun to plˆtoc

kai thn diaforˆ fˆshc tou kˆje ìrou.

Parˆdeigma 3.5.2:

Ac doÔme t¸ra èna ˆllo parˆdeigma ìpou èqoume dÔo bagonèta.

To èna èqei mˆza 2 kg kai

taxideÔei me taqÔthta 3 m/c proc to deÔtero to opoÐo èqei mˆza 1 kg. Sto deÔtero bagonèto upˆrqei ènac profulakt rac o opoÐoc aposbaÐnei bebaÐwc thn sÔgkroush kai tautìqrona ta sundèei (en¸nei ta dÔo bagonèta) kai metˆ ta af nei na kinhjoÔn eleÔjera. O profulakt rac mporeÐ na jewrhjeÐ san èna elat rio stajerˆc

k = 2 N/m.

To deÔtero bagonèto

apèqei apì ènan toÐqo 10 mètra. DeÐte to Sq ma 3.14.

k

m1

m2 10 mètra

Σχήμα 3.14: Σύγκρουση δύο βαγονέτων.

MporoÔme na jèsoume diˆfora erwt mata.

Se pìso qrìno metˆ thn ènws  touc ta bagonèta ja sugkroustoÔn

ston toÐqo? Poia ja eÐnai h taqÔthta tou deÔterou bagonètou ìtan ja sugkrousteÐ ston toÐqo? Ac kataskeuˆsoume to sÔsthma pr¸ta.

x1 h sunˆrthsh pou stigm  t = 0, kai èstw x2 h 'Estw

Ac upojèsoume ìti ta bagonèta en¸nontai thn qronik  stigm 

sunˆrthsh pou mac dÐnei thn apomˆkrunsh tou deÔterou swmatidÐou apì thn arqik  jèsh

tou. EÐnai safèc ìti h sÔgkroush ja epèljei akrib¸c ìtan taqÔthta bebaÐwc eÐnai

t = 0.

mac dÐnei thn apomˆkrunsh tou pr¸tou swmatidÐou apì thn jèsh pou eÐqe thn qronik 

x02 (t).

paradeÐgmatoc qwrÐc ìmwc to

x2 (t) = 10.

Gia thn qronik  stigm 

t

thc sÔgkroushc, h

To sÔsthma sumperifèretai akrib¸c ìpwc sumperiferìtan to sÔsthma tou prohgoumènou

k1 .

'Ara h exÐswsh eÐnai



  0 00 −2 ~ x = 1 2

2 0

 2 ~ x. −2

 

~ x 00 =



−1 2

 1 ~ x. −2 A eÐnai √ 0 kai −3 (ˆskhsh) en¸ ta idiodianÔsmata ω2 = 3 kai qrhsimopoioÔme to deÔtero mèroc tou

Den eÐnai dÔskolo na upologÐsoume ìti oi idiotimèc tou pÐnaka eÐnai

[ 11 ]

kai



1 −2



antÐstoiqa (ˆllh ˆskhsh).

Shmei¸noume ìti

jewr matoc gia na broÔme ìti h genik  lÔsh eÐnai

~x =

    √ √  1 1 (a1 + b1 t) + a2 cos 3 t + b2 sin 3 t = 1 −2

√ √  a1 + b1 t + a2 cos √3 t + b2 sin √ 3t a1 + b1 t − 2a2 cos 3 t − 2b2 sin 3 t

 =

Ac efarmìsoume t¸ra tic arqikèc sunj kec. Ta bagonèta xekinˆne apì to shmeÐo 0 dhlad  To pr¸to taxideÔei me taqÔthta 3 m/c, ˆra

x01 (0) = 3

x1 (0) = 0 kai x2 (0) = 0. x02 (0) = 0.

kai to deÔtero xekinˆ qwrÐc arqik  taqÔthta, dhlad 

Oi pr¸tec sunj kec mac dÐnoun ìti

~0 = ~ x(0) =



 a1 + a2 . a1 − 2a2

82

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

Den eÐnai dÔskolo na diapist¸soume ìti to parapˆnw mac dÐnei kai

a2

a1 = a2 = 0.

AntikajistoÔme tic timèc autèc twn

a1

kai paragwgÐzoume gia na pˆroume ìti

√ √  b1 + √3 b2 cos √3 t ~ x (t) = . b1 − 2 3 b2 cos 3 t 

0

'Ara

√     3 b1 + √3 b2 =~ x0 (0) = . 0 b1 − 2 3 b2

Den eÐnai dÔskolo na broÔme lÔnontac thn parapˆnw ìti

b1 = 2

na sugkrousjoÔn ston toÐqo)

"

2t + ~ x= 2t −

√1 3 √2 3

kai

b2 =

√1 . 'Ara h jèsh twn bagonètwn eÐnai (mèqri 3

√ # sin 3 t √ . sin 3 t

Shmei¸ste pwc h parousÐa thc mhdenik c idiotim c eÐqe san apotèlesma ènan ìro pou perièqei to

t.

Autì shmaÐnei ìti

ta bagonèta ja taxideÔoun sthn jetik  kateÔjunsh ìso o qrìnoc auxˆnei, prˆgma to opoÐo anamènoume bebaÐwc. Autì pou pragmatikˆ mac endiafèrei eÐnai h deÔterh èkfrash, aut  pou aforˆ to

2 √ 3

sin

√ 3 t.

DeÐte sto Sq ma 3.15 thn grafik  parˆstash tou

0

1

2

3

x2

x2 .

'Eqoume ìti

x2 (t) = 2t −

wc proc ton qrìno.

4

5

6

12.5

12.5

10.0

10.0

7.5

7.5

5.0

5.0

2.5

2.5

0.0

0.0 0

1

2

3

4

5

6

Σχήμα 3.15: Η θέση του δεύτερου βαγονέτου σαν συνάρτηση του χρόνου (αγνοώντας τον τοίχο). Parathr¸ntac to grˆfhma mporoÔme na doÔme ìti h sÔgkroush ja epèljei se perÐpou 5 deuterìlepta metˆ thn

√ 10 = x2 (t) = 2t− √23 sin 3 t ja broÔme ìti tsÔgkroush ≈ 5.22 √ 0 'Oson aforˆ thn taqÔthta èqoume ìti x2 = 2 − 2 cos 3 t. Thn stigm  thc sÔgkroushc (5.22 deuterìlepta apì thn 0 stigm  t = 0) brÐskoume ìti x2 (timpact ) ≈ 3.85. √ Epiprìsjeta parathroÔme ìti h mègisth dunat  taqÔthta isoÔtai me thn mègisth tim  thc parˆstashc 2−2 cos 3 t, qronik  stigm  mhdèn. Prˆgmati eˆn lÔsoume thn exÐswsh

dhlad  me 4. Sunep¸c h sÔgkroush gÐnetai me sqedìn thn mègisth taqÔthta. Ac upojèsoume t¸ra ìti èqoume thn dunatìthta na apomakrÔnoume to deÔtero bagonèto apì ton toÐqo (  na plhsiˆsoume ston toÐqo) qwrÐc ìmwc na mporoÔme na apofÔgoume thn epaf  me to pr¸to bagonèto. na apofÔgoume thn sÔgkroush me ton toÐqo metakin¸ntac to bagonèto?

MporoÔme

Pìso prèpei na to metakin soume gia na

katafèroume kˆti tètoio? Parathr¸ntac to Sq ma 3.15, blèpoume èna plat¸ metaxÔ twn qronik¸n stigm¸n

t=3

kai

t = 4.

EkeÐ upˆrqei

èna shmeÐo sto opoÐo h taqÔthta eÐnai mhdèn. Gia na broÔme autì to shmeÐo prèpei na lÔsoume thn exÐswsh

√ Dhlad  cos 3 t = 1 opìte èqoume t =

2π √ √ , 4π3 , . . . . Antikajist¸ntac thn pr¸th tim  paÐrnoume 3

x2



2π √ 3



=

x02 (t) = 0. 4π √ 3

≈ 7.26.

'Ara h asfal s apìstash eÐnai perÐpou 7,30 mètra apì ton toÐqo. Aut  bebaÐwc eÐnai kai h mikrìterh asfal c apìstash mia kai eˆn antikatast soume tic upìloipèc timèc tou tic opoÐec br kame ìti antistoiqoÔn se

x02 (t) = 0

8π 3

kai ˆllec asfaleÐc apostˆseic ìpwc h √

≈ 14.51.

t

gia

BebaÐwc kai

3.5. ΣΥΣԟΗΜΑΤΑ ΔşΥΤΕΡΗΣ ΤŸΑΞΗΣ ΚΑΙ ΕΦΑΡΜΟßΕΣ h

t=0

gia thn opoÐa èqoume

x2 = 0

83

kˆti pou antistoiqeÐ sthn perÐptwsh pou to deÔtero bagonèto akoumpˆei ston

toÐqo.

3.5.3

Exanagkasmènec Talant¸seic

Telei¸nontac ac asqolhjoÔme me thn perÐptwsh twn exanagkasmènwn talant¸sewn. 'Estw loipìn ìti to sÔsthmˆ mac eÐnai to ex c

~ cos ω t. ~ x 00 = A~ x+F

(3.3)

Dhlad  èqoume prosjèsei sto sÔsthma mia periodik  dÔnamh sthn kateÔjunsh tou dianÔsmatoc Profan¸c arkeÐ na broÔme mia sugkekrimènh lÔsh tontac sthn genik  lÔsh

~ xc

~ xp

~. F

tou parapˆnw mh omogenoÔc probl matoc thn opoÐa prosjè-

tou omogenoÔc probl matoc ja pˆroume thn genik  lÔsh tou parapˆnw mh-omogenoÔc

ω

probl matoc (3.3). An upojèsoume ìti to

den eÐnai Ðso me kˆpoia apì tic fusikèc suqnìthtec tou

~ x 00 = A~ x,

tìte

mporoÔme na mantèyoume ìti

~ xp = ~c cos ω t, ìpou

~c eÐnai kˆpoio ˆgnwsto stajerì diˆnusma.

Parathr ste ìti h manteyiˆ mac den perièqei hmÐtono mia kai h exÐsws 

mac emplèkei mìnon thn deÔterh parˆgwgo. Eˆn upologÐsoume to

~c èqoume

brei thn

~ xp .

Sthn ousÐa loipìn èqoume thn

mèjodo twn aprosdiìristwn suntelest¸n gia sust mata. ParagwgÐzoume dÔo forèc thn

~ xp

kai èqoume

~ xp 00 = −ω 2~c cos ω t. AntikajistoÔme t¸ra sthn exÐswsh

~ cos ω t −ω 2~c cos ω t = A~c cos ω t + F ApaleÐfontac to sunhmÐtono èqoume

~. (A + ω 2 I)~c = −F Opìte

~ ). ~c = (A + ω 2 I)−1 (−F (A + ω 2 I) = (A − (−ω 2 )I) eÐnai antistrèyimoc. O en lìgw pÐnakac 2 idiotim  tou A den eÐnai Ðsh me −ω . Autì sumbaÐnei an kai mìnon an to ω

Profan¸c prèpei na upojèsoume ìti o pÐnakac eÐnai antistrèyimoc an kai mìnon an kamiˆ

den eÐnai mia fusik  suqnìthta tou sust matoc.

Parˆdeigma 3.5.3:

Ac jewr soume to parˆdeigma tou Sq matoc 3.12 sth selÐda 80 me tic Ðdiec ìpwc kai pro-

hgoumènwc paramètrouc, dhlad : mia dÔnamh

2 cos 3t

m1 = 2, m2 = 1, k1 = 4,

kai

k2 = 2.

Upojètoume t¸ra ìti èqoume epiprìsjeta kai

h opoÐa dra sto deÔtero bagonèto.

H exÐswsh loipìn eÐnai h ex c

~ x

00

 −3 = 2

   1 0 ~ x+ cos 3t. −2 2

'Opwc èqoume  dh dei lÔnontac to analogoÔn omogenèc prìblhma h sumplhrwmatik  lÔsh eÐnai

~ xc =

    1 1 (a1 cos t + b1 sin t) + (a2 cos 2t + b2 sin 2t) . 2 −1

ParathroÔme ìti mia kai oi fusikèc suqnìthtec eÐnai 1 kai 2 kai bebaÐwc den eÐnai Ðsec me thn exwterik  suqnìthta h opoÐa isoÔtai me 3, mporoÔme na dokimˆsoume thn

~c cos 3t.

ParagwgÐzontac kai antikajist¸ntac katal goume sto

ex c grammikì algebrikì sÔsthma

~ (A + ω 2 I)~c = −F Dhlad 

"

−3

1

2

−2

#

! 2

+3 I

" ~c =

6

1

2

7

LÔnontac to parapˆnw sÔsthma paÐrnoume

" ~c =

1 20 −3 10

# .

#

" ~c =

0 −2

# .

84

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ EÔkola t¸ra katal goume sthn parakˆtw genik  lÔsh thc exÐswshc

~ x=~ xc + ~ xp = Oi stajerèc An to

ω

a1 , a2 , b1 ,

kai

b2

" # 1 2

" (a1 cos t + b1 sin t) +

1

−1

#

~ cos ω t ~ x 00 = A~ x+F " #

(a2 cos 2t + b2 sin 2t) +

1 20 −3 10

cos 3t.

ja prosdioristoÔn bebaÐwc qrhsimopoi¸ntac tic arqikèc sunj kec tou sust matoc.

sumpèsei me mia fusik  suqnìthta tou sust matoc tìte èqoume to fainìmeno tou suntonismoÔsuntonismìc

epeid  ja prèpei na prospaj soume mia sugkekrimènh lÔsh thc morf c (upojètontac ìti ìlec oi idiotimèc tou pÐnaka twn suntelest¸n eÐnai diaforetikèc metaxÔ touc)

~ xp = ~c t sin ω t + d~ cos ω t. Shmei¸ste ìti ìso to

3.5.4

3.5.3

t

auxˆnei tìso kai to plˆtoc talˆntwshc thc lÔshc aut c auxˆnei, qwrÐc kanèna ìrio.

Ask seic

BreÐte mia sugkekrimènh lÔsh tou sust matoc

~ x 00 = 3.5.4

−3 2

   1 0 ~ x+ cos 2t. −2 2

Jewr ste to parˆdeigma tou Sq matoc 3.12 sth selÐda 80 me tic Ðdiec timèc twn paramètrwn ìpwc kai pro-

hgoumènwc:

cos 5t



m1 = 2, k1 = 4,

kai

k2 = 2 ,

ektìc apì thn

dra sto pr¸to swmatÐdio. BreÐte mia tim  thc

m2

m2

thn tim  thc opoÐac den gnwrÐzoume. 'Estw ìti h dÔnamh

tètoia ¸ste na upˆrqei sugkekrimènh lÔsh gia thn opoÐa to

pr¸to swmatÐdio paramènei akÐnhto. ShmeÐwsh: H parapˆnw idèa eÐnai gnwst  me ton ìro dunamik  apìsbesh. Sthn prˆxh bebaÐwc pˆnta upˆrqei mia èstw kai mikr  apìsbesh h opoÐa ja akinhtopoi sei to pr¸to swmatÐdio se kˆje perÐptwsh, endeqomènwc metˆ apì arketˆ megˆlo qronikì diˆsthma.

3.5.5

Ac jewr soume to parˆdeigma 3.5.2 sth selÐda 81, upojètontac t¸ra ìti thn stigm  thc emplok c twn dÔo

bagonètwn to deÔtero bagonèto kineÐtai proc ta aristerˆ me taqÔthta 3 metˆ thn emplok .

m/c.

a) BreÐte thn sumperiforˆ tou sust matoc

b) Ja sugkrousjeÐ to deÔtero bagonèto ston toÐqo   ìso pernˆ o kairìc ja apomakrÔnetai apì

autìn? g) Me poia taqÔthta prèpei na kineÐtai to pr¸to ìqhma ètsi ¸ste to sÔsthma na parameÐnei akÐnhto metˆ thn emplok  twn dÔo oqhmˆtwn?

3.5.6

Ac jewr soume to parˆdeigma tou Sq matoc 3.12 sth selÐda 80 me paramètrouc

m1 = m2 = 1, k1 = k2 = 1.

Upˆrqei sunduasmìc arqik¸n sunjhk¸n gia ton opoÐo to pr¸to ìqhma kineÐtai en¸ to deÔtero paramènei akÐnhto? Eˆn upˆrqei breÐte ton kai eˆn den upˆrqei exhg ste giatÐ sumbaÐnei autì.

3.6. ΙΔΙΟΤΙ̟ΕΣ ΜΕ ΠΟΛΛΑΠ˟ΟΤΗΤΑ 3.6

85

Idiotimèc me pollaplìthta

MporeÐ fusikˆ na tÔqei ì pÐnakˆc mac na èqei idiotimèc pou epanalambˆnontai.

det(A − λI) = 0

mporeÐ na èqei epanalambanìmenec rÐzec.

na sumbeÐ gia kˆpoion tuqaÐo pÐnaka. tou

A)

Dhlad , h qarakthristik  exÐswsh

'Opwc  dh anafèrame, kˆti tètoio eÐnai mˆllon apÐjano

Eˆn diatarˆxoume lÐgo ton pÐnaka

A

(eˆn dhlad  allˆxoume lÐgo ta stoiqeÐa

ja pˆroume ènan pÐnaka me diaforetikèc metaxÔ touc idiotimèc. 'Opwc kai se kˆje sÔsthma ìmwc, jèloume na

elègxoume ti ja mporoÔse na sumbeÐ se oriakèc katastˆseic, anexˆrthta apì to gegonìc ìti stic katastˆseic autèc bebaÐwc mìnon asumptwtikˆ plhsiˆzoume.

3.6.1

Gewmetrik  pollaplìthta

Jewr ste ton ex c diag¸nio pÐnaka

 A= o opoÐoc profan¸c èqei idiotim  3 me pollaplìthta 2. qarakthristik c exÐswshc algebrik  pollaplìthta. anexˆrthta idiodianÔsmata, ta

[ 10 ]

[ 01 ].

kai

 0 . 3

3 0

Sun jwc onomˆzoume thn pollaplìthta twn idiotim¸n miac

Sthn perÐptwsh pou exetˆzoume t¸ra upˆrqoun dÔo grammikˆ

Autì shmaÐnei ìti h onomazìmenh gewmetrik  pollaplìthta tic idiotim c

aut c eÐnai 2. Se ìsa apì ta parapˆnw jewr mata apaitoÔsame o pÐnakac na èqei jèlame na èqoume

n

n diaforetikèc metaxÔ touc idiotimèc, ousiastikˆ ~ x 0 = A~ x èqei thn ex c genik 

grammikˆ anexˆrthta idiodianÔsmata. Gia parˆdeigma, to sÔsthma

lÔsh

~ x = c1

    1 3t 0 3t e + c2 e . 0 1

Ac epanadiatup¸soume to je¸rhma pou aforˆ pragmatikèc idiotimèc.

Sto parakˆtw je¸rhma ja epanalambˆnoume

kˆje idiotim  ìsec forèc eÐnai h (algebrik ) pollaplìthtˆ thc. Dhlad  gia ton parapˆnw pÐnaka

A

ja lème ìti èqei

idiotimèc to 3 kai to 3.

Je¸rhma 3.6.1.

'Estw

~ x 0 = P~ x.

An

P

eÐnai èna

n × n pÐnakac o opoÐoc èqei tic ex c n pragmatikèc idiotimèc (oi λ1 , . . . , λn , kai an se autèc antistoiqoÔn ta ex c n grammikˆ

opoÐec den eÐnai aparaÐthta diaforetikèc metaxÔ touc), anexˆrthta idiodianÔsmata

~v1 ,

...,

~vn ,

tìte h genik  lÔsh tou sust matoc SDE mporeÐ na grafjeÐ wc ex c

~ x = c1~v1 eλ1 t + c2~v2 eλ2 t + · · · + cn~vn eλn t . H gewmetrik  pollaplìthta miac idiotim c algebrik c pollaplìthtac

n

isoÔtai me to pl joc twn anexˆrthtwn

idiodianusmˆtwn thc pou mporoÔme na broÔme. EÐnai logikì na jewr soume ìti h gewmetrik  pollaplìthta eÐnai pˆnta mikrìterh   Ðsh me thn algebrik  pollaplìthta. Parapˆnw antimetwpÐsame thn perÐptwsh ìpou oi dÔo pollaplìthtec eÐnai Ðsec.

Sthn perÐptwsh aut  pou h algebrik  pollaplìthta isoÔtai me thn gewmetrik  pollaplìthta h en lìgw

idiotim  eÐnai pl rhc. 'Ara to parapˆnw je¸rhma mporeÐ na epanadiatupwjeÐ apait¸ntac ìlec oi idiotimèc tou ta

n

P

na eÐnai pl reic opìte kai

idiodianÔsmata ja eÐnai grammikˆ anexˆrthta kai sunep¸c ja èqoume thn genik  lÔsh pou perigrˆfei to je¸rhma.

Shmei¸ste ìti an h gewmetrik  pollaplìthta mia idiotim c eÐnai megalÔterh   Ðsh me 2, tìte to sÔnolo twn grammikˆ anexˆrthtwn idiodianusmˆtwn den eÐnai monadikì (wc proc kˆpoia stajerˆ) ìpwc anafèrame prohgoumènwc. Gia parˆdeigma gia ton diag¸nio pÐnaka

A

ja mporoÔsame na epilèxoume tic idiotimèc

[ 11 ]

kai



1 −1 ,   sthn pragmatikìthta



opoiad pote dÔo grammikˆ anexˆrthta dianÔsmata.

3.6.2 Eˆn ènac

AteleÐc idiotimèc

n×n

pÐnakac èqei ligìtera apì

n

grammikˆ anexˆrthta idiodianÔsmata, tìte lègete atel c. Sthn perÐptwsh

aut  upˆrqei toulˆqiston mia idiotim  thc opoÐac h algebrik  pollaplìthta eÐnai megalÔterh apì thn gewmetrik . Mia tètoia idiotim  lègetai atel c thn de diaforˆ twn pollaplot twn atèleia.

Parˆdeigma 3.6.1:

O pÐnakac



3 0

1 3



86

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

èqei thn idiotim  3 me algebrik  pollaplìthta 2. Ac prospaj soume na upologÐsoume ta idiodianÔsmata.

 0 0 Prèpei na èqoume

v2 = 0.

1 0

  v1 = ~0. v2

'Ara kˆje idiodiˆnusma prèpei na eÐnai thc morf c

[ v01 ].

Opoiad pote dÔo tètoia dianÔsmata

eÐnai grammikˆ exarthmèna kai sunep¸c h gewmetrik  pollaplìthta eÐnai 1. Sunep¸c, h atèleia eÐnai 1 kai den mporoÔme na efarmìsoume thn parapˆnw mejodologÐa gia na broÔme thn lÔsh tou sust matoc twn SDE ta opoÐa èqoun pÐnaka suntelest¸n me aut  thn idiìthta.

λ eÐnai mia idiotim  enìc pÐnaka A algebrik c polm grammikˆ anexˆrthta idiodianÔsmata lÔnontac thn exÐswsh (A − λI)m~v = ~0.

Mia kaÐria parat rhsh, kombik c spoudaiìthtac, eÐnai ìti eˆn laplìthta

m,

tìte mporoÔme na broÔme

Ta dianÔsmata autˆ onomˆzontai genikeumèna idiodianÔsmata.

A = [ 30 13 ]

Ac suneqÐsoume me to parˆdeigma

~ x 0 = A~ x. 'Eqoume mia idiotim  λ = 3 (algebrik c) idiodiˆnusma ~ v1 = [ 10 ] apì thn opoÐa paÐrnoume thn lÔsh

kai thn exÐswsh

pollaplìthtac 2 kai atèleia 1. 'Eqoume  dh brei èna

~ x1 = ~v1 e3t . Sthn perÐptwsh aut , ac dokimˆsoume (sto pneÔma twn epanalambanìmenwn riz¸n thc qarakthristik c exÐswshc mia SDE) mia lÔsh thc ex c morf c

~ x2 = (~v2 + ~v1 t) e3t . ParagwgÐzontac èqoume

~ x2 0 = ~v1 e3t + 3(~v2 + ~v1 t) e3t = (3~v2 + ~v1 ) e3t + 3~v1 te3t . To

~x2 0

prèpei na isoÔtai me

A~ x2 ,

opìte

A~ x2 = A(~v2 + ~v1 t) e3t = A~v2 e3t + A~v1 te3t . Parathr¸ntac touc suntelestèc twn

e3t

kai

te3t

(A − 3I)~v1 = ~0, ~v1

~ x2

3~v1 = A~v1 .

Autì shmaÐnei ìti

eÐnai mÐa lÔsh.

GnwrÐzoume ìti h pr¸th apì tic exis¸seic

eÐnai idiodiˆnusma. Antikajist¸ntac thn deÔterh exÐswsh sthn pr¸th brÐskoume ìti

(A − 3I)(A − 3I)~v2 = ~0, An mporoÔme loipìn na broÔme èna

(A − 3I)~v2 = ~v1 ,

kai

(A − 3I)~v2 = ~v1 .

kai

Eˆn autèc oi dÔo exis¸seic ikanopoioÔntai, tìte to ikanopoieÐtai epeid  to

3~v2 + ~v1 = A~v2

blèpoume ìti

~v2

tìte telei¸same.

(A − 3I)2~v2 = ~0.

or

to opoÐo na apoteleÐ lÔsh tou

(A − 3I)2~v2 = ~0,

kai na ikanopoieÐ thn sqèsh

Prèpei loipìn na lÔsoume dÔo algebrikˆ grammikˆ sust mata, prˆgma arketˆ

eÔkolo.

(A − 3I)2

Parathr ste ìti gia thn apl  perÐptwsh pou asqoloÔmaste o Sunep¸c, kˆje diˆnusma

~v2

eÐnai lÔsh tou

(A − 3I) ~v2 = ~0. 2

eÐnai ènac mhdenikìc pÐnakac (ˆskhsh).

'Ara arkeÐ na sigoureutoÔme ìti

(A − 3I)~v2 = ~v1 .

Ac

grˆyoume

 0 0 a = 0 (to a mporeÐ sust matoc ~ x 0 = A~ x eÐnai

Ac jèsoume tou

1 0

    a 1 = . b 0

na pˆrei ìpoia tim  jèloume) kai

~ x = c1

~v1

pou antistoiqeÐ sto

Opìte èqoume

~v2 = [ 01 ].

'Ara h genik  lÔsh

        3t  1 3t 0 1 c1 e + c2 te3t e + c2 + t e3t = . 3t 0 1 0 c2 e

Ac perigrˆyoume t¸ra ton genikì algìrijmo. Pr¸ta gia idiodiˆnusma

b = 1.

λ.

λ

pollaplìthtac 2 kai atèleiac 1. Pr¸ta brÐskoume to

Metˆ prèpei na broÔme èna

~v2

(A − 3I)2~v2 = ~0, (A − 3I)~v2 = ~v1 .

tètoio ¸ste

3.6. ΙΔΙΟΤΙ̟ΕΣ ΜΕ ΠΟΛΛΑΠ˟ΟΤΗΤΑ

87

Me ton trìpo autì paÐrnoume dÔo grammikˆ anexˆrthtec lÔseic

~ x1 = ~v1 eλt , ~ x2 = (~v2 + ~v1 t) eλt . H parapˆnw mejodologÐa mporeÐ na genikeujeÐ gia megalÔterouc pÐnakec kai megalÔterec atèleiec. Parìlo pou den ja asqolhjoÔme susthmatikˆ me to jèma ac d¸soume thn genik  idèa. 'Estw ìti o

m.

A èqei mia idiotim  λ pollaplìthtac

BrÐskoume dianÔsmata tètoia ¸ste

(A − λI)k ~v = ~0,

(A − λI)k−1~v 6= ~0.

allˆ

Ta dianÔsmata autˆ lègontai genikeumèna idiodianÔsmata. Gia kˆje idiodiˆnusma idiodianusmˆtwn

~v2 . . . ~vk

~v1

brÐskoume mia seirˆ genikeumènwn

tètoia ¸ste:

(A − λI)~v1 = ~0, (A − λI)~v2 = ~v1 , . . .

(A − λI)~vk = ~vk−1 . Kataskeuˆzoume tic grammikˆ anexˆrthtec lÔseic

~ x1 = ~v1 eλt , ~ x2 = (~v2 + ~v1 t) eλt , . . .

~ xk =

  tk−2 tk−1 ~vk + ~vk−1 t + · · · + ~v2 + ~v1 eλt . (k − 2)! (k − 1)!

ProqwroÔme sthn eÔresh seir¸n èwc ìtou kataskeuˆsoume

m grammikˆ anexˆrthtec lÔseic (m eÐnai h pollaplìthta).

Endeqomènwc na qreiasjeÐ na brejoÔn arketèc seirèc gia kˆje idiotim .

3.6.3

Ask seic

3.6.1

'Estw

A=

3.6.2

'Estw

A =

 5 −3 

3 −1 . LÔste thn exÐswsh

h

'Estw

A=

i

a) Poiec eÐnai oi idiotimèc?

b) Poia/poièc eÐnai oi atèleiec twn idiodianusmˆtwn?

c)

~ x = A~ x. h2 1 0i

LÔste thn exÐswsh

3.6.3

5 −4 4 0 3 0 . −2 4 −1 0

~ x 0 = A~ x.

0 2 0 . a) Poiec eÐnai oi idiotimèc? b) Poia/poièc eÐnai oi atèleiec twn idiodianusmˆtwn? g) LÔste 0 0 2

~x 0 = A~ x me dÔo diaforetikoÔc trìpouc kai epibebai¸ste ìti prˆgmati br kate swstˆ thn lÔsh. h 0 1 2 i −1 −2 −2 . a) Poiec eÐnai oi idiotimèc? b) Poia/poièc eÐnai oi atèleiec twn idiodianusmˆtwn? 'Estw A =

thn exÐswsh

3.6.4

−4 4

LÔste thn exÐswsh

3.6.5

'Estw

A =

LÔste thn exÐswsh

3.6.6

'Estw

A =

LÔste thn exÐswsh

3.6.7

'Estw

7

~ x 0 = A~ x. h 0 4 −2 i

g)

−1 −4 1 . 0 0 −2 0

a) Poiec eÐnai oi idiotimèc?

b) Poia/poièc eÐnai oi atèleiec twn idiodianusmˆtwn?

g)

−1 0 2 −1 −2 4 0

a) Poiec eÐnai oi idiotimèc?

b) Poia/poièc eÐnai oi atèleiec twn idiodianusmˆtwn?

g)

~ x = A~ x. h 2 1 −1 i

.

~ x = A~ x.

A ènac 2 × 2 pÐnakac me idiotim  λ pollaplìthtac 2.

'Estw epÐshc ìti upˆrqoun dÔo grammikˆ anexˆrthta

idiodianÔsmata. ApodeÐxte ìti o pÐnakac eÐnai diag¸nioc kai sugkekrimèna

A = λI .

88

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

3.7

Ekjetikˆ Pinˆkwn

3.7.1

OrismoÐ

Sthn parˆgrafo aut  ja parousiˆsoume ènan enallaktikì trìpo eÔreshc twn jemeliwd¸n lÔsewn enìc sust matoc. Wc sun jwc, jewr ste to ex c sÔsthma stajer¸n suntelest¸n

~ x 0 = P~ x, An to parapˆnw sÔsthma apoteleÐto apì mÐa mìnon exÐswsh (to

P

1×1

eÐnai ènac

pÐnakac, dhlad  ènac arijmìc) tìte

h lÔsh ja eÐqe thn morf 

~ x = eP t . Parìmoia mporoÔme na genikeÔsoume orÐzontac katˆllhla thn èkfrash thc sunˆrthshc

at

e

gia kˆpoio arijmì

eP t .

Ac jumhjoÔme o anˆptugma seirˆc

T aylor

a. ∞

eat = 1 + at +

X (at)k (at)2 (at)3 (at)4 + + + ··· = . 2 6 24 k! k=0

JumhjeÐte ìti

k! = 1 · 2 · 3 · · · k, a + a2 t +

kai

0! = 1.

ParagogÐzontac èqoume

  (at)2 (at)3 a t a t + + · · · = a 1 + at + + + · · · = aeat . 2 6 2 6 3 2

4 3

Ac kˆnoume kˆti parìmoio gia pÐnakec. Ac orÐsoume pr¸ta gia kˆje

def

eA = I + A +

n×n

pÐnaka

A

ton ekjetikì pÐnaka wc ex c

1 2 1 3 1 A + A + · · · + Ak + · · · 2 6 k!

Fusikˆ kai prèpei h parapˆnw seirˆ na sugklÐnei allˆ ac mhn mac apasqol sei to jèma autì t¸ra. Ac upojèsoume dhlad  ìti ìlec oi seirèc pou ja akolouj soun sugklÐnoun. Profan¸c gia kˆje pÐnaka

P

èqoume

P t = tP

opìte kai

paÐrnoume

d  tP  e = P etP . dt tP O P kai katˆ sunèpeia kai o e eÐnai ènac pÐnakac n × n. Autì pou mènei na broÔme eÐnai apl¸c èna diˆnusma. Sugkekrimèna, sthn perÐptwsh pou o pÐnakac eÐnai 1 × 1 ousiastikˆ mac mènei na pollaplasiˆsoume me mia tuqaÐa stajerˆ gia na broÔme thn genik  lÔsh. Sthn perÐptwsh twn n × n pinˆkwn aplˆ pollaplasiˆzoume thn lÔsh pou br kame parapˆnw me èna stajerì diˆnusma ~ c. Je¸rhma 3.7.1.

Eˆn

P

eÐnai ènac

n×n

pÐnakac, tìte h genik  lÔsh tou

~ x 0 = P~ x

eÐnai

~ x = etP ~c, ìpou

~c

eÐnai èna opoiod pote stajerì diˆnusma. Mˆlista isqÔei h sqèsh

~ x(0) = ~c.

Ac to epibebai¸soume.

d d  tP  ~ x= e ~c = P etP ~c = P ~ x. dt dt 'Ara o

etP

eÐnai o jemeli¸dhc pÐnakac lÔsewn tou omogenoÔc sust matoc. Eˆn broÔme ènan trìpo upologismoÔ tou

ekjetikoÔ enìc pÐnaka tìte ja èqoume mia ˆllh mèjodo epÐlushc omogen¸n susthmˆtwn me stajeroÔc suntelestèc. H en lìgw mèjodoc mˆlista antimetwpÐzei thn diadikasÐa epilog c thc sugkekrimènhc lÔshc pou ikanopoieÐ dojeÐsec arqikèc sunj kec me polÔ pio eÔkolo trìpo. Prˆgmati, gia na lÔsoume to

~ x 0 = A~ x, ~ x(0) = ~b,

paÐrnoume thn lÔsh

~ x = etA~b. Autì prokÔptei apì to gegonìc ìti

e0A = I ,

opìte

~ x(0) = e0A~b = ~b.

Upˆrqei èna mikrì prìblhma me ta ekjetikˆ pinˆkwn.

Genikˆ èqoume ìti

eA+B 6= eA eB .

Autì ofeÐletai sto

gegonìc ìti upˆrqei sobarì endeqìmeno dÔo pÐnakec na mhn antimetatÐjentai, dhlad  na èqoume antimetajetikìthta aut  twn pinˆkwn sunepˆgetai ìti

eA+B 6= eA eB

AB 6= BA.

H mh-

gegonìc pou mac dhmiourgeÐ sobarˆ probl mata

3.7. ΕΚΘΕΤΙʟΑ ΠΙ͟ΑΚΩΝ

89 T aylor.

ìtan prospajoÔme na lÔsoume èna sÔsthma qrhsimopoi¸ntac seirèc

A

B

kai

antimetatÐjentai, tìte

eA+B = eA eB

Fusikˆ ìtan

AB = BA,

ìtan dhlad  oi

prˆgma idiaÐtera bolikì ìpwc ja doÔme. Ac diatup¸soume ta parapˆnw

sumperˆsmata se morf  jewr matoc.

Je¸rhma 3.7.2.

3.7.2

Eˆn

AB = BA

tìte

eA+B = eA eB .

Eidˆllwc èqoume

eA+B 6= eA eB .

Aplèc peript¸seic

Se merikèc peript¸seic arkeÐ na antikatast soume timèc stic metablhtèc twn seir¸n. Mia tètoia perÐptwsh eÐnai kai

D = [ a0 0b ]. Tìte  k  a 0 Dk = , 0 bk

aut  pou o pÐnakac eÐnai diag¸nioc. Gia parˆdeigma,

kai

e

 1 2 1 3 1 = I + D + D + D + ··· = 0 2 6

D

  1 a2 0 + b 2 0

  0 a + 1 0

  a e 0 + ··· = b3 0

  1 a3 0 + b2 6 0

 0 . eb

Me autì ton trìpo katal goume sto

eI =



e 0

0 e

 kai

eaI =



ea 0

 0 a . e

Ta parapˆnw apotelèsmata mac bohjˆne sto na upologÐsoume eÔkola ta ekjetikˆ ˆllwn pio genik¸n pinˆkwn.

Gia  −3   3 −3  A = 35 −1 mporeÐ na grafjeÐ wc 2I + B ìpou B = . Shmei¸ste ìti oi 3 −3 B 2 = [ 00 00 ]. Opìte B k = 0 gia kˆje k ≥ 2. Sunep¸c, eB = I + B . 'Estw ìti tB Oi 2tI kai tB antimetatÐjentai (epibebai¸ste to san ˆskhsh) kai e = I + tB ,

parˆdeigma shmei¸ste ìti o pÐnakac

2I

kai

B

antimetatÐjentai, kai ìti

etA .

jèloume na upologÐsoume ton since

(tB)2 = t2 B 2 = 0.

tA

e

2tI+tB

=e

'Eqoume loipìn

2tI tB

=e

e

e2t = 0 

 0 (I + tB) = e2t  =

e2t 0

0 e2t



1 + 3t 3t

Br kame loipìn ton jemeli¸dh pÐnaka twn lÔsewn tou sust matoc

  −3t (1 + 3t) e2t = 3te2t 1 − 3t

~ x 0 = A~ x.

 −3te2t 2t . (1 − 3t) e

Shmei¸ste ìti o pÐnakac

A

èqei mia

atel  idiotim  pollaplìthtac 2, upˆrqei dhlad  mìnon èna idiodiˆnusma gia thn en lìgw idiotim . 'Eqoume loipìn brei mia en dunˆmei mèjodo antimet¸pishc tètoiwn peript¸sewn. Prˆgmati, eˆn o pÐnakac

λ

pollaplìthtac 2, tìte eÐte eÐnai diag¸nioc, eÐte

A = λI + B

ìpou

B 2 = 0.

A

eÐnai

2×2

kai èqei mia idiotim 

H parakˆtw eÐnai mia polÔ kal  kai

endeqomènwc lÐgo dÔskolh ˆskhsh.

3.7.1

2 × 2 pÐnakac A èqei A = λI + B , ìpou B 2 = 0.

'Estw ìti o

grˆyete ìti

mìnon mia idiotim , thn

ApodeÐxte ìti

(A − λI)2 = 0.

Tìte mporeÐte na

Upìdeixh: Pr¸ta grˆyte thn exÐswsh pou prokÔptei apì to gegonìc ìti h

idiotim  èqei pollaplìthta 2 kai metˆ upologÐste to

3.7.3

λ.

B2.

GenikoÐ pÐnakec

Genikˆ o upologismìc tou ekjetikoÔ enìc pÐnaka den eÐnai tìso eÔkolo ìso eÐdame parapˆnw. Den mporoÔme sun jwc na grˆyoume ènan pÐnaka san ˆjroisma antimetijìmenwn pinˆkwn ìpote kai mporoÔme na suneqÐsoume thn diadikasÐa epÐlushc eÔkola.

Mhn apogohteÔeste ìmwc eidikˆ an mporoÔme na broÔme arketˆ idiodianÔsmata.

sto ex c ergaleÐo thc grammik c ˆlgebrac. Gia kˆje tetragwnikoÔc pÐnakec

eBAB

−1

A

kai

B

Ja sthriqjoÔme

èqoume ìti

= BeA B −1 .

MporoÔme na doÔme thn orjìthta thc parapˆnw sqèshc qrhsimopoiìntac anaptÔgmata

T aylor.

ìti

(BAB −1 )2 = BAB −1 BAB −1 = BAIAB −1 = BA2 B −1 .

Prin ìmwc parathr ste

90

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

Katèpèktash isqÔei ìti

(BAB −1 )k = BAk B −1 . eBAB

−1

T¸ra ac grˆyoume ton pÐnaka tai diagwnopoÐhsh.

A

'Ara ac grˆyoume to anˆptugma

T aylor

gia thn

eBAB

−1

1 1 (BAB −1 )2 + (BAB −1 )3 + · · · 2 6 1 1 = BB −1 + BAB −1 + BA2 B −1 + BA3 B −1 + · · · 2 6
−1 1 2 1 3 = B I + A + A + A + ··· B 2 6 = BeA B −1 . = I + BAB −1 +

san

EDE −1 ,

ìpou

D

eÐnai ènac diag¸nioc pÐnakac. H diadikasÐa aut  onomˆze-

Eˆn mporoÔme na thn fèroume eic pèrac tìte o upologismìc tou ekjetikoÔ mac gÐnete eÔkola.

Prosjètontac katˆllhla to

t

parathroÔme ìti

etA = EetD E −1 . Gia na broÔme ta

E

kai

D

qreiazìmaste

n

grammikˆ anexˆrthta idiodianÔsmata tou

A.

Sthn perÐptwsh pou den ta

èqoume ja prèpei na prospaj soume kˆti poio perÐploko. Den axÐzei ìmwc na asqolhjoÔme sto mˆjhma autì me mia tètoia perÐptwsh. 'Estw

~v1 ,

idiotimèc kai

...,

~vn

E

A. ~vn ].

o pÐnakac o opoÐoc èqei san st lec ta idiodianÔsmata tou

ta idiodianÔsmata tou

A.

E = [ ~v1

SofÐ èqoume

A.

pÐnakac pou èqei san diag¸nia stoiqeÐa tic idiotimèc tou



···

~v2

'Estw epÐshc 'Estw tèloc

λ1 , . . . , λn oi D o diag¸nioc

Tìte èqoume

··· ···

 0 0  . . .  .

λ1 0  D= .  ..

0 λ2 . . .

.

0

0

···

λn

AE = A[ ~v1

~v2

···

~vn ]

.

.

'EÔkola blèpoume ìti

= [ A~v1 = [ λ1~v1 = [ ~v1

···

A~v2

···

λ2~v2 ···

~v2

A~vn ] λn~vn ]

~vn ]D

= ED. E

Oi st lec tou 'Ara o

E

eÐnai grammikˆ anexˆrthtec epeid  upojèsame oti ta idiodianÔsmata tou

eÐnai antistrèyimoc. Epeid 

AE = ED,

pollaplasiˆzontac me

E −1

A

eÐnai grammikˆ anexˆrthta.

èqoume

A = EDE −1 . Dhlad 

eA = EeD E −1 .

Prosjètontac kai to

t

katal goume ìti

etA = EetD E −1

 λ1 t e  0  =E .  ..

0 λ2 t

e

0

··· ···

. . .

.

0

···

.

.

 0 0   −1 . . E .  .

(3.4)

eλn t

Sunep¸c h sqèsh (3.4) mac prosfèrei ènan tÔpo gia ton upologismì tou pÐnaka twn jemeliwd¸n lÔsewn sust matoc

0

~x = A~ x

sthn perÐptwsh pou o pÐnakac

A

èqei

n

etA

tou

grammikˆ anexˆrthta idiodianÔsmata.

Shmei¸ste ìti h parapˆnw diadikasÐa mporeÐ ˆmesa na efarmosjeÐ kai sthn perÐptwsh pou oi idiotimèc (kai ta idiodianÔsmata profan¸c) eÐnai migadikèc. Aplˆ oi prˆxeic mac ja gÐnontai me migadikoÔc arijmoÔc. Den eÐnai dÔskolo na doÔme ìti eˆn o pÐnakac ton tÔpo tou

Euler

A eÐnai pragmatikìc, tìte kai o etA

ja eÐnai pragmatikìc. Tèloc jumhjeÐte ìti efarmìzontac

mporoÔme na aplopoi soume ta migadikˆ apotelèsmatˆ mac metatrèpontac ton pÐnaka

autìc na mhn perièqei kˆpoion migadikì arijmì.

A

ètsi ¸ste

3.7. ΕΚΘΕΤΙʟΑ ΠΙ͟ΑΚΩΝ Parˆdeigma 3.7.1:

91

UpologÐste ton pÐnaka twn jemeliwd¸n lÔsewn tou sust matoc qrhsimopoi¸ntac ekjetikˆ

pinˆkwn

 0  1 x = 2 y

2 1

  x . y

UpologÐste epÐshc kai thn sugkekrimènh lÔsh h opoÐa ikanopoieÐ tic ex c arqikèc sunj kec 'Estw

A

o pÐnakac twn suntelest¸n

ta antÐstoiqa idiodianÔsmata eÐnai

tA

e

[ 11 ]

 1 = 1  1 = 1 −1 2

=

−1 = 2

kai

[12 21 ]. 1

x(0) = 4

kai

y(0) = 2. −1 kai ìti

Pr¸ta brÐskoume (ˆskhsh) ìti oi idiotimèc tou eÐnai 3 kai

−1 . 'Ara mporoÔme na doÔme ìti

−1 1 e−t −1   3t    1 e 0 −1 −1 −1 −t −1 0 e 1 2 −1   3t  −t e e −1 −1 1 e3t −e−t −1  " e3t +e−t  3t −t 3t −e − e −e + e−t 2 = e3t −e −t −e3t + e−t −e3t − e−t 1 −1



e3t 0

0



1 1

e3t −e−t 2 e3t +e−t 2

2

# .

x(0) = 4 kai y(0) = 2. 'Ara me bˆsh thn idiìthta e0A = I brÐskoume ìti h sugkekrimènh b ìpou ~b eÐnai [ 42 ]. H sugkekrimènh lÔsh pou yˆqnoume loipìn eÐnai h ex c eÐnai h e   " e3t +e−t e3t −e−t #    3t    3t x 3e + e−t 4 2e + 2e−t + e3t − e−t 2 2 . = = e3t −e = 3t −t −t 3t −t 3t −t 3t −t e +e y 3e − e 2e − 2e + e + e 2

Oi arqikèc sunj kec eÐnai lÔsh pou yˆqnoume

tA~

2

3.7.4

2

PÐnakac jemeliwd¸n lÔsewn

Shmei¸ste ìti eˆn katafèrete na upologÐsete ton pÐnaka twn jemeliwd¸n lÔsewn me kˆpoion diaforetikì trìpo, mporeÐte na ton qrhsimopoi sete gia na breÐte ton

etA .

pÐnaka twn jemeliwd¸n lÔsewn enìc sust matoc SDE den

t = 0 gÐnetai o tautotikìc X eÐnai ènac opoiosd pote

eÐnai monadikìc. O ekjetikìc pÐnakac eÐnai o pÐnakac twn jemeliwd¸n lÔsewn o opoÐoc gia pÐnakac.

Prèpei loipìn na broÔme ton katˆllhlo pÐnaka twn jemeliwd¸n lÔsewn.

pÐnakac twn jemeliwd¸n lÔsewn tou sust matoc

~ x 0 = A~ x

Eˆn

tìte mporoÔme na isqurisjoÔme ìti

etA = X(t) [X(0)]−1 . Profan¸c eˆn jèsoume

t=0

sto

X(t) [X(0)]−1

paÐrnoume ton tautotikì.

Den eÐnai dÔskolo na diapist¸soume ìti

mporoÔme na pollaplasiˆsoume ènan pÐnaka jemeliwd¸n lÔsewn apì ta dexiˆ me opoiond pote stajerì antistrèyimo pÐnaka kai na katal xoume se ènan ˆllo pÐnaka twn jemeliwd¸n lÔsewn. me allag  twn tuqaÐwn stajer¸n thc genik c lÔshc

3.7.5

Dhlad  h parapˆnw diadikasÐa isodunameÐ

~ x(t) = X(t)~c.

Ask seic

x0 = 3x + y , y 0 = x + 3y .

3.7.2

BreÐte ton pÐnaka twn jemeleiwd¸n lÔsewn tou sust matoc

3.7.3

UpologÐste ton pÐnaka

3.7.4

BreÐte ton pÐnaka twn jemeleiwd¸n lÔsewn tou sust matoc

x03 = −3x1 − 2x2 − 5x3 .

eAt

ìpou

A = [ 20 32 ].

BreÐte metˆ thn lÔsh pou ikanopoieÐ tic

A

UpologÐste ton pÐnaka

3.7.6

'Estw

3.7.7

Qrhsimopoi ste thn ˆskhsh 3.7.6 gia na deÐxete ìti

AB = BA

ìtan

−2

A = [ 10 21 ].

3.7.5

e

x01 = 7x1 + 4x2 + 12x3h, x02i = x1 + 2x2 + x3 , 0 ex c arqikèc sunj kec ~ x= 1 .

(antimetatijìmenoi pÐnakec). DeÐxte ìti

strèyimoc akìma kai sthn perÐptwsh pou o

A

den eÐnai.

eA+B = eA eB .

(eA )−1 = e−A

gegonìc pou shmaÐnei ìti o

eA

eÐnai anti-

92 3.7.8

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ 'Estw ìti o pÐnakac

A

èqei idiotimèc

−1,

1, kai antÐstoiqa ididianÔsmata

[ 11 ], [ 01 ].

a) BreÐte ènan pÐnaka

ikanopoieÐ tic parapˆnw idiìthtec. b) BreÐte ton pÐnaka twn jemeleiwd¸n lÔsewn tou sust matoc to sÔsthma me tic ex c arqikèc sunj kec

3.7.9

'Estw ìti

A

DeÐxte ìti o pÐnakac

ènac

A

n×n

~ x(0) = [ 23 ]

pÐnakac me idiotim 

~ x0 = A~ x.

A

pou

c) LÔste

.

λ

pollaplìthtac

eÐnai diag¸nioc, kai sugkekrimèna

A = λI .

n

kai

n

grammikˆ anexˆrthta idiodianÔsmata.

Upìdeixh: Qrhsimopoi ste diagwnopoÐhsh kai to

gegonìc ìti o tautotikìc pÐnakac antimetatÐjetai me opoiond pote ˆllo pÐnaka.

3.8. ΜΗ-ΟΜΟΓΕΝŸΗ ΣΥΣԟΗΜΑΤΑ 3.8

93

Mh-omogen  sust mata

3.8.1

Pr¸thc tˆxhc sust mata me stajeroÔc suntelestèc

Ολοκληρωτικός παράγοντας Ac epikentrwjoÔme pr¸ta sthn mh-omogen  exÐswsh pr¸thc tˆxhc

~ x 0 (t) = A~ x(t) + f~(t), ìpou o

A

eÐnai stajerìc (den exartˆtai apì to

t).

H pr¸th mèjodoc pou ja exetˆsoume eÐnai h mèjodoc tou

oloklhrwtikoÔ parˆgonta. Aplˆ xanagrˆfoume thn exÐswsh wc ex c.

~ x 0 (t) + P ~ x(t) = f~(t), ìpou

P = −A.

Pollaplasiˆzoume kai ta dÔo mèlh thc exÐswshc me

etP

etP ~ x 0 (t) + etP P ~ x(t) = etP f~(t). Shmei¸ste ìti

P etP = P

P etP = etP P .

 I + tP +

Autì prokÔptei eÔkola eˆn grˆyoume ton orismì tou

1 (tP )2 + · · · 2



= P + tP 2 +

d dt


etP = P etP .

san seirˆ,

1 2 3 t P + ··· = 2 =

'Eqoume  dh diapist¸sei ìti

etP

  1 I + tP + (tP )2 + · · · P = etP P. 2

Sunep¸c,

 d  tP e ~ x(t) = etP f~(t). dt MporoÔme t¸ra na oloklhr¸soume. Oloklhr¸nontac bebaÐwc kˆje sunist¸sa twn dianusmˆtwn xeqwristˆ.

etP ~ x(t) = Qrhsimopoi¸ntac to gegonìc ìti

(etP )−1 = e−tP

Z

etP f~(t) dt + ~c.

èqoume

~ x(t) = e−tP

Z

etP f~(t) dt + e−tP ~c.

'Iswc ìla ta parapˆnw na eÐnai pio katanohtˆ an qrhsimopoi soume orismèna oloklhr¸mata.

Sthn perÐptwsh

aut  mˆlista ja mporèsoume na broÔme tic lÔseic pou ikanopoioÔn tic arqikèc sunj kec eukolìtera. 'Estw loipìn h parakˆtw exÐswsh kai arqikèc sunj kec.

~ x 0 (t) + P ~ x(t) = f~(t),

~ x(0) = ~b.

H lÔsh t¸ra mporeÐ na grafjeÐ sthn ex c morf 

~ x(t) = e−tP

t

Z

esP f~(s) ds + e−tP ~b.

(3.5)

0

Mhn xeqnˆte ìti to olokl rwma tou dianÔsmatoc

esP f~(s)

prokÔptei oloklhr¸nontac kˆje sunist¸sa tou xeqwristˆ.

Den eÐnai dÔskolo na doÔme ìti h (3.5) pragmatikˆ ikanopoieÐ tic arqikèc sunj kec

~ x(0) = e−0P

0

Z 0

esP f~(s) ds + e−0P ~b = I~b = ~b.

~ x(0) = ~b.

94

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

Parˆdeigma 3.8.1:

Jewr ste to sÔsthma

x01 + 5x1 − 3x2 = et , x02 + 3x1 − x2 = 0, kai tic ex c arqikèc sunj kec

x1 (0) = 1, x2 (0) = 0.

Ac diatup¸soume to sÔsthma sthn ex c morf 

~ x0 + etP

'Eqoume  dh upologÐsei to arnhtikì prìshmo sto

gia

P =



5 3

  t e −3 ~ x= , −1 0

~ x(0) =

  1 . 0

 5 −3 

3 −1 . EÔkola diapist¸noume ìti mporoÔme na upologÐsoume to

e−tP , dÐnontac

t.

etP =

 (1 + 3t) e2t 3te2t

 −3te2t 2t , (1 − 3t) e

e−tP =



(1 − 3t) e−2t −3te−2t

 3te−2t −2t . (1 + 3t) e

AntÐ na upologÐsoume ìlon to tÔpo monomiˆc ac ton upologÐsoume stadiakˆ. ArqÐzoume me

t

Z

esP f~(s) ds =

0

Z

t



t



0

(1 + 3s) e2s 3se2s

(1 + 3s) e3s = 3se3s 0 " # te3t = (3t−1) e3t +1 . Z

−3se2s (1 − 3s) e2s

  s e ds 0

 ds

3 Then

~ x(t) = e−tP

Z

t

esP f~(s) ds + e−tP ~b

0

#  " te3t (1 − 3t) e−2t 3te−2t (1 − 3t) e−2t = −2t −2t (3t−1) e3t +1 + −3te (1 + 3t) e −3te−2t 3     te−2t (1 − 3t) e−2t
−2t + = −2t et 1 −3te − 3 + 3 +t e   −2t (1 − 2t) e
. = t − e3 + 31 − 2t e−2t 

3te−2t (1 + 3t) e−2t

  1 0

OÔf! Ac epibebai¸soume thn orjìthta autoÔ pou mìlic br kame.

x01 + 5x1 − 3x2 = (4te−2t − 4e−2t ) + 5(1 − 2t) e−2t + et − (1 − 6t) e−2t = et . ParomoÐwc mporoÔme na doÔme ìti (ˆskhsh)

x02 + 3x1 − x2 = 0.

Tèloc eÔkola diapist¸noume (ˆllh ˆskhsh) ìti kai

oi arqikèc sunj kec ikanopoioÔntai.

H mèjodoc twn oloklhrwtik¸n paragìntwn gia sust mata eÐnai apotelesmatik  mìnon ìtan o pÐnakac exartˆtai apì to

t,

dhlad  o

P

eÐnai stajerìc. To prìblhma bebaÐwc eÐnai ìti en gènei èqoume

R d R P (t) dt e 6= P (t) e P (t) dt , dt epeid  oi dÔo pÐnakec mporoÔn kˆllista na mhn antimetatÐjentai.

P

den

3.8. ΜΗ-ΟΜΟΓΕΝŸΗ ΣΥΣԟΗΜΑΤΑ

95

Παραγοντοποίηση Ιδιοδιανυσμάτων H jewrÐa allˆ kai h praktik  thc parakˆtw mejìdou basÐzetai sto gegonìc ìti ta idiodianÔsmata enìc pÐnaka mac dÐnoun tic kateujÔnseic stic opoÐec o en lìgw pÐnakac dra san arijmìc. Eˆn lÔsoume to sÔsthmˆ mac stic kateujÔnseic autèc tìte oi lÔseic autèc ja eÐnai aploÔsterec mia kai ja mporoÔme na diaqeirizìmaste ton pÐnaka san arijmì. MporoÔme na sunduˆsoume tic aplèc autèc lÔseic me katˆllhlo trìpo kai na pˆroume thn genik  lÔsh. Jewr ste thn exÐswsh

~ x 0 (t) = A~ x(t) + f~(t). Upojèste ìti o pÐnakac

A

èqei

n

grammikˆ anexˆrthta idiodianÔsmata

(3.6)

~v1 , . . . , ~vn .

Ac grˆyoume

~ x(t) = ~v1 ξ1 (t) + ~v2 ξ2 (t) + · · · + ~vn ξn (t).

(3.7)

A. Gia na upologÐsoume f~ sunart sei twn idiodianusmˆtwn.

Jèloume dhlad  na grˆyoume thn lÔsh mac san grammikì sunduasmì twn idiodianusmˆtwn tou thn lÔsh mac

~x, arkeÐ na broÔme tic sunart seic ξ1

èwc

ξn .

Ac upologÐsoume kai thn

Ac grˆyoume

f~(t) = ~v1 g1 (t) + ~v2 g2 (t) + · · · + ~vn gn (t). Jèloume loipìn na broÔme tic

g1

èwc

gn

(3.8)

oi opoÐec ikanopoioÔn thn sqèsh (3.8). Shmei¸ste ìti epeid  ìla ta idiodianÔs-

E = [ ~v1 ~v2 · · · ~vn ] eÐnai antistrèyimoc. Blèpoume ìti f~ = E~g , ìpou oi sunist¸sec thc ~g eÐnai oi sunart seic g1 èwc gn . Tìte èqoume ~g = E −1 f~. Sunep¸c eÐnai pˆnta efiktì na broÔme to ~g arkeÐ na upˆrqoun n grammikˆ anexˆrthta idiodianÔsmata. AntikajistoÔme thn (3.7) sthn (3.6), kai parathroÔme ìti A~ vk = λk ~vk . mata tou pÐnaka

A

eÐnai grammikˆ anexˆrthta, o pÐnakac

mporoÔme na grˆyoun thn (3.8) wc ex c

~ x 0 = ~v1 ξ10 + ~v2 ξ20 + · · · + ~vn ξn0 = A (~v1 ξ1 + ~v2 ξ2 + · · · + ~vn ξn ) + ~v1 g1 + ~v2 g2 + · · · + ~vn gn = A~v1 ξ1 + A~v2 ξ2 + · · · + A~vn ξn + ~v1 g1 + ~v2 g2 + · · · + ~vn gn = ~v1 λ1 ξ1 + ~v2 λ2 ξ2 + · · · + ~vn λn ξn + ~v1 g1 + ~v2 g2 + · · · + ~vn gn = ~v1 (λ1 ξ1 + g1 ) + ~v2 (λ2 ξ2 + g2 ) + · · · + ~vn (λn ξn + gn ). Eˆn anakalÔyoume touc suntelestèc twn dianusmˆtwn

ξ10 ξ20

~v1

èwc

~vn

katal goume stic exis¸seic

= λ1 ξ1 + g1 , = λ2 ξ2 + g2 , . . .

ξn0 = λn ξn + gn . H kˆje mia apì tic exis¸seic autèc eÐnai anexˆrthth apì tic ˆllec. EÐnai ìlec grammikèc exis¸seic pr¸thc tˆxhc kai mporoÔme eÔkola na tic lÔsoume me thn basik  mèjodo twn oloklhrwtik¸n paragìntwn. Gia parˆdeigma gia thn

kth

exÐswsh èqoume

ξk0 (t) − λk ξk (t) = gk (t). Qrhsimopoi¸ntac ton oloklhrwtikì parˆgonta

e−λk t

paÐrnoume

i d h ξk (t) e−λk t = e−λk t gk (t). dx Oloklhr¸nontac kai lÔnontac wc proc

ξk

katal goume sto

ξk (t) = eλk t

Z

e−λk t gk (t) dt + Ck eλk t .

Shmei¸ste ìti eˆn endiafèreste gia mia opoiad pote sugkekrimènh lÔsh tìte ja bìleue polÔ na jèsete to me to mhdèn.

Eˆn den d¸soume timèc stic stajerèc, tìte aplˆ èqoume thn genik  lÔsh.

Sugkekrimèna thn

Ck Ðso ~ x(t) =

~v1 ξ1 (t) + ~v2 ξ2 (t) + · · · + ~vn ξn (t). eÐnai Ðswc protimìtero an, ìpwc kai prohgoumènwc, grˆyoume ta oloklhr¸mata san orismèna oloklhr¸mata. Ac upojèsoume ìti èqoume thn ex c arqik  sunj kh

~ x(0) = ~b.

96

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ PaÐrnoume

~c = E −1~b

kai parathroÔme ìti, ìpwc akrib¸c kai prin, èqoume

ξk (t) = eλk t

Z

t

~b = ~v1 a1 + · · · + ~vn an .

Metˆ grafoume

e−λk s gk (s) dt + ak eλk t ,

0 kai ètsi sthn ousÐa ja pˆroume thn sugkekrimènh lÔsh

~x(0) = ~b,

epeid 

~ x(t) = ~v1 ξ1 (t) + ~v2 ξ2 (t) + · · · + ~vn ξn (t)

h opoÐa ikanopoieÐ thn

ξk (0) = ak .

Parˆdeigma 3.8.2:

'Estw

A = [ 13 31 ].

LÔste to sÔsthma

~ x0 = A~ x + f~ ìpou f~(t) =  1 

A eÐnai −2 kai 4 kai ta antÐstoiqa idiodianÔsmata eÐnai −1 E kai ac upologÐsoume ton antÐstrofì tou.     1 1 −1 1 1 −1 . E= , E = 1 −1 1 2 1  1  AnazhtoÔme mia lÔsh thc morf c ~ x = −1 ξ1 + [ 11 ] ξ2 . Jèloume na na ekfrˆsoume  t  1  ~ 2e idiodianusmˆtwn. Jèloume dhlad  na grˆyoume f = = −1 g1 + [ 11 ] g2 . 'Ara 2t      t    t 1 1 −1 2et e −t g1 2e = t . = = E −1 2t e +t 1 2t g2 2 1 Oi idiotimèc tou

h

i

 3/16 2et gia ~ x(0) = −5/16 . 2t 1 kai [ 1 ] antÐstoiqa (ˆskhsh). 

Ac

grˆyoume ton pÐnaka

'Ara

g1 = et − t

kai

3/16 −5/16

i

=



1 −1



a1 +

~ x(0)

sunart sei twn idiodianusmˆtwn. Jèloume dhlad  na grˆyoume

a1 a2

a1 = 1/4

kai

sunart sei twn

~ x(0) =

[ 11 ] a2 . Sunep¸c "

'Ara

f~

g2 = et + t.

Jèloume epiplèon na ekfrˆsoume to

h

kai to

#

" =E

−1

3 16 −5 16

#

" =

1 4 −1 16

# .

a2 = −1/16. AntikajistoÔme to ~ x sthn exÐswsh kai paÐrnoume             1 1 0 1 1 1 1 ξ10 + ξ2 = A ξ1 + A ξ2 + g1 + g −1 1 −1 1 −1 1 2         1 1 1 1 t = (−2ξ1 ) + 4ξ2 + (e − t) + (et − t). −1 1 −1 1

PaÐrnoume loipìn tic ex c exis¸seic

ξ10 = −2ξ1 + et − t,

where

ξ20 = 4ξ2 + et + t,

where

1 , 4 −1 . ξ2 (0) = a2 = 16

ξ1 (0) = a1 =

Eˆn lÔsoume me oloklhrwtikì parˆgonta (ˆskhsh, prosoq  ja prèpei na qrhsimopoi sete olokl rwsh katˆ mèrh) èqoume

Z

ξ1 = e−2t C1

eÐnai h stajerˆ olokl rwshc Epeid 

et t 1 − + + C1 e−2t . 3 2 4 1/4 = 1/3 + 1/4 + C kai sunep¸c C = −1/3. 1 1

e2t (et − t) dt + C1 e−2t = ξ1 (0) = 1/4

èqoume ìti

Parìmoia

èqoume

et t 1 − − + C2 e4t . 3 4 16 ξ2 (0) = 1/16 èqoume ìti −1/16 = −1/3 − 1/16 + C2 kai sunep¸c C2 = 1/3. H lÔsh eÐnai #   t     4t  " e4t −e−2t 1 − 2t e − et 4t + 1 e − e−2t 1 1 + 3−12t 3 16 + = e−2t +e4t +2et . ~x(t) = + − −1 1 3 4 3 16 + 4t−5 3 16 ξ2 = e4t

Epeid 

Dhlad ,

3.8.1

x1 =

e4t −e−2t 3

+

3−12t kai 16

Epibebai¸ste ìti ta

thn arqik  sunj kh.

x1

kai

Z

e−4t (et + t) dt + C2 e4t = −

x2 =

x2

e−2t +e4t +2et 3

+

4t−5 . 16

eÐnai lÔseic tou sust matoc, ikanopoioÔn dhlad  kai thn diaforik  exÐswsh kai

3.8. ΜΗ-ΟΜΟΓΕΝŸΗ ΣΥΣԟΗΜΑΤΑ

97

Απροσδιόριστοι συντελεστές MporoÔme bebaÐwc na epekteÐnoume thn mèjodo twn aprosdiìristwn suntelest¸n gia sust mata. Autì mporeÐ na gÐnei idiaÐtera eÔkola mia kai h mình diaforˆ ja eÐnai to ìti ja prospajoÔme na prosdiorÐsoume thn tim  enìc dianÔsmatoc kai ìqi enìc mìnon arijmoÔ. Profan¸c ìti probl mata antimetwpÐsame sthn perÐptwsh thc miac exÐswshc perimènoume na antimetwpÐsoume kai sthn perÐptwsh twn susthmˆtwn. H mèjodoc aut  loipìn den mporeÐ na antimetwpÐsei ìla ta probl mata en¸ eÐnai idiaÐtera polÔplokh ìtan to dexiì mèroc thc exÐswshc eÐnai kˆpwc perÐploko. Epeid  h mèjodoc den diafèrei ousiastikˆ kajìlou apì thn antÐstoiqh mèjodoc miac exÐswshc ac asqolhjoÔme amèswc me èna parˆdeigma.

Parˆdeigma 3.8.3:

f~(t) =



t 

e t

.

A = tou A

'Estw

Oi idiotimèc

 −1 0 

−2 1 .

eÐnai

UpologÐste mia sugkekrimènh lÔsh tou sust matoc

−1

1 kai 1 kai ta idiodianÔsmata eÐnai [ 1 ] kai

~ x0 = A~ x + f~

[ 01 ] antÐstoiqa.

ìpou

Sunep¸c h

sumplhrwmatik  lÔsh mac eÐnai

    1 −t 0 t ~ xc = α1 e + α2 e, 1 1 gia kˆpoiec stajerèc

α1

kai

α2 .

T¸ra jèloume na mantèyoume mia sugkekrimènh lÔsh tou

~ x = ~aet + ~bt + ~c. 'Omwc, kˆpoioc ìroc thc morf c eˆn to diˆnusma

~a

~aet

faÐnetai na upˆrqei sthn sumplhrwmatik  lÔsh.

eÐnai grammikˆ exarthmèno (pollaplˆsio dhlad ) tou

[ 01 ]

Epeid  den gnwrÐzoume akìma

den gnwrÐzoume eˆn ja upˆrxei prìblhma.

Endeqomènwc bèbaia na mhn upˆrxei prìblhma. Gia na eÐmaste asfal c ìmwc ja prèpei na dokimˆsoume kai thn Ac dokimˆsoume loipìn kai thn

~aet

Thuc we have 8 unknownc. We write

kai thn

~btet

san manteyièc, kai ìqi mìnon thn

~ ~ x = ~aet + ~btet + ~ct + d. h i h i ~a = [ aa12 ], ~b = bb12 , ~c = [ cc12 ], kai d~ = dd12 ,

sthn exÐswsh. Pr¸ta ac upologÐsoume

~ x

0

~btet .

~btet .

Sunep¸c èqoume

Ac antikatast soume tic parapˆnw

.

  ~ x 0 = ~a + ~b et + ~btet + ~c. T¸ra h

~x 0

prèpei na eÐnai Ðsh me

A~ x + f~ ˆra

A~x + f~ = A~aet + A~btet + A~ct + Ad~ + f~ =          t −a1 −b1 −c1 −d1 e . = et + tet + t+ + t −2a1 + a2 −2b1 + b2 −2c1 + c2 −2d1 + d2 Ac exis¸soume t¸ra touc suntelestèc twn

et , tet , t

kai touc stajeroÔc ìrouc.

a1 + b1 = −a1 + 1, a2 + b2 = −2a1 + a2 , b1 = −b1 , b2 = −2b1 + b2 , 0 = −c1 , 0 = −2c1 + c2 + 1, c1 = −d1 , c2 = −2d1 + d2 . MporoÔme na grˆyoume tic parapˆnw exis¸seic san èna

8×9

pÐnaka kai na lÔsoume to sÔsthma me apaloif . Sthn

perÐptws  mac bèbaia to sÔsthma eÐnai tìso aplì pou mporoÔme eÔkola na diapist¸soume ìti Antikajist¸ntac tic timèc autèc stic exis¸seic brÐskoume ìti

c2 = −1

eÐnai oi ex c

a1 = −a1 + 1, a2 + b2 = −2a1 + a2 .

kai

d2 = −1.

b1 = 0, c1 = 0, d1 = 0.

Oi upìloipec qr simec exis¸seic

98

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

'Ara

1 kai b2 = −1. 2 a2 = 0. Tìte,

a1 =

epilèxoume

To

a2

mporeÐ na pˆrei opoiad pote tim .

~ x = ~aet + ~btet + ~ct + d~ = Dhlad ,

x1 =

1 2

et , x2 = −tet − t − 1.

1 2

0

et +



Mia kai yˆqnoume mia opoiad pote lÔsh ac

       1 t e 0 0 0 2 tet + t+ = . −1 −1 −1 −tet − t − 1

H genik  lÔsh tou sust matoc eÐnai ìpwc gnwrÐsoume to ˆjroisma tic

sugkekrimènhc lÔshc pou mìlic br kame kai tic sumplhrwmatik c lÔshc. Parathr ste ìti mac qreiˆsjhkan kai h kai h

~aet

~btet .

3.8.2

Exetˆste eˆn oi

x1

kai

x2

eÐnai prˆgmati lÔseic.

EpÐshc dokimˆste na jèsete

a2 = 1

kai dokimˆste eˆn oi

prokÔptousec sunart seic eÐnai lÔseic. Poia eÐnai h diaforˆ twn dÔo lÔsewn pou upologÐzoume me ton trìpo autì

3.8.2

Exis¸seic pr¸thc tˆxhc me metablhtoÔc suntelestèc

Upˆrqei bebaÐwc kai h mèjodoc twn metablht¸n paramètrwn , thn opoÐa  dh qrhsimopoi same gia na lÔsoume aplèc exis¸seic.

Gia thn perÐptwsh susthmˆtwn me stajeroÔc suntelestèc h mèjodoc aut  ousiastikˆ tautÐzetai me thn

mèjodo twn oloklhrwtik¸n paragìntwn me thn opoÐa asqolhj kame parapˆnw.

H mèjodoc aut  apoktˆ idiaÐtero

endiafèron gia sust mata me mh-stajeroÔc suntelestèc kai idiaÐtera sthn perÐptwsh pou  dh èqoume thn lÔsh tou antÐstoiqou omogenoÔc probl matoc. Ac jewr soume thn exÐswsh

~ x 0 = A(t) ~ x + f~(t)

(3.9)

kai ac upojèsoume ìti èqoume  dh lÔsei thn antÐstoiqh omogen  exÐswsh jemeliwd¸n lÔsewn diˆnusma

~c.

X(t).

0

~ x = A(t) ~ x

kai br kame ton pÐnaka twn

H genikeumènh lÔsh tou omogenoÔc probl matoc eÐnai bebaÐwc

X(t)~c

gia kˆpoio stajerì

'Opwc akrib¸c sthn perÐptwsh thc mÐac exÐswshc ac prospaj soume na broÔme thn lÔsh thc mh-omogenoÔc

qrhsimopoi¸ntac mia manteyiˆ thc morf c

~ xp = X(t) ~ u(t), ìpou

~ u(t)

eÐnai èna diˆnusma ta stoiqeÐa tou opoÐou antÐ gia stajerèc eÐnai sunart seic. Antikajist¸ntac sthn (3.9)

èqoume

~ xp 0 (t) = X 0 (t) ~ u(t) + X(t) ~ u 0 (t) = A(t) X(t) ~ u(t) + f~(t). Epeid  o

X

eÐnai o pÐnakac twn jemeliwd¸n lÔsewn tou omogenoÔc èqoume ìti

X 0 (t) = A(t)X(t),

kai sunep¸c

X 0 (t) ~ u(t) + X(t) ~ u 0 (t) = X 0 (t) ~ u(t) + f~(t). 'Ara

X(t) ~ u 0 (t) = f~(t). Eˆn upologÐsoume to [X(t)]−1 , tìte ~ u 0 (t) = [X(t)]−1 f~(t). Ac oloklhr¸soume ~ u kai katèpèktash thn ~ xp = X(t) ~ u(t). MporoÔme bebaÐwc na grˆyoume thn lÔsh wc ex c

t¸ra gia na

pˆroume to

Z ~ xp = X(t) Shmei¸ste ìti an o

A

[X(t)]−1 f~(t) dt.

eÐnai stajerìc kai prospaj soume thn manteyiˆ

kai sunep¸c paÐrnoume thn lÔsh

~ xp = e

tA

R

−tA

e

f~(t) dt

X(t) = etA ,

èqoume ìti

[X(t)]−1 = e−tA

dhlad  ìti akrib¸c p rame qrhsimopoi¸ntac thn mèjodo tou

oloklhrwtikoÔ parˆgonta.

Parˆdeigma 3.8.4:

BreÐte mia sugkekrimènh lÔsh tou

~ x0 =

 1 t t2 + 1 1

   −1 t ~ x+ (t2 + 1). t 1

A = t21+1 [ 1t −1 t ] profan¸c den eÐnai stajerìc pÐnakac. Ac upojèsoume ìti gnwrÐzoume (h 0 X = [ 1t −t ] apoteleÐ lÔsh tou X (t) = A(t)X(t). Apì thn stigm  pou èqoume thn sumplhrwmatik  1

Sthn perÐptwsh aut  o manteÔoume) ìti o

(3.10)

lÔsh eÐnai eÔkolo na upologÐsoume thn genikeumènh lÔsh tou (3.10). Pr¸ta diapist¸noume ìti

[X(t)]−1 =

 1 1 t2 + 1 −t

 t . 1

3.8. ΜΗ-ΟΜΟΓΕΝŸΗ ΣΥΣԟΗΜΑΤΑ

99

Mia sugkekrimènh lÔsh tou (3.10) ìpwc gnwrÐsoume eÐnai h

Z ~ xp = X(t) [X(t)]−1 f~(t) dt  Z    1 1 −t 1 t t = (t2 + 1) dt t 1 t2 + 1 −t 1 1  Z   1 −t 2t = dt 2 t 1 −t + 1    1 −t t2 = 1 3 t 1 −3 t + t  1 4  t = 2 33 . t +t 3 Prosjètontac se aut n thn sumplhrwmatik  lÔsh paÐrnoume thn ex c genikeumènh lÔsh tou (3.10).

 1 ~ x= t 3.8.3

Epibebai¸ste ìti prˆgmati oi

3.8.3

−t 1 x1 =

    1 4   t c1 − c2 t + 13 t4 c1 3 = . + 2 3 t +t c2 + (c1 + 1) t + 23 t3 c2 3 1 3

t4

kai

x2 =

2 3

t3 + t

apoteloÔn lÔsh tou (3.10).

Sust mata deÔterhc tˆxhc me stajeroÔc suntelestèc

Απροσδιόριστοι συντελεστές 'Eqoume  dh epilÔsei sust mata deÔterhc tˆxhc me stajeroÔc suntelestèc qrhsimopoi¸ntac thn mèjodo twn aprosdiìristwn suntelest¸n sto

§

3.5. H mèjodoc eÐnai ousiastik  h Ðdia me thn mèjodo twn aprosdiìristwn suntelest¸n

gia sust mata pr¸thc tˆxhc. MporoÔme na kˆnoume kˆpoiec qr simec aplopoi seic, san autèc pou kˆname sthn

§

3.5.

Jewr ste thn exÐswsh

ìpou o pÐnakac

A

eÐnai stajerìc.

An to

~ (t), ~ x 00 = A~ x+F ~ (t) eÐnai thc morf c F ~0 cos ω t, diˆnusma F

tìte mporoÔme na pˆroume san

manteyiˆ thc lÔshc thn ex c

~ xp = ~c cos ω t, kai profan¸c den qreiˆzetai na sumperilˆboume hmÐtona. An to

~ F

~ (t) = F ~0 cos ω0 t + F ~1 cos ω1 t, F ~0 cos ω0 t, mporoÔme na dokimˆsoume ~a cos ω0 t gia na lÔsoume to ~ x 00 = A~ x+F ~0 cos ω1 t. Metˆ mporoÔme apl¸c na prosjèsoume tic lÔseic. to ~ x 00 = A~ x+F

eÐnai ˆjroisma sunhmitìnwn, tìte me bˆsh thn arq  thc upèrjeshc, an

mporoÔme na dokimˆsoume thn manteyiˆ thn manteyiˆ

~b cos ω1 t

gia na lÔsoume

Thn perÐptwsh na upˆrqei  dh kˆpoioc ìroc thc manteyiˆc mac kai sthn sumplhrwmatik  lÔsh,   h exÐswsh na eÐnai thc morf c

~ (t), ~x 00 = A~ x 0 + B~ x+F

mporoÔme na thn antimetwpÐsoume me ton Ðdio akrib¸c trìpo pou qrhsimopoi same

gia ta sust mata pr¸thc tˆxhc. 'Opwc èqoume tèloc  dh diapist¸sei den ja qˆsoume tÐpote eˆn sthn manteyiˆ mac sumperilˆboume kai ìrouc thc sumplhrwmatik c lÔshc, aplˆ oi suntelest¸n twn ìrwn aut¸n ja diapist¸soume ìti èqoun tim  mhdèn kai apl¸c ja èqoume talaipwrhjeÐ qwrÐc lìgo me kˆpoiec epiprìsjetec prˆxeic pou den  tan anagkaÐec.

Ανάλυση ιδιοδιανυσμάτων Gia na lÔsoume to sÔsthma

~ (t), ~ x 00 = A~ x+F mporoÔme na qrhsimopoi soume anˆlush idiodianusmˆtwn, me akrib¸c ton Ðdio trìpo me ta sust mata pr¸thc tˆxhc. 'Estw oi idiotimèc

λ1 ,

...,

λn

kai ta idiodianÔsmata

~v1 ,

...,

~vn .

'Opwc kai prin me bˆsh to

~ x(t) = ~v1 ξ1 (t) + ~v2 ξ2 (t) + · · · + ~vn ξn (t). AnalÔoume to

~ F

me ìrouc idiodianusmˆtwn

~ (t) = ~v1 g1 (t) + ~v2 g2 (t) + · · · + ~vn gn (t). F

E = [ ~v1 · · · ~vn ]

èqoume.

100

ΚΕ֟ΑΛΑΙΟ 3. ΣΥΣԟΗΜΑΤΑ ΣΔΕ

opìte èqoume

~. ~g = E −1 F

AntikajistoÔme kai ìpwc kai prin èqoume

~ x 00 = ~v1 ξ100 + ~v2 ξ200 + · · · + ~vn ξn00 = A (~v1 ξ1 + ~v2 ξ2 + · · · + ~vn ξn ) + ~v1 g1 + ~v2 g2 + · · · + ~vn gn = A~v1 ξ1 + A~v2 ξ2 + · · · + A~vn ξn + ~v1 g1 + ~v2 g2 + · · · + ~vn gn = ~v1 λ1 ξ1 + ~v2 λ2 ξ2 + · · · + ~vn λn ξn + ~v1 g1 + ~v2 g2 + · · · + ~vn gn = ~v1 (λ1 ξ1 + g1 ) + ~v2 (λ2 ξ2 + g2 ) + · · · + ~vn (λn ξn + gn ). Exis¸nontac touc suntelestèc twn idiodianusmˆtwn èqoume tic exis¸seic

ξ100 = λ1 ξ1 + g1 , ξ200 = λ2 ξ2 + g2 , . . .

ξn00 = λn ξn + gn . Kˆje mia apì tic parapˆnw exis¸seic eÐnai anexˆrthth apì tic ˆllec kai sunep¸c mporoÔme na thn lÔsoume me mejìdouc tou kefalaÐou 2. Grˆfoume loipìn

~ x(t) = ~v1 ξ1 (t) + · · · + ~vn ξn (t),

kai èqoume sunep¸c upologÐsei mia sugkekrimènh

lÔsh.

Parˆdeigma 3.8.5:

Ac lÔsoume to sÔsthma tou paradeÐgmatoc

§

3.5 qrhsimopoi¸ntac thn parapˆnw mèjodo.

'Eqoume loipìn thn exÐswsh

~ x 00 = Oi idiotimèc eÐnai

−1

kai

−4,

 −3 2

kai ta idiodianÔsmata

" # g1 g2

=E

−1

   1 0 ~ x+ cos 3t. −2 2  1  1 1  [ 12 ] kai −1 . 'Ara E = 2 −1

" 1 1 ~ (t) = F 3 2

1 −1

#"

0 2 cos 3t

#

" =

kai

E −1 =

1 3

1

1 2 −1 . Sunep¸c,



#

2 cos 3t 3 −2 cos 3t 3

.

Tèloc antikajistoÔme kai èqoume

2 cos 3t, 3 2 ξ200 = −4 ξ2 − cos 3t. 3

ξ100 = −ξ1 +

Efarmìzontac thn mèjodo twn aprosdiìristwn suntelest¸n, manteÔontac ìti shc kai

C2 cos 3t

C1 cos 3t

eÐnai h lÔsh thc pr¸thc exÐsw-

eÐnai h lÔsh thc deÔterhc exÐswsh kai antikajist¸ntac èqoume

2 cos 3t, 3 2 −9C2 cos 3t = −4C2 cos 3t − cos 3t. 3

−9C1 cos 3t = −C1 cos 3t +

MporoÔme na lÔsoume kˆje mia apì tic parapˆnw exis¸seic xeqwristˆ kai na pˆroume ìti

−9C2 = −4C2 − 2/3.

'Ara

C1 = −1/12 ~ x=

kai

C2 = 2/15.

Sunep¸c h sugkekrimènh lÔsh mac eÐnai

        1/20 −1 2 1 1 cos 3t + cos 3t = −3 cos 3t. 2 −1 /10 12 15

BebaÐwc h lÔsh mac aut  tautÐzetai me aut n pou br kame sthn

§

3.5.

−9C1 = −C1 + 2/3

kai

3.8. ΜΗ-ΟΜΟΓΕΝŸΗ ΣΥΣԟΗΜΑΤΑ 3.8.4

3.8.4

101

Ask seic

BreÐte mia sugkekrimènh lÔsh tou sust matoc

x0 = x+2y+2t, y 0 = 3x+2y−4, a) qrhsimopoi¸ntac thn mèjodo

tou oloklhrwtikoÔ parˆgonta, b) qrhsimopoi¸ntac thn mèjodo thc anˆlushc idiodianusmˆtwn, g) qrhsimopoi¸ntac thn mèjodo twn aprosdiìristwn suntelest¸n.

3.8.5

BreÐte mia sugkekrimènh lÔsh tou sust matoc

x0 = 4x+y −1, y 0 = x+4y −et , a) qrhsimopoi¸ntac thn mèjodo

tou oloklhrwtikoÔ parˆgonta, b) qrhsimopoi¸ntac thn mèjodo thc anˆlushc idiodianusmˆtwn, g) qrhsimopoi¸ntac thn mèjodo twn aprosdiìristwn suntelest¸n.

3.8.6

BreÐte mia sugkekrimènh lÔsh tou sust matoc

x001 = −6x1 + 3x2 + cos t, x002 = 2x1 − 7x2 + 3 cos t,

a) qrhsi-

mopoi¸ntac thn mèjodo thc anˆlushc idiodianusmˆtwn, b) qrhsimopoi¸ntac thn mèjodo twn aprosdiìristwn suntelest¸n.

3.8.7

BreÐte mia sugkekrimènh lÔsh tou sust matoc

x001 = −6x1 + 3x2 + cos 2t, x002 = 2x1 − 7x2 + 3 cos 2t,

a)

qrhsimopoi¸ntac thn mèjodo thc anˆlushc idiodianusmˆtwn, b) qrhsimopoi¸ntac thn mèjodo twn aprosdiìristwn suntelest¸n.

3.8.8

Jewr ste thn exÐswsh

0

1

~ x =

t

1

−1 1 t



 t2 . ~ x+ −t 

a) Epibebai¸ste ìti h

 ~ xc = c1 eÐnai h sumplhrwmatik  lÔsh. kekrimènh lÔsh.

   t cos t t sin t + c2 −t cos t t sin t

b) Qrhsimopoi ste thn mèjodo twn metablht¸n paramètrwn gia na breÐte mia sug-

Kefˆlaio 4

Seirèc F ourier kai MDE 4.1 4.1.1

Probl mata sunoriak¸n tim¸n Probl mata sunoriak¸n tim¸n

Prin asqolhjoÔme me tic seirèc èstw ìti gia kˆpoia stajerˆ

λ

F ourier,

prèpei na melet soume ta probl mata sunoriak¸n tim¸n Gia parˆdeigma,

èqoume

x00 + λx = 0, ìpou to

x(t)

èqei pedÐo orismoÔ to

[a, b].

x(a) = 0,

x(b) = 0,

Se antÐjesh me prohgoumènwc, ìpou kajorÐzame thn tim  kai thn parˆgwgo

thc lÔshc se èna shmeÐo, t¸ra orÐsoume thn tim  thc lÔshc se dÔo diaforetikˆ shmeÐa. Shmei¸ste ìti h profan¸c mia lÔsh thc exÐswshc, opìte kai to jèma thc Ôparxhc lÔsewn den mac apasqoleÐ. lÔshc ìmwc eÐnai èna ˆllo endiafèron jèma. Perimènoume ìti sthn genik  lÔsh tou

x00 + λx = 0

x=0

eÐnai

H monadikìthta thc ja emplèkontai dÔo

tuqaÐec stajerèc. EÐnai loipìn logikì (allˆ lˆjoc!) na pisteÔoume ìti ikanopoi¸ntac tic dÔo sunoriakèc sunj kec ja odhghjoÔme se monadik  lÔsh.

Parˆdeigma 4.1.1:

'Estw

λ = 1, a = 0, b = π .

Dhlad  èqoume,

x00 + x = 0,

x(0) = 0,

x(π) = 0.

x = sin t eÐnai mia ˆllh lÔsh (epiprìsjeta thc x = 0) h opoÐa bebaÐwc ikanopoieÐ tic sunoriakèc sunj kec. Ac grˆyoume thn genik  lÔsh thc exÐswshc h opoÐa gnwrÐsoume ìti eÐnai h x = A cos t + B sin t. H sunj kh x(0) = 0 mac anagkˆzei na jèsoume A = 0. H ˆllh sunj kh x(π) = 0 den mac prosfèrei kˆpoia epiprìsjeth plhroforÐa mia kai h x = B sin t thn ikanopoieÐ gia opoiad pote tim  tou B . Upˆrqoun dhlad  ˆpeirec lÔseic thc morf c x = B sin t, ìpou B eÐnai mia tuqaÐa stajerˆ. Profan¸c h

Upˆrqoun ìmwc kai ˆllec lÔseic?

Parˆdeigma 4.1.2:

Ac exetˆsoume thn perÐptwsh tou

x00 + 2x = 0, H genik  lÔsh eÐnai sunj kh prèpei na

λ = 2.

x(0) = 0,

x(π) = 0.

√ √ t. Jètontac x(0) = x = A cos 2 t + B sin 2 √ √ 0 èqoume A = 0. Gia na ikanopoi soume thn deÔterh èqoume 0 = x(π) = B sin 2 π . Epeid  sin 2 π 6= 0 paÐrnoume ìti B = 0. 'Ara h x = 0 eÐnai h

monadik  lÔsh tou probl matoc.

Ti sumbaÐnei loipìn me ta probl mata autˆ?

Mac endiafèrei profan¸c na anakalÔyoume gia poiec stajerèc

èqoume mh-mhdenikèc lÔseic, kai fusikˆ na upologÐsoume tic lÔseic autèc. prìblhma eÔreshc twn idiotim¸n kai twn idiodianusmˆtwn enìc pÐnaka.

102

λ

To prìblhma autì eÐnai anˆlogo me to

4.1. ΠΡΟΒ˟ΗΜΑΤΑ ΣΥΝΟΡΙΑʟΩΝ ΤΙ̟ΩΝ 4.1.2

103

Probl mata idiotim¸n

Gia na anaptÔxoume ta basikˆ stoiqeÐa twn seir¸n

F ourier

ja qreiasteÐ na melet soume ta parakˆtw trÐa probl mata

idiotim¸n.

x00 + λx = 0, 00

x(a) = 0,

x(b) = 0,

0

x + λx = 0,

(4.1)

0

x (a) = 0,

x (b) = 0,

(4.2)

kai

x00 + λx = 0, 'Ena arijmìc

x0 (a) = x0 (b),

x(a) = x(b),

(4.3)

λ lègetai idiotim  tou (4.1) (antÐstoiqa tou (4.2)   (4.3)) an kai mìnon an upˆrqei mh-mhdenik  lÔsh (lÔsh λ. H mh-mhdenik 

pou den eÐnai tautotikˆ mhdèn) tou (4.1) (antÐstoiqa tou (4.2)   (4.3)) me dedomèno to sugkekrimèno aut  lÔsh pou brÐskoume lègetai to antÐstoiqo idiodiˆnusma.

Shmei¸ste thn omoiìthta me tic idiotimèc kai ta idiodianÔsmata pinˆkwn. H omoiìthta aut  den eÐnai kajìlou tuqaÐa. Eˆn fantastoÔme tic algebrikèc exis¸seic san diaforikoÔc telestèc, tìte akoloujoÔme akrib¸c th Ðdia diadikasÐa. d2 Gia parˆdeigma jètoume oi opoÐec na ikanopoioÔn sugkekrimènec 2 . AnazhtoÔme mh-mhdenikèc sunart seic

L = − dt

f

sunoriakèc sunj kec kai apoteloÔn lÔseic thc

(L − λ)f = 0.

Megˆlo mèroc rou formalismoÔ thc grammik c ˆlgebrac

mporeÐ eÔkola na efarmosjeÐ kai gia ta probl mata tou kefalaÐou autoÔ. Den ja epektajoÔme ìmwc sto jèma autì.

Parˆdeigma 4.1.3:

Ac upologÐsoume tic idiotimèc kai ta idiodianÔsmata tou ex c probl matoc

x00 + λx = 0,

x(0) = 0,

x(π) = 0.

Gia lìgouc pou sÔntoma ja katano soume, ac asqolhjoÔme me tic peript¸seic exetˆsoume pr¸ta thn perÐptwsh

λ > 0,

√ x = A cos H sunj kh

x(0) = 0

λ > 0, λ = 0 , λ < 0 x00 + λx = 0 eÐnai

xeqwristˆ. Ac

gia thn opoÐa h genik  lÔsh thc exÐswshc

amèswc sunepˆgetai ìti

A = 0.

√ λ t + B sin

λ t.

EpÐshc

√ 0 = x(π) = B sin

λ π.

√ B eÐnai mhdèn tìte √ h x den eÐnai mh-mhdenik  lÔsh. 'Ara gia na pˆroume mia mh-mhdenik  lÔsh prèpei na√èqoume λ π = 0. 'Ara, h λ π prèpei na eÐnai akèraio pollaplˆsio tou π . Me ˆlla lìgia, prèpei na isqÔei λ = k 2 gia kˆpoio jetikì akèraio k . Sunep¸c oi jetikèc idiotimèc eÐnai k gia ìlouc tou akèraiouc k ≥ 1. MporoÔme na epilèxoume eÔkola san antÐstoiqec idiosunart seic tic x = sin kt. Profan¸c, ìpwc kai sthn grammik  ˆlgebra, kˆje An to

sin

pollaplˆsio enìc idiodianÔsmatoc eÐnai kai autì idiodiˆnusma, ˆra arkeÐ na epilèxoume mìnon èna apì autˆ. Eˆn

λ=0

h exÐswsh ekfulÐzetai sthn

sunepˆgetai ìti

B = 0,

x(π) = 0 ìti ìti λ < 0. Sthn

kai h

Tèloc ac upojèsoume

x00 = 0 kai bebaÐwc h genik  lÔsh eÐnai x = At + B . A = 0. Autì shmaÐnei ìti λ = 0 den eÐnai idiotim .

A=0

√ √ −λ t + B sinh −λ t.

cosh 0 = 1 kai sinh 0 = 0). 'Ara h lÔsh mac prèpei na eÐnai thc x(π) = 0. Kˆti tètoio eÐnai dunatìn mìnon eˆn to B eÐnai ξ = 0, kˆnte thn grafik  parˆstash thc sunˆrthshc sinh gia na

Jètontac

ìti

morf c

kai na ikanopoieÐ thn sunj kh

mhdèn.

x(0) = 0

perÐptwsh aut  èqoume thn ex c genik  lÔsh

x = A cosh x(0) = 0 √ paÐrnoume x = B sinh −λ t kai GiatÐ? Epeid  h sinh ξ

H sunj kh

(jumhjeÐte ìti

eÐnai mhdèn mìnon gia

epibebai¸sete thn parat rhsh aut . Enallaktikˆ mporoÔme na thn epibebai¸soume qrhsimopoi¸ntac ton orismì tou t −t et −e−t me bˆsh ton opoÐo èqoume . 'Ara , to opoÐo mac dÐnei kai to opoÐo isqÔei mìnon

sinh ìtan

0 = sinh t =

t = 0.

2

t = −t

e =e

'Ara den upˆrqoun arnhtikèc idiotimèc.

Sumperasmatikˆ, oi idiotimèc kai ta antÐstoiqa idiodianÔsmata eÐnai

λk = k 2 Parˆdeigma 4.1.4:

me idiodiˆnusma

xk = sin kt

gia kˆje akèraio

k ≥ 1.

Ac upologÐsoume tic idiotimèc kai ta idiodianÔsmata tou ex c probl matoc

x00 + λx = 0,

x0 (0) = 0,

x0 (π) = 0.

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

104

= 0, λ < 0 √ λ > 0, λ √ λ t + B sin λ t. 'Ara √ √ x0 = −A sin λ t + B cos λ t.

Ac exetˆsoume kai pˆli kˆje mia apì tic peript¸seic

λ > 0.

H genik  lÔsh thc

H sunj kh

x0 (0) = 0

x00 + λx = 0

sunepˆgetai

eÐnai

B = 0.

Epiprìsjeta èqoume

0 = x0 (π) = −A sin Epeid  o A √ λ = k gia

den mporeÐ na eÐnai mhdenikìc eˆn jèloume to kˆpoio jetikì akèraio

k.

xeqwristˆ. Pr¸ta ac upojèsoume ìti

x = A cos

λ

√ λ π.

na eÐnai idiotim , kai to

Sunep¸c oi jetikèc idiotimèc eÐnai kai pˆli oi

antÐstoiqa idiodianÔsmata mporoÔme na pˆroume tic

k2

sin

√ λπ

eÐnai mhdèn mìnon ìtan

gia kˆje akèraio

k≥1

en¸ san

x = cos kt.

λ = 0. Sthn perÐptwsh aut  h exÐswsh ekfulÐzetai sthn x00 = 0 kai h genik  lÔsh eÐnai x = At + B opìte èqoume x = A. H x0 (0) = 0 sunepˆgetai ìti A = 0. Profan¸c jètontac x0 (π) = 0 den paÐrnoume kˆpoia epiprìsjeth plhroforÐa sqetikˆ me thn lÔsh. Autì shmaÐnei ìti to B mporeÐ na pˆrei opoiad pote tim  (ac epilèxoume loipìn thn tim  1). 'Ara h λ = 0 eÐnai mia idiotim  kai h x = 1 eÐnai to antÐstoiqo idiodiˆnusma. √ √ Tèloc, an λ < 0 h genik  lÔsh èqei thn morf  x = A cosh −λ t + B sinh −λ t kai sunep¸c √ √ x0 = A sinh −λ t + B cosh −λ t. Ac upojèsoume t¸ra ìti

0

EÐdame  dh ìti (ta

A = B = 0.

A

kai

B

antallˆssoun touc rìlouc touc) gia na eÐnai h lÔsh mhdèn gia

t=0

kai

t=π

prèpei

Den upˆrqoun loipìn arnhtikèc idiotimèc.

Sumperasmatikˆ, oi idiotimèc kai ta antÐstoiqa idiodianÔsmata eÐnai

λk = k 2

me idiodiˆnusma

xk = sin kt

gia kˆje akèraio

k ≥ 1,

kai upˆrqei kai mia akìma idiotim 

λ0 = 0

x0 = 1.

me idiodiˆnusma

To parakˆtw prìblhma eÐnai autì pou mac odhgeÐ stic genikèc seirèc

Parˆdeigma 4.1.5:

F ourier.

Ac upologÐsoume tic idiotimèc kai ta idiodianÔsmata tou ex c probl matoc

x00 + λx = 0, Ac paraleÐyoume thn perÐptwsh

λ< 0

x(−π) = x(π),

x0 (−π) = x0 (π).

thn opoÐa mporoÔme eÔkola na antimetwpÐsoume ìpwc parapˆnw katal -

gontac sto sumpèrasma ìti den upˆrqoun arnhtikèc idiotimèc.

λ = 0, h genik  lÔsh eÐnai x = At + B . H sunj kh x(−π) = x(π) sunepˆgetai ìti A = 0 (h Aπ + B = −Aπ + B A = 0). H deÔterh sunj kh x0 (−π) = x0 (π) den epibˆlei kˆpoion periorismì stic timèc tou B kai sunep¸c h λ = 0 eÐnai idiotim  me antÐstoiqo idiodiˆnusma x = 1. √ √ Gia λ > 0 èqoume ìti x = A cos λ t + B sin λ t. T¸ra √ √ √ √ A cos − λ π + B sin − λ π = A cos λ π + B sin λ π. Gia

sunepˆgetai ìti

sin −θ = − sin θ paÐrnoume √ √ √ √ A cos λ π − B sin λ π = A cos λ π + B sin λ π. √ kai sunep¸c eÐte B = 0   sin λ π = 0. Parìmoia √ (ˆskhsh) an paragwgÐsoume thn x kai antikatast soume sthn deÔterh sunj kh brÐskoume ìti eÐte A = 0   sin λ π = 0. 'Ara gia na èqoume kai to A kai to B mhdèn prèpei na √ √ isqÔei sin λ π = 0. Sunep¸c, h λ eÐnai kˆpoioc akèraioc kai ìpwc kai prohgoumènwc oi idiotimèc eÐnai xanˆ λ = k2 gia kˆje akèraio k ≥ 1. Sthn perÐptwsh aut  ìmwc h x = A cos kt + B sin kt eÐnai idiosunˆrthsh gia kˆje tim  tou A kai tou B . 'Ara èqoume tic ex c dÔo grammikˆ anexˆrthtec idiosunart seic sin kt kai cos kt. An jumhjoÔme ìti

cos −θ = cos θ

kai

Sumperasmatikˆ, oi idiotimèc kai ta antÐstoiqa idiodianÔsmata eÐnai

λk = k 2

me idiodianÔsmata

cos kt

λ0 = 0

me idiodiˆnusma

x0 = 1.

kai

sin kt

gia kˆje akèraio

k ≥ 1,

4.1. ΠΡΟΒ˟ΗΜΑΤΑ ΣΥΝΟΡΙΑʟΩΝ ΤΙ̟ΩΝ 4.1.3

105

Orjogwniìthta idiodianusmˆtwn

Je¸rhma 4.1.1.

Eˆn

x1 (t) kai x2 (t) eÐnai dÔo idiosunart seic tou probl matoc (4.1), (4.2)   (4.3) pou antistoiqoÔn λ1 kai λ2 tìte autèc eÐnai orjog¸niec me thn ex c ènnoia Z b x1 (t)x2 (t) dt = 0.

se dÔo diaforetikèc metaxÔ touc idiotimèc

a To je¸rhma autì èqei thn ex c polÔ apl , komy  kai diafwtistik  apìdeixh. XekinoÔme me tic ex c dÔo exis¸seic

x001 + λ1 x1 = 0 Pollaplasiˆzoume thn pr¸th me

x2

kai thn deÔterh me

x002 + λ2 x2 = 0.

kai

x1

kai afairoÔme gia na katal xoume sthn sqèsh

(λ1 − λ2 )x1 x2 = x002 x1 − x2 x001 . Ac oloklhr¸soume t¸ra kai ta dÔo mèlh thc exÐswshc.

b

Z (λ1 − λ2 )

b

Z x1 x2 dt =

a

x002 x1 − x2 x001 dt

a

Z

b


d x02 x1 − x2 x01 dt a dt h ib = x02 x1 − x2 x01 = 0.

=

t=a

H teleutaÐa isìthta prokÔptei ex aitÐac twn sunoriak¸n sunjhk¸n. Gia parˆdeigma, eˆn jewr soume thn (4.1) èqoume

x1 (a) = x1 (b) = x2 (a) = x2 (b) = 0

kai ˆra

x02 x1 − x2 x01

eÐnai mhdèn kai sto

a

kai sto

b.

Mia kai

λ1 6= λ2 ,

h apìdeixh

èqei oloklhrwjeÐ.

4.1.1(eÔkolh)

Oloklhr¸ste pl rwc thn apìdeixh tou jewr matoc (elègxte thn teleutaÐa isìthta thc apìdeixhc) gia

tic peript¸seic (4.2) kai (4.3). 'Opwc diapist¸same prohgoumènwc h 'Ara èqoume to olokl rwma

sin nt

eÐnai idiodiˆnusma tou probl matoc

x00 + λx = 0, x(0) = 0, x(π) = 0.

π

Z

(sin mt)(sin nt) dt = 0,

ìtan

m 6= n.

(cos mt)(cos nt) dt = 0,

ìtan

m 6= n.

(sin mt)(sin nt) dt = 0,

ìtan

m 6= n,

(cos mt)(cos nt) dt = 0,

ìtan

m 6= n,

0 Parìmoia èqoume

π

Z 0 kai telikˆ èqoume

Z

π

Z

−π π −π

kai

π

Z

(cos mt)(sin nt) dt = 0. −π To parapˆnw je¸rhma isqÔei kai gia diaforetikèc sunoriakèc sunj kec. (deÐte to thn apìdeixh) ìti isqÔei gia tic

x0 (a) = x0 (b) = 0,

  gia tic

Gia parˆdeigma, eÐnai eÔkolo na doÔme

x(a) = x0 (b) = 0

  gia tic

x0 (a) = x(b) = 0.

MporoÔme, me bˆsh ta parapˆnw, na efarmìsoume to je¸rhma gia na broÔme tic timèc twn ex c oloklhrwmˆtwn

Z

π

Z (sin mt)(sin nt) dt = 0

−π ìtan

m 6= n,

π

(cos mt)(cos nt) dt = 0,

kai

−π

kai

Z

π

(sin mt)(cos nt) dt = 0, −π gia kˆje

m

kai

n.

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

106 4.1.4

F redholm

Enallaktikì je¸rhma tou

Ac asqolhjoÔme gia lÐgo me èna idiaÐtera qr simo je¸rhma twn diaforik¸n exis¸sewn. To je¸rhma autì isqÔei gia polÔ pio genikˆ probl mata apì autì pou ja asqolhjoÔme emeÐc. H apl  morf  tou, pou ja parousiˆsoume parakˆtw, ìmwc kalÔptei pl rwc tic anˆgkec mac.

Je¸rhma 4.1.2

(enallaktikì je¸rhma tou

F redholm

∗

). 'Etsw

p

kai

q

suneqeÐc sunart seic sto

[a, b].

Tìte eÐte

to prìblhma

x00 + λx = 0,

x(a) = 0,

x(b) = 0

(4.4)

èqei mia mh-mhdenik  lÔsh, eÐte to prìblhma

x00 + λx = f (t), èqei mia monadik  lÔsh gia kˆje suneq  sunˆrthsh

x(a) = 0,

x(b) = 0

(4.5)

f.

To je¸rhma isqÔei kai gia tic upìloipec (diaforetikoÔ tÔpou) sunoriakèc sunj kec pou èqoume sunant sei parapˆnw.

To je¸rhma ousiastikˆ mac dhl¸nei ìti eˆn to

monadik  lÔsh gia kˆje dexiì mèloc. mhn èqei lÔsh gia kˆpoiec

4.1.5

f,

λ

den eÐnai idiotim , tìte to mh-omogenèc prìblhma (4.5) èqei

Mac dhl¸nei epÐshc ìti an h

λ

eÐnai idiotim  tìte to prìblhma (4.5) mporeÐ na

kai epiprìsjeta, akìma kai eˆn èqei lÔsh tìte aut  den ja eÐnai monadik .

Ask seic

Upìdeixh: Gia ìlec tic parakˆtw ask seic shmei¸ste ìti oi

cos

√

λ (t − a)

kai

sin

√

λ (t − a)

eÐnai epÐshc lÔshc tou

omogenoÔc probl matoc.

4.1.2

UpologÐste ìlec tic idiotimèc kai ta idiodianÔsmata tou probl matoc

• x00 + λx = 0, x(a) = 0, x(b) = 0. • x00 + λx = 0, x0 (a) = 0, x0 (b) = 0. • x00 + λx = 0, x0 (a) = 0, x(b) = 0. • x00 + λx = 0, x(a) = x(b), x0 (a) = x0 (b). 4.1.3

Elègxte analutikˆ thn perÐptwsh tou

0

0

x(π), x (−π) = x (π).

∗ P re

λ<0

gia to ex c prìblhma sunoriak¸n tim¸n

Eidikìtera deÐxte ìti den upˆrqoun arnhtikèc idiotimèc.

to ìnomˆ tou apì ton Souhdì majhmatikì Erik Ivar F redholm (1866

1927).

x00 + λx = 0, x(−π) =

4.2. ΤΡΙΓΩΝΟΜΕΤΡΙʟΕΣ ΣΕΙџΕΣ 4.2

107

Trigwnometrikèc seirèc

4.2.1

Periodikèc sunart seic

H melèth tou probl matoc

x00 + ω02 x = f (t), ìpou

f (t)

(4.6)

eÐnai kˆpoia periodik  sunˆrthsh ja mac d¸sei èna safèc kÐnhtro gia na melet soume tic seirèc

F ourier.

'Hdh èqoume lÔsei thn exÐswsh

x00 + ω02 x = F0 cos ω t. 'Enac trìpoc gia na lÔsoume thn (4.6) eÐnai na analÔsoume thn

(4.7)

f (t) san ˆjroisma hmitìnwn (endeqomènwc kai sunhmitìn-

wn) kai katìpin na lÔsoume pollˆ probl mata thc morf c (4.7).

Metˆ arkeÐ na qrhsimopoi soume thn arq  thc

upèrjeshc, gia na ajroÐsoume ìlec tic lÔseic pou p rame kai ètsi na apokt soume thn lÔsh thc (4.6). Prin proqwr soume parapèra, ac asqolhjoÔme gia lÐgo me merikˆ jèmata twn periodik¸n sunart sewn. Lème ìti

P an isqÔei f (t) = f (t + P ) gia kˆje t. En suntomÐa ja lème ìti h f (t) eÐnai P -periodik  sunˆrthsh eÐnai tautìqrona kai 2P -periodik , 3P -periodik  kai oÔtw kaj' cos t kai sin t eÐnai 2π -periodik  ìpwc bebaÐwc eÐnai kai oi cos kt kai sin kt gia kˆje akèraio

mia sunˆrthsh eÐnai periodik  me perÐodo

P -periodik .

Shmei¸ste ìti mia

ex c. Gia parˆdeigma, oi arijmì

k.

Oi stajerèc sunart seic apoteloÔn akraÐa paradeÐgmata periodik¸n sunart sewn. EÐnai periodikèc wc proc

opoiad pote perÐodo (giatÐ?).

f (t) orismènh se kˆpoio diˆsthma [−L, L] thn opoÐa epijumoÔme na epekteÐnoume 2L-periodik  sunˆrthsh. MporoÔme na kˆnoume thn epèktash aut  orÐzontac mia nèa sunˆrthsh F (t) tètoia ¸ste gia t sto diˆsthma [−L, L], F (t) = f (t). Gia t sto diˆsthma [L, 3L], orÐzoume F (t) = f (t − 2L), gia t sto [−3L, −L], F (t) = f (t + 2L), kai oÔtw to kajex c. Ac jewr soume mia sunˆrthsh

periodikˆ ètsi ¸ste na thn kˆnoume

Parˆdeigma 4.2.1:

OrÐste thn

f (t) = 1−t2

sto diˆsthma

[−1, 1] kai epekteÐnete thn periodikˆ ètsi ¸ste na pˆrete

mia 2-periodik  sunˆrthsh. DeÐte to Sq ma 4.1. -3

-2

-1

0

1

2

3

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5 -3

-2

-1

0

1

2

3

Σχήμα 4.1: Περιοδική επέκταση της συνάρτησης 1 − t2 .

Prosoq !

H

f (t)

den tautÐzetai me thn epèktas  thc.

ˆllh perÐodo apì thn epèktas  thc

4.2.1

OrÐste thn

f (t) = cos t

Gia parˆdeigma h

sto diˆsthma

[−π/2, π/2].

mporeÐ na eÐnai epÐshc periodik  me

π -periodik  cos t.

Kataskeuˆste thn

grafik  thc parˆstash. SugkrÐnete thn me thn grafik  parˆstash thc

4.2.2

f (t)

F (t). epèktas  thc kai kˆnte thn

Eswterikì ginìmeno kai anˆlush idiodianusmˆtwn

Gia na analÔsoume to diˆnusma

~v

se sunduasmì dianusmˆtwn

w ~1

kai

~v = a1 w ~ 1 + a2 w ~ 2.

w ~2

orjog¸niwn metaxÔ touc èqoume

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

108 Ac upologÐsoume ènan tÔpo gia ta

a1

kai

a2 .

Pr¸ta ac upologÐsoume

h~v , w~1 i = ha1 w ~ 1 + a2 w ~ 2 , w~1 i = a1 hw ~ 1 , w~1 i + a2 hw ~ 2 , w~1 i = a1 hw ~ 1 , w~1 i. 'Ara,

a1 =

h~v , w~1 i . hw ~ 1 , w~1 i

a2 =

h~v , w~2 i . hw ~ 2 , w~2 i

Parìmoia èqoume

Pijan¸c na jumˆste ton parapˆnw tÔpo apì to mˆjhma tou LogismoÔ.

Parˆdeigma 4.2.2:

 1  ~v = [ 23 ] san grammikì sunduasmì twn w~1 = −1 kai w ~2 = [ 11 ]. ta w ~ 1 kai w ~ 2 eÐnai orjog¸nia metaxÔ touc mia kai hw ~ 1, w ~ 2 i = 1(1) + (−1)1 = 0.

Grˆyte to

Xekin ste shmei¸nontac ìti

a1 =

Tìte

2(1) + 3(−1) h~v , w~1 i −1 = = . hw ~ 1 , w~1 i 1(1) + (−1)(−1) 2 a2 =

h~v , w~2 i 2+3 5 = = . hw ~ 2 , w~2 i 1+1 2

Hence

      5 1 −1 1 2 + . = 3 2 −1 2 1 4.2.3

Trigwnometrikèc seirèc

T¸ra, antÐ na analÔsoume èna diˆnusma wc proc ta idiodianÔsmata enìc pÐnaka, ac analÔsoume mia sunˆrthsh wc proc tic idiosunart seic enìc probl matoc idiotim¸n. Sugkekrimèna, ac jewr soume to ex c prìblhma

x00 + λx = 0,

x(−π) = x(π),

tou opoÐou ta idiodianÔsmata xèroume ìti eÐnai 1, sunˆrthsh

f (t)

cos kt

kai

x0 (−π) = x0 (π)

sin kt.

Jèloume na parast soume mia

2π -periodik 

wc ex c

∞

f (t) = H parapˆnw seirˆ lègete seirˆ

F ourier†

qrhsimopoi same thn idiosunˆrthsh na asqolhjoÔme mìnon me tic

cos kt

a0 X an cos nt + bn sin nt. + 2 n=1

  trigwnometrik  seirˆ tou

kai

h f (t) , cos nt i.

def

Gia parˆdeigma, gia na broÔme to

1 π

Z

† Apì

kai sunep¸c

an

prèpei na up-

π

f (t)g(t) dt. −π

Qrhsimopoi¸ntac ton orismì autìn, blèpoume ìti oi idiosunart seic

sin kt

1 = cos 0t,

OrÐsoume loipìn to eswterikì ginìmeno wc ex c

h f (t) , g(t) i =

dianÔsmatoc), kai

Shmei¸ste ìti, gia dikiˆ mac eukolÐa,

sin kt.

Ac orÐsoume t¸ra to eswterikì ginìmeno sunart sewn. ologÐsoume to

f (t).

1 antÐ gia thn 1. MporoÔme bebaÐwc na jewr soume ìti 2

cos kt

(sumperilambanomènou tou stajeroÔ idio-

eÐnai, ìpwc apodeÐxame sthn prohgoÔmenh parˆgrafo, orjog¸niec me thn ex c ènnoia

h cos mt , cos nt i = 0

gia

m 6= n,

h sin mt , sin nt i = 0

gia

m 6= n,

h sin mt , cos nt i = 0

gia kˆje

m

kai

to ìnoma tou Gˆllou majhmatikoÔ Jean Baptiste Joseph F ourier (1768

n. 1830).

4.2. ΤΡΙΓΩΝΟΜΕΤΡΙʟΕΣ ΣΕΙџΕΣ

109

h cos nt , cos nt i = 1 (ektìc apì thn perÐptwsh n = 0) h 1 , 1 i = 2. Oi suntelestèc loipìn dÐnontai apì tic sqèseic

EÔkola diapist¸noume ìti stajerˆ èqoume

an = bn = O parapˆnw tÔpoc isqÔei kai gia

n = 0,

h f (t) , cos nt i 1 = h cos nt , cos nt i π h f (t) , sin nt i 1 = h sin nt , sin nt i π

Z

kai

h sin nt , sin nt i = 1.

Gia thn

π

f (t) cos nt dt, Z

−π π

f (t) sin nt dt. −π

opìte èqoume

a0 =

1 π

Z

π

f (t) dt. −π

Ac elègxoume touc tÔpou pou p rame qrhsimopoi¸ntac tic idiìthtec orjogwniìthtac. Ac upojèsoume proc stigm  ìti

∞

f (t) = Tìte gia

m≥1

a0 X + an cos nt + bn sin nt. 2 n=1

èqoume

h f (t) , cos mt i = =

∞ E Da X 0 an cos nt + bn sin nt , cos mt + 2 n=1 ∞ X a0 an h cos nt , cos mt i + bn h sin nt , cos mt i h 1 , cos mt i + 2 n=1

= am h cos mt , cos mt i. kai sunep¸c

4.2.2

am =

h f (t) , cos mt i . h cos mt , cos mt i

UpologÐste analutikˆ ta

Parˆdeigma 4.2.3:

a0

kai

bm .

'Estw h sunˆrthsh

f (t) = t gia

t

sto diˆsthma

(−π, π].

EpekteÐnete periodikˆ thn

f (t)

kai grˆyte thn se morf  seirˆc

aut  eÐnai gnwst  me to ìnoma prionwt . -5.0

-2.5

0.0

2.5

5.0

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3 -5.0

-2.5

0.0

2.5

5.0

Σχήμα 4.2: Γραφική παράσταση της πριονωτής συνάρτησης.

F ourier.

H sunˆrthsh

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

110

H grafik  parˆstash thc epektamènhc periodik c sunˆrthshc dÐnetai sto Sq ma 4.2 sthn prohgoÔmenh selÐda. Ac upologÐsoume t¸ra touc suntelestèc xekin¸ntac apì to

Z

1 π

a0 =

a0

π

t dt = 0. −π

Ja kˆnoume suqn  qr sh tou gegonìtoc ìti to olokl rwma miac peritt c sunˆrthshc se èna summetrikì diˆsthma

ϕ(t)

eÐnai mhdèn. JumhjeÐte ìti lème ìti mia sunˆrthsh

t, sin t,

kai en prokeimènw

t cos mt

am = Ac upologÐsoume t¸ra ta

bm .

eÐnai peritt  an

ϕ(−t) = −ϕ(t).

Gia parˆdeigma oi sunart seic

eÐnai ìlec perittèc.

1 π

π

Z

t cos mt dt = 0. −π

Ed¸ ja qrhsimopoi soume to gegonìc ìti to olokl rwma miac ˆrtiac sunˆrthshc se

èna summetrikì diˆsthma isoÔtai me to diplˆsio tou oloklhr¸matoc sto misì diˆsthma. sunˆrthsh

ϕ(t)

eÐnai ˆrtia an

JumhjeÐte ìti lème ìti mia

ϕ(−t) = ϕ(t). Gia parˆdeigma h t sin mt eÐnai ˆrtia. Z 1 π bm = t sin mt dt π −π Z π 2 = t sin mt dt π 0  π  Z π 2 −t cos mt 1 = cos mt dt + π m m 0 t=0  2  −π cos mπ = +0 π m 2 (−1)m+1 −2 cos mπ = = . m m

Qrhsimopoi same to gegonìc ìti

( cos mπ = (−1)

m

=

1 −1

an to an to

m m

,

eÐnai ˆrtio

.

eÐnai perittì

'Ara oi seirèc eÐnai

f (t) =

∞ X 2 (−1)n+1 sin nt. n n=1

Ac grˆyoume tic 3 pr¸tec armonikèc twn seir¸n tou

f (t).

f (t) = 2 sin t − sin 2 t +

2 sin 3 t + · · · 3

H grafik  parˆstash twn pr¸twn tri¸n aut¸n ìrwn twn seir¸n, mazÐ me aut  twn pr¸twn 20 dÐdetai sto Sq ma 4.3 sthn epìmenh selÐda.

Parˆdeigma 4.2.4:

Na epektajeÐ periodikˆ h sunˆrthsh

( 0 f (t) = π kai na grafjeÐ san seirˆ

F ourier.

an an

−π < t ≤ 0, 0 < t ≤ π.

H en lìgw sunˆrthsh, kai oi parallagèc thc, emfanÐzetai suqnˆ se pollèc

efarmogèc kai eÐnai gnwst  me to ìnoma tetragwnikì kÔma. H grafik  parˆstash thc epektamènh periodik c sunˆrthshc dÐnetai sto Sq ma 4.4 sthn paroÔsa selÐda. upologÐsoume t¸ra touc suntelestèc xekin¸ntac apì to

a0 =

1 π

Z

a0

π

f (t) dt = −π

1 π

π

Z

π dt = π. 0

Ac

4.2. ΤΡΙΓΩΝΟΜΕΤΡΙʟΕΣ ΣΕΙџΕΣ -5.0

-2.5

0.0

111

2.5

5.0

-5.0

-2.5

0.0

2.5

5.0

3

3

3

3

2

2

2

2

1

1

1

1

0

0

0

0

-1

-1

-1

-1

-2

-2

-2

-2

-3

-3

-3 -5.0

-2.5

0.0

2.5

-3

5.0

-5.0

-2.5

0.0

2.5

5.0

Σχήμα 4.3: Οι πρώτες 3 (αριστερό γράφημα) και οι πρώτες 20 (δεξιό γράφημα) αρμονικές της πριονωτής συνάρτησης. -5.0

-2.5

0.0

2.5

5.0

3

3

2

2

1

1

0

0

-5.0

-2.5

0.0

2.5

5.0

Σχήμα 4.4: Γραφική παράσταση της συνάρτησης του τετραγωνικού κύματος.

AkoloÔjwc,

am =

1 π

Z

π

f (t) cos mt dt = −π

1 π

Z

π

π cos mt dt = 0. 0

kai telikˆ

bm =

1 π

Z

π

Z

−π π

f (t) sin mt dt

1 π sin mt dt π 0  π − cos mt = m t=0 =

1 − (−1)m 1 − cos πm = = = m m

(

2 m

an to

0

an to

m m

,

eÐnai perittì

.

eÐnai ˆrtio

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

112 'Ara oi seirèc eÐnai

f (t) =

∞ X

π + 2

n=1

n

∞

π X 2 2 sin nt = + sin (2k − 1) t. n 2 2k − 1 k=1

perittì

Ac grˆyoume tic 3 pr¸tec armonikèc twn seir¸n tou

f (t) =

f (t).

2 π + 2 sin t + sin 3t + · · · 2 3

H grafik  parˆstash twn pr¸twn tri¸n aut¸n ìrwn twn seir¸n, mazÐ me aut  twn pr¸twn 20 dÐdetai sto Sq ma 4.5. -5.0

-2.5

0.0

2.5

5.0

-5.0

-2.5

0.0

2.5

5.0

3

3

3

3

2

2

2

2

1

1

1

1

0

0

0

0

-5.0

-2.5

0.0

2.5

5.0

-5.0

-2.5

0.0

2.5

5.0

Σχήμα 4.5: Οι πρώτες 3 (αριστερό γράφημα) και οι πρώτες 20 (δεξιό γράφημα) αρμονικές της συνάρτησης του τετραγωνικού κύματος.

Den asqolhj kame, oÔte ja asqolhjoÔme, me thn sÔgklish twn seir¸n pou upologÐsame.

Ac rÐxoume ìmwc mia

matiˆ sto katˆ pìso aut  exartˆtai apì endeqìmenec asunèqeiec. Ac estiˆsoume sto shmeÐo asunèqeiac tou tetragwnikoÔ kÔmatoc kai ac kˆnoume thn grafik  parˆstash twn pr¸twn 100 armonik¸n sto Sq ma 4.6 sthn paroÔsa selÐda. Parathr ste ìti h seirˆ apoteleÐ mia kal  prosèggish thc sunˆrthshc makriˆ apì ta shmeÐa asunèqeiac. To sfˆlma (uper-ektÐmhsh) kontˆ sto asuneqèc shmeÐo san fainìmeno tou

Gibbs.

t=π

den faÐnetai na elatt¸netai. H sumperiforˆ aut  eÐnai gnwst  kai

H perioq  ìpou to sfˆlma eÐnai megˆlo ìmwc surrikn¸nesai ìso auxˆnoume touc ìrouc thc

seirˆc pou lambˆnoume upoyin mac.

4.2.4

4.2.3

Ask seic

EpekteÐnete periodikˆ kai upologÐste tic seirèc twn sunart sewn

• f (t) = sin 5t + cos 3t • f (t) = |t| • f (t) = |t|3 ( −1 • f (t) = 1 • f (t) = t3 • f (t) = t2

an an

−π < t ≤ 0, 0 < t ≤ π.

f (t)

oi opoÐec orÐzontai sto

[−π, π]

wc ex c

4.2. ΤΡΙΓΩΝΟΜΕΤΡΙʟΕΣ ΣΕΙџΕΣ

1.75

113

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.50

3.25

3.25

3.00

3.00

2.75

2.75

1.75

2.00

2.25

2.50

2.75

3.00

3.25

Σχήμα 4.6: Η δράση του φαινομένου του Gibbs.

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

114 4.3 4.3.1

Epiprìsjeta jèmata seir¸n

2L-periodikèc

diaforetikèc periìdouc. loipìn ìti èqoume mia

sunart seic

F ourier

'Eqoume asqolhjeÐ me seirèc

2π -periodikèc

gia

sunart seic.

2L-periodik  f (t)

L

sunˆrthsh (to

onomˆzetai hmi-perÐodoc). 'Estw



2π -periodik .

Ac asqolhjoÔme t¸ra kai me sunart seic me

Mia tètoia epèktash eÐnai idiaÐtera eÔkolh kai apaiteÐ aplˆ mia allag  metablht¸n.

g(s) = f eÐnai

F ourier

π L

s=

t,

'Estw

tìte h sunˆrthsh

 L s π

Prèpei bebaÐwc na allˆxoume kai ta hmÐtona kai ta sunhmÐtona wc ex c.

∞

f (t) =

Eˆn allˆxoume thn metablht 

t

se

s

nπ a0 X nπ + t + bn sin t. an cos 2 L L n=1

blèpoume ìti

∞

g(s) = 'Ara mporoÔme na upologÐsoume ta

an

kai

bn

a0 X + an cos ns + bn sin ns. 2 n=1

ìpwc kai prohgoumènwc. Grˆfoume loipìn ta oloklhr¸mata kai allˆzoume

thn metablht  mac epistrèfontac ètsi pÐsw sto

a0 = an = bn =

π

1 π

Z

1 π

Z

1 π

t.

g(s) ds = −π π

1 L

Z

L

f (t) dt, −L

g(s) cos ns ds = −π Z π

g(s) sin ns ds = −π

1 L

Z

1 L

Z

L

f (t) cos −L L

f (t) sin −L

nπ t dt, L

nπ t dt. L π F ourier genik¸n 2L-

Oi dÔo sunart seic me tic opoÐec asqoloÔntai, lìgw thc aplìthtac touc, ta paradeÐgmatˆ mac èqoun hmi-perÐodo kai 1. Na tonÐsoume ìti aplˆ apaiteÐtai allag  metablht¸n kai tÐpote ˆllo. Eˆn èqoume katano sei tic seirèc gia

2π -periodikèc

sunart seic den ja sunant soume kanèna apolÔtwc prìblhma me thn perÐptwsh twn

periodik¸n sunart sewn.

Parˆdeigma 4.3.1:

'Estw ìti h

f (t) = |t|

gia

−1 < t < 1,

èqei epektajeÐ periodikˆ. H grafik  parˆstash thc periodik c aut c epèktashc dÐdetai sto Sq ma 4.7 sthn epìmenh selÐda. UpologÐste thn seirˆ

F ourier thc. P f (t) = a20 + ∞ n=1 an cos nπt + bn sin nπt.

Jèloume na pˆroume

Gia

n≥1

èqoume ìti h

|t| cos nπt

sunep¸c

Z

1

an =

f (t) cos nπt dt −1

Z

1

t cos nπt dt

=2 0

1 Z 1 t 1 sin nπt −2 sin nπt dt nπ nπ 0 t=0 (
i1 2 (−1)n − 1 0 1 h = 0 + 2 2 cos nπt = = −4 n π n2 π 2 t=0 n2 π 2 

=2

an to an to

n n

,

eÐnai ˆrtio

.

eÐnai perittì

eÐnai ˆrtia kai

4.3. ΕΠΙΠџΟΣΘΕΤΑ ȟΕΜΑΤΑ ΣΕΙџΩΝ F OU RIER -2

-1

115

0

1

2

1.00

1.00

0.75

0.75

0.50

0.50

0.25

0.25

0.00

0.00

-2

-1

0

1

2

Σχήμα 4.7: Περιοδική επέκταση της συνάρτησης f (t).

O

a0

upologÐzetai eÔkola wc ex c

Z

1

|t| dt = 1.

a0 = −1 Tèloc ac upologÐsoume ta

bn .

|t| sin nπt

ParathroÔme ìti h

eÐnai peritt  kai sunep¸c èqoume

1

Z bn =

f (t) sin nπt dt = 0. −1

'Ara h seirˆ pou yˆqnoume eÐnai h

f (t) =

1 + 2

−4 cos nπt. n2 π 2

X n=1

perittìc

n

Ac grˆyoume analutikˆ merikoÔc apì touc pr¸touc ìrouc thc seirˆc. Sugkekrimèna, ac asqolhjoÔme me tic treic pr¸tec armonikèc.

f (t) ≈

1 4 4 − 2 cos πt − 2 cos 3πt − · · · 2 π 9π

H grafik  parˆstash thc parapˆnw seirˆc ìpwc kai aut c pou apoteleÐtai apì tic pr¸thc 20 armonikèc dÐnetai sto Sq ma 4.8 sthn epìmenh selÐda.

•

AxÐzei na parathr soume to pìso eÔkola proseggÐzei h seirˆ thn pragmatik 

sunˆrthsh kai thn apousÐa tou fainomènou tou

4.3.2

4.3.1

Gibbs

mia kai den upˆrqoun shmeÐa asunèqeiac.

Ask seic

'Estw ìti h

( f (t) = èqei epektajeÐ periodikˆ. a) UpologÐste thn seirˆ

0 t

−1 < t < 0, 0 ≤ t < 1,

if if

F ourier

thc

f (t).

b) Grˆyte thn seirˆ analutikˆ mèqri kai thn

3h

thc armonik .

4.3.2

'Estw ìti h

( f (t) = èqei epektajeÐ periodikˆ. a) UpologÐste thn seirˆ thc armonik .

−t t2

if if

F ourier

−1 < t < 0, 0 ≤ t < 1,

thc

f (t).

b) Grˆyte thn seirˆ analutikˆ mèqri kai thn

3h

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

116 -2

-1

0

1

2

-2

-1

0

1

2

1.00

1.00

1.00

1.00

0.75

0.75

0.75

0.75

0.50

0.50

0.50

0.50

0.25

0.25

0.25

0.25

0.00

0.00

0.00

0.00

-2

-1

0

1

2

-2

-1

0

1

2

Σχήμα 4.8: Σειρές F ourier της f (t) με τις 3 πρώτες αρμονικές (αριστερά) και τις 20 πρώτες αρμονικές (δεξιά).

4.3.3

'Estw ìti h

( f (t) = èqei epektajeÐ periodikˆ. a) UpologÐste thn seirˆ thc armonik .

−t 10 t 10

if if

F ourier

−10 < t < 0, 0 ≤ t < 10, thc

f (t).

b) Grˆyte thn seirˆ analutikˆ mèqri kai thn

3h

4.4. ΣΕΙџΕΣ ΗΜΙԟΟΝΩΝ ΚΑΙ ΣΥΝΗΜΙԟΟΝΩΝ 4.4

117

Seirèc hmitìnwn kai sunhmitìnwn

4.4.1

Perittèc kai ˆrtiec periodikèc sunart seic

Ja parathr sate Ðswc ìti oi perittèc sunart seic den èqoun ìrouc sunhmitìnou sta anaptÔgmata touc se seirèc

F ourier

oi ˆrtiec den èqoun ìrouc hmitìnwn. Kˆti tètoio den eÐnai tuqaÐo ìpwc ja doÔme parakˆtw.

JumhjeÐte ìti h parˆdeigma, h

cos nt

f (t)

eÐnai mia peritt  an

eÐnai ˆrtia kai h

kai peritt  gia perittèc timèc tou

4.4.1

h(t)

Mia sunˆrthsh

f (t) tk

eÐnai ˆrtia an

f (−t) = f (t).

Gia

eÐnai ˆrtia gia ˆrtiec timèc tou

k

f (t)

kai

g(t)

kai orÐste ton ginìmenì touc

h(t) = f (t)g(t). a) Eˆn kai oi dÔo eÐnai h(t) peritt    ˆrtia? c) Eˆn kai oi

peritt ? b) Eˆn eÐnai mia peritt  kai h ˆllh ˆrtia, eÐnai h

dÔo eÐnai ˆrtiec, eÐnai kai h An h

f (−t) = −f (t).

eÐnai peritt . Parìmoia h sunˆrthsh

k.

Jewr ste dÔo sunart seic

perittèc, eÐnai kai h

sin nt

h(t)

ˆrtia?

f (t) eÐnai kai h g(t) eÐnai den mporoÔme na apofanjoÔme gia to eˆn h f (t) + g(t) eÐnai kai aut  ˆrtia. Mˆlista, F ourier miac opoiasd pote sunˆrthshc apoteleÐte apì to ˆjroisma miac peritt c sunˆrthshc

to anˆptugma miac seirˆc

(oi ìroi hmitìnwn) kai miac ˆrtiac sunˆrthshc (oi ìroi sunhmitìnwn). Sthn parˆgrafo aut  ja asqolhjoÔme mìnon me sunart seic pou eÐnai eÐte perittèc   ˆrtiec. orÐsame

2L-periodikèc

epektˆseic miac sunˆrthshc orismènhc se èna diˆsthma thc morf c

endiafèrei h sumperiforˆ thc sunˆrthshc mìno sto diˆsthma

[0, L].

[−L, L].

Prohgoumènwc

Arketèc forèc mac

Eˆn h en lìgw sunˆrthsh eÐnai peritt  (  ˆrtia)

tìte den ja upˆrqoun oi ìroi twn hmitìnwn (  antÐstoiqa twn sunhmitìnwn). 'Estw loipìn ìti h sunˆrthsh sto diˆsthma

f (t)

eÐnai orismènh sto diˆsthma

[0, L].

MporoÔme na orÐsoume tic ex c sunart seic

(−L, L] ori

(

ori

(

Fperitt  (t) = Fˆrtia (t) = kai epekteÐnoume tic

f (t) −f (−t) f (t) f (−t)

0 ≤ t ≤ L, −L < t < 0,

an an an an

0 ≤ t ≤ L, −L < t < 0.

Fperitt  (t) kai Fˆrtia (t) ètsi ¸ste na eÐnai 2L-periodik . Tìte h Fperitt  (t) f (t), kai h Fˆrtia (t) onomˆzetai ˆrtia periodik  epèktash thc f .

onomˆzetai peritt 

periodik  epèktash thc

4.4.2

Epibebai¸ste ìti h

Parˆdeigma 4.4.1:

Fperitt  (t)

eÐnai peritt  kai h

'Estw h sunˆrthsh

Fˆrtia (t)

f (t) = t(1 − t)

ˆrtia.

orismènh sto diˆsthma

tic grafikèc parastˆseic twn peritt¸n kai ˆrtiwn epektˆsewn thc -2

-1

0

1

2

-2

[0, 1].

To sq ma 4.9 perilambˆnei

f (t). -1

0

1

2

0.3

0.3

0.3

0.3

0.2

0.2

0.2

0.2

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

-0.1

-0.1

-0.1

-0.1

-0.2

-0.2

-0.2

-0.2

-0.3

-0.3

-0.3 -2

-1

0

1

2

-0.3 -2

-1

0

1

Σχήμα 4.9: Περιττή και άρτια 2-περιοδική επέκταση της f (t) = t(1 − t), 0 ≤ t ≤ 1.

2

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

118 4.4.2 'Estw ìti

Seirèc hmitìnwn kai sunhmitìnwn

f (t)

eÐnai mia peritt 

2L-periodik  sunˆrthsh. An upologÐsoume to anˆptugma an (sumperilambanomènou kai tou a0 ) èqoume

thc

f (t)

se seirˆ

F ourier,

upologÐzontac ìlouc touc suntelestèc

an =

1 L

L

Z

f (t) cos −L

nπ t dt = 0. L

Den upˆrqoun bebaÐwc ìroi sunhmitìnwn. To olokl rwma eÐnai mhdèn epeid  h

f (t) cos nπL t

eÐnai peritt  sunˆrthsh

(ginìmeno peritt c me ˆrtia sunˆrthsh eÐnai peritt  sunˆrthsh) kai to olokl rwma miac peritt c sunˆrthshc se èna summetrikì diˆsthma eÐnai pˆntote mhdèn. diˆsthma

[−L, L]

Epiprìsjeta, to olokl rwma mia ˆrtiac sunˆrthshc se èna summetrikì

eÐnai to diplˆsio tou oloklhr¸matoc thc sunˆrthshc sto diˆsthma

[0, L].

H sunˆrthsh

f (t) sin

nπ L

t

eÐnai peritt  mia kai eÐnai ginìmeno dÔo peritt¸n sunart sewn.

bn =

1 L

MporoÔme t¸ra na grˆyoume thn seirˆ

L

Z

f (t) sin −L

F ourier

nπ 2 t dt = L L f (t)

thc

∞ X

f (t) eÐnai bn = 0 kai ìti

Parìmoia, an h

peritt 

2L-periodik  an =

O parapˆnw tÔpoc isqÔei kai gia

n=0

2 L

F ourier

0

nπ t dt. L

nπ t. L

bn sin

sunˆrthsh gia akrib¸c touc Ðdiouc me touc parapˆnw lìgouc

L

Z

f (t) cos 0

nπ t dt. L

opìte kai èqoume

2 L

a0 = 'Ara to anˆptugma se seirˆ

f (t) sin

wc ex c

n=1

brÐskoume ìti

L

Z

L

Z

f (t) dt. 0

eÐnai

∞ a0 X nπ t. an cos 2 n=1 L

Je¸rhma 4.4.1.

Eˆn

ex c anˆptugma se seirˆ

f (t) eÐnai F ourier.

mia tmhmatikˆ suneq c sto

Fperitt  (t) =

∞ X

[0, L]

bn sin

n=1

sunˆrthsh tìte h peritt  thc epèktash èqei to

nπ t, L

ìpou

bn = H ˆrtia epèktash thc

f (t)

2 L

L

Z

f (t) sin 0

èqei to ex c anˆptugma se seirˆ

nπ t dt. L

F ourier ∞

Fˆrtia (t) =

a0 X nπ + an cos t, 2 L n=1

ìpou

an =

2 L

L

Z

f (t) cos 0

nπ t dt. L

4.4. ΣΕΙџΕΣ ΗΜΙԟΟΝΩΝ ΚΑΙ ΣΥΝΗΜΙԟΟΝΩΝ Parˆdeigma 4.4.2:

f (t) = t2

gia

119 F ourier

UpologÐste to anˆptugma se seirˆ

thc ˆrtiac periodik c epèktashc thc sunˆrthshc

0 ≤ t ≤ π.

'Eqoume

∞

a0 X + an cos nt, 2 n=1

f (t) = ìpou

a0 =

2 π

π

Z

t2 dt = 0

2π 2 , 3

kai

π  Z π 4 2 21 sin nt − t sin nt dt t π n nπ 0 0 0 Z π iπ 4(−1)n 4 4 h = 2 t cos nt + 2 cos nt dt = . n π n π 0 n2 0

an =

2 π

Z

π

t2 cos nt dt =

Ac grˆyoume analutikˆ merikoÔc ìrouc thc seirˆc

4 π2 − 4 cos t + cos 2t − cos 3t + · · · 3 9 4.4.3

Efarmogèc

'Opwc  dh anafèrame oi seirèc

F ourier

eÐnai sundedemènec me ta probl mata sunoriak¸n tim¸n pou melet same

prohgoumènwc. Ac doÔme to pwc sundèontai ìmwc me perissìterh lÐgh leptomèreia. Jewr ste to parakˆtw prìblhma sunoriak¸n tim¸n gia

0 < t < L,

x00 (t) + λ x(t) = f (t), kai tic ex c sunoriakèc sunj kec

Dirichlet x(0) = 0, x(L) = 0. Qrhsimopoi¸ntac to λ den eÐnai idiotim  tou antÐstoiqou

(Je¸rhma 4.1.2 sth selÐda 106) shmei¸noume ìti eˆn to tìte to parapˆnw prìblhma èqei monadik  lÔsh. parapˆnw prìblhma idiotim¸n eÐnai oi sunart seic thn

f (t)

se seirˆ

F ourier.

Grˆfoume kai to

je¸rhma tou

F redholm

omogenoÔc probl matoc,

Shmei¸ste epÐshc ìti oi idiosunart seic pou antistoiqoÔn sto

sin

nπ t. 'Ara, gia na upologÐsoume thn lÔsh, pr¸ta anaptÔsoume L

x san anˆptugma seir¸n hmitìnwn. BebaÐwc to en lìgw anˆptugma F ourier touc opoÐouc fusikˆ eˆn upologÐsoume kai antikatast soume

sumperilambˆnei ˆgnwstouc suntelestèc autoÔc sto anˆptugma tìte ja èqoume to to anˆptugma thc

x

x, dhlad  ja èqoume thn lÔsh.

Gia na katafèroume to parapˆnw ac antikatast soume

sthn diaforik  exÐswsh kai ja lÔsoume.

Sthn perÐptwsh twn sunoriak¸n sunjhk¸n

N eumann x0 (0) = 0, x0 (L) = 0 akoloujoÔme thn parapˆnw diadikasÐa

qrhsimopoi¸ntac ìmwc t¸ra seirèc sunhmitìnou. Se kˆje perÐptwsh o pio katˆllhloc trìpoc gia na katalˆboume tic mejìdouc autèc eÐnai efarmìzontac tec se paradeÐgmata.

Parˆdeigma 4.4.3:

Jewr ste to prìblhma sunoriak¸n tim¸n gia

0 < t < 1,

x00 (t) + 2x(t) = f (t), f (t) = t gia 0 < t < 1. Jèloume na broÔme mia lÔsh x pou na x(0) = 0, x(1) = 0. Ac anaptÔxoume thn f (t) se seirˆ hmitìnwn ìpou

f (t) =

∞ X

ikanopoieÐ tic ex c sunoriakèc sunj kec

cn sin nπt,

n=1 ìpou

Z cn = 2

1

t sin nπt dt = 0

Ac grˆyoume kai thn

x(t)

2 (−1)n+1 . nπ

wc ex c

x(t) =

∞ X n=1

bn sin nπt.

Dirichlet

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

120 Antikajist¸ntac sthn exÐswsh èqoume

∞ X

x00 (t) + 2x(t) =

n=1 ∞ X

=

∞ X

−bn n2 π 2 sin nπt + 2

bn sin nπt

n=1

bn (2 − n2 π 2 ) sin nπt

n=1

= f (t) =

∞ X 2 (−1)n+1 sin nπt. nπ n=1

Sunep¸c,

bn (2 − n2 π 2 ) =

2 (−1)n+1 nπ

 or

bn =

2 (−1)n+1 . nπ(2 − n2 π 2 ) F ourier

'Eqoume loipìn upologÐsei thn lÔsh san anˆptugma seirˆc

x(t) =

∞ X n=1

Parˆdeigma 4.4.4:

2 (−1)n+1 sin nπt. nπ (2 − n2 π 2 )

MporoÔme na antimetwpÐsoume sunj kec

soume thn Ðdia exÐswsh gia

N eumann

me ton Ðdio ousiastikˆ trìpo. Ac jewr -

0 < t < 1, x00 (t) + 2x(t) = f (t),

ìpou

f (t) = t

gia

Ac grˆyoume thn

0 < t < 1. Aut  thn forˆ ja èqoume tic f (t) san anˆptugma seirˆc sunhmitìnwn

ex c sunoriakèc sunj kec

N eumann x0 (0) = 0, x0 (1) = 0.

∞

f (t) =

c0 X + cn cos nπt, 2 n=1

ìpou

1

Z c0 = 2

t dt = 1, 0

kai

1

Z cn = 2 0 Ac grˆyoume kai thn

x(t)

2((−1)n − 1) t cos nπt dt = = π 2 n2

(

−4 π 2 n2

gia

0

gia

n n

,

perittì

.

ˆrtio

san anˆptugma seirˆc sunhmitìnwn

∞

x(t) =

a0 X + an cos nπt. 2 n=1

Antikajist¸ntac sthn exÐswsh èqoume

x00 (t) + 2x(t) =

∞ h X

∞ h i i X −an n2 π 2 cos nπt + a0 + 2 an cos nπt

n=1

= a0 +

n=1 ∞ X

an (2 − n2 π 2 ) cos nπt

n=1

= f (t) =

∞ X

1 + 2

n=1

n

perittì

−2 cos nπt. π 2 n2

4.4. ΣΕΙџΕΣ ΗΜΙԟΟΝΩΝ ΚΑΙ ΣΥΝΗΜΙԟΟΝΩΝ 'Ara,

1 , 2

a0 =

an = 0

gia

n

n

ˆrtio kai gia

121

n ≥ 1)

perittì (

an (2 − n2 π 2 ) =

−4 π 2 n2

 

an =

−4 . n2 π 2 (2 − n2 π 2 ) F ourier

'Eqoume loipìn upologÐsei thn lÔsh san anˆptugma seir¸n

x(t) =

∞ X n=1 n odd

4.4.4

4.4.3

4.4.4

'Estw h sunˆrthsh

f.

gia

gia

orismènh gia

0 ≤ t ≤ 1.

a) D¸ste thn grafik  parˆstash thc ˆrtiac

b) D¸ste thn grafik  parˆstash thc ˆrtiac periodik c epèktashc thc

f.

F ourier gia thn ˆrtia kai gia thn peritt  periodik  epèktash thc sunˆrthsh-

0 ≤ t ≤ 1.

UpologÐste ta anaptÔgmata seir¸n

f (t) = t

4.4.6

f (t) = (t − 1)2

UpologÐste ta anaptÔgmata seir¸n

f (t) = (t − 1)2

4.4.5 c

−4 cos nπt. n2 π 2 (2 − n2 π 2 )

Ask seic

periodik c epèktashc thc

c

wc ex c

F ourier gia thn ˆrtia kai gia thn peritt  periodik  epèktash thc sunˆrthsh-

0 ≤ t ≤ π.

UpologÐste to anˆptugma seir¸n

F ourier

gia thn ˆrtia periodik  epèktash thc sunˆrthshc

f (t) = sin t

gia

0 ≤ t ≤ π. 4.4.7

ìpou

UpologÐste thn lÔsh thc exÐswshc

x00 (t) + 4x(t) = f (t),

f (t) = 1

gia

0 < t < 1 h opoÐa ikanopoieÐ a) tic ex c N eumann x0 (0) = 0, x0 (1) = 0.

Dirichlet x(0) = 0, x(1) = 0.

sunoriakèc sunj kec

b) tic

ex c sunoriakèc sunj kec

4.4.8

ìpou

UpologÐste thn lÔsh thc exÐswshc

x00 (t) + 9x(t) = f (t),

f (t) = sin 2πt

sunoriakèc sunj kec

4.4.9

0 < t < 1. a) tic ex c sunoriakèc N eumann x0 (0) = 0, x0 (1) = 0.

gia

sunj kec

Dirichlet x(0) = 0, x(1) = 0.

b) tic ex c

'Estw ìti

f (t) =

ìpou

sunart sei

x00 (t) + 3x(t) = f (t), P∞

n=1 bn sin nπt. twn ìrwn bn .

D¸ste to anˆptugma se seirˆ

x(0) = 0, F ourier

x(1) = 0, thc lÔshc

x(t),

ìpou oi suntelestèc na dÐnontai

ΚΕ֟ΑΛΑΙΟ 4. ΣΕΙџΕΣ F OU RIER ΚΑΙ ΜΔΕ

122 4.5

MDE, qwrismìc metablht¸n, kai h exÐswsh jermìthtac

Mia exÐswsh lègetai merik  diaforik  exÐswsh   MDE eˆn perilambˆnei merikèc parag¸gouc wc proc perissìterec

‡

apì mia anexˆrthtec metablhtèc . H epÐlush MDE apoteleÐ Ðswc thn pio shmantik  kai basik  efarmog  twn seir¸n

F ourier. Mia MDE lègetai grammik  an h anexˆrthth metablht  kai oi parˆgwgoÐ thc den emfanÐzontai se kˆpoia dÔnamh (megalÔterh tou 1) oÔte san ìrisma kˆpoiac sunˆrthshc.

Ja asqolhjoÔme mìno me grammikèc MDE. MazÐ me mia

MDE, sun jwc orÐzoume kai kˆpoiec sunoriakèc sunj kec, me tic opoÐec kajorÐzoume thn tim  thc lÔshc  /kai thc parag¸gou thc sto sÔnoro enìc qwrÐou. Epiprìsjeta,   enallaktikˆ, mporeÐ na orÐsoume kˆpoiec arqikèc sunj kec me tic opoÐec kajorÐzoume thn tim  thc lÔshc  /kai thc parag¸gou thc se kˆpoia qronik  stigm  thn opoÐa jewroÔme arqik .

Merikèc forèc oi dÔo parapˆnw tÔpoi sunjhk¸n sumplèkontai kai tìte lème ìti èqoume genikˆ pleurikèc

sunj kec. Ja melet soume treic tÔpouc merik¸n diaforik¸n exis¸sewn, h kˆje mia apì tic opoÐec eÐnai antiproswpeutik  miac genikìterhc klˆshc diaforik¸n exis¸sewn.

Pr¸ta ja melet soume thn exÐswsh thc jermìthtac, h opoÐa eÐnai

èna parˆdeigma parabolik c MDE. Katìpin ja melet soume thn exÐswsh tou kÔmatoc, h opoÐa eÐnai èna parˆdeigma uperbolik c MDE. Tèloc ja melet soume thn exÐswsh tou

Laplace,

h opoÐa eÐnai èna parˆdeigma elleiptik c MDE.

H sumperiforˆ kai h praktik  epÐlushc kajemiˆc apo tic parapˆnw exis¸seic eÐnai antiproswpeutikèc ìlhc thc klˆshc twn exis¸sewn sthn opoÐa an koun.

4.5.1

Jermìthta se èna monwmèno kal¸dio

Ac melet soume pr¸ta thn exÐswsh thc jermìthtac 'Estw ìti èqoume èna kal¸dio (  mia lept  metallik  rˆbdo) h opoÐa eÐnai monwmènh pantoÔ ektìc apì ta dÔo ˆkra thc. 'Estw ìti me ìti me

t

x

sumbolÐsoume thn jèsh pˆnw sthn rˆbdo kai

sumbolÐzoume ton qrìno. DeÐte to Sq ma 4.10.

jermokrasÐa u

0

L

mìnwsh

x

Σχήμα 4.10: Μονωμένο καλώδιο.

Ac sumbolÐsoume t¸ra me

u(x, t)

thn jermokrasÐa tou kalwdÐou sto shmeÐo

x

thn qronik  stigm 

t.

H exÐswsh

pou dièpei to sÔsthma autì onomˆzetai mono-diˆstath exÐswsh jermìthtac:

∂u ∂2u = k 2, ∂t ∂x ìpou

k>0

eÐnai mia stajerˆ.

kaiBibliografÐaBibliografÐa

‡ EÐnai, katˆ thn gn¸mh mou polÔ pio orjì, allˆ dustuq¸c kajìlou sunhjismèno oi exis¸seic autèc na lègontai exis¸seic me merikèc parag¸gouc

BibliografÐa [BM] Paul W. Berg kai James L. McGregor, Elementary Partial Differential Equations, Holden-Day, San Francisco, CA, 1966. [BD] William E. Boyce kai Richard C. DiPrima Differential Equations and Boundary Value Problems, 9th edition, Laurie Rosatone, 2009. [EP]

C.H. Edwards kai D.E. Penney, Differential Equations kai Boundary Value Problems: Computing and Modeling, 4th edition, Prentice Hall, 2008.

[F]

Stanley J. Farlow, An Introduction to Differential Equations and Their Applications, McGraw-Hill, Inc., Princeton, NJ, 1994.

[EP]

C.H. Edwards kai D.E. Penney, Differential Equations kai Boundary Value Problems: Computing and Modeling, 4th edition, Prentice Hall, 2008.

[I]

E.L. Ince, Ordinary Differential Equations, Dover Publications, Inc., New York, NY, 1956.

123

124

ΒΙΒΛΙΟΓΡΑ֟ΙΑ

Euret rio ˆrtia periodik  epèktash, 117

diaqwrÐsimh, 19

ˆrtia sunˆrthsh, 110, 117

dunamik  apìsbesh, 84

èmmesh lÔsh, 20 Ôparxh kai monadikìthta, 16, 36

eikìna fˆshc, 32

Ôparxhc kai monadikìthtac, 42

ekjetikì enìc pÐnaka, 88

MDE, 6

ekjetikì pÐnaka, 88 eleÔjerh kÐnhsh, 46

aìristo olokl rwma, 10

elleiptik c MDE, 122

algebrik  pollaplìthta, 85

elleiptikì (dianusmatikì pedÐo), 75

anˆlush idiodianusmˆtwn, 99

enallaktikì je¸rhma tou

analogoÔsa omogen c exÐswsh, 52

F redholm

apl  perÐptwsh, 106

anexˆrthth metablht , 5

epekteÐnoume periodikˆ, 107

antÐstoiqo idiodiˆnusma, 103

epitˆqunsh, 12

antiparˆgwgoc, 10

eswterikì ginìmeno sunart sewn, 108

apìstash, 12

eustajèc krÐsimo shmeÐo, 31

apl  armonik  kÐnhsh, 48

eustaj  periodik  lÔsh, 60

aprosdiìristoi suntelestèc, 52

eustaj c kìmboc, 74

Bernoulli, 28 Chebychev 1hc tˆxhc, 37 ExÐswsh Hermite 2hc tˆxhc, 37

gia sust mata, 83

exÐswsh

sust mata, 97

exÐswsh

sust mata deÔterhc tˆxhc, 99 arqikèc sunj kec, 7

exÐswsh thc jermìthtac, 122

arqikèc sunj kec miac MDE, 122

exÐswsh tou

asjen c pÐnakac, 85

exÐswsh tou

asjen¸c fjÐnon sÔsthma, 50

exÐswsh tou kÔmatoc, 122

astajèc krÐsimo shmeÐo, 31

exanagkasmènec talant¸seic, 83

astaj c kìmboc, 74

exanagkasmènh kÐnhsh, 46, 49

atèleia, 85

exarthmènh metablht , 5

atel c idiotim , 85

exis¸seic

autìnomh exÐswsh, 31

exis¸seic

Euler, 41 Laplace, 122

Cauchy − Euler, Euler, 37

37

autìnomo sÔsthma, 65 fainìmeno tou

Gibbs,

112

deÔtero nìmo tou NeÔtwna, 46, 47, 64

fantastikì mèroc, 40

deÔteroc nìmoc tou NeÔtwna, 78

fusikèc suqnìthtec, 80

diˆgramma fˆshc, 32

fusik  (gwniak ) suqnìthta, 48

diˆnusma metatìpishc, 78

fusik  suqnìthta, 57

diˆnusma sunart sewn, 67

fusik  suqnìthta suntonismoÔ, 60

diaforik  exÐswsh, 5

fusikì plˆtoc suntonismoÔ, 60

diaforik  exÐswsh deÔterhc tˆxhc, 8

fusikìc suntonismìc, 58, 60

diaforik c exÐswshc pr¸thc tˆxhc, 5

fusikoÐ trìpoi talˆntwshc, 80

diag¸nioc pÐnakac ekjetikì pÐnaka enìc, 89

genik  lÔsh, 7

diagwnopoÐhsh, 90

genikeumèna idiodianÔsmata, 86, 87

diakÔmansh twn paramètrwn, 54

gewmetrik  pollaplìthta, 85

dianusmatikì pedÐo, 65

grammikˆ anexˆrthtec, 36, 42

125

126

ΕΥΡΕԟΗΡΙΟ

grammikˆ exarthmènec, 42

migadikèc rÐzec, 40

grammikèc exis¸seic, 23

migadikìc arijmìc, 39

grammikèc exis¸seic pr¸thc tˆxhc, 23

mono-diˆstath exÐswsh jermìthtac, 122

grammik  exÐswsh, 35

montèlo ekjetik c aÔxhshc, 7

grammik  MDE, 122

morf 

Leibniz ,

19

grammik c diaforik c exÐswshc deÔterhc tˆxhc, 35 grammikì sÔsthma pr¸thc tˆxhc, 65

nìmo tou Qouk, 46

grammikì sÔsthma SDE pr¸thc tˆxhc, 67

Nìmoc diˆdoshc thc jermìthtac tou NeÔtwna, 26

grammikìc telest c, 52

nìmoc thc diˆdoshc tou NeÔtwna, 31

gwniak  suqnìthta, 48

nìmoc tou

hmi-perÐodoc, 114

oloklhrwtikìc parˆgontac, 23

Hooke,

78

omogenèc sÔsthma, 67 idiˆzousec lÔseic, 20

omogen  grammik  exÐswsh, 35

idiodiˆnusma, 70, 103

Omogen c exÐswsh, 29

idiotim , 70

orjog¸niec

idiotim  enìc probl matoc sunoriak¸n tim¸n, 103

sunart seic, 105, 108

isqurˆ fjÐnwn, 49 pÐnakac akamyÐac, 78 je¸rhma tou

P icard,

16

pÐnakac mˆzac, 78

jemeli¸dhc pÐnakac, 68

pÐnakac sunart sewn, 67

jemeli¸dhc pÐnakac epÐlushc, 68

parˆstash fˆsewn, 65

jemeli¸dhc pÐnakac lÔsewn, 88

parabolik c MDE, 122 paragontopoÐhsh idiodianusmˆtwn, 95

kèntro, 76

pedÐo kateujÔnsewn, 14

kÐnhsh me apìsbesh, 46

pedÐou dieujÔnsewn, 65

kÐnhsh qwrÐc apìsbesh, 46

perÐodoc, 48

kÐnhsh qwrÐc ustèrhsh

periodik , 107

sust mata, 78 kÔklwma

RLC ,

46

peritt  periodik  epèktash, 117 peritt  sunˆrthsh, 110, 117

kampÔlh epÐlushc, 65

phg , 74

katabìjra, 74

plˆtoc, 48

katanˆlwsh, 32

pl rhc idiotim , 85

krÐsima fjÐnon, 50

pleurikèc sunj kec miac MDE, 122

krÐsimo shmeÐo, 31

pollaplìthta, 44 pollaplìthta miac idiotim c, 85

lÔsh, 5 lÔsh isorropÐac, 31 logistik  exÐswsh, 31 logistik  exÐswsh me katanˆlwsh, 32

mèjodoc elˆttwshc thc tˆxhc, 37

pollaplasiasmìc migadik¸n arijm¸n, 39 prìblhma sunoriak¸n tim¸n, 102 pragmatikì mèroc, 40 prionwt , 109

qarakthristik  exÐswsh, 38

mèjodoc oloklhrwtikoÔ parˆgonta, 23 mèjodoc tou oloklhrwtikoÔ parˆgonta sust mata, 93

rÐzec pollaplìthtac, 43 realistikˆ probl mata, 6

majhmatik  lÔsh, 6 majhmatikì montèlo, 6

sÔsthma diaforik¸n exis¸sewn, 63

MDE, 122

sagmatikì shmeÐo, 74

merikèc diaforikèc exis¸seic, 6

SDE, 6

merik  diaforik  exÐswsh, 122

seirˆ

metabatik  lÔsh, 60

speiroeid c katabìjra, 76

metablhtèc parˆmetroi

speiroeid c phg , 76

sust mata, 98

F ourier,

108

stajeroÐ suntelestèc, 38

metatìpish fˆshc, 48

stajeroÔc suntelestèc, 67

mh-exanagkasmènh kÐnhsh, 46

sugkekrimènh lÔsh, 7, 52

ΕΥΡΕԟΗΡΙΟ sumbolismì

127

Leibniz ,

11

sumplhrwmatik  lÔsh, 52 sun jeic diaforikèc exis¸seic, 6 sunoriakèc sunj kec

Dirichlet,

119

sunoriakèc sunj kec miac MDE, 122 sunoriak¸n sunjhk¸n

N eumann,

suntonismìc, 58, 84 superposition, 42 sust mata tri¸n swmatidÐwn, 78 suzug  migadikì, 72

tÔpo tou

Euler,

40

tÔpo tou triwnÔmou, 38 taqÔthta, 12 tetrˆgwno kÔma, 61 tetragwnikì kÔma, 110 trigwnometrik  seirˆ, 108 troqiˆ, 65

upèrjesh, 35, 68 uperbolik c MDE, 122

119