JËmata Diaforik∏n Exis∏sewn - University of...

JËmata Diaforik∏n Exis∏sewn

Tm†ma HlektrolÏgwn Mhqanik∏n kai Mhqanik∏n Upologist∏nPanepist†mio Jessal–ac

17 Febrouar–ou 2015

2

Eisagwg†

To parÏn s‘gramma apotele– mia sullog† jemàtwn sqetik∏n me diàforec exe-tàseic tou ant–stoiqou maj†matoc, leptomËriec gia to opo–o ja bre–te sthn sel–dahttp://de.inf.uth.gr. Ta en lÏgw jËmata epËlexe kai daktulogràfhse se LatËqo M. Bàbalhc. Kàpoiec apo autËc epil‘jhkan kai daktulograf†jhkan apÏ ton–dio kai kàpoiec àllec apo touc K. Qo‘no kai G. MalakatsÏpoulo. Parakal∏shmei∏ste Ïti den Ëqei akÏma endeleq∏c elegqje– h poiÏthta (kai se merikËcpeript∏seic o‘te h orjÏthta) twn l‘sewn. Sthn per–ptwsh pou upopËsei sthnant–lhy† sac kàpoio làjoc parakal∏ enhmer∏ste me ([email protected]) †kànte thn allag† ap' euje–ac sto latËq ke–meno pou ja bre–te ed∏.Kàje jËma fËrei kai mia etikËta thc morf†c EXETOS.AA Ïpou

• EX pa–rnei tim† TE an prÏkeitai gia Telik† ExËtash, EE gia Epanalhptik†ExËtash, EP gia ExËtash ProoÏdou, VE gia EmbÏlimh ExËtash kai AE gia'Allh exËtash

• ETOS dhl∏nei to Ëtoc pou Ëlabe q∏ra h exËtash kai

• AA ton a‘xonta arijmÏ tou jËmatoc sthn exËtash.

Dustuq∏c ta jËmata pou akoloujo‘n den e–nai organwmËna se enÏthtec. El-p–zw na brw qrÏno († ejelont†/ejelÏntria) kàpote gia thn en lÏgw paràlhyh,qwr–c na parablËpw pollËc àllec paral†yeic.

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3

To parÏn morfopoi†jhke me qr†sh tou sust†matoc L

A

T

E

X.

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c�2015

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'Eqete to dika–wma na qrhsimopoi†sete, ektup∏sete, antigràyete kai na dianËmete

tic shmei∏seic autËc Ïsec forËc jËlete. Mpore–te na bas–sete tic dikËc sac shmei∏seic

se autËc †/kai na qrhsimopoi†sete olÏklhra mËrh touc, arke– na diathr†sete thn paro-

‘sa àdeia qr†shc anallo–wth. Eàn skope‘ete na tup∏sete tic paro‘sec shmei∏seic gia

didaktiko‘c skopo‘c tÏte mpore–te na qre∏sete Ïpou epijume–te to posÏ pou apaite–tai

gia thn ekt‘pwsh, thn diàjesh kai thn dianom† tou ektupwmËnou pakËtou.

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PerieqÏmena

1 Sun†jeic DiaforikËc Exis∏seic 7

2 SDE An∏terhc Tàxhc 25

6 PERIEQOMENA

Kefàlaio 1

Sun†jeic DiaforikËc

Exis∏seic

JËma TE2014.1. Kukl∏ste Ïpoia apÏ tic parakàtw sunart†seic piste‘eteÏti e–nai l‘sh thc ex–swshc

y00 + ky = 0.

Apànthsh TE2014.1. Ja l‘soume thn 2ou bajmo‘ omogen† ex–swsh me sta-jero‘c suntelestËc:

y00 + ky = 0

JËtw y = ery ) y0 = rery ) y00 = r2(ery). Antikajist∏ sthn ex–swsh mou kaiËqw:

r2 + k = 0 ) r2 = �k ) r = ±p�k

H l‘sh mac e–nai h:r = 0± i

pk

'Otan Ëqoume migadikËc r–zec xËroume pwc h exisws† mac ja Ëqei l‘sh thc morf†c

y = C1e↵tcos�t+ C2e

↵tsin�t

Antikajisto‘me tic r–zec pou br†kame kai Ëqoume:

y = C1e0cos

pkt+ C2e

0sinpkt )

y = C1cospkt+ C2sin

pkt

Ïpwc ep–shc an prospaj†soume thn tetrimmËnh 0 blËpoume pwc epalhje‘ei (0 = 0)thn arqik† ex–wsh mac àra e–nai kai aut† l‘sh.

( apànthsh (b),(g),(e) ).

8 Sun†jeic DiaforikËc Exis∏seic

JËma TE2014.2. Sqediàste to dianusmatikÏ ped–o tou sust†matoc x0=

�2 �1

�1 �2

�x kai sqoliàste thn sumperiforà twn l‘sewn se bàjoc qrÏnou.

Apànthsh TE2014.2.

�!x 0(t) =

�2 �1

�1 �2

��!x (t)

Xekinàme th l‘sh br–skontac pr∏ta tic IdiotimËc tou arqiko‘ p–naka, (ac ton o-nomàsoume A):

det(A� �I) = 0

) det

�2� � �1

�1 �2� �

�= 0

) (�2� �)2 � (�1)(�1) = 0

) �2+ 4�+ 4� 1 = 0

) �2+ 4�+ 3 = 0

) �1 = �1 , �2 = �3

Afo‘ br†kame tic IdiotimËc, sth sunËqeia ja Ëprepe na bro‘me kai to ant–stoiqoIdiodiànusma gia thn kàje Idiotim†.

Gia �1 = �1:

L‘noume thn ex–swsh (A� �1I)�!u1 =

�!0 )

)�2� (�1) �1

�1 �2� (�1)

� u11

u12

�=

0

0

�

)�1 �1

�1 �1

� u11

u12

�=

0

0

�

)�1 �1

0 0

� u11

u12

�=

0

0

�(apaloif† Gauss)

EfarmÏzontac proc ta p–sw antikatàstash ston teleuta–o p–naka, pa–rnoume:

�u11 � u12 = 0 ) �u11 = u12 ) u11, u12 ele‘jerec

9

'Ara, to Idiodiànusma �!u1 e–nai:

�!u1 =

�CC

�, 8 C 2 < † �!u1 =

�1

1

�, gia C = 1

Gia �2 = �3:

L‘noume thn ex–swsh (A� �2I)�!u2 =

�!0 )

)�2� (�3) �1

�1 �2� (�3)

� u21

u22

�=

0

0

�

)1 �1

�1 1

� u21

u22

�=

0

0

�

)1 �1

0 0

� u21

u22

�=

0

0

�(apaloif† Gauss)


u21 � u22 = 0 ) u21 = u22 ) u21, u22 ele‘jerec


�!u2 =

CC

�, 8 C 2 < † �!u2 =

1

1

�, gia C = 1

Sunep∏c, Genik† L‘sh:

�!x (t) = C1�!u1e

�1t+ C2

�!u2e�2t

) �!x (t) = C1

�1

1

�e�t

+ C2

1

1

�e�3t


Kai to ped–o kateuj‘nsewn thc l‘shc gia diaforetikà C:

Sq†ma 1.1: Ped–o kateuj‘nsewn thcl‘shc

JËma TE2014.3. Qrhsimopoi†ste metasqhmatismo‘c Laplàc gia na upolo-g–sete thn l‘sh tou probl†matoc

y00 + y =

(0 gia t < ⇡

3 gia t � ⇡., y(0) = 1, y0(0) = �3.

Apànthsh TE2014.3. Koitàzontac prosektikà to prÏblhma arqik∏n tim∏npou mac d–netai sthn àskhsh, parathro‘me Ïti h kladik† sunàrthsh sta dexiàthc ex–swshc mpore– na grafe– san sunàrthsh heaviside wc 3u(t�⇡). Sunep∏c,mporo‘me na xanagràyoume to prÏblhma wc ex†c:

y00 + y = 3u(t� ⇡), y(0) = 1, y0(0) = �3

OpÏte h S.D.E. metasqhmat–zetai apÏ to ped–o tou qrÏnou sto ped–o thc su-qnÏthtac mËsw tou orjo‘ metasqhmatismo‘ Laplace wc ex†c:

L{y00}+ L{y} = 3L{u(t� ⇡)}

Qrhsimopoi∏ntac ton metasqhmatismÏ 2hc paràgwgou Laplace

L{y00(t)} = s2Y (s)� sy(0)� y0(0)

11

Ëqoume:

s2Y (s)� sy(0)� y0(0) + Y (s) = 3

e�⇡s

s

) s2Y (s)� s+ 3 + Y (s) = 3

e�⇡s

s(antikatàstash arqik∏n sunjhk∏n)

) s3Y (s)� s2 + 3s+ sY (s) = 3e�⇡s

) Y (s)(s3 + s)� s2 + 3s = 3e�⇡s

) Y (s)(s3 + s) = s2 � 3s+ 3e�⇡s

) Y (s) =s2 � 3s+ 3e�⇡s

(s3 + s)

) Y (s) =s2 � 3s+ 3e�⇡s

s(s2 + 1)

=

⇤ !

Anàlush tou s‘njetou klàsmatoc se aplà klàsmata:

⇤ ! s

2�3s+3e�⇡s

s(s2+1) =

A

s

+

Bs+C

s

2+1 = ... =

=

(3e�⇡s)s

+

(1�3e�⇡s)s+(�3)s

2+1 =

=

3e�⇡s

s

+

s�3se�⇡s�3s

2+1

) Y (s) = 3e�⇡s

s

+

s�3se�⇡s�3s

2+1

) Y (s) = 3e�⇡s

s

+

s

s

2+1 � 3se�⇡s

s

2+1 � 3s

2+1

) Y (s) = 3e�⇡s

(

1

s) + (

s

s2 + 1

)� 3e�⇡s

(

s

s2 + 1

)� 3(

1

s2 + 1

)

'Etsi, me ant–strofo metasqhmatismÏ Laplace pernàme pàli p–sw sto ped–o touqrÏnou (apÏ to ped–o thc suqnÏthtac) kai br–skoume thn l‘sh thc arqik†c S.D.E.:


y(t) = L�1{Y (s)}

= 3L�1{e�⇡s

(

1

s)}+ L�1{ s

s2 + 1

}� 3L�1{e�⇡s

(

s

s2 + 1

)}� 3L�1{ 1

s2 + 1

}

= 3L�1{e�⇡s}(1) + cos(t)� 3L�1{e�⇡s}cos(t)� 3sin(t)

Sto shme–o autÏ kai gia na mporËsoume na suneq–soume ton ant–strofo metasqh-matismÏ, ja basisto‘me sthn idiÏthta metatÏpishc tou heaviside (kefàlaio 6.2.3twn shmei∏sewn), h opo–a mac lËei Ïti:

L{f(t� a)u(t� a)} = e�asL{f(t)}

EfarmÏzontac ant–strofo metasqhmatismÏ laplace sthn idiÏthta aut† (gia nathn prosarmÏsoume sthn per–ptws† mac), sunepàgetai Ïti:

f(t� a)u(t� a) = L�1{e�as}f(t)

OpÏte l‘sh:

y(t) = 3u(t� ⇡) + cos(t)� 3cos(t� ⇡)u(t� ⇡)� 3sin(t)

JËma TE2014.4. Kukl∏ste Ïpoion apo touc parakàtw arijmo‘c piste‘e-te Ïti antistoiqe– sthn tim† tou A6 Ëtsi ∏ste h sunàrthsh u(x, t) = A0 +P1

n=1 An

cos(nx)e�n

2t na apotele– l‘sh tou probl†matoc

uxx

= ut

gia 0 < x < ⇡, t > 0,

ux

(0, t) = ux

(⇡, t) = 0 gia t > 0,

u(x, 0) =⇣x� ⇡

2

⌘2gia 0 < x < ⇡.

Apànthsh TE2014.4. Ed∏ Ëqoume mono-diàstath ex–swsh jermÏthtac - Fourier.

ApÏ thn ekf∏nhsh sumpairËnoume Ïti L = ⇡ kai k = 1. Mac zhte–tai na upolo-g–soume to A

n

. H f(x) sthn per–ptws† mac e–nai h u(x, 0) = f(x) = (x� ⇡

2 )2.

'Ara,

13

An

=

2

L

ZL

0f(x)cos(

n⇡x

L)dx

=

2

⇡

Z⇡

0(x� ⇡

2

)

2cos(nx)dx

=

2

⇡

Z⇡

0(x2

+

⇡2

4

� ⇡x)cos(nx)dx

=

2

⇡

Z⇡

0x2cos(nx)dx+

⇡

2

Z⇡

0cos(nx)dx� 2

Z⇡

0xcos(nx)dx

=

4(�1)

n

n2+ 0 +

2(�1)

n+1+ 2

n2

=

4(�1)

n

+ 2(�1)

n+1+ 2

n2

Sunep∏c,

A6 =

4� 2 + 2

36

=

1

9

( apànthsh (e) ).

JËma TE2014.5. Kukl∏ste Ïpoia apo tic parakàtw sunart†seic piste‘eteÏti e–nai l‘sh thc ex–swshc

y00 + 2y0 + y = sin(2t).

Apànthsh TE2014.5. L‘nw pr∏ta thn omogen† ex–swsh:

y00 + 2y0 + y = 0

JËtw y = ery ) y0 = rery ) y00 = r2ery. Antikajist∏ sthn ex–swsh mou kaiËqw:

r2 + 2r + 1 = 0

H diakr–nousa auto‘ e–nai: � = �2 � 4↵� ) � = 0 àra Ëqoume m–a dipl† r–za hopo–a e–nai h

r =

��

2↵= �2

2

) r = �1


H genikeumËnh l‘sh thc ex–swshc mou ja e–nai thc morf†c:

yc

= C1e�t

+ C2te�t

gia opoiesd†pote C1 kai C2. Gia to dex– mh omogenËc mËloc ja kànoume manteyiàtou t‘pou: y = Asin(2t) + Bcos(2t). Kànw thn antikatàstash sthn arqik† giana brw ta A kai B. Paragwg–zontac thn Ëqoume:

y0 = 2Acos(2t)� 2Bsin(2t)

y00 = �4Asin(2t)� 4Bcos(2t)

An kànoume antikatàstash sthn arqik† ex–swsh:

y00 + 2y0 + y = sin(2t) )

�4Asin(2t)�4Bcos(2t)+4Acos(2t)�4Bsin(2t)+Asin(2t)+Bcos(2t) = sin(2t) )

�3Asin(2t)� 3Bcos(2t) + 4Acos(2t)� 4Bsin(2t) = sin(2t)

ApÏ thn parapànw sumpera–noume pwc �3A�4B = 1 kai 4A�3B = 0. L‘nontacthn 2h Ëqw: 4A = 3B ) A =

3B4 .

'Ara �3(

3B4 )� 4B = 1 ) �9(

B

4 )� 4B = 1 ) �25B = 4 ) B = � 4

25

'Etsi Ëqoume kai 4A+

1225 = 0 ) A = � 3

25

'Ara h sugkekrimËnh mac l‘sh e–nai h

yp

= � 3

25

sin(2t)� 4

25

cos(2t)

( apànthsh (g),(e),(st) ).

JËma TE2014.6. Kukl∏ste Ïpoia apo tic parakàtw sunart†seic piste‘eteÏti e–nai l‘sh tou probl†matoc

dy

dt+ y = �2et, y(0) = 0.

Apànthsh TE2014.6. L‘nw pr∏ta thn omogen† ex–swsh:

y0 + y = 0

JËtw y = ery ) y0 = rery. Antikajist∏ sthn ex–swsh mou kai Ëqw:

r + 1 = 0 ) r = �1

15

H genikeumËnh l‘sh thc ex–swshc mou ja e–nai thc morf†c:

yc

= C1e�t

me arqik† tim† to y(0) = 0. EfarmÏzw thn arqik† tim† gia na brw thn stajeràC1.

0 = C1e0 ) C1 = 0

kai telikà h genikeumËnh l‘sh mou e–nai thc morf†c:

yc

= e�t

Gia to dex– mh omogenËc mËloc ja kànoume manteyià tou t‘pou: y = Det ) y0 =Det. Kànw thn antikatàstash sthn arqik† gia na brw to D kai Ëqw:

Det +Det = �2et ) 2Det = �2et ) D = �1


yp

= �et

XËrontac Ïti h l‘sh thc diaforik†c ex–swshc e–nai thc morf†c:

y = yc

+ yp

y = e�t � et

( apànthsh (g) ).

JËma TE2014.7. Qrhsimopoi†ste thn mËjodo diaqwrismo‘ twn metablht∏ngia na upolog–sete tic l‘seic thc diaforik†c ex–swshc yu

xy

+ u = 0

Apànthsh TE2014.7. ApÏ ton orismÏ thc mejÏdou diaqwrismo‘ metablht∏nËqoume:

u(x, y) = X(x)Y (y)

'Ara,


yuxy

+ u = 0

) yX 0(x)Y 0

(y) +X(x)Y (y) = 0

) yX 0(x)Y 0

(y) = �X(x)Y (y)

) yX 0(x)Y 0

(y)

X(x)= �Y (y)

) X 0(x)

X(x)= � Y (y)

yY 0(y)

Parathro‘me Ïti to aristerÏ mËroc thc ex–swshc den exartàtai apÏ to y, en∏ todex– mËroc den exartàtai apÏ to x. 'Ara ta d‘o mËrh prËpei na iso‘ntai me m–astajerà, Ëstw thn stajerà �� (Ïpwc lËei kai o orismÏc). OpÏte:

X 0(x)

X(x)= �� = � Y (y)

yY 0(y)

'Etsi prok‘ptoun oi parakàtw d‘o exis∏seic tic opo–ec ja l‘soume xeqwristà:

X 0(x) + �X(x) = 0

��yY 0(y) + Y (y) = 0

H pr∏th ex–swsh apotele– m–a 1hc tàxhc grammik† S.D.E.

X 0(x) + �X(x) = 0

JËtw X(x) = erx ) X 0(x) = rerx

Antikajist∏ sthn ex–swsh kai pa–rnw:

rerx + �erx = 0

Diair∏ me erx kai ta d‘o mËrh thc ex–swshc (qwr–c periorismÏ, afo‘ to erx dengineta– potË mhdËn) kai pa–rnw:

r + � = 0

) r = ��

'Ara l‘sh 1hc:

X(x) = Ce��x

17

H de‘terh ex–swsh apotele– m–a 1hc tàxhc Euler grammik† S.D.E.

��yY 0(y) + Y (y) = 0

JËtw Y (y) = yr ) Y 0(y) = ryr�1

Antikajist∏ sthn ex–swsh kai pa–rnw:

��yryr�1+ yr = 0

) ��ryr + yr = 0

JËlw na diairËsw me yr kai ta d‘o mËrh thc ex–swshc. Gia na mporËsw na tokànw autÏ prËpei na pàrw periorismÏ Ïti to y 6= 0.

OpÏte:

Gia y = 0 (kai qwr–c na diairËsw) h ex–swsh g–netai 0 = 0

Gia y 6= 0 diair∏ me yr kai ta d‘o mËrh kai h ex–swsh g–netai:

��r + 1 = 0

) ��r = �1

) �r = 1

) r =

1

�

'Ara l‘sh 2hc:

Y (y) = Cy1�

ApÏ ton orismÏ thc mejÏdou diaqwrismo‘ metablht∏n (pou p†rame sthn arq†)Ëqoume:

u(x, y) = X(x)Y (y)

Sunep∏c, genik† l‘sh:

u(x, y) = Ce��xy1�

JËma TE2014.8. D∏ste mia grammik† diaforik† ex–swsh h opo–a Ëqei thnex†c genikeumËnh l‘sh

y(t) = c1e2tcos(3t) + c2e

2tsin(3t) + 3e3t.


Apànthsh TE2014.8. Parathro‘me pwc to 1o kommàti thc l‘shc mac C1e2tcos(3t)+

C2e2tsin(3t) proËrqetai apÏ m–a ex–swsh h opo–a bàsh t‘pou Ëqei tic ex†c miga-

dikËc r–zec:

r1 = 2 + 3i

kair2 = 2� 3i

XËroume pwc upolog–zontai autËc oi r–zec, Ëtsi mporo‘me na kànoume kai thnant–strofh diadikas–a.

r1,2 =

��

2↵±

p4↵� � �2

2↵

'Estw ↵ = 1 ja Ëqoume:

��

2↵= 2 ) �� = 4↵ ) � = �4

kaip

4↵� � �2

2↵= 3 )

p4↵� � �2

= 6a ) 4↵��2= (6a)2 ) 4� = 36+16 ) � = 13

'Eqontac brei touc suntelestËc mporo‘me na gràyoume to omogenËc mËloc apothn ex–swsh pou yàqnoume.

y00 � 4y0 + 13y = 0

To de‘tero kommàti thc l‘shc mac 3e3t proËrqetai apÏ to mh omogenËc kommàtithc arqik†c ex–swshc. Sun†jwc Ïtan Ëqoume sto mh omogenËc mËloc ekjetik†sunàrthsh gia na bro‘me thn l‘sh mac kànoume manteyià thc morf†c y = Dex.Se aut† thn per–ptwsh pou Ëqoume †dh thn y

p

, ja kànoume to ant–strofo.

yp

= 3e3t ) y0p

= 9e3t ) y00p

= 27e3t

An kànoume antikatàstash:

27e3t�4(9e3t)+13(3e3t) = De3t ) 27e3t�36e3t+39e3t = De3t ) D = 27�36+39 ) D = 30

'Ara sunolikà h ex–swsh pou yàqnoume e–nai:

y00 � 4y0 + 13y = 30e3t

19

JËma TE2014.9. Apode–xte Ïti gia m 6= n Ëqoume Ïti

Z⇡

0(sinmt)(sinnt) dt =

Z⇡

0(cosmt)(cosnt) dt =

Z⇡

�⇡

(sinmt)(sinnt) dt =

Z⇡

�⇡

(cosmt)(cosnt) dt = 0.

Apànthsh TE2014.9. S‘mfwna me to je∏rhma OrjogwniÏthtac Idiodianu-smàtwn (paràgrafoc 4.1.3 twn shmei∏sewn), eàn x1(t), x2(t) e–nai d‘o idiosunar-t†seic tou probl†matoc x00

+�x = 0 kai �1, �2 h ant–stoiqh idiotim† thc kajemiàc(�1 6= �2), tÏte oi idiosunart†seic e–nai orjog∏niec metax‘ touc sto diàsthma[a, b], dhlad† prok‘ptei Ïti

Rb

a

x1(t)x2(t)dt = 0.

'Opwc e–dame kai sthn paràgrafo 4.1.2 twn shmei∏sewn, oi ant–stoiqec idiosu-nart†seic tou porbl†matoc x00

+ �x = 0 e–nai oi cos(kt) kai sin(kt) me idiotim†� = k2, (k � 1) kai oi d‘o.

Sunep∏c, oi sin(mt), sin(nt), cos(mt), cos(nt) apotelo‘n ep–shc idiosunart†seictou –diou probl†matoc kai e–nai kai diaforetikËc metax‘ touc afo‘ �1 = m,�2 =

n,m 6= n. OpÏte to olokl†rwma tou ginomËnou touc sto diàsthma [0,⇡] allàkai sto [�⇡,⇡] ja iso‘tai me mhdËn Ïpwc apodeikn‘etai kai apÏ tic parapànwparagràfouc twn shmei∏sewn.

Gia pereta–rw epibeba–wsh Ïmwc, ac l‘soume ta oloklhr∏mata autà:

Z⇡

0sin(mt)sin(nt)dt =

nsin(m⇡)cos(n⇡)�msin(n⇡)cos(m⇡)

m2 � n2

= 0 gia n,m 2 Z

Z⇡

0cos(mt)cos(nt)dt =

msin(m⇡)cos(n⇡)� ncos(m⇡)sin(n⇡)

m2 � n2

= 0 gia n,m 2 Z

Z⇡

�⇡

sin(mt)sin(nt)dt =2nsin(m⇡)cos(n⇡)� 2msin(n⇡)cos(m⇡)

m2 � n2

= 0 gia n,m 2 Z


Z⇡

�⇡

cos(mt)cos(nt)dt =2msin(m⇡)cos(n⇡)� 2ncos(m⇡)sin(n⇡)

m2 � n2

= 0 gia n,m 2 Z

'Ara apode–xame akÏma kai me pràxeic Ïti ta parapànw idiodian‘smata e–nai orjo-g∏nia e–te sto [0,⇡], e–te sto [�⇡,⇡].

JËma TE2014.10. Diatup∏ste to prÏblhma MDE pou aforà thn

1. Diàdosh thc jermÏthtac se mia ràbdo m†kouc 2 mËtrwn Ïtan h jermokras–ase kàje shme–o thc opo–ac thn arqik† qronik† stigm† e–nai 20 bajmo–, toaristerÏ àkro thc e–nai monwmËno kai to dexiÏ diathre–tai pànta se jermo-kras–a 20 bajm∏n.

2. MetatÏpish mia elastik†c qord†c m†kouc 4 thc opo–ac h arister† pleuràparamËnei ak–nhth, h dexià thc e–nai ele‘jerh na kinhje– kai t–jetai se k–nhshapo thn jËsh isorrop–ac thc me arqik† taq‘thta g(x) = 2x� 3.

Apànthsh TE2014.10. S‘mfwna me tic perigrafËc,

Gia thn (a') per–ptwsh, Ëqoume:

L = 2m (to m†koc thc ràbdou)

ut

= kuxx

, 0 < x < 2, t > 0 (monodiàstath ex–swsh jermÏthtac)

u(x, 0) = 20

oC, 0 < x < 2 (arqik† katanom† jermokras–ac)

ux

(0, t) = 0

oC, t > 0 (to aristerÏ àkro e–nai monwmËno)

u(2, t) = 20

oC, t > 0 (to dex– àkro se stajer† jermokras–a)

Gia thn (b') per–ptwsh, Ëqoume:

L = 4m (to m†koc thc qord†c)

utt

= a2uxx

, 0 < x < 4, t > 0 (monodiàstath ex–swsh k‘matoc)

ut

(x, 0) = 2x� 3 m/sec, 0 < x < 4 (arqik† taq‘thta)

21

u(x, 0) = 0 m, 0 < x < 4 (arqik† jËsh thc qord†c)

u(0, t) = 0, t > 0 (arister† pleurà paktwmËnh)

ux

(4, t) = 0, t > 0 (dexià pleurà ele‘jerh)

JËma TE2014.11.

D∏ste tic timËc tou ! gia tic opo–ec to prÏblhma y00+(2⇡)2y = cos(!t), y(0) =y0(0) = 0

1. Ëqei l‘sh h opo–a paramËnh fragmËnh gia kàje t > 0.

2. sumba–nei suntonismÏc.

kai d∏ste tic ant–stoiqec l‘seic y(t).

Apànthsh TE2014.11. To prÏblhma y00 + (2⇡)2y = cos(!t) e–nai mia exa-nagkasmËnh talàntwsh.

Ac anagnwr–soume tic paramËtrouc tou probl†matoc mac.

m = 1, F0 = 1,!0 = 2⇡,! =?

H genik† l‘sh aut∏n twn problhmàtwn e–nai:

y = C1cos(!0t) + C2sin(!0t) +F0

m(!20 � !2

)

cos(!t)

Gia kàje ! 6= !0 6= 2⇡ mporo‘me na ulog–soume thn eidik† mac l‘sh qrhsimopoi-∏ntac tic arqikËc mac sunj†kec y(0) = 0, y0(0) = 0 kai Ëtsi na upolog–soume kaithn anàlogh l‘sh.

Gia ! = !0 h diadikas–a diaforopoie–tai l–go.

Ja dokimàsoume manteyià tou t‘pou yp

= Atcos(!t) + Bsin(!t) miac kai 2hparàgwgoc tou tcos(!t) periËqei hm–tona.Gràfoume loipÏn thn ex–swsh:

y00 + !2y =

F0

mcos(!t)

Kànoume antikatàstash sto aristerÏ mËloc kai Ëqoume:

2B!cos(!t)� 2A!sin(!t) =F0

mcos(!t)


AutÏ mac kànei to A = 0 kai to B =

F02m!

, sthn dik–a mac per–ptwsh e–nai B =

14⇡

H sugkekrimËnh l‘sh e–nai:F0

2m!tsin(!t)

kai h genikeumËnh l‘sh:

y = C1cos(!t) + C2sin(!t) +F0

2m!tsin(!t)

y = C1cos(2⇡t) + C2sin(2⇡t) +1

4⇡tsin(2⇡t)

An qrhsimopoi†soume tic arqikËc sunj†kec ja pàroume: C1 = 0, C2 = 0

'Ara h telik† mac l‘sh Ïtan ! = !0 = 2⇡ e–nai:

y =

1

4⇡tsin(2⇡t)

JËma TE2014.12.

Upolog–ste thn genikeumËnh l‘sh tou mh-omogeno‘c probl†matoc

dx1

dt= �2x1 + 3x2 + et

dx2

dt= �x1 + 2x2 + t

an gnwr–zoume Ïti h genikeumËnh l‘sh tou ant–stoiqou omogeno‘c probl†matoce–nai †

x

o

(t) = c1e�t

3

1

�+ c2e

t

1

1

�.

Apànthsh TE2014.12. EfÏson xËroume thn genikeumËnh l‘sh jËloume namantËyoume l‘sh thc morf†c: �!x =

�!↵ et +�!b t+�!c .

'Omwc den gnwr–zoume an to �!↵ e–nai grammikà exarthmËno sto1

1

�,Ëtsi ja doki-

màsoume ep–shc to:�!b tet. Sunolikà ja Ëqoume mia l‘sh thc morf†c:

�!x =

�!↵ et +�!b tet +�!c t+

�!d

Pr∏ta ac upolog–soume to:

�!x0

= (

�!↵ +

�!b )et +

��!btet +�!c

23

to opo–o prËpei na e–nai –so me

A�!x +

�!f )

A�!x +

�!f = A�!↵ et +A

�!b tet +A�!c t+A

�!d +

�!f =

�2↵1 + 3↵2

�↵1 + 2↵2

�et +

�2b1 + 3b2�b1 + 2b2

�tet +

�2c1 + 3c2�c1 + 2c2

�t+

�2d1 + 3d2�d1 + 2d2

�+

et

t

�

Ac exis∏soume t∏ra touc suntelestËc twn et,tet, t kai touc stajero‘c Ïrouc.

8>>>>>>>>>>>>><

>>>>>>>>>>>>>:

↵1 + b1 = �2↵1 + 3↵2

↵2 + b2 = �↵1 + 2↵2

b1 = �2b1 + 3b2

b2 = �b1 + 2b2

0 = �2c1 + 3c2

0 = �c1 + 2c2 + 1

c1 = �2d1 + 3d2

c2 = �d1 + 2d2

)

8>>>>>>>>>>>>><

>>>>>>>>>>>>>:

↵1 = 1

↵2 = 1

b1 = 1

b2 = 1

c1 =

34

c1 =

12

d1 = 0

d2 =

312

'Ara sunolikà Ëqoume:

�!x =

�!↵ et +�!b tet +�!c t+

�!d =

1

1

�et +

1

1

�tet +

3/41/2

�t+

0

3/12

�=

et + tet + 3

4 tet + tet + 1

2 t+14

�

Kefàlaio 2

SDE An∏terhc Tàxhc

JËma EP2014.1.1. D∏ste Ïlec tic l‘seic thc ex–swshc y000 = �y.

Apànthsh EP2014.1.1. Pr∏ta ja prospaj†soume thn tetrimmËnh 0 blËpou-me pwc epalhje‘ei (0 = 0) thn arqik† ex–wsh mac àra apotele– l‘sh.

An pàme to y sto aristerÏ mËloc Ëqoume thn ex–swsh:

y000 + y = 0

JËtw y = ery ) y0 = rery ) y00 = r2(ery) ) y000 = r3(ery). Antikajist∏ sthnex–swsh mou kai Ëqw:

r3 + r = 0 ) (r + 1)(r2 � r + 1 = 0)

m–a l‘sh e–nai h r = �1 oi àllec d‘o bgoun apÏ thn ep–lush tou parapànwpoluwn‘mou.H diakr–nousa auto‘ e–nai: � = �2 � 4↵� ) � = �3 àra Ëqoume migadikËc r–zecoi opo–ec e–nai:

r2,3 =

��

2↵±

p4↵� � �2

2↵= �1

2

± ip3

2

)

r2 = �1

2

+

ip3

2

kai

r3 = �1

2

� ip3

2

'Otan Ëqoume migadikËc r–zec xËroume (je∏rhma 2.2.3) pwc h exisws† mac ja Ëqeil‘sh thc morf†c

y = C1e↵tcos�t+ C2e

↵tsin�t

Antikajisto‘me tic r–zec pou br†kame kai Ëqoume:

y = C1e�t/2cos

p3

2

t+ C2e�t/2sin

p3

2

t

26 SDE An∏terhc Tàxhc

'Ara sunolikà Ëqoume:

y = C1e�t/2cos

p3

2

t+ C2e�t/2sin

p3

2

t+ C3e�t

JËma EP2014.1.2. D∏ste thn l‘sh tou probl†matoc y

0

y

+ 2x + xy3 = 0,y(0) = 1.

Apànthsh EP2014.1.2. An pollaplasiàsoume me y thn ex–swsh mac Ëqou-me:

y0 + 2xy + xy4 = 0

h opo–a e–nai mia ex–swsh Bernulli me p(x) = 2x kai q(x) = �x.

JËtoume u = y1�4 ) u = y�3 ep–shc Ëqoume u0= �3y�4y0 ) y0 = � u

0

3y�4 =

�y

4

3 u0 AutÏ mac kànei:

y0 + 2xy + xy4 = 0 ) �y4

3

u0+ 2xy + xy4 = 0

Ac pollaplasiàsoume me y�4 thn ex–swsh mac.

�u0

3

+ 2xy�3+ x = 0 ) �u0

3

+ 2xu+ x = 0 )

�u0+ 6ux+ 3x = 0 ) u0 � 6ux� 3x = 0

BlËpoume pwc h exisws† mac Ëqei thn morf† oloklhrwtik∏n paragÏntwn mep(x) = �6x , f(x) = 3x kai r(x) = e

Rp(x)dx

= eR�6xdx

= e�3x2

.

u = e�Rp(x)dx

[

ZeRp(x)dxf(x)dx+ C] )

u = e3x2

[

Ze�3x2

3xdx+ C] = e3x2

(�1

2

e�3x2

+ C) = Ce3x2

� 1

2

Kànoume antikatàstash 'proc ta p–sw':

u = y�3= Ce3x

2

� 1

2

=

2Ce3x2 � 1

2

)

y =

3p2

3p2Ce3x2 � 1

ApÏ thn arqik† sunj†kh Ëqw Ïti y(0) = 1 ) 1 =

3p23p

2Ce

3x2�1) 2 = 2C � 1 )

C =

32

'Ara h l‘sh mac e–nai h:

27

y =

3p2

3p3e3x2 � 1

JËma EP2014.1.3. D∏ste Ïlec tic l‘seic thc ex–swshc 2t2y00+ ty0�3y = 0

(UpÏdeixh: 1/t).

Apànthsh EP2014.1.3. Pr∏ta ja prospaj†soume thn tetrimmËnh 0 blËpou-me pwc epalhje‘ei (0 = 0) thn arqik† ex–wsh mac àra apotele– l‘sh.

'Epeita ja dokimàsoume an h upÏdeixh apotele– l‘sh thc ex–swshc.'Eqoune y =

1t

) y0 = � 1t

2 ) y00 = 2t

3 kai ja kànoume antikatàstash gia na do‘me ean epa-lhje‘ei thn ex–swsh.

2t22

t3+ t(� 1

t2)� 3(

1

t) = 0 ) 4

t� 1

t� 3

t= 0 ) 0 = 0

'Ara kai h 1/t apotele– l‘sh thc ex–swshc. Mporo‘me e‘kola na parathr†soumepwc h exisws† mac e–nai t‘pou Euler.Se aut† thn per–ptwsh jËtoume y = tr )y0 = rtr�1 ) y00 = r(r � 1)tr�2. Kànoume antikatàstash sthn ex–swsh mac kaiËqoume:

2t2r(r � 1)tr�2+ trtr�1 � 3tr = 0 ) 2r(r � 1)tr + rtr � 3tr = 0

Diair∏ me tr

2r(r � 1) + r � 3 = 0 ) 2r2 � 2r + r � 3 = 0 ) 2r2 � r � 3 = 0

ApÏ thn ep–lush tou parapànw poluwn‘mou. � = �2 � 4↵� ) � = 25 Ëqoume:

r1,2 =

�� ±p�

2↵=

1± 5

4

oi r–zec mac e–nai r1 =

3

2

kai r2 = �1

H l‘sh mac e–nai thc morf†c:

y = C1t32+ C2t

�1

JËma EP2014.1.4. Suntairiàxte ta parakàtw ped–a kateuj‘nsewn me tic e-x†c exis∏seic (1) y0 = 2�y, (2) y0 = x(2�y), (3) y0 = x+y�1, (4) y0 = sinx sin y.

Apànthsh EP2014.1.4. Antisto–qish ped–wn me tic exis∏seic touc.


Sq†ma 2.1: y0 = 2� y Sq†ma 2.2: y0 = x(2� y)

Sq†ma 2.3: y0 = x+ y � 1 Sq†ma 2.4: y0 = sin(x)sin(y)

29

JËma EP2014.1.5. Se Ëna iqjuotrofe–o o plhjusmÏc twn yari∏n auxàneikatà 80% kàje m†na. An yare‘oume 40 apÏ autà ta yària kàje mËra:

(↵0) Dwste thn ex–swsh pou aforà twn plujhsmÏ twn yari∏n tou iqjuo-

trofe–ou.

(�0) Perigràyte ti ja sumbe– sto iqjuotrofe–o se bàjoc qrÏnou.

Apànthsh EP2014.1.5. 'Estw y(t) o plujhsmÏc twn yari∏n thn qronik†stigm† t, kai t o qrÏnoc se m†nec.O rujmÏc metabol†c tou plujhsmo‘ anà m†naja e–nai a‘xhsh 80% ep– tou arqiko‘ kai stajer† me–wsh 30⇤40 = 1200 apÏ autàpou yare‘oume se Ëna m†na, dhlad†:

y0(t) = 0.8y(t)� 1200

Parathro‘me Ïti prÏkeitai gia m–a ex–swsh 1hc tàxhc grammik† S.D.E.:

y0(t) = 0.8y(t)� 1200 ) y0(t)� 0.8y(t) = �1200

Thn fËrame sthn morf† y0 + p(t)y = f(t), àra mporo‘me na thn l‘soume me thnmËjodo oloklhrwtik∏n paragÏntwn:

p(t) = �0.8

f(t) = �1200

r(t) = eRp(t)dt

= e�0.8Rdt

= e�0.8t

'Ara Genik† L‘sh:

y(t) = e0.8t[

Ze�0.8t

(�1200)dt+ C]

) y(t) = Ce0.8t + 1500

Mh Ëqontac arqik† sunj†kh gia ton plujhsmÏ twn yari∏n den mporo‘me na u-polog–soume thn stajerà C thc ex–swshc mac. AutÏ pou mporo‘me na parath-r†soume e–nai pwc h stajerà C e–nai pollaplasiasmËnh me Ënan ekjetikÏ Ïro oopo–oc sto bàjoc tou qrÏnou ja megal∏nei suneq∏c. Ep–shc ËpiplËon Ëqoumekai Ëna stajerÏ Ïro a‘xhshc kata 1500. 'Ara analÏgwc to prÏshmo tou C jag–netai a‘xhsh † me–wsh tou plujhsmo‘ twn yari∏n ant–stoiqa.

JËma EP2014.1h(sunËqeia).1. Perigràyte thn sumperiforà thc l‘shcthc ex–swshc y0 = x3

(x+ 2)(x� 1)

2 gia megàla x.

Apànthsh EP2014.1h(sunËqeia).1. H ex–swswsh apotele– mia 1hc tàxhcgrammik† S.D.E.. Mac zhte–tai na perigràyoume thn sumperiforà thc l‘shc gia


megàla x, dhlad† thn l‘sh kaj∏c to x te–nei sto àpeiro.

Arqikà parathro‘me Ïti h sugkekrimËnh S.D.E. mhden–zetai gia x = �2, x =

0 kai x = 1. Sunep∏c sto ped–o kateuj‘nse∏n thc (kai sta shme–a autà) jaupàrqoun diakekomËnec euje–ec me kl–sh mhdËn mo–rec.

Sq†ma 2.5: Ped–o kateuj‘nsewn thcS.D.E. y0 = x3

(x+ 2)(x� 1)

2

Sth sunËqeia, gia na perigràyoume thn sumperiforà thc l‘shc, prËpei pr∏ta nal‘soume thn ex–sws† mac. H sugkekrimËnh S.D.E. ap Ïti parathro‘me apotele–diaqwr–simh ex–swsh. OpÏte, Ëqoume:

dy

dx= x3

(x+ 2)(x� 1)

2

) dy = x3(x+ 2)(x� 1)

2dx

) dy = (x6 � 3x4+ 2x3

)dx

)Z

dy =

Z(x6 � 3x4

+ 2x3)dx

)Z

dy =

Zx6dx� 3

Zx4dx+ 2

Zx3dx

31

) y(x) =x7

7

� 3x5

5

+

x4

2

+ C

'Ara gia na do‘me pwc sumperifËretai h l‘sh gia megàla x, ja pàroume to Ïriothc l‘shc kaj∏c to x te–nei sto àpeiro:

lim

x!+1

x7

7

� 3x5

5

+

x4

2

+ C = +1

'Ara h l‘sh te–nei sto àpeiro kaj∏c to x te–nei sto àpeiro, Ïpwc mporo‘me nadiapist∏soume kai apÏ to parakàtw sq†ma (gia C = 0):

Sq†ma 2.6: Grafik† paràstash thcl‘shc y(x) =

x

7

7 � 3x5

5 +

x

4

2 (gia x a-pÏ 0 wc 1000)

JËma EP2014.1h(sunËqeia).2. D∏ste thn l‘sh tou probl†matoc y0 =

1+3t2

3y2�6y , y(0) = 1.

Apànthsh EP2014.1h(sunËqeia).2. To prÏblhma apotele– m–a 1hc tàxhc,diaqwr–simh, mh-grammik† S.D.E.

'Ara:


y0 =1 + 3t2

3y2 � 6y

) dy

dt=

1 + 3t2

3y2 � 6y

) (3y2 � 6y)dy = (1 + 3t2)dt

)Z

(3y2 � 6y)dy =

Z(1 + 3t2)dt

) 3

Zy2dy � 6

Zydy =

Zdt+ 3

Zt2dt+ C

) y3 � 3y2 = t+ t3 + C ('Emmesh L‘sh)

ApÏ thn arqik† tim† Ëqoume:

y(0) = 1

) y2(0) = 1

2= 1

) y3(0) = 1

3= 1

Sunep∏c:

1� 3 = 0 + 0 + C

) C = �2

'Ara SugkekrimËnh ('Emmesh) L‘sh gia C = �2:

y3 � 3y2 = t+ t3 � 2

JËma EP2014.1h(sunËqeia).3. D∏ste thn l‘sh tou probl†matoc y00 �ey(y0)3 = 0, y(2) = 0, y0(2) = 1/2.

Apànthsh EP2014.1h(sunËqeia).3. Ed∏ Ëqoume mia 2hc tàxhc mh-grammik†S.D.E.

y00 � ey(y0)3 = 0

33

Sth sugkekrimËnh S.D.E., lÏgw thc idia–terhc morf†c pou Ëqei, ja prËpei naergasto‘me (arqikà) me allag† metablht†c.

JËtw y0 = z(y) sunep∏c y00 = [z(y)]0 = z0(y)y0 = z0(y)z(y). OpÏte h S.D.E.metatrËpetai sthn:

z0z � eyz3 = 0

) �z(�z0 + eyz2) = 0

) z(�z0 + eyz2) = 0

PrËpei (a) z = 0 kai (b) �z0 + eyz2 = 0

(a):

z = 0 h l‘sh (a)

(b):

�z0 + eyz2 = 0

) z0 = eyz2

) dz

dy= eyz2

) 1

z2dz = eydy

)Z

1

z2dz =

Zeydy

) �1

z= ey + C1

) z =

�1

ey + C1h l‘sh (b)

Kànoume proc ta p–sw antikatàstash sthn metablht† pou jËsame sthn arqh


(z = y0) kai Ëqoume:

(a):

z = 0 ) y0 = 0 ) y = 0 h l‘sh (a)

(b):

z =

�1

ey + C1

) y0 =�1

ey + C1

) dy

dx=

�1

ey + C1

) (ey + C1)dy = �dx

)Z(ey + C1)dy = �

Zdx

)Z

eydy +

ZC1dy = �

Zdx

) ey + C1y = �x+ C2 Ëmmesh l‘sh (b)

ApÏ arqik† tim† y(2) = 0 Ëqoume:

y(2) = 0

) e0 + C10 = �2 + C2

) 1 = �2 + C2

) C2 = 3

'Etsi:

35

ey + C1y = �x+ C2

) ey + C1y = �x+ 3

JËma EP2014.1h(sunËqeia).4. Upolog–ste katà prosËggish thn tim† twny(5) kai y(8) Ïpou y e–nai h l‘sh miac diaforik†c ex–swshc thc opo–ac to ped–okateuj‘nsewn e–nai to parakàtw gia thn opo–a l‘sh Ëqoume (a) y(0) = 2 kai (b)y(�2) = 2.

Apànthsh EP2014.1h(sunËqeia).4.

JËma EP2014.1h(sunËqeia).5. 'Eqete mÏlic apokt†sei mia klhronomià e-nÏc ekatommur–ou eur∏ kai skope‘ete na topojet†sete to posÏ autÏ se trapezikÏlogariasmÏ me et†sio tÏko 5%. Perigràyte, dikaiog∏ntac pl†rwc thn apànths†sac, ti ja sumbe– se bàjoc qrÏnou an kàje m†na xode‘ete tËssereic qiliàdeceur∏.

Apànthsh EP2014.1h(sunËqeia).5. 'Estw y(t) to kefàlaio thn qronik†stigm† t, kai t o qrÏnoc se Ëth.

y(0) = 1.000.000 eur∏, to arqikÏ kefàlaio

O rujmÏc metabol†c anà Ëtoc ja e–nai a‘xhsh 5% ep– tou kefala–ou kai paràl-lhla stajer† me–wsh 12 ⇤ 4.000 = 48.000 eur∏, dhlad†:

y0(t) = 0.05y(t)� 48000

Sunep∏c, afo‘ sqhmat–same th diaforik† ex–dwsh tou rujmo‘ metabol†c, denËqoume parà na th l‘soume kai na bro‘me thn arqik† sunàrthsh tou kefala–ouy(t). Ap Ïti parathro‘me prÏkeitai gia m–a 1hc tàxhc grammik† S.D.E.:

y0 = 0.05y � 48000

) y0 � 0.05y = �48000

Thn fËrame sthn morf† y0 + p(t)y = f(t), àra mporo‘me na thn l‘soume me thn


mËjodo oloklhrwtik∏n paragÏntwn:

p(t) = �0.05

f(t) = �48000

r(t) = eRp(t)dt

= e�0.05Rdt

= e�0.05t


y(t) = e0.05t[

Ze�0.05t

(�48000)dt+ C]

) y(t) = Ce0.05t + 960000

ApÏ arqik† sunj†kh Ëqoume:

y(0) = 1000000

) Ce0.05⇤0 + 960000 = 1000000

) C = 40000

'Ara SugkekrimËnh L‘sh gia C = 40000:

y(t) = 40000e0.05t + 960000

'Etsi, to 1o, 2o kai 3o Ëtoc ant–stoiqa, to kefàlaio ja kumanje– wc ex†c:

y(1) = 40000e0.05⇤1 + 960000 = 1, 002, 050.844 eur∏

y(2) = 40000e0.05⇤2 + 960000 = 1, 004, 206.837 eur∏

y(3) = 40000e0.05⇤3 + 960000 = 1, 006, 473.370 eur∏

Sunep∏c se bàjoc qrÏnou to kefàlaio Ïlo kai ja auxànetai...

JËma EP2014.2h.1. Upolog–ste Ïlec tic l‘seic tou parakàtw sust†matocdiaforik∏n exis∏sewn, metatrËpontàc to pr∏ta se mia isod‘namh diaforik† e-x–swsh. x0

= �4x+ 2y, y0 = � 52x+ 2y.

37

Apànthsh EP2014.2h.1. Pr∏ta ja paragwg–soume thn 2h ex–swsh:

y00 = �5

2

x0+ 2y0 ) y00 = �5

2

(�4x+ 2y) + 2y0 )

y00 = 10x� 5y + 2y0 ) y00 � 2y0 + 5y = 10x

ApÏ thn 2h ex–swsh Ëqoume:

�5

2

x = y0 � 2y ) 10x = �4y0 + 8y

y00 � 2y0 + 5y = �4y0 + 8y ) y00 + 2y0 � 3y = 0

JËtw y = ery ) y0 = rery ) y00 = r2(ery). Antikajist∏ sthn ex–swsh mou kaiËqw:

r2 + 2r � 3 = 0

H diakr–nousa auto‘ e–nai: � = �2 � 4↵� ) � = 16

r1,2 =

�� ±p�

2↵=

�2± 4

2

oi r–zec mac e–nai r1 = 1 kai r2 = �3


y = C1et

+ C2e�3t

t∏ra mpor∏ na upolog–sw kai thn l‘sh gia to x

x =

y00 � 2y0 + 5y

10

=

C1et

+ 9C2e�3t � 2(C1e

t � 3C2e�3t

) + 5(C1et

+ C2e�3t

)

10

=

=

C1et

+ 9C2e�3t � 2C1e

t

+ 6C2e�3t

+ 5C1et

+ 5C2e�3t

10

=

4C1et

+ 20C2e�3t

10

x =

1

5

C1et

+ 2C2e�3t

JËma EP2014.2h.2. Qrhsimopoi†ste thn mËjodo diak‘manshc twn paramËtrwngia na upolog–ste Ïlec tic l‘seic thc ex–swshc x2y00 � xy0 + y = 2x. (UpÏdeixh:y1 = x kai y2 = x lnx e–nai l‘seic thc ant–stoiqhc omogeno‘c ex–swshc.)


Apànthsh EP2014.2h.2. H ex–swsh x2y00�xy0+y = 2x apotele– miac 2hctàxhc, mh-omogen†, Euler grammik† S.D.E. thn opo–a kalo‘maste na thn l‘sou-me mËsw thc mejÏdou Diak‘manshc ParamËtrwn. OpÏte:

JËtw wc Eidik† L‘sh:

yp

= u1y1 + u2y2

) yp

= u1x+ u2xlnx

Sth sunËqeia, jËlw na diairËsw me x2 kai ta d‘o mËrh thc S.D.E. outwc ∏steo Ïroc pou br–sketai sth mËgisth paràgwgo na e–nai ¨kajarÏc’ apÏ suntelestËc.Gia na mporËsw na to kànw autÏ prËpei na pàrw periorismÏ Ïti to x 6= 0.

OpÏte:

Gia x = 0 (kai qwr–c na diairËsw) h ex–swsh katal†gei sthn l‘sh y = 0

Gia x 6= 0 diair∏ me x2 kai ta d‘o mËrh kai h ex–swsh g–netai:

y00 � 1

xy0 +

1

x2y =

2

x

T∏ra, mpor∏ na sqhmat–sw kai na l‘sw to s‘sthma s‘mfwna me ton orismÏ thcDiak‘manshc ParamËtrwn:

(u01y1 + u0

2y2 = 0

u01y

01 + u0

2y02 = f(x)

(1) )(u01x+ u0

2xlnx = 0

u01 + u0

2(lnx+ 1) =

2x

(2) )(u01x+ u0

2xlnx = 0

u01 = �u0

2(lnx+ 1) +

2x

(3) )((�u0

2(lnx+ 1) +

2x

)x+ u02xlnx = 0

⇤ !u01 = �u0

2(lnx+ 1) +

2x

⇤ ! (�u02lnx� u0

2 +2x

)x+ u02xlnx = 0

⇤ ! �u02xlnx� u0

2x+ 2 + u02xlnx = 0

⇤ ! u02 =

2x

(4) )(u02 =

2x

u01 = �u0

2(lnx+ 1) +

2x

(5) )(u02 =

2x

u01 = � 2

x

(lnx+ 1) +

2x

(6) )(u02 =

2x

u01 = � 2lnx

x

Br†ka ta u01, u

02, opÏte ja oloklhr∏sw gia na pàrw kai ta ant–stoiqa u1 kai u2:

39

u1 = �2

Zlnx

xdx = �ln2

(x) + C1

u2 = 2

Z1

xdx = 2ln(x) + C2

ApÏ thn s‘mbash pou e–qa kànei sthn arq† gia thn eidik† l‘sh yp

, Ëqw:

yp

= u1x+ u2xlnx = �xln2(x) + 2xln2

(x) = xln2(x)

'Ara, Ïlec oi l‘seic thc arqik†c D.E. (kai s‘mfwna me to je∏rhma thc upËrje-shc) e–nai oi:

y(x) = yc

(x) + yp

(x)

) y(x) = C1x+ C2xln(x) + xln2(x)

JËma EE2013.1. PoiËc apo tic parakàtw sunart†seic apotelo‘n l‘sh thcex–swshc @

2u

@x

2 = 9

@

2u

@y

2

(a') cos(3x+ y) (b') x2+ y2 (g') sin(3x+ y) (d') e�3⇡x

sin(⇡y) (e') kamm–a apo ticparapànw

Apànthsh EE2013.1. Sthn per–ptwsh aut†, ant– na l‘soume thn merik† dia-forik† ex–swsh, mporo‘me e‘kola na paragwg–soume merik∏c thn kajem–a apÏ ticproteinÏmenec l‘seic, na thn antikatast†soume sthn arqik† ex–swsh kai na do‘mean thn epalhje‘ei. An thn epalhje‘ei, tÏte apotele– l‘sh thc ex–swshc.

Paragwg–zontac thn (a') wc proc @

2

@x

2 :


@2

@x2cos(3x+ y) =

= � @

@xsin(3x+ y)

@

@x(3x+ y)

= �3

@

@xsin(3x+ y)

= �3cos(3x+ y)@

@x(3x+ y)

= �9cos(3x+ y)

Paragwg–zontac thn (a') wc proc 9 @

2

@y

2 :

9

@2

@y2cos(3x+ y) =

= �9

@

@ysin(3x+ y)

@

@y(3x+ y)

= �9

@

@ysin(3x+ y)

= �9cos(3x+ y)@

@y(3x+ y)

= �9cos(3x+ y)

'Ara h (a') apotele– l‘sh!

Suneq–zoume, paragwg–zontac thn (g') wc proc @

2

@x

2 :

41

@2

@x2sin(3x+ y) =

=

@

@xcos(3x+ y)

@

@x(3x+ y)

= 3

@

@xcos(3x+ y)

= �3sin(3x+ y)@

@x(3x+ y)

= �9sin(3x+ y)

Paragwg–zontac thn (g') wc proc 9 @

2

@y

2 :

9

@2

@y2sin(3x+ y) =

= 9

@

@ycos(3x+ y)

@

@y(3x+ y)

= 9

@

@ycos(3x+ y)

= �9sin(3x+ y)@

@y(3x+ y)

= �9sin(3x+ y)

'Ara kai h (g') apotele– l‘sh!

Paragwg–zontac kai tic upÏloipec proteinÏmenec l‘seic parathro‘me Ïti den e-palhjeuoun thn ex–swsh, opÏte den apotelo‘n l‘seic.

JËma EE2013.3. PoiËc apo tic parakàtw sunart†seic apotele– genikeumËnhl‘sh thc ex–swshc 6y(4) + 11y00 + 4y = 0

(a') y = c1 cos(x

2 ) + c2 sin(x

2 ) + c3 cos(4x3 ) + c4 sin(

4x3 )

(b') y = c1 cos(x

2 ) + c2 sin(x

2 ) + c3 cos(2x3 ) + c4 sin(

2x3 )


(g') y = c1 cos(xp2) + c2 sin(

xp2) + c3 cos(

2xp3) + c4 sin(

2xp3)

(d') y = c1e� 4

3x+ c2e

� 12x

(e') y = c1e� 4

3x+ c2xe

� 43x

+ c3e� 1

2x+ c4xe

� 12x (st') kamm–a apo tic parapànw

Apànthsh EE2013.3. H ex–swsh:

6y(4) + 11y00 + 4y = 0

apotele– miac uyhlÏterhc tàxhc grammik† S.D.E. OpÏte:

JËtw

y(x) = erx

) y0(x) = rerx

) y00(x) = r2erx

) y000(x) = r3erx

) y(4)(x) = r4erx

Antikajist∏ sthn S.D.E. kai pa–rnw:

6r4erx + 11r2erx + 4erx = 0

Diair∏ me erx kai ta d‘o mËrh thc ex–swshc (qwr–c periorismÏ, afo‘ to erx dengineta– potË mhdËn) kai pa–rnw:

6r4 + 11r2 + 4 = 0

) (2r2 + 1)(3r2 + 4) = 0

) (2r2 + 1) = 0 kai (3r2 + 4) = 0

(a)

2r2 + 1 = 0

) 2r2 = �1

) r2 =

�1

2

) r = ±i

r1

2

) r1 = 0± i1p2

43

(b)

3r2 + 4 = 0

) 3r2 = �4

) r2 =

�4

3

) r = ±i

r4

3

) r2 = 0± i2p3

Parathro‘me Ïti kai oi d‘o r–zec twn qarakthristik∏n exis∏sewn e–nai migadikËcthc morf†c ↵ ± i�. Se aut†n thn per–ptwsh (upojËtontac Ïti h anexàrththmetablht† thc S.D.E. e–nai h x, Ïpwc sthn per–ptws† mac) h Genik† L‘sh jae–nai thc morf†c y(x) = C1e

↵xcos�x+ C2e↵xsin�x gia thn kàje migadik† r–za.


y(x) = C1e0⇤xcos(

1p2

x) + C2e0⇤xsin(

1p2

x) + C3e0⇤xcos(

2p3

x) + C4e0⇤xsin(

2p3

x)

) y(x) = C1cos(xp2

) + C2sin(xp2

) + C3cos(2xp3

) + C4sin(2xp3

)

JËma EE2013.4. Qrhsimopoi†ste metasqhmatimso‘c Laplàc gia na upolo-g–sete thn l‘sh tou probl†matoc

y00 � 3y0 + 2y = 2x+ 1, y(0) = 2, y0(0) = �1.

Apànthsh EE2013.4. H arqik†, 2hc tàxhc, grammik† S.D.E.:

y00 � 3y0 + 2y = 2x+ 1

metasqhmat–zetai apÏ to ped–o tou qrÏnou, sto ped–o thc suqnÏthtac, mËsw touorjo‘ metasqhmatismo‘ Laplace wc ex†c:


L{y00}� 3L{y0}+ 2L{y} = 2L{x}+ L{1}

) s2Y (s)� sy(0)� y0(0)� 3[sY (s)� y(0)] + 2Y (s) =2

s2+

1

s

) s2Y (s)� 2s+ 1� 3sY (s) + 6 + 2Y (s) =2

s2+

1

s(antikatàstash arqik∏n sunjhk∏n)

) Y (s)[s2 � 3s+ 2]� 2s+ 7 =

2

s2+

1

s

) Y (s)[s2 � 3s+ 2]� 2s+ 7 =

2s+ s2

s3

) Y (s)[s2 � 3s+ 2] = 2s� 7 +

2s+ s2

s3

) Y (s)[s2 � 3s+ 2] =

2s4 � 7s3 + s2 + 2s

s3

) Y (s) =2s4 � 7s3 + s2 + 2s

s3(s2 � 3s+ 2)

) Y (s) =2s4 � 7s3 + s2 + 2s

s3(s� 2)(s� 1)

=

⇤ !

Anàlush tou s‘njetou klàsmatoc se aplà klàsmata:

⇤ ! 2s4�7s3+s

2+2ss

3(s�2)(s�1) =

A

s

+

B

s

2 +

C

s

3 +

D

s�2 +

E

s�1 = ... =

=

2s

+

1s

2 +

0s

3 � 2s�2 +

2s�1

) Y (s) =2

s+

1

s2� 2

s� 2

+

2

s� 1

'Etsi, me ant–strofo metasqhmatismÏ Laplace, pernàme pàli p–sw sto ped–otou qrÏnou (apÏ to ped–o thc suqnÏthtac) kai br–skoume thn l‘sh thc arqi-k†c S.D.E.:

45

y(x) = L�1{Y (s)}

= 2L�1{1s}+ L�1{ 1

s2}� 2L�1{ 1

s� 2

}+ 2L�1{ 1

s� 1

}

= 2 + x� 2e2x + 2ex

JËma EE2013.5. Upolog–ste thn tim† thc l‘shc thc ex–swshc t2 d

2y

dt

2 �4tdydt

+

6y = 0 pou ikanopoie– tic arqikËc sunj†kec y(1) = 2 kai y0(1) = �1 thn qronik†stigm† t = 2.

Apànthsh EE2013.5. H ex–swsh:

t2d2y

dt2� 4t

dy

dt+ 6y = 0

apotele– mia 2hc tàxhc Euler grammik† S.D.E.

Sunep∏c, jËtw y(t) = tr ) y0(t) = rtr�1 ) y00(t) = r(r � 1)tr�2

Antikajist∏ sthn S.D.E. kai pa–rnw:

t2r(r � 1)tr�2 � 4trtr�1+ 6tr = 0

) r(r � 1)tr � 4rtr + 6tr = 0

JËlw na diairËsw me tr kai ta d‘o mËrh thc ex–swshc. Gia na mporËsw na tokànw autÏ prËpei na pàrw periorismÏ Ïti to t 6= 0.

OpÏte:

Gia t = 0 (kai qwr–c na diairËsw) h ex–swsh g–netai 0 = 0

Gia t 6= 0 diair∏ me tr kai ta d‘o mËrh kai h ex–swsh g–netai:

r(r � 1)� 4r + 6 = 0

) r2 � 5r + 6 = 0

) r1 = 3, r2 = 2


y(t) = C1t3+ C2t

2


Suneq–zontac, apÏ arqikËc sunj†kec Ëqoume:

y0(t) = (C1t3+ C2t

2)

0= C13t

2+ C22t

y(1) = 2

) C1(1)3+ C2(1)

2= 2

) C1 + C2 = 2

) C1 = 2� C2

y0(1) = �1

) C13(1)2+ C22(1) = �1

) 3C1 + 2C2 = �1

) 3(2� C2) + 2C2 = �1

) 6� 3C2 + 2C2 = �1

) C2 = 7

C1 = 2� C2

) C1 = �5

'Ara, SugkekrimËnh L‘sh gia C1 = �5 kai C2 = 7:

y(t) = �5t3 + 7t2

TËloc, h tim† thc l‘shc gia thn qronik† stigm† t = 2:

y(2) = �5(2)

3+ 7(2)

2= �12

JËma EE2013.6. Bre–te Ïlec tic l‘seic tou sust†matoc x0(t) =

2

4�2 1 1

0 1 1

0 0 5

3

5x(t)

gia tic opo–ec isq‘ei limt!1 x(t) =

2

40

0

0

3

5 .

47

Apànthsh EE2013.6.

�!x 0(t) =

2

4�2 1 1

0 1 1

0 0 5

3

5�!x (t)

Xekinàme th l‘sh br–skontac pr∏ta tic IdiotimËc tou arqiko‘ p–naka, (ac ton o-nomàsoume A):

det(A� �I) = 0

) det

2

4�2� � 1 1

0 1� � 1

0 0 5� �

3

5= 0

ApÏ ton kanÏna tou Sarrus Ëqoume:

(�2� �)(1� �)(5� �) + (1)(1)(0) + (1)(0)(0)

�(0)(1� �)(1)� (0)(1)(�2� �)� (5� �)(0)(1) = 0

) (�2� �)(1� �)(5� �) = 0

) (�2 + 2�� + �2)(5� �) = 0

) (�2+ �� 2)(5� �) = 0

) �1 = 1 , �2 = �2 , �3 = 5

Afo‘ br†kame tic IdiotimËc, sth sunËqeia ja Ëprepe na bro‘me kai to ant–stoiqoIdiodiànusma gia thn kàje Idiotim†. 'Omwc, s‘mfwna me thn ekf∏nhsh tou pro-bl†matoc, emàc mac endiafËrei mÏno h l‘sh h opo–a mhden–zetai gia pol‘ megàla t.L‘sh h opo–a mhden–zetai kaj∏c to t te–nei sto àpeiro mac d–noun mÏno arnhtikËcIdiotimËc, sthn per–ptws† mac mÏno m–a, h �2 = �2.

Sunep∏c, gia �2 = �2 l‘noume thn ex–swsh (A� �2I)�!u2 =

�!0 )

)

2

4�2� (�2) 1 1

0 1� (�2) 1

0 0 5� (�2)

3

5

2

4u21

u22

u23

3

5=

2

40

0

0

3

5

)

2

40 1 1

0 3 1

0 0 7

3

5

2

4u21

u22

u23

3

5=

2

40

0

0

3

5



7u23 = 0 ) u23 = 0

3u22 + u23 = 0 ) 3u22 = 0 ) u22 = 0

u22 + u23 = 0 ) 0 = 0 ) u21 ele‘jerh


�!u2 =

2

4C0

0

3

5 , 8 C 2 < † �!u2 =

2

41

0

0

3

5 , gia C = 1


�!x (t) = C2�!u2e

�2t

) �!x (t) = C2

2

41

0

0

3

5 e�2t

Parathro‘me Ïti h �!x (t) iso‘te me to diànusma

2

40

0

0

3

5 kaj∏c to t te–nei sto àpei-

ro...

JËma EE2013.7. Qrhsimopoi†ste seirËc FouriË gia na upolog–sete thn l‘shtou ex†c probl†matoc

ut

= uxx

, 0 < x < L, t > 0

u(0, t) = 0, ux

(L, t) = 0, t > 0

u(x, 0) = u0 0 < x < L.

49

Apànthsh EE2013.7. Sthn àskhsh aut† Ëqoume prÏblhma jermÏthtac memonodiàstath ex–swsh jermÏthtac me k = 1, 0 < x < L, t > 0.

To aristerÏ àkro br–sketai se stajer† jermokras–a [u(0, t) = 0], en∏ to dex–àkro e–nai monwmËno [u

x

(L, t) = 0].

H arqik† katanom† jermokras–ac e–nai u(x, 0) = f(x) = u0, stajer† se Ïlo tom†koc thc ràbdou.

Ja xekin†soume thn l‘sh mac basizÏmenoi sth mËjodo diaqwrismo‘ metablht∏nkai sth sunËqeia ja bro‘me idiotimËc kai idiodian‘smata.

ApÏ ton orismÏ Ëqoume:

u(x, t) = X(x)T (t)

ut

(x, t) = X(x)T 0(t)

uxx

(x, t) = X 00(x)T (t)

Sunep∏c,

ut

= uxx

) X(x)T 0(t) = X 00

(x)T (t)

) T 0(t)

T (t)=

X 00(x)

X(x)

Parathro‘me Ïti to aristerÏ mËroc thc ex–swshc den exartàtai apÏ to x, en∏to dex– mËroc den exartàtai apÏ to t. 'Ara ta d‘o mËrh prËpei na iso‘ntai me m–astajerà, Ëstw thn stajerà �� (Ïpwc lËei kai o orismÏc). OpÏte:

) T 0(t)

T (t)= �� =

X 00(x)

X(x)

'Etsi prok‘ptoun oi parakàtw d‘o exis∏seic tic opo–ec kalo‘mste na l‘soumexeqwristà:

(1) T 0(t) + �T (t) = 0

(2) X 00(x) + �X(x) = 0

Pr∏ta Ïmwc prËpei na asqolhjo‘me me tic sunoriakËc sunj†kec diÏti ja macqrhsimËuseoun sth l‘sh thc ex–swshc (2). Sunep∏c, apÏ thn pr∏th sunoriak†sunj†kh Ëqoume u(0, t) = 0 ) X(0)T (t) = 0.

Me bàsh to gegonÏc Ïti anazhto‘me mh-tetrimmËnec l‘seic, upojËtoume Ïti h T (t)


den e–nai tautotikà mhdËn. 'Ara o mÏnoc trÏpoc gia na bgei to ginÏmeno –so memhdËn e–nai h X(0) = 0. 'Etsi katal†goume sto Ïti h X(0) = 0 apotele– thnpr∏th sunoriak† sunj†kh gia thn ex–swsh (2).

Suneq–zontac sthn deuterh sunoriak† sunj†kh Ëqoume ux

(L, t) = 0 ) X 0(L)T (t) =

0. Me thn –dia logik† katal†goume sto Ïti h X 0(L) = 0 apotele– thn de‘terh

sunoriak† sunj†kh gia thn ex–swsh (2).

Proqwr∏ntac stic l‘seic twn exis∏sewn, h ex–swsh (1) apotele– mia 1hc tàxhcgrammik† S.D.E. h opo–a mpore– e‘kola na luje– jËtontac kai antikajistÏntacsthn ex–swsh T (t) = ert kai T 0

(t) = rert

(1-l‘sh) Tn

(t) = e��

n

t

H ex–swsh (2) apotele– mia 2hc tàxhc grammik† S.D.E. h opo–a mpore– ki au-t† e‘kola na luje– jËtontac kai antikajistÏntac sthn ex–swsh X(x) = erx kaiX 00

(x) = r2erx. OpÏte:

An � > 0:

Metà thn antikatàstash katal†goume sto gegonÏc Ïti r1,2 = ±ip�, opÏte (kai

s‘mfwna me thn anàlush idiotim∏n tou kefala–ou 4.1.2 twn shmei∏sewn) Ëqoumegenik† l‘sh:

X(x) = C1cos(p�x) + C2sin(

p�x)

ApÏ sunoriak† sunj†kh X(0) = 0 ) C1cos(0) + C2sin(0) = 0 ) C1 = 0.

'Ara X(x) = C2sin(p�x)

Paragwg–zoume th l‘sh m–a forà kai ËqoumeX 0(x) = [C2sin(

p�x)]0 = C2

p�cos(

p�x)

ApÏ sunoriak† sunj†khX 0(L) = 0 ) C2

p�cos(L

p�) = 0 ) C2cos(L

p�) =

0

OpÏte katal†goume sto gegonÏc Ïti C2 = 0 † cos(Lp�) = 0 . S‘mfwna me ton

parapànw sullogismÏ gia tic mh-tetrimmËnec l‘seic, jËloume to C2 6= 0, opÏtesunep∏c to cos(L

p�) prËpei na iso‘tai me mhdËn. DedomËnou Ïti to cos(�) Ëqei

r–zec gia � =

(2n�1)⇡2 gia n > 0 2 Z sunepàgetai Ïti sthn per–ptws† mac prËpei

to Ïrisma tou sunhmitÏnou Lp� na iso‘tai me (2n�1)⇡

2 . 'Ara:

51

Lp� =

(2n� 1)⇡

2

)p� =

(2n� 1)⇡

2L

) � = (

(2n� 1)⇡

2L)

2

) �n

= (

(n� 12 )⇡

L)

2 oi idiotimËc gia n > 0.

En∏ ta ant–stoiqa idiodian‘smata Xn

(x) e–nai ta:

(2-l‘sh) Xn

(x) = sin((n� 1

2 )⇡

Lx), n > 0

Oi jemeli∏deic l‘seic mac e–nai oi

un

(x, t) = Xn

(x)Tn

(t) = sin((n� 1

2 )⇡

Lx)e�

(n� 12)2⇡

2

L

2 t

TËloc, mac mËnei na anapt‘xoume tic l‘seic mac se Seirà Fourier. LÏgw touÏti ta idiodian‘smata X

n

(x) pou br†kame parapànw Ëqoun hm–tona ja qrhsimo-poi†soume peritt† Seirà Fourier († alli∏c Seirà HmitÏnwn). OpÏte, qrhsimo-poi∏ntac to je∏rhma thc upËrjeshc, h l‘sh mac gràfetai:

u(x, t) =

1X

n=1

bn

un

(x, t) =

1X

n=1

bn

sin((n� 1

2 )⇡

Lx)e�

(n� 12)2⇡

2

L

2 t

'Opou bn

=

2L

RL

0 u0sin((n� 1

2 )⇡L

x)dx.

An � = 0:

H ex–swsh (2) katal†gei sthn X 00(x) = 0 kai sunep∏c h genik† l‘sh thc e–nai h

X(x) = C1x+ C2.

ApÏ thn pr∏th sunoriak† sunj†kh X(0) = 0 sunepàgetai Ïti C2 = 0. SthsunËqeia br–skoume thn paràgwgo thc X(x), X 0

(x) = [C1x]0= C1. Dokimàzoume

kai thn 2h sunoriak† sunj†kh X 0(L) = 0 kai sunepàgetai Ïti C1 = 0. 'Ara h

l‘sh pou pa–rnoume e–nai mhdenik†, àra h � = 0 den apotele– idiotim† thc ex–swshc


(2).

An � < 0:

JËtoume kai antikajisto‘me sthn ex–swsh (2) thnX(x) = erx kaiX 00(x) = r2erx,

opÏte:

X 00(x) + �X(x) = 0

) r2erx + �erx = 0

) r2 + � = 0

) r2 = ��

) r1,2 = ±p�� (stamatàme ed∏, afo‘ o � � e–nai jetikÏc arijmÏc)

OpÏte, genik† l‘sh:

X(x) = C1ep��x

+ C2e�p��x

MËsw thc sqËshc twn ekjetik∏n tou euler kai twn uperbolik∏n hmitÏnwn kaisunhmitÏnwn Ëqoume

ex = cosh(x) + sinh(x)

e�x

= cosh(x)� sinh(x)

'Ara

C1ep��x

= C1cosh(p��x) + C1sinh(

p��x)

C2e�p��x

= C2cosh(p��x)� C2sinh(

p��x)

OpÏte, h genik† l‘sh se pio praktik† morf† g–netai

X(x) = (C1 + C2)cosh(p��x) + (C1 � C2)sinh(

p��x)

= Acosh(p��x) +Bsinh(

p��x)

'Eqontac up Ïyin to gegonÏc Ïti cosh(0) = 1 kai sinh(0) = 0 Ïpwc kai Ïti[cosh(x)]0 = sinh(x) kai [sinh(x)]0 = cosh(x) epexergazÏmaste tic sunoriakËcsunj†kec:

53

X(0) = 0 ) Acosh(0) +Bsinh(0) = 0 ) A = 0

X 0(x) = [Bsinh(

p��x)]0 = (

p��)(Bcosh(

p��x))

X 0(L) = 0 ) (

p��)(Bcosh(

p��L)) = 0

) Bcosh(p��L) = 0

OpÏte katal†goume sto gegonÏc Ïti B = 0 † cosh(p��L) = 0. S‘mfwna

me ton parapànw sullogismÏ gia tic mh-tetrimmËnec l‘seic, jËloume to B 6= 0,opÏte sunep∏c to cosh(

p��L) prËpei na iso‘tai me mhdËn. DedomËnou Ïmwc Ïti

to cosh(�) den kànei potË mhdËn ex orismo‘, tÏte o‘te to cosh(p��L) mpore–

na kànei potË mhdËn. 'Ara h l‘sh pou pa–rnoume e–nai mhdenik†, àra arnhtikËcidiotimËc den apotelo‘n l‘seic thc ex–swshc (2).

JËma EE2013.9. D–nontai d‘o dexamenËc h pr∏th apo tic opo–ec periËqei 200l–tra krasio‘ kai h de‘terh 100 l–tra nero‘. UpojËste Ïti rËei nerÏ apo thnde‘terh dexamen† sthn pr∏th me rujmÏ 1 l–tro to leptÏ kai Ïti nerwmËno kras–rËei apo thn pr∏th dexamen† sthn de‘terh me ton –dio rujmÏ. UpojËtontac Ïti hme–xh nero‘ kai krasio‘ g–nete stigmia–a(a') Diatup∏ste to majhmatikÏ montËlo tou parapànw probl†matoc. (b') Upolo-g–ste thn posÏthta x(t) krasio‘ pou br–sketai sthn pr∏th dexamen† thn qronik†stigm† t > 0.(g') Upolog–ste thn posÏthta y(t) krasio‘ pou br–sketai sthn de‘terh dexamen†thn qronik† stigm† t > 0.(d') Upolog–ste thn mËgisth posÏthta krasio‘ pou mpore– na periËqei h de‘terhdexamen†

Apànthsh EE2013.9. Sto prÏblhmà mac Ëqoume d‘o dexamenËc. Ac ono-màsoume x thn dexamen† A kai y thn dexamen† B.

(a') - To majhmatikÏ montËlo tou probl†matoc:

x(t) e–nai ta l–tra pou ugro‘ pou periËqei thn qronik† stigm† t h dexamen† A

y(t) e–nai ta l–tra pou ugro‘ pou periËqei thn qronik† stigm† t h dexamen† B

x(0) = 200 l–tra kras– Ëqei arqikà h dexamen† A

y(0) = 100 l–tra nerÏ Ëqei arqikà h dexamen† B

ApÏ thn dexamen† x ekrËei 1 l–tro apÏ ta 200 anà leptÏ, àra o rujmÏc metabol†c


tou ugro‘ thc dexamen†c ja e–nai �1/200 tou ugro‘, dhlad†:

x0(t) =

�1

200

x(t)

ApÏ thn dexamen† y ekrËei 1 l–tro apÏ ta 100 anà leptÏ, allà paràllhla eisËr-qetai kai 1 l–tro apÏ ta 200 thc dexamen†c x. 'Ara o rujmÏc metabol†c tou ugro‘thc dexamen†c y ja e–nai �1/100 tou ugro‘ apÏ aut†n sun 1/200 tou ugro‘ apÏthn x, dhlad†:

y0(t) =�1

100

y(t) +1

200

x(t)

(b') - UpologismÏc thc x(t):

ApÏ ton rujmÏ metabol†c xËroume Ïti x0=

�1200x ) x0

+

1200x = 0. 'Ara Ëqoume

m–a 1hc tàxhc grammik† S.D.E..

JËtoume loipÏn x(t) = ert ) x0(t) = rert kai antikajisto‘me:

rert +1

200

ert = 0

) r +1

200

= 0

) r =

�1

200

OpÏte, Genik† L‘sh thc x(t):

x(t) = C1ert

= C1e�1200 t

ApÏ arqik† sunj†kh Ëqoume x(0) = 200 ) C1e0= 200 ) C1 = 200

OpÏte, SugkekrimËnh L‘sh thc x(t) gia C1 = 200:

x(t) = 200e�1200 t

(g') - UpologismÏc thc y(t):

55

ApÏ ton rujmÏ metabol†c thc y(t)xËroume Ïti:

y0(t) =�1

100

y(t) +1

200

x(t)

) y0(t) +1

100

y(t) =1

200

x(t)

XËroume thn l‘sh thc x(t), àra thn antikajisto‘me

) y0(t) +1

100

y(t) =1

200

200e�1200 t

) y0(t) +1

100

y(t) = e�1200 t

kai parathro‘me Ïti Ëqoume m–a 1hc tàxhc, mh-omogen†, grammik† S.D.E. thn o-po–a mporo‘me pol‘ Ëukola na l‘soume me oloklhrwtiko‘c paràgontec.

y0 + p(t)y = f(t)

p(t) = 1100

f(t) = e�1200 t

r(t) = eRp(t)dt

= e1

100

Rdt

= e1

100 t


y(t) = e�1

100 t[

Ze

1100 te

�1200 tdt+ C]

) y(t) = e�1

100 t[

Ze

1200 tdt+ C]

) y(t) = Ce�1

100 t+ 200e�

1200 t

ApÏ arqik† sunj†kh Ëqoume y(0) = 100 ) Ce0+200e0 = 100 ) C = �100

'Ara, SugkekrimËnh L‘sh thc y(t) gia C = �100:

y(t) = 200e�1

200 t � 100e�1

100 t


(d') - UpologismÏc mËgisthc posÏthtac sthn dexamen† y:

Gia na bro‘me thn mËgisth posÏthta sthn dexamen† y prËpei na paragwg–soumethn y(t) kai na bro‘me tic r–zec thc parag∏gou thc. OpÏte:

y0(t) = [200e�1

200 t � 100e�1

100 t]

0

= e�1

100 t � e�1

200 t

R–zec:

y0(t) = 0

) e�1

100 t � e�1

200 t= 0

) e�1

100 t= e�

1200 t (logarijm–zoume aristerà kai dexià afo‘ t > 0 )

) ln(e�1

100 t) = ln(e�

1200 t

)

) � 1

100

t = � 1

200

t

) t =1

2

t

) t� 1

2

t = 0

) t(1� 1

2

) = 0

) t = 0

'Ara h mËgisth posÏthta sth dexamen† y ja e–nai:

y(0) = 200e0 � 100e0 = 100 l–tra kras–

57

JËma TE2012.1. Qrhsimopoie–ste metasqhmatismo‘c Laplace gia na upolo-g–sete thn l‘sh tou parakàtw probl†matoc arqik∏n tim∏n.

x00= �6x+ 4y, , y0 = 5x, x(0) = 6, x0

(0) = �6, y(0) = 6.

Apànthsh TE2012.1. An metatrËyoume thn 1h ex–swsh me Laplace Ëqoume:

s2X(s)� 6s+ 6 + 6X(s) = 4Y (s)

Kànoume to –dio kai gia thn 2h:

sY (s)� 6 = 5X(s) ) X(s) =sY (s)� 6

5

Kànontac antikatàstash Ëqoume:

s3Y (s)� 6s2

5

� 6s+ 6 +

6sY (s)� 36

5

= 4Y (s) )

s3Y (s) + 6sY (s)� 20Y (s) = 6s2 + 30s� 30 + 36 )

Y (s)(s3 + 6s� 20) = 6s2 + 30s+ 6 )

Y (s) =6s2 + 30s+ 6

(s3 + 6s� 20)

Ja kànoume anàlush merik∏n klasmàtwn:

6s2 + 30s+ 6

(s� 2)(s2 + 2s+ 10)

=

A

(s� 2)

+

Bs+ �

(s2 + 2s+ 10)

)

6s2 + 30s+ 6 = A(s2 + 2s+ 10) + (Bs+ �)(s� 2) =

= As2 + 2As+ 10A+Bs2 � 2Bs+ �s� 2� )6s2 + 30s+ 6 = s2(A+B) + s(2A� 2B + �) + (10A� 2�) )

8><

>:

A+B = 6

2A� 2B + � = 30

10A� 2� = 6

)

8><

>:

A = 5

B = 1

� = 22

'Ara Ëqoume:

6s2 + 30s+ 6

(s� 2)(s2 + 2s+ 10)

=

5

(s� 2)

+

s+ 22

(s2 + 2s+ 10)

=


=

5

(s� 2)

+

s+ 1

((s+ 1)

2+ 9)

+

21

((s+ 1)

2+ 9)

L�1= { 5

(s� 2)

+

s+ 1

((s+ 1)

2+ 9)

+

21

((s+ 1)

2+ 9)

)}

y(t) = 5e2t + e�tcos3t+ 7e�tsin3t

ApÏ thn ex–swsh y0 = 5x mporo‘me na upolog–soume kai to x(t).

x =

y0

5

=

10e�t

(e3t � sin3t+ 2cos3t)

5

)

x(t) = 2e2t � 2e�tsin3t+ 4e�tcos3t

JËma TE2012.2. D∏ste thn l‘sh tou parakàtw sust†matoc.

~x0=

0 1

�1 0

�~x+

cos t� sin t

�, ~x(0) =

0

0

�.

Apànthsh TE2012.2.

�!x 0(t) +

�0 �1

1 0

��!x (t) =

cos(t)�sin(t)

�,�!x (0) =

0

0

�

Br–skoume tic idiotimËc tou P :

det(P � �I) = 0

) det

�� 1

1 ��

�= 0

(��)(��)� (�1) = 0 )

�2= �1 ) � = ±i

Br–skoume to idiodiànusma gia l=iL‘noume thn ex–swsh (P � �1I)

�!u1 =

�!0 )

(

0 �1

1 0

��i 0

0 i

�)

u11

u12

�=

0

0

�

59

�i �1

1 �i

� u11

u12

�=

0

0

�

(�iu11 � u12 = 0

u11 � iu12 = 0

)(iu11 + u12 = 0

�u11 + iu12 = 0

) �!u =

i1

�

Br–skoume to idiodiànusma gia l=�i

L‘noume thn ex–swsh (P � �1I)�!u1 =

�!0 )

(

0 �1

1 0

��i 0

0 �i

�)

u21

u22

�=

0

0

�

i �1

1 i

� u21

u22

�=

0

0

�

(�iu21 + u22 = 0

�u21 � iu22 = 0

) �!u =

�i1

�

etP = EetDE�1=

i �i1 1

� eit 0

0 e�it

� i �i1 1

��1

=

1

2i

i �i1 1

� eit 0

0 e�it

� 1 i�1 i

�=

1

2i

ieit �ie�it

eit e�it

� 1 i�1 i

�=

1

2i

ieit + ie�it i2eit � (i)2e�it

eit � e�it ieit + ie�it

�=

"ie

it+ie

�it

2ii

2e

it�(i)2e�it

2ie

it�e

�it

2iie

it+ie

�it

2i

#=

cos(t) sin(t)�sin(t) cos(t)

�

'Ara Ëqoume:

etP =

cos(t) sin(t)�sin(t) cos(t)

�


kai

e�tP

=

cos(�t) sin(�t)�sin(�t) cos(�t)

�=

cos(t) �sin(t)sin(t) cos(t)

�

T∏ra upolog–soume to: Zt

0esP

��!f(s)ds =

Zt

0

cos(s) sin(s)�sin(s) cos(s)

� cos(s)�sin(s)

�ds =

Zt

0

cos2(s)� sin2

(s)�cos(s)sin(s)� cos(s)sin(s)

�=

sin(t)cos(t)�sin2

(t)

�

H l‘sh mac mpore– na grafje– sthn ex†c morf†:

��!x(t) = e�tP

Zt

0esP

��!f(s)ds+ e�tP

�!b =

cos(t) �sin(t)sin(t) cos(t)

� sin(t)cos(t)�sin2

(t)

�+ 0 =

cos2(t)sin(t) + sin3

(t)sin2

(t)cos(t)� sin2(t)cos(t)

�=

sin(t)(cos2(t) + sin2

(t))0

�=

sin(t)

0

�

JËma TE2012.3. Bre–te timËc twn paramËtrwn c0, c1, c2, c3 tËtoiec s∏ste hsunàrthsh y(x) = c0 + c1x + c2x

2+ c3x

3+ . . . na e–nai l‘sh tou parakàtw

probl†matoc arqik∏n tim∏n.

y00 + (x+ 1)y = 0, y(0) = 2, y0(0) = 3.

Apànthsh TE2012.3. 'Eqoume:

y(x) = C0 + C1x+ C2x2+ C3x

3

y0(x) = 0 + C1 + 2C2x+ 3C3x2

y00(x) = 0 + 0 + 2C2 + 6C3x

y000(x) = 0 + 0 + 0 + 6C3

ApÏ tic arqikËc sunj†kec mpÏrw na upolog–sw tic stajerËc.

y(x) = C0 + C1x+ C2x2+ C3x

3 ) y(0) = 2 ) C0 = 2

y0(x) = 0 + C1 + 2C2x+ 3C3x2 ) y0(0) = 3 ) C1 = 3

61

An kànoume antikatàstash tic arqikËc sunj†kec sthn exisws† mac:

y00(x) + (x+ 1)y(x) = 0 ) y00(0) + (0 + 1)y(0) = 0 ) y00(0) = �2

y00(0) = 0 + 0 + 2C2 + 0 ) �2 = 2C2 ) C2 = �1

An paragwg–sw thn ex–swsh:

y00(x) + (x+ 1)y(x) = 0 ) y000(x) + (x+ 1)y0(x) + y(x) = 0

) y000(0) + (0 + 1)y0(0) + y(0) = 0 ) y000(0) = �5

y000(0) = 0 + 0 + 0 + 6C3 = �5 ) C3 = �5

6

y(x) = 2 + 3x� x2 � 5

6

x3

JËma TE2012.4. D∏ste to limt!1 x(t) ean dx

dt

= x3 � 6x2+ 11x� 6.

Apànthsh TE2012.4. Pr∏ta ja l‘soume to dex– polu∏numo x3 � 6x2+

11x� 6. EfarmÏzontac sq†ma Horner:

2

41 �6 11 �6 1

1 �5 6

1 �5 6 0

3

5

'Ara blËpoume pwc to polu∏numo mac pa–rnei thc morf†:

(x� 1)(x2 � 5x+ 6)

An l‘soume thn ex–swsh 2ou bajmo‘ ja Ëqoume:

� = �2 � 4↵� ) � = 1

x2,3 =

�� ±p�

2↵=

5± 1

2

x2 = 2

kaix3 = 3


'Ara se telik† morf†:(x� 1)(x� 2)(x� 3)

'Opwc blËpoume h ex–swsh mac mhden–zetai sta shme–a 1,2,3 kai Ëqei thn ex†cgrafik† anaparàstash:

Mporo‘me na do‘me Ïti Ïtan –squei 1 < x(0) < 3 to lim

t!1 = 2 oi upÏloipectimËc apeir–zontai.

JËma TE2012.6a. Upolog–ste Ïlec tic l‘seic tou probl†matoc: y00 � y0 �12y = 20e2t, y(0) = 1, y0(0) = �1.

Apànthsh TE2012.6a. L‘nw pr∏ta thn omogen† ex–swsh:

y00 � y0 � 12y = 0

JËtw y = ery ) y0 = rery. Antikajist∏ sthn ex–swsh mou kai Ëqw:

r2 � r � 12 = 0


r1,2 =

�� ±p�

2↵=

1± 7

2

oi r–zec mac e–nai r1 = 4 kai r2 = �3

63


yc

= C1e4t+ C2e

�3t

Qrhsimopoi∏ tic arqikËc sunj†kec gia na upolog–sw ta C1 kai C2.

y(0) = 1 ) 1 = C1 + C2 ) C1 = 1� C2

y0(0) = �1 ) �1 = 4C1�3C2 ) �1 = 4(1�C2)�3C2 ) �1 = 4�4C2�3C2 ) C2 =

5

7

) C1 =

2

7

H genikeumËnh l‘sh thc ex–swshc mou e–nai:

yc

=

2

7

e4t +5

7

e�3t

Gia to dex– mh omogenËc mËloc ja kànoume manteyià tou t‘pou: y = De2t ) y0 =2De2t ) y00 = 4De2t. Kànw thn antikatàstash sthn arqik† gia na brw to Dkai Ëqw:

4De2t � 2De2t � 12De2t = 20e2t ) �10De2t = 20e2t ) D = �2


yp

= �2e2t

XËrontac Ïti h l‘sh thc diaforik†c ex–swshc e–nai thc morf†c:

y = yc

+ yp

y =

2

7

e4t +5

7

e�3t � 2e2t

JËma TE2012.6b. Upolog–ste Ïlec tic l‘seic tou probl†matoc:t2y00 + 5ty0 + 4y = 0.

Apànthsh TE2012.6b. Parathro‘me pwc h exisws† mac e–nai t‘pouEuler.Seaut† thn per–ptwsh jËtoume y = tr ) y0 = rtr�1 ) y00 = r(r� 1)tr�2. Kànoumeantikatàstash sthn ex–swsh mac kai Ëqoume:

t2r(r � 1)tr�2+ 5trtr�1

+ 4tr = 0 ) r(r � 1)tr + 5rtr + 4tr = 0

Diair∏ me tr

r(r � 1) + 5r + 4 = 0 ) r2 � r + 5r + 4 = 0 ) r2 + 4r + 4 = 0


r1,2 = �4

2

) r1,2 = �2



y = C1t�2

+ C2t�2ln(t)

JËma TE2012.7. 'Estw Ïti Ëqoume mia lept† metallik† ràbdo m†kouc ⇡ kaijermik†c stajeràc k =

14 h opo–a e–nai monwmËnh panto‘ ektÏc apÏ to aristerÏ

àkro thc Ïpou diathro‘me thn jermokras–a stajer† kai –sh me 10 bajmo‘c. 'Estwep–shc Ïti h arqik† katanom† thc jermokras–ac se kàje shme–o thc ràbdou e–naikatà 10 bajmo‘c megal‘terh tou hm–tonou twn 3/2 thc apÏstas†c tou apo toaristerÏ àkro thc ràbdou.(↵0

) Diatup∏ste to prÏblhma merik∏n diaforik∏n exis∏sewn pou antistoiqe–sto parapànw fusikÏ prÏblhma.(�0

) D∏ste thn tim† thc jermokras–ac se opoiod†pote shme–o thc ràbdou thnopoiad†pote qronik† stigm† (sto mËllon).

Apànthsh TE2012.7.

ut

=

14uxx

, 0 < x < ⇡, t > 0,

u(0, t) = 10, t > 0,

ux

(⇡, t) = 0, t > 0,

u(x, 0) = sin(3x/2) + 10, 0 < x < ⇡.

Gia ton upologismÏ thc l‘shc mac ja qreiasto‘me ta ↵0,↵n

,�n

.

↵0 =

1

L

ZL

�L

f(x)dx =

1

⇡

Z⇡

�⇡

sin(3

2

x) + 10dx =

1

⇡

Z⇡

�⇡

sin(3

2

x)dx+

10

⇡

Z⇡

�⇡

dx =

1

⇡

1

3/2[�cos(

3

2

x)]L�L

=

2

3⇡[�cos(

3

2

⇡) + cos(�3

2

⇡)] )

↵0 = 0

↵n

=

1

⇡

Z⇡

�⇡

(sin(3

2

x) + 10)cos(nx)dx =

1

⇡

Z⇡

�⇡

sin(3

2

xcos(nx))dx+

10

⇡

Z⇡

�⇡

cos(nx)dx =

65

↵n

= 0

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