diferansiyel denklemler - yaşar Üniversitesi ders notları

8/10/2019 Diferansiyel Denklemler - Yaar niversitesi Ders Notlar

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Dnahrfksnyhc

Dhkoch`chrUrma.Dr.fjfkHZHK^fsfr knvhrsnthsn Ahk-Hdhjnyft

Afocthsn

Jc` 6

6.6. Gnrn

6.:. Dnahrfksnyhc dhkoch` jnn`chrn

6.2. tnpn


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:

OF^KFOCFZ

Fydk, @., Gkdz, G., Ouryhc, J. (6>>:). Hch`hktfry DnaahrhktnfcHquftnmks fkd Jmukdfry Qfcuh Urmjch`s. Anatl hdntnmk. ImlkWnchy & Rmks, Nkb.

Blnrgwnk, J.L., Ucu`ptmk, B. (6>14). F bmursh ma @ftlh`ftnbs

amr Hkgnkhhrs fkd Rbnhktnsts. Uhrgf`mk Urhss. Qmcu`h :. Blnrgwnk, J.L., Ucu`ptmk, B. (6>14). F bmursh ma @ftlh`ftnbs

amr Hkgnkhhrs fkd Rbnhktnsts. Uhrgf`mk Urhss. Qmcu`h 8.


3/199

2

OF^KFOCFZ

Hr, X. (6>14). Dnaahrhktnfc Hquftnmks amrHkgnkhhrs fkd Rbnhktnsts. Tlh Hkgcnsl Xknvhrsntnhs Urhss Ctd.

Zfnkvncch, H.D., Jhdnhkt, U.H. (6>). Hch`hktfry DnaahrhktnfcHquftnmks. Rhvhktl hdntnmk. @fb`nccfk Uujcnslnkg Bm`pfky.

Rpnhghc, @.Z. (6>18). Tlhmry fkd Urmjch`s ma CfpcfbhTrfksamr`s. Rblfu` Uujcnslnkg Bm`pfky.


4/199

4

Jc` 6 Dnahrfksnyhc Dhkoch`chr

6.6. Gnrn

Ju jc`dh dnahrfksnyhc dhkoch` ofvrf`focfkfbfo vh omkuyf foco ghtnr`ho f`fbycfjncnkhk jnr rkho hch fckfbfo vh ju rkho yfrd`ycfdnahrfksnyhc dhkoch` mcuturuc`fs sfcfkfbfotr.


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1


rkho9 6.6.

Otchsn ` mcfk jnr bnsn` yhrhon`n htonsn fctkdf

shrjhst d`h yfp`fotfdr. Rhrjhst mcfrfo dhkju otchknk zhrnkh lfvfkk dnrhkbn hton ht`hothdnr.Lfvfkk dnrhkbn dhk bns`nk lzkk ofrhsnych dmrumrfktcdr. Jnr t zf`fk dtkdh bns`nk (v) lzkvh dt `hsfahyn jucukuz.


7/199

7


honc 6.6.dhk dh grchbhn gnjn otchknk zhrnkh nonouvvht hton ht`hothdnr9

hrhon`n ouvvhtn 9 `g

Lfvfkk dnrhkbn 9 o`v:

Jurfdf o jnr sfjnt vh g yhrhon`n ouvvhtndnr.


8/199


Bnsn` yhrh dmru dtkdhk `g ; o`v:dnr.Dmcfysycf bns`h hton hdhk kht ouvvht,

A 0 g o`v: (6.:) dnr.

A 0 ` f (6.2)

mcduukdfk (`, otch vh f df nv`hyn jhcnrt`hothdnr).


10/199

65


(6.4)


11/199

66


(6.4)

vh dmcfysycf

(6.8)

hcdh hdncnr.


12/199

6:


Ju dhkoch`dh (hntcnoth) dnahrfksnyhc

oftsfys jucukduukdfk dhkoch` dnahrfksnyhc

dhkoch` mcfrfo jncnk`hothdnr. Ju dnahrfksnyhc

dhkoch`nk z` smkubukdf jnr t fkkdfon v

lzk hcdh ht`ho ``ok mcfbfotr.


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62


snk bns`nk t zf`fk ofdfr d`hsn smkubufckfk `hsfahyn jhcnrttnnkn vfrsfyfc`. Ju`hsfahyn juc`fo nnk

nafdhsn (6.8) hntcnnkdhyhrnkh omkursf ju hntcno,

(6.1) hocnkh dkr.


14/199

64


Jnr dnahrfksnyhc dhkoch`dh hk yosho dhrhbhdhk trhvy(k) nsh dhkoch`h knkbn dhrhbhdhk dnahrfksnyhc dhkoch`dhknr. rkhnk, (6.8) hntcnnkdh hk yosho trhv

jnrnkbn dhrhbhdhk mcduukdfk ju hntcnh jnrnkbndhrhbhdhk dnahrfksnyhc dhkoch` vh (6.1) hntcnnkdh hkyosho dhrhbhdhk trhv

nonkbn dhrhbhdhk mcduukdfk ju hntcnh nonkbndhrhbhdhk dnahrfksnyhc dhkoch` dhknr.


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68


Qhrnchk rkhoth jf`c dhnohkchr ymc s,s 0 a(t)

vh lz v,

v 0 g (t)sfdhbh tho jnr t jf`sz dhnohknknk amkosnymkcfrmcduukdfk

mrdnkfry dnahrfksnyhc oftsfycfrkdfk dmcfy (6.8) vh(6.1) dnahrfksnyhc dhkoch`chrn fdn (mrdnkfry) dnahrfksnyhcdhkoch`chr mcfrfo jncnk`hothdnr.


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61


Hhr z jf`c dhnohkn x vh y gnjn non jf`szdhnohknk amkosnymku nsh rkhnk,

z 0 a (x, y)

nsh, zknk x vh y dhnohkchrnkh grh trhvchrn fckrsf,

vh jhkzhr honcdh,

os`n dnahrfksnyhc oftsfycfr hcdh hdncnr. Dmcfysycf jutr oftsfycfr nhrhk dnahrfksnyhc dhkoch`chrh dh os`ndnahrfksnyhc dhkoch`chr dhknr.


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67


rkhnk,

(6.7)

vh(6.


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6)

dnahrfksnyhc dhkoch`nknk z`k hcdhhdnknz. f jnr sfjnt dhhrn nafdh ht`hothdnr.


22/199

::


nafdhsnknk xh grh nkthgrfcn fckrsf,


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:2



(6.65)

(F jnr sfjnt) hcdh hdncnr. Ju nafdhknk xh grh

nkthgrfcn fckrsf,


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:4



(6.65)

(F jnr sfjnt) hcdh hdncnr. Ju nafdhknk xh grh

nkthgrfcn fckrsf,(6.66)

(J jnr sfjnt) jucukur.


25/199

:8


Hcdh hdnchk ju nafdhknk xh grh jnr ohz dflfnkthgrfcn fckrsf,

(6.6:)

hcdh hdncnr. Ju nafdh vhrnchk (6.>) dnahrfksnyhcdhkoch`nknk z`dr vh ju z` F, J, B

gnjn sfjntchrn nhrdnnkdhk ghkhc z` mcfrfojncnknr.


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F, J, B sfjntchrnkn hcdh ht`ho nnk x 0 5 nohk

frtcfr (jfcfkg frtcfr) vhrncnrsh, ju dhhrchr (6.65),(6.66) vh (6.6:) kmcu hntcnochrdh yhrnkh omkursf,

(6.62)


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F, J, B sfjntchrnkn hcdh ht`ho nnk x 0 5 nohk

frtcfr (jfcfkg frtcfr) vhrncnrsh, ju dhhrchr (6.65),(6.66) vh (6.6:) kmcu hntcnochrdh yhrnkh omkursf,

(6.62)

(6.64)


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hntcnochrn hcdh hdncnr. Ju hntcnochrdhk

jucukur.


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25


hntcnochrn hcdh hdncnr. Ju hntcnochrdhk

jucukur.

Hcdh hdnchk ju dhhrchr (6.6:) kmcu hntcnoth yhrnkhomkursf,

hcdh hdncnr. Ju (6.>) kmcudnahrfksnyhc dhkoch`nk zhc

z`dr. zhc z` mcfrfojhcnrtnc`hsnknk khdhkn, jfcfkgomuccfr dhntnrncdnnkdh F, J, B dhhrchrnknk dh juomuccfrf grh dhn`hoth mc`fskdfk dmcfydr.


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rkho 6.8.

Os`n nkthgrfc ykth`n uygucfkrsf


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rkho 6.8.



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rkho 6.8.



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:

:

( 6)

( :)

( :)

( :)( ):

( 2)( ):

x x

x

x

x x

x

dyh x h Fx J

dx

h x Fx J

dy h x Fx J dx

xy x h h F Jx B

xy x h F Jx B


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6.4.Dhnohkchrnkh fyrcfjncnr dnahrfksnyhc dhkoch`tnpn9


45/199

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6.4.Dhnohkchrnkh fyrcfjncnrdnahrfksnyhc dhkoch` tnpn9

(6.67)

a(y) sfdhbh yknk, g(x) nsh sfdhbh xnk jnramkosnymkudur.


46/199

4


6.4.Dhnohkchrnkh fyrcfjncnrdnahrfksnyhc dhkoch` tnpn9

(6.67)

a(y) sfdhbh yknk, g(x) nsh sfdhbh xnk jnramkosnymkudur. ^uofrdfon nafdh

a (y)dy 0 g (x)dx (6.6


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8


(6.6


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(6.6


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rkho 6.)

dnahrfksnyhc dhkoch`nknk zhc z`kx0 5,y0 5 frt nnk hcdh hdnknz.


50/199

) hntcnn dzhkchknrsh

hcdh hdncnr.


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>


Grcd gnjn ydhnohkn jnr tfrfatf, xdhnohkndnhr tfrfatf yhr fc`fotfdr. Lhr non tfrfak nkthgrfcnfckrsf,

(6.:5)

ghkhc z` hcdh hdncnr. Jurfdf ydnrho mcfrfo xnkjnr amkosnymku hocnkdh nafdh hdnc`h`hothdnr (y,xnk ofpfc (n`pcnbnt) jnr amkosnymkudur).


52/199

65


x0 5,y0 5 (6.:5) kmcu hntcnoth yhrnkh omkursf,-6 06 + Fhntcnnkdhk F0 -: hcdh hdncnr. Ju dhhr(6.:5)dh yhrnkh omkursf zhc z`,

(6.:6)mcfrfo jucukur.

:)6( :

6:

xh y


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66


x0 5,y0 5 (6.:5) kmcu hntcnoth yhrnkh omkursf,-6 06 + Fhntcnnkdhk F0 -: hcdh hdncnr. Ju dhhr(6.:5)dh yhrnkh omkursf zhc z`,

(6.:6)mcfrfo jucukur. (6.:6) kmcu nafdh dzhkchknrsh zhcz`,

hocnkdh hcdh hdncnr.

:)6( :

6:

xh y


54/199

6:


Dhnohkchrnkh fyrcfjnchk dnahrfksnyhc dhkoch`trkh ncnonk rkhochr ffdf sukuc`utur.

rkho9 6.>.

dnahrfksnyhc dhkoch`nknk x 0 6, y 0 5frt nnk zhcz`k hcdh hdnknz.


55/199

62


hntcnnkn kbh dhnohkchrnkh fyrcfjnchk tr lfcnkhdktrhcn`.


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68


Lhr non tfrfak nkthgrfcn fckrsf,


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:6


mcduukdfk,


64/199

::


mcduukdfk,


65/199

:2


Fxy

y

uu

du

y

y

duxdxxdxduxu

)6ck(:

6

6

6ck

:

6

ck:

6

:

6

6

6ck

:

6

:,:,6

:

:


66/199

:4


x0 6 nnk y0 5 omucu ouccfkcrsf


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hcdh hdncnr.


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Ju dhhr ghkhc z`dh yhrnkh omkursf,


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zhc z`hcdh hdncnr.


78/199

Dnahrfksnyhc

Dhkoch`chrUrma.Dr.fjfkHZHK^fsfr knvhrsnthsn Ahk-HdhjnyftAfocthsn

Jc` 6

6.8.Lm`mihk Hntcnochr9


79/199

:



tnpnkdhon jnrnkbn dhrhbhdhk dnahrfksnyhc

dhkoch`chr Lm`mihk Hntcnochr mcfrfoskacfkdrc`tr.


80/199

2



tnpnkdhon jnrnkbn dhrhbhdhk dnahrfksnyhc

dhkoch`chr Lm`mihk Hntcnochr mcfrfoskacfkdrc`tr.Dnahrfksnyhc dhkoch`nk z` nnk

y 0 vx (6.::)

dnyhcn`. Jurfdf yvh vdhnohkchrn xnk amkosnymkcfrdr.


81/199

4



tnpnkdhon jnrnkbn dhrhbhdhk dnahrfksnyhcdhkoch`chr Lm`mihk Hntcnochr mcfrfo

skacfkdrc`tr.Dnahrfksnyhc dhkoch`nk z` nnk

y 0 vx (6.::)

dnyhcn`. Jurfdf yvh vdhnohkchrn xnk amkosnymkcfrdr.Lhr non tfrfak xh grh trhvn fckrsf,

(6.:2)hcdh hdncnr.


82/199

8


(6.::) vh (6.:2) kmcu nafdhchr hntcnnkdh

yhrnkh omkursf(6.:4)

jucukur.


83/199

1




jucukur. (6.:4) kmcu nafdh dzhkchknrsh

(6.:8)

hcdh hdncnr.


84/199

7




jucukur. (6.:4) kmcu nafdh dzhkchknrsh

(6.:8)

hcdh hdncnr. Jukuk smkubu mcfrfo dhnohkchrnkh fyrcfkhntcnn hcdh hdncnr. Lhr non tfrfaknkthgrfcn fckr vh ncgncn dhnohkchryhrnkh omkursf vhrnchk lm`mihk

dnahrfksnyhc dhkoch`nk ghkhc z` jucukur.


85/199


rkho 6.6:.dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.

kbh ju dnahrfksnyhc dhkoch` trk jhcnrch`hyhfcfc`.


87/199

65


mcduukdfk ju lm`mihkdnahrfksnyhc trdr.


88/199

66


mcduukdfk ju lm`mihkdnahrfksnyhc trdr.

nafdhsnknk xh grh trhvn fckrsf(v, xnk jnr amkosnymkudur)

hcdh hdncnr.


89/199

6:


Ju hntcno vhrnchk dnahrfksnyhc dhkoch`dh yhrnkhomkursf,


90/199

62



Jc 6 Dna n c D oc c


91/199

64



(6.:1)

jucukur.


92/199


93/199

Jc 6 Dna n c D oc c


94/199

67


Ju nafdhyn dhnohkchrnkh fyrcfjnchk dnahrfksnyhcdhkoch` trkh dktr`hyh fcfc`.

(6.:7)

hntcnnknk lhr non tfrfakk nkthgrfcn fckrsf,(6.:)

hcdh hdncnr. (6.:>) kmcu hntcno vhrnchk dnahrfksnyhcdhkoch`nk ghkhc z`dr.

Jc 6 Dna n c D oc c


95/199

6


rkho 6.62.dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.

(6.25)

mcduukdfk, y 0 v x dnyhcn`.

nafdhsn (6.25) kmcu hntcnoth yhrnkh omkursf,



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107/199

25


ju smk hntcnoth yhrnkh omkursf,



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22


hcdh hdncnr. Ju nafdh vhrnchk dnahrfksnyhc dhkoch`nkghkhc z`dr.

Dnahrfksnyhc


111/199

Dnahrfksnyhc

Dhkoch`chrUrma.Dr.fjfk HZHK

Hgh knvhrsnthsn @lhkdnscno AfocthsnJncgnsfyfr @lhkdnscnn Jc`

Jc` 6

6.1.



112/199

:

y

6.1.

Qhrnchk dnahrfksnyhc dhkoch`dh bvhyf bsfjntchrnkdhkhk fz jnrnknk sar mc`fd vfrsfyc`fotfdr. Lhr nonsnknk

sar mcduu duru`df hntcno lm`mihk dnahrfksnyhc trkhdkr.

Ju dnahrfksnyhc dhkoch`nk z`kdh non duru`uk

gzkkh fck`fs ghrhonr.


113/199



114/199

4

y

honc9 6.:.U kmotfskk Mxy vh M\^ ommrdnkft hoshkchrnkh grh pmznsymku.



115/199

8

y

(6.26)

dnahrfksnyhc dhkoch`nkdh,x 0 l+\, y0o + ^omkursf dhkoch`



116/199

1

y

(6.26)


(6.2:)

hocnkh dkr.



117/199

7

y

(6.26)


(6.2:)

hocnkh dkr.



118/199

rkho 6.67.dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.


120/199



121/199

66

Ju hntcnochrdhkl0 : vho0 -6 dhhrchrn jucukur.

Qhrnchk dnahrfksnyhc dhkoch`dh,

x 0 \+lvhy 0 ^+o omkursf dnahrfksnyhc dhkoch`

(6.24)

lm`mihk dhkoch` trkh dkr.



122/199

6:

(6.24) kmcu hntcnoth,

omkursf



123/199

62

(6.24) kmcu hntcnoth,

omkursf (6.24) kmcu hntcno

hocnkh dkr vh jurfdfk,



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6

hcdh hdncnr.



130/199

:5

(6.28)

dhhrchrn (6.28) kmcu hntcnoth yhrnkh omkursf ghkhc

z`


131/199



132/199

::

rkho 6.6


133/199

:2

Hn`chrn hnt mc`fdkdfk pfy vh pfydfdfondmrucfrk ohsn` kmotfs(l, o) mcsuk.

l o 0 5

l


134/199

:4

Hn`chrn hnt mc`fdkdfk pfy vh pfydfdfondmrucfrk ohsn` kmotfs(l, o) mcsuk.

l o 0 5

l


135/199

:8

x 0 \ +l vh ^ 0 ^ +ohntcnochrn vhrnchk dnahrfksnyhc dhkoch`dh yhrnkhomkursf, dnahrfksnyhc dhkoch`n`nz

(6.21)




136/199

:1

x 0 \ +l vh ^ 0 ^ +ohntcnochrn vhrnchk dnahrfksnyhc dhkoch`dh yhrnkhomkursf, dnahrfksnyhc dhkoch`n`nz

(6.21)


^ 0 Q\

nafdhchrn (6.21)df yhrnkh omkursf,



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148/199

2

(6.27)jucukur.



150/199

45

\ 0 (x 6), ^ 0 (y 6) dhhrchrn (6.27)dh yhrnkh

omkursf



151/199

46

\ 0 (x 6), ^ 0 (y 6) dhhrchrn (6.27)dh yhrnkh

omkursf

ghkhc z` hcdh hdnc`n mcur.


o 6 6>


152/199

4:

rkho 6.6>.

dnahrfksnyhc dhkoch`nknkz`k hcdh hdnknz.


Ufy vh pfydfdfon dmrucfrk hn`n hnt dhncdnr


153/199

42

Ufy vh pfydfdfon dmrucfrk hn`n hnt dhncdnr.Dmcfysycf ju dmrucfr ohsnnr vh ohsn` kmotfskk(l, o) mcduuku vfrsfyfc`.

:l o 0 5l +:o 8 0 5

hntcnochrnkdhkl 0 6,o 0 :jucukur.Dmcfysycf\ 0 x 6vh^ 0 y :dnr.



154/199

44

x 0 \ + l vh y 0 ^ + o hntcnochrn vhrnchkdnahrfksnyhc dhkoch`dh yhrnkh omkursf dnahrfksnyhcdhkoch`n`nz,

(6.2


155/199

48

^ 0 Q\

hntcnochrn yfrd`ycf (6.2


156/199

41

^ 0 Q\

hntcnochrn yfrd`ycf (6.2


157/199


Jurfdfk


158/199

4

Jurfdfk,


Jurfdfk,


160/199

85

Jurfdfk,


Jurfdfk,


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,



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Dmrucfrk pfrfchc mc`fs duru`u.

dnahrfksnyhc dhkoch`nkdh pfyvh pfydfdfon dmrucfrk

hn`chrn hnt yfkn dmrucfrpfrfchc nsh dhkoch`nk z` nnk ffdfonfd`cfr uygucfkr.


Duru` NN.


170/199

15

Dmrucfrk pfrfchc mc`fs duru`u.

dnahrfksnyhc dhkoch`nkdh pfyvh pfydfdfon dmrucfrk

hn`chrn hnt yfkn dmrucfrpfrfchc nsh dhkoch`nk z` nnk ffdfonfd`cfr uygucfkr.

dhrsho


171/199


172/199


n cnn d cd dncn


173/199

12

hntcnnkdhk hcdh hdncnr.

hntcnn (6.2>)df yhrnkh omkursf


ntcnn d cd dncn


174/199

14

hntcnnkdhk hcdh hdncnr.

hntcnn (6.2>)df yhrnkh omkursf

(6.45)


175/199


vh ju hntcnnk dzhkchk`hsn smkubukdf


176/199

11

(6.46)

jucukur. Grcd gnjn vhrnchk dnahrfksnyhc

dhkoch` dhnohkchrnkh fyrc` mcduukdfk

hntcnnknk nkthgrfcn fckfrfo smkubf gndncnr.


rkho 6.:5.


177/199

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75

mcduukdfk dmrucfr pfrfchcdnr.

z 0 x :y dhrsho

Dmcfysycf,

mcur.


Ju hntcno vhrnchk dnahrfksnyhc dhkoch`dh yhrnkh

omkursf


181/199

76

omkursf


Ju hntcno vhrnchk dnahrfksnyhc dhkoch`dh yhrnkh

omkursf


182/199

7:

omkursf

vh ju hntcnnk dzhkchk`hsn smkubukdf

(6.4:)

hcdh hdncnr.


(6.4:) kmcu hntcno dhnohkchrnkh fyrcfjncnr duru`f

dzhkchkhjncnr


183/199

72

dzhkchkhjncnr.

Ju hntcnnk lhr non yfkkk nkthgrfcn fckrsf,

nafdhsnkdhk



184/199

74

hntcnn jucukur. Ju hntcnothz 0 x :yomkursf

(6.42)

ghkhc z` hcdh hdncnr.


rkho 6.:6.

dna n c d oc n n


185/199

78



Hntcnothk dh grcd gnjn


186/199

71

g g

mcduukdfk dmrucfr pfrfchcdnr.




187/199

77

g g

mcduukdfk dmrucfr pfrfchcdnr. Qhrnchk dhkoch`dh,




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