Sayısal Analiz (MATLAB) - Sakarya Üniversitesi Çözümlü Final Soruları

8/10/2019 Saysal Analiz (MATLAB) - Sakarya niversitesi zml Final Sorular

1/52

Uur Jrlcogu

XJ\LXJG JKJG] \J] ONUGU CKJG XLKJS XOPUGJPL- (2:/26/0225)Xoru :)

S0)2(Sh , J:)2(l :G

0)t(ln J P:1: ofa, G:12.5 F, H1: C

P01: ofa, G012.5 F, J5.2)2(l 0G

enlg :

j)

enlg :'de verlgek devreye lglnlk durua dekngeagerl jjdj verlgaltlr. Devreye lglnlk x(t) duruadelnekgerl ogjk )t(l),t(S :Gh ve )t(l 0G dellagerlkl dsogve noautuku nuggjkjrjn (sea`ogln

ortjadj) ugukuz.

0*

2

2H

:

l

l

S

0G

0P2

0G

:

22:G

:H

:

H

:

:P*H

:

l

l

S

dt

d

0G

:G

h

0G

:G

h

0*

2

2

:

l

l

S

020

220

:::

l

l

S

dt

d

0G

:G

h

0G

:G

h

`) ezpgotnoautuku yjrda lge t=^2 :2Q sjklye jrjgkdj ve yjk yjkj ogjhjn enlgde )t(l),t(S :Gh ve

)t(l 0G dellagerlkl lzdlrehen proirja sjtrgjrk yjzkz. (su`pgotnoautuku nuggjkk)

h)

enlg :'de verlgek devrede )t(ln dellalenlg 0'de isterlgdll il`l ogursj gslanoautuku nuggjkjrjn

)t(l),t(S :Gh ve )t(l 0G dellagerlkl `ugukuz ve^2 :2Q sjklye jrjgkdjlzdlrlklz .

enlg 0za

j);;^y x zQ1dsogve('Dy+y+x+z1-0','Dx-0*y12','Dz-0*y+0*z12','y(2)10',...

'x(2)1:','z(2)12.5) ('ekter')y 1-0-:/0*exp(-t)*(-3-:4/8*slk(8[(:/0)*t)*8[(:/0)) % vh(t)x 1

exp(-t)*(-8/0-9/3*slk(8[(:/0)*t)*8[(:/0)+9/0*hos(8[(:/0)*t)) %lG:(t)z 1:/0*exp(-t)*(-3+9/8*slk(8[(:/0)*t)*8[(:/0)+9*hos(8[(:/0)*t)) %lG0(t)

`);;su`pgot(8,:,:),ezpgot(y,^2 :2Q),su`pgot(8,:,0),ezpgot(x,^2 :2Q),...

su`pgot(8,:,8),ezpgot(z,^2 :2Q) ('ekter')

0 J

t(sk)

4 J

)t(ln

4*slkt

0.3:

G:P: H

P0

G0


2/52

Uur Jrlcogu

h) j 1 ^-: -: -:>0 2 2>0 2 -0Q>` 1 ^:>2>2Q>h1eye(8,8)> %uh jdet hlnls lstekdlil lhlkd 1 2>x2 1 ^0 : 2.5Q> % lgn nosuggjrt 1 2=2.22:=:2> % djfj fjssjs hlzla lhlk degtj t 12.2: yjplglyoru1okes(:,gekitf(t))> % t lge u(t) klk `oyutu eslt yjplglyor%pjrhjgl u(t) tjkltlglyorcor f1:=gekitf(t)>

lc t(f)72>u(f) 1 2>

ekdlc 271t(f) & t(f)710.3:>u(f)14*slk(t(f))>

ekdlc t(f);0.3:u(f)10>

ekdekd%pjrhjgl u(t) tjkltlal sokj erdlgsla(j,`,h,d,u,t,x2)>irld>


3/52

Uur Jrlcogu

Xoru 0)

enlg 8'de c: erlsl 4. derehedek `lr poglkoa ogup njtsjy ventr ^: 2.6 -3.:5 -0.:5 3.5Q englkdedlr. c0

erlsl lse y1 0x coknslyoku lge verlgaentedlr. Fer lnl erlyl ^-8 8Q jrjgkdj lzdlrek ve enlg :8.03'djisterlgek tjrjg jgjk fesjpgjyjk proirja yjzkz.

enlg 8

za

j1-8=2.2:=8>y1^: 2.6 -3.:5 -0.:5 3.5Q>c:1pogyvjg(y,j)>c01j.[0>pgot(j,c:,j,c0) % fer lnl eirl de hlzdlrlglyor%sogve('c:-c0') fesj`l yjplglyor ve lnl erlklknesla kontjgjr% ùgukuyor(seaògln ortjadj)

t1sogve('t[4+2.6*t[8-3.:5*t[0 -0.:5*t+3.5-t[012')% nonger seaògln njrjnterdek sjysjg deeredktrgyordeier1dou`ge(t)jgjk1qujd('t.[4+2.6*t.[8-3.:5*t.[0-0.:5*t+3.5-t.[0',deier(0),deier(8))

% d(8)st skr,d(0) jgt skr\unjrdj verlgek proirja sjtrgjrkk jgtrgajs sokukdj jjdjnl deerger egde edlglr=;;t 1-0.96-:.:5.c:1t.[0>y1^: 2.6 -3.:5 -0.:5 3.5Q>c01pogyvjg(y,t)

dc1dlcc(c0)./dlcc(t)> % òyutu (k-:) dlrxd1t(0=gekitf(t))> % òyutu (k-:) dlrhjrpla1dc(:=gekitf(dc)-:).*dc(0=gekitf(dc))> nrltln1xd(clkd(hjrpla72))% lnl erlklk hjrplakk ljret delstlrdlil

% yerger ùgukuyor\unjrdj verlgek proirjak jgtrgajs sokukdj egde edlgek deerger jjdj isterlgaltlr= ;;nrltln 1

-0.2222 -2.:922 :.5922

Xoru 4))t(@

)t(J

5t6

dt)8t5t8((

0t8t3

8t4))t8t8(dt

d(

0

0

0

80

\jkdj verlgek lcjdede J(t) ve @(t) poglkoagjrk

yjgkzhjseaògln jrj nutusunoautgjrnuggjkjrjn ùgjk AJTGJ@proirja yjzkz

zasyas tpjy:18*t[0-8*t>pjy::1dlcc(pjy:)>pjy014*t[8-8>pjydj:13*t[0-8*t+0>pjy81lkt(8*t[0+5*t-8)>pjydj016*t[0+5>neslr:1(pjy::+pjy0)/pjydj:>neslr01pjy8/pjydj0>neslr1neslr:+neslr0>^J @Q1kuadek(neslr)

\unjrdj verlgek proirja sjtrgjrkk uyiugjkajs sokukdj egde edlgek deerger jjdj verlgaltlr=;;J 16


5/52

Uur Jrlcogu

nonger(j)dlsp('`u non njtg')

egseekd

ekdekd

Xoru 3) @lr egentrln devreslkdek jnjk jnaj lglnlk etnlk jna deerl(l(t) jkl jna deerl ogajn zere)>

00wt

2:wt

0etnlk dwt)wt(l

0

:L

corag lge fesjpgjkr. Jjdj verlgek>

l(t)1 w/wt:22e*6:.


6/52

:9/25/0223Uur Jrlcogu

XJ AF.CJN. EGN.-EGK. [email protected]\LXJG JKJG]CKJG XLKJS XOPUGJPLXoru :)

enlg :enlg :'de verlgek devreye lglnlk ierlgla ve jna dekngeagerl jjdj isterlgaltlr (devreye lglnlk ta lgn nougdeergerl 2 jgkjhjntr). Devrede fer lnl jkjftjr jn duruadjdr ve t12 jkkdj fer lnl jkjftjr dj njpjtgajntjdr. @uduruadj egde edlgek devre eltglngerl jjdj verlgaltlr=

dttdlGPtlPtlE l )()()( 0000:: (:)

dt

tdlG

dt

tdlGPtlE

)()()( 88

00000 (0)

)()()( 08: tltltl (8)

j) \unjrdj verlgek 8 jdet dekngeal jyr jyr s doaeklke evlrlklz. )(:tl 'klk gjpgjhe trjkscorauku L:, )(0 tl 'klk

gjpgjhe trjkscorauku L0, )(8tl 'klk gjpgjhe trjkscorauku L8jgkz.

`) Jjdj verlgek AJTGJ@proirja, jnnkdj `ugukjk s doaekl jna eltglngerlkl nuggjkjrjn )(:tl ,

)(0 tl ve )(8 tl jna dekngeagerl, ^2 :2Qsjklye jrjgkdj lzdlrlgehentlr. Jjdj Hoaajkd Ylkdow

ortjakdj yunjrdj lstekek lgerlk yjpgd proirja isterlgaltlr. @u proirjadj irgek `ogungjrdogdurukuz. t doaeklkde egde edlgek jna dekngeagerlkde, kontjdjk sokrj ajnslaua 0 jdet sjy

ogjhjntr.;;syas L: L0 L8 s>;;H1sogve('','.',')>;;L:1L: 1

:2*(-0+s)/s/(9*s+0*s[0+0);;L01L0 1

02*(s+:)/s/(9*s+0*s[0+0);;L81L8 1

:2/s*(4+s)/(9*s+0*s[0+0);;L:_t1

L:_t 1-:2.+:2.*exp(-:.95*t)*(hosf(:.44*t)+:.59*slkf(:.44*t))

;;L0_t1L0_t 1

:2.+:2.*exp(-:.95*t)*(-:.*hosf(:.44*t)-.500*slkf(:.44*t));;L8_t1L8_t 1

02.+02.*exp(-:.95*t)*(-:.*hosf(:.44*t)-:.24*slkf(:.44*t));;cliure(:);;ezpgot(,^2 :2Q)>;;cliure(0);;ezpgot(,^2 :2Q)>;;cliure(8)

;;ezpgot(,^2 :2Q)>


7/52


Xoru 0) t8''' e0y3y5y , :y ' > y(2)10

j) Dlcerjkslyeg dekngealkl '0: y)t(x>y)t(x dkak nuggjkjrjn gslanoautu lge

ze`lgaen llk durua uzjy coraukj ietlrlklz.J,@,H,D ajtrlsgerlkl egde edlklz.

`)\jzjhjkz `lr proirja lge ve gslanoautuku nuggjkjrjn, y(t) ve 'y (t) erlgerlkl ^2 3Q sjklye

jrjgkdj jgt jgtj lzdlrlklz. lzla pekhereslkde stte, y(t) jgttj lse trevl lzdlrlgehentlr.

Xoru 8)

enlg 0 enlg 8

enlg 0de isterlgek erlyl lzdlrek `lr proirja yjzkz. @u proirjaj yge proirja sjtrgjr lgjve edlk nl, enlg 0de irgek erl, enlg 8de isterlgek coraj dksk.

Xoru 4) 6x5x4x8 80:

8xx0x 80:

:3x4x9x0 80:

glkeer dekngea slstealkl >:)2(x: >0)2(x0 8)2(x8 lgn nouggjr jgtkdj, Ijuss-Xeldjg lterjsyok

yktealkl nuggjkjrjn(fesjpgjajgjr egge yjpjrjn) jjdj verlgek tj`goyu dogdurukuz.

terjsyoksjys

:x 0x

8x

:0

...

..

...

Xoru 5) 8j*c-0d*`+5c1 ;;H1sogve(':2/s1L:+0*L0+s*L0','02/s1L0*0+s*L0+0*s*L8','L:+L81L0',L:,L0,L8)>

;;L:1H.L:L: 1

:2*(-0+s)/s/(0+9*s+0*s[0);;L01H.L0L0 1

02*(:+s)/s/(0+9*s+0*s[0);;L81H.L8L8 1

:2/s*(4+s)/(0+9*s+0*s[0);;L:_t1vpj(lgjpgjhe(L:),0)L:_t1-:2.+:2.*exp(-:.6*t)*(hosf(:.4*t)+:.5*slkf(:.4*t))

;;L0_t1vpj(lgjpgjhe(L0),8)

L0_t 1:2.+:2.*exp(-:.95*t)*(-:.*hosf(:.44*t)-.500*slkf(:.44*t));;L8_t1vpj(lgjpgjhe(L8),8)L8_t 1

02.+02.*exp(-:.95*t)*(-:.*hosf(:.44*t)-:.24*slkf(:.44*t));;cliure(:);;ezpgot(L:_t,^2 :2Q);;cliure(0);;ezpgot(L0_t,^2 :2Q);;cliure(8);;ezpgot(L8_t,^2 :2Q)

0) j)t8

:0'0 e0x3x5x

0': xx > :xy

t8

0

:'0

': e

0

2

x

x

53

:2

x

x

t80

:e2

x

x2:y

`) j1^2 :>3 -5Q> `1^2>-0Q> h1^: 2Q> d 1 2>x2 1 ^0 :Q> t 1 2=2.:=3> u 1 -0*exp(-8*t)>^hlnls xQ1gsla(j,`,h,d,u,t,x2)>su`pgot(0,:,:),pgot(t,x(=,:)),xgjèg( 't'),ygjèg('y(t)'),irldsu`pgot(0,:,0),pgot(t,x(=,0)),xgjèg( 't'),ygjèg('y(t)'),irld>


9/52


8) t12=2.22:=2.24>cor n1:=gekitf(t)>J(n)1slk(0*pl*52*t(n))>J:(n)1J(n)>pgot(n,J(n),'n')lc J(n)712J:(n)12>

egselc J(n);12.3J:(n)12.3>

ekdekdpgot(t,J:,'n')

4)8

x5x46x

)2(8

)2(0):(

:

>

0

xx8x

)2(8

)2(:):(

0

>

4

x9x0:3x

)2(0

)2(:):(

8

58

8*50*46x ):(:

>0

858x ):(0

15.5 >

4

5.5*95*0:3x ):(8

18.:05 (`lrlkhl lter.soku)

8

x5x46x

):(8

):(0)0(

:

>

0

xx8x

):(8

):(:)0(

0

>

4

x9x0:3x

):(0

):(:)0(

8

54:9.28

:05.8*55.5*46

x

)0(

:

> 888.80

:05.854:9.28

x

)0(

0

53:;; J.d

jks 1-(-


10/52


jks 10/9*(-3:*h*`+8*h*j[0+4:*h*j+093*h-09*h*j*`+::0*`-0:*j[0-85*j)/(-:3*`+8*j[0+5*j)

;; J.cjks 1

(-3*`[0+;; H10*h+3*h*d-9*e-:4>;; M1mjho`ljk(^J>@>HQ,^j ` hQ)M 1

^ 8*c, 2, 2Q^ d, 8, 2Q^ 2, 2, 0+3*dQ


11/52

XJ\LXJG JKJG] \J] ONUGU CKJG XLKJS XOPUGJPL- (2:/26/0223)Xoru :)

enlg :enlg :'de verlgek l(t) erlslkl t12=2.222:=5 sjklye jrjgkdj :4. derehedek `lr poglkoa lge tenrjrlzdlraen lstlyoruz. @u poglkoauk njtsjygjrk ùgdurjk ajtgj`proirja yjzkz. (02 p)

zat12=2.22:=5>uzuk1gekitf(t)>cor n1:=uzuk

lc t(n)70.3:6jnla(n)14*slk(t(n))>

egsejnla(n)10>

ekdekd`1pogyclt(t,jnla,:4)>j1pogyvjg(`,t)>pgot(t,j)

Xoru 0) enlg :de verlgek l(t) erlslklk t12=2.222:=5sjklye jrjgkdj>j) ortjgjajsktrjpznoautu yjrda lge `) ortjgjajsk aejknoautu yjrda lge h) l(t) jnakk

etnlk deerlkl ùgukuz. (02 p)

zat12=2.22:=5>uzuk1gekitf(t)>cor n1:=uzuk

lc t(n)70.3:6jnla(n)14*slk(t(n))>

egsejnla(n)10>

ekdekdortjgjaj1(:/5)*trjpz(t,jnla) % j nnortjgjaj1aejk(jnla) % ` nnetnlk1sqrt(aejk(jnla.[0)) % h nn

Xoru 8) enlg :deverlgek l(t) jnakk iekgllklk 8Jdek `yn ogduu zjajk jrjgk ùgukuz. (:2 p)

zaùg1clkd(jnla;8)>`jsgjkilh1t(ùg(:))sok1t(ùg(ekd))

Xoru 4) enlg 0de verlgek devrede Jjkjftjr t12 jkkdj njpjtgajntjdr. Devreye lglnlk NJ\ (: jdet) veNI\ (0 jdet) dekngeagerlkl nuggjkjrjn dsogvenoautu lge )t(l8 jnakk t'ye `jg lcjdeslkl (seaògln

ortjadj) ùgjk ve ezpgot noautu lge ^2,2.2:Q jrjgkdj lzdlrek AJTGJ@proirjak yjzkz.Jnagjrj lglnlk lgn nouggjrscr jgkjhjntr. (02 p)

0 J

t(sk)

4 J

)t(ln

4*slkt

0.3:6


12/52

enlg 0za% x1l:(t), y1l0(t), z1l8(t)

;;J1dsogve(':2+y*0+z+Dz*(:2[(-8))12','8*x+Dx*(:2[(-8))-:2-0*y12','Dz1Dx+Dy','x(2)12','y(2)','z(2)12')>

;;l8_t1J.z>;;ezpgot(l8_t,^2 2.2:Q)

Xoru 5) tslk8y0y8y ''' > ( 2)2(y>2)2(y ' )

Dlcerjkslyeg dekngealkl, delnek dka yjrda lge durua dekngeagerl coraukj ietlreren y(t)n deerlkl t12=2.222:=5 sjklye jrjgndj lzdlrlklz (gslanuggjkkz). (02 p)

zay(t)1 :x

0

'

:

' xx)t(y

'0

'' x)t(y

)tslk(8x0x8x :0'0 )tslk(8x0x8x :0

'0

0': xx

)tslk(8

2

x

x

80

:2

x

x

0

:'0

':

> )tslk(*2

x

x2:)t(y

0

:

;;J1^2 :>-0 8Q> @1^2 >8Q>H1^: 2Q> D12> x21^2 2Q>;;t12=2.22:=5> u1 slk(t)>;;gsla(J,@,H,D,u,t,x2)>

Xoru 3) 8x-4y-jz12> 0z-8`10> -x+0y-z1h

Glkeer dekngea slstealkde j, `ve hyl x, y, zhlkslkdek `ugjk ajtgj`proirja yjzkz.Kot= gilgl noautuk sog tjrjckdj negl pjrjktez (nltjptjnlklk jnslke) ogajyjhjntr. (:2 p)

za;;J1sogve(8x-4y-jz12,0z-8`10,-x+0y-z1h,j,`,h);;J.j>;;J.`>;;J.h>

Kotgjr jntr(ajsj zerlkde djkn ogajyjhjntr)

Xkjv sresl 62 djnlnjdr


13/52

U.Jrlcogu :5/25/0229XJ.AF.CJN.EGN-EGK.AF.@G. XJ\LXJG JKJG] CKJG XLKJS XOPUGJPL

enlg :Xoru :) enlg :de verlgek devreye lglnlk durua dekngeagerl jjdj verlgaltlr= S5)2(Sh , J2)2(lG

)t(l

2H

:

t(l

)t(v

G

:

G

:H

I:

H

I

t(l

)t(v

dt

dn

G

h

G

h

)(

:

2

2

2

2

2

2

2

:

2

)(

)(

22

2:

:

:2

::

:2

:

:

2:

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

tltl

tv

II

II

G

II

tl

tv

tl

tv

tl

tv

tl

tv

tl

tv

nG

h

n

jn

in

in

P

P

G

G

hh

j) gsla noautu yjrda lge P dlrekhlke lglnlk jnalzdlrlklz.(t12=2.2:=0 sjklye jgkz)

`) Xeaògln toogòx noautgjrk nuggjkjrjn )t(lG jnakk dekngealkl ugukuz.

Xoru 0) )t(r5y8y0y '''''' > y1y(t)1 )t(x: , )t(xy 0' , )t(xy 8

''

y1y(2)1 :)2(x: , :)2(x)2(y 0' , 0)2(x)2(y 8

''

dlcerjkslyeg dekngealkde odenoautu yjrda lge ^2 3Qsjklye jrjgkdj, )t(x0

dellalklùgukuz ve lzdlrlklz.

enlg 0 (nltj`j jtgd)Xoru 8) enlg 8de irgekdoru jna aotorukuk durua dekngeal jjdj verlgaltlr=

)t(v

2G

:

w

l

M

N

M

NG

N

G

P

'w

'lj

ca

`

> )t(v2

w

l:2)t(y j

:295.2

28.24j > `

2

0> l(2)12> w(2)12 (t12 jkkdj aotor duruyor deaentlr)

w(t)> aotoruk jsjg fz(rjdyjk/sjklye)l(t)> aotor jna (Japer)

2:.2t jgkz. JPNJ sjflceye ielklz

H1:C, G12.5F

I1:/P10 afoJ8)t(ln

t

r t

0 4

:

8

2


14/52

U.Jrlcogu :5/25/0229

enlg 8de verlgek doru jna aotorukuk `esgeae ierlglal Sogt052)t(vj ogduukj ire, aotor fzkk ^2 0Q

sjklye jrjgkdjnl dellalkl lzdlreren, fzk 8 (rjdyjk/sjklye)dek 6 (rjdyjk/sjklye) fzkj ugjajs llk ieekzjajk `ugukuz.

enlg 8Xoru 4) slk(x)lk Tjygor serlsl jga>

J1)!:k0(

x):(.........

! x>`lrlkhl terla,!8

x8

>lnlkhl terla,!5

x5

>kh terla..

Xoru 5) Jjdj verlgek proirjak jgtrgajs sokukdj egde edlgek Jajtrlslklk @ve Hye `jg lcjdeslkl eglrgeylklz=

@1^Q> H1^Q> % @ ve H dlndrtiekajtrlsgerdlr

cor k1:= :2>

cor a1:=02>

J(k,a)12>

cor n1:=:5>

J(k,a)1J(k,a)+@(k,n)*H(n,a)

ekd

ekd

ekd

J

Xkjv sresl


15/52

U.Jrlcogu :5/25/0229]AGEP

Hevjp :)

j) j1^-0 :>0 -0Q> `1^:>2Q> h1^0 -0Q> d12t12=2.2:=0>jj1gekitf(t)>u18*okes(:,jj)>x21^5 2Q>^hlnls xQ1gsla(j,`,h,d,u,t,x2)

pgot(t,hlnls),xgj`eg('t'),ygj`eg('LP'),irld ok

` ;; J1dsogve('Dx1-0*x+y+8','Dy10*x-0*y','x(2)15','y(2)12')>;; lG1vpj(J.y,4)

lG 1

8-2.6464e-:*exp(-2.563*t)-0.


16/52


Hevjp 8) j1^-4 -2.28>2.95 -:2Q> `1^0>2Q> h1^2 :Q> d12>x21^2 2Q> % lgn nosuggjrt12=2.2:=0> % djfj fjssjs hlzla lhlk degtj t 12.2: yjplglyoru1052*okes(:,gekitf(t))> % t lge u(t) klk òyutu eslt yjplglyor^hlnls xQ 1 gsla(j,`,h,d,u,t,x2)>% x1^x: x0Q ogduiu fjtlrgjkajgldlrpgot(t,x(=,0))>

tltge('w(t) deilslal')>xgjèg('t')>ygjèg('w(t)')>irld>slklr1clkd( 871 x(=,0)& x(=,0)716)>t(slklr(:))t(slklr(ekd))

Hevjp 4) x18*pl/0>J12>cor n12=:222>

J1J+((-:)[n)*(x[(0*n+:))/prod(:=0*n+:)> lc j`s(slk(x)-J)71 2.22:

x % terla deerln % terla sjylsl`rejn

egseekd

ekd

Hevjp 5)Serlgek proirja @*H lgealkl yjpjr, dler `lr lcjde lge @ ve H ajtrlsgerlkl jrpjr.


17/52



18/52

U.Jrlcogu 2:/26/0229

[email protected]\LXJG JKJG] \J] ONUGU CKJGXLKJS XOPUGJPL

rekhlye lgjk edlgdl

Xoru :)j) Jjdj verlgek dlcerjklyeg dekngealkl gslanoautu nuggjkjrjn zkz. \jgkzhj

y(t)dellalkl t12=2.2:=8 sjklye jrjgkdj lzdlrlklz.

)t(uy4y6y5y '''''' > :)2(y'' , :)2(y' , 2)2(y

`) Jyk soruyu ode45noautu lge yjpkz.

enlg 0Xoru 0) enlg 0de verlgek tjrjg yzeyl ajtgj` noautgjr nuggjkjrjn lzdlrek `lr proirja

yjzkz.(ensekgere dlnnjt edlklz). Nltj`j jttaXoru 8) Jjdj verlgek J ajtrlslklk deteralkjktkk -5822 ogduu `lglklyor. J ajtrlslkde 8.sjtr,

0. stuk egeajkk deerl (x) lse `lglkeaeaentedlr. \jzdkz `lr proirja yjrda lge

`lglkaeyek ù egeajkk deerlkl ùgukuz. x lk deer jrjg -:22=:22 ogjrjn jgkj`lglr.

x `lr tjasjydr.

J 1: 5 < ::

-0 3 9 4

2 x : 6:: 35 6< 0:

Xoru 4)

......! (f(t) enlg :de verlgaltlr) y1y(t)1 )t(x: , )t(xy 0' , )t(xy 8

''

dlcerjkslyeg dekngealkde> y1y(2)1 :)2(x: , :)2(x)2(y 0'

, 0)2(x)2(y 8''

nougjr jgtkdj

j) `lrlkhl aerteèdek jdet dlcerjkslyeg dekngeae dktreren gsla noautu yjrda lge )t(x8

dellalkllzdlrlklz.

t12=2.2:=0

`)

\unjrdj verlgekkh aerteèdek dlcerjkslyeg dekngealkde ode45noautu yjrda lge )t(x8

dellalkl lzdlrlk.

t12=2.2:=0 (nltj`j jtgd)

Xoru 8) jrhos(x) 1 ........9

x*

3*4*0

5*8*:

5

x*

4*0

8*:

8

x*5.2x

0

pl 958 ( :x: )

ogduukj ire fesjp ajnlkjslkdj jrhoslgeal ùgukajyjk `lr rekhlklk yunjrdjnl eltgll nuggjkjrjn jrhos(2.9)

deerlkl ùgajn llk (fesjp fjtjsk j`s(:e-6) dek nn tutajn jrt lge) yunjrdjnl eltglnte xlk njkh nuvvetlke

njdjr fesjpgjajs ierentllkl ugjk lr ajtgj proirja oguturukuz.

Xoru 4) cor x1:=acor y1:=k

@(x,y)1 x*(0[y)ekd

ekd@

Rroirjak ljret jn dlyjirjalge isterlklz.

Xoru 5) j) enlg :de verlgek f(t) erlslklk t12=2.2:=0sjklye jrjskdj lkteirjglkl fesjpgjykz. (5 pujk)

enlg :denl f(t) erlslklk t12=2.2:=0sjklye llkierl cjrn yjngja lge trevlkl fesjpgjykz ve lzlklz (5 pujk)

h) serl PGH devreslkl èsgeyek `lr ierlgla njykjS(t)1 t*:.2e*8 ogduukj ire, P1: ofa,H1:C,G1:F llkù devreye

serl `jg `lr japeraetreklknj japer isterehelkl ùgjk `lr ajtgj`proirja yjzkz. Devredenl ta lgn nouggjrscr jgkjhjntr. T10 sjklye jgkjhjntr.(:2 pujk)

Xkjv sresl S5)2(vh ,

J2)2(lG , ò`lk ierlglaldt

)t(dlG)t(v GG lse

(f(t) dellal enlg :de verlgaltlr.(G18 fekry)

j) ò`lk jnakk t12=2.2:=0 sjklye jrjgkdj

dellalkl lzdlrek `lr ajtgj` proirjak gslanoautu yjrda lge oguturukuz.

`)

ò`lk ierlglalklk t12=2.2:=0 sjklye jrjgkdj

dellalkl lzdlrek `lr ajtgj` proirjak gslanoautu yjrda lge oguturukuz. (Nltj`j jtgd ` nn)


23/52

U.Jrlcogu 5/23/0226Hevjp :) G15>

j1^-8 -:>-0 2.0Q> `1^:>0Q> h1^2 :Q> d12> G18>x21^5 2Q>t12=2.2:=0>f1zeros(:,gekitf(t))>cor n1:=gekitf(t)

lc t(n)712.9580f(n)1:2*slk(pl*t(n))>

egsef(n)19>ekd

ekd^hlnls xQ1gsla(j,`,h,d,f,t,x2)>cliure(:)pgot(t,x(=,0)),xgjèg('t'),ygjèg('ò`lk jnlal'),irld ok % j nn `lttlcliure(0)

SG1G*(-0*x(=,:)+2.0*x(=,0)+0*f')> % ò`lk ierlglal dekngeal-` nnpgot(t,SG),xgjèg('t'),ygjèg('ò`lk ierlglal'),irld ok %` nn `lttlveyjh1^-3 2.3Q> d1^3Q>y1gsla(j,`,h,d,f,t,x2)>

cliure(8)pgot(t,y),xgjèg('t'),ygjèg('ò`lk ierlglal'),irld ok %` nn `lttl

Hevjp 0)j) )t(f8xx5x :8'8

8'0 xx y1y(t)1 )t(x: , )t(xy 0

' , )t(xy 8

''

0

'

: xx

)8)t(f(

:

2

2

)t(x

)t(x

)t(x

52:

:22

2:2

)t(x

)t(x

)t(x

dt

d

8

0

:

8

0

:

J1^ 2 : 2>2 2 :>: 2 -5Q>@1^2>2>:Q> H1^2 2 :Q> D12>x21^-: : 0Q>t12=2.2:=0>f1zeros(:,gekitf(t))>cor n1:=gekitf(t)

lc t(n)712.9580f(n)1:2*slk(pl*t(n))-8>

egsef(n)19-8>

ekdekd^hlnls xQ1gsla(J,@,H,D,f,t,x2)>

pgot(t,x(=,8)), xgjèg('t'), ygjèg(x8(t)'), irld ok

`) ;; x21^-: : 0Q>;; ^t xQ1ode45(hevjp0226,2,0, x2);; pgot(t,x(=,8)),xgjèg(t),ygjèg(x8),irld ok

Jgt proirja dosyjs jjdj verlgaltlr>cukhtlok dx1hevjp0226(t,x)t12=2.2:=0>uzuk1gekitf(t)>f1zeros(uzuk,:)>cor n1:=uzuk

lc t(n)712.9580

f(n)1:2*slk(pl*t(n))>egse

f(n)19>ekd

dx1^x(0)>x(8)>-5*x(8)+x(:)-8+f(n)Q >


24/52

U.Jrlcogu 5/23/0226ekd

Hevjp 8)n1:>x12.9>J1pl/0-x>cjrn1:>wflge cjrn;1:e-6

n1n+0>J:1 -prod(:=0=n-0)*(x[n)/(n*prod(0=0=n-:))>J1J+J:cjrn1j`s(jhos(x)-J)

ekdn % x lk nuvvetl

Hevjp 4)

@JGJ

k ve a deerlkl ilrlklz

x1: > y1:

y7ky1y+:@(x,y)1 x*(0[y)

y1:

x7ax1x+:

DUP

F

E

E

F


25/52

U.Jrlcogu 5/23/0226

Hevjp 5) hght12=2.2:=0>f1zeros(:,gekitf(t))>cor n1:=gekitf(t)

lc t(n)712.9580f(n)1:2*slk(pl*t(n))>egse

f(n)19>ekd

ekdlkteirjg1trjpz(t,f) % j nn `lttl

turevf1dlcc(f)./dlcc(t)>zjajk1t(0=gekitf(t))>pgot(zjajk,turevf)> % ` nn `lttl

jnla1dsogve('2.8*exp(2.:*t)1Dx+D0x+x','x(2)12, Dx(2)')>

jnla_ort1dou`ge(2.5*lkt(jnla,2,0)) % h nn `lttljnla_ort 1

2.::38


26/52

U.Jrlcogu 06/29/0226

XJ.AF.CJN.EGN-EGK.AF.@G.\J] ONUGU XJ\LXJG JKJG] CKJG XLKJS XOPUGJPL

Xoru :)

)thos(

)tslk(

62

20

l

v

64

0:

l

v

dt

d

G

H

G

H > :)2(vH > 2)2(lG

\unjrdj verlgek durua dekngealkde su`pgotyjrda lge )t(vH ve )t(lG dellagerlkl t1:=0 sk

zjajk jrjgkdj, stte )t(vH , jgttj lse )t(lG ogjhjn enlgde lzdlrlklz. (02 pujk)

Xoru 0) )t(fy3y::y3y ''''''

> (f(t) enlg :de verlgaltlr) y1y(t)1 )t(x: , )t(xy 0' , )t(xy 8

''

dlcerjkslyeg dekngealkde> y1y(2)1 :)2(x: , :)2(x)2(y 0'

, 0)2(x)2(y 8''

nouggjr jgtkdj>

j)

Serlgek dlc.dekngeal `lrlkhl aerteèdek jdet dlcerjkslyeg dekngeae dktreren gsla noautu yjrda lge

n(t)1 )t(x0)t(x8:

dellalkllzdlrlklz. t12=2.2:=3 (:5 pujk)

`) \unjrdj verlgek kh aerteèdek dlcerjkslyeg dekngealkde ode45 noautu yjrda lge n(t)1 )t(x0)t(x8:

dellalkl lzdlrlk.t12=2.2:=3 (:5 pujk)Xoru 8)j)enlg :de verlgekf(t) erlsl`lr jnaj lglnlk della lse ù jna doru jna japeraetreslkdekietllkde japeraetreklk

isterdll deerl fesjpgjyjk `lr ajtgj`proirja yjzkz.t12=2.2:=3sjklye . (5 pujk)`) enlg :denl f(t) erlsl `lr jna dellalkl isterdll njùg lge jgterkjtlc jna gek `lr japeraetredek

ietllkde japeraetreklk isterdll deerl fesjpgjyjk `lr ajtgj proirja yjzkz.t12=2.2:=3sjklye(5 pujk)

h) enlg :de verlgek f(t) erlslkl 8.aerteèdek lr poglkoaj uydurjk lr ajtgj`proirja yjzkz.(5 pujk)d) enlg :de verlgek f(t) coknslyoku serl PH devreslkl èsgeyek `lr jnanjykjogduukj ire, P18

ofa, H10C llk, jna njykjkk ugjr jrjskdjnl ierlglalk zjajkj ire dellalkl lzdlrek`lr

ajtgj`proirja yjzkz. (:5 pujk)e) enlg :deverlgek erlklk lgerl cjrn yjngjakj ire trevlkl jgjk ve trev erlslklk zjajk ire dellalkl

lzdlrek`lrajtgj`proirja yjzkz. (5 pujk)8.sorudj ta ngjr jgt jgtj ten `lr proirja llkde yjzj`lglrslklz (yjkgjrkj jngjaj yjpkz)

enlg 0Xoru 4) enlg 0de`lr aerdlveklk yjkdjk irk verlgaltlr. @u aerdlveklk slze jnjk yjk yzeyl tjftj lge njpgjkjhjntr. @u

aerdlveklk yjkdjk irkk lzdlrek ve nj `lrla njre tjftj fjrhjkjhjk ùgjk `lr ajtgj`proirja yjzkz.(:5 pujk)-jgjk fesj` egge yjpgajyjhjn ajtgj` noautgjr lge yjpgjhjntr

Xkjv sresl


27/52


zaHevjp :)

;; J1dsogve('Dx1-x-0*y+0*slk(t)','Dy14*x-6*y-6*hos(t)','x(2)1:','y(2)12');; su`pgot(0::),ezpgot(J.x,^: 0Q),tltge(Sh(t) dellal),irld;; su`pgot(0:0),ezpgot(J.y,^: 0Q),tltge(lG(t) dellal),irld

Hevjp 0)j) )t(fy3y::y3y ''''''

)t(fx3x::x3x :08'8

8'0 xx y1y(t)1 )t(x: , )t(xy 0

' , )t(xy 8

''

0

'

: xx

)t(f

:

2

2

)t(x

)t(x

)t(x

3::3

:22

2:2

)t(x

)t(x

)t(x

dt

d

8

0

:

8

0

:

J1^ 2 : 2>2 2 :>-3 -:: -3Q>@1^2>2>:Q> H1^: 2 0Q> D12>x21^-: : 0Q>t12=2.2:=3>f1zeros(:,gekitf(t))>cor n1:=gekitf(t)

lc t(n)710f(n)14*slk(2.9654*t(n))>

egsef(n)1-t(n)+3>

ekdekd^hlnls xQ1gsla(J,@,H,D,f,t,x2)>pgot(t,hlnls), xgjèg('t'), ygjèg(x:(t)+0*x8(t)'), irld ok

`) ;; x21^-: : 0Q>;; ^t xQ1ode45(hevjp0226_yjz,2,3, x2);; pgot(t,x(=,:)+0*x(=,8)),xgjèg(t),ygjèg(x:+0*x8),irld ok

Jgt proirja dosyjs jjdj verlgaltlr>cukhtlok dx1hevjp0226_yjz(t,x)t12=2.2:=3>uzuk1gekitf(t)>f1zeros(uzuk,:)>cor n1:=uzuklc t(n)710

f(n)10*slk(2.9654*t(n))>egse

f(n)1-t(n)+3>ekddx1^x(0)>x(8)>-3*x(:)-::*x(0)-3*x(8)+f(n)Q >ekd

Hevjp 8) j) t12=2.2:=3>cor n1:=gekitf(t)>

lc t(n)710f(n)14*slk(2.9654*t(n))>

egsef(n)1-t(n)+3>

ekdekddh_ort1aejk(f)> % d.j. japeraetreslklk isterdll deer

`) jh_etnlk1sqrt(aejk(f.[0))> % j.j. japeraetreslklk isterdll deer

h) J1pogyclt(t,f,8)>

d) 3

2n dt)t(l

H

:)t(l*P)t(v

;; Sn18*f+(:/0)*huatrjpz(t,f)> pgot(t,Sn)

e);; turev1dlcc(f)./dlcc(t)> pgot(t(:=ekd-:),turev)


28/52


Hevjp 4) @12=2.2:=5.5> % @ yjtjy ensekl, J lse dey ensekl isteraentedlrcor n1:=gekitf(@)

lc @(n);12 & @(n)7:J(n)1:>

egselc @(n);1: & @(n)70J(n)10>

egselc @(n);10 & @(n)78

J(n)18>egselc @(n);18 & @(n)74

J(n)14>egselc @(n);14 & @(n)74.5

J(n)15>egselc @(n);14.5 & @(n)715.5

J(n)13>ekd

ekdpgot(@,J) % aerdlveklk yjkdjk irk lzdlrlglyortrjpz(@,J) % nuggjkgjhjn tjftj yzeyl


29/52

U.Jrlcogu 5/26/022 8.4 :.6 9 6> -:.4 4.0 -4 3> 8 -5 9 0Q>

ajtrlsl reeg (pozltlc+keijtlc) sjygjrdjk oguajntjdr. yge `lr proirja yjzkz nl> J ajtrlsl stk stuktjrjksk ve fer yekl egeajk onukduukdj o jkj njdjr tjrjkjk ta sjygjrk ortjgjajs jgksk. @ugukjkortjgjaj sjykk 9 njt, o jkj njdjr tjrjkjk ek `yn sjydjk `yn ogduukdjproirja dursuk ve ek sokonukjk sjy deerl ve ù sjykk jdresl enrjkj yjzdrgsk. (:5 pujk)

Xoru 8) x1 2=pl/52=:2*pl>

y1 0*slk(x)>z1 (hos(x)).[8+slk(8*x)>

ogduukj ire òyutgu z1c(x,y) erlslklk verlgek jrjgntj tepe kontjskk (x,y,z) noordlkjtgjrk ùgjkajtgj`proirja yjzkz. (:2 pujk)

Xoru 4) t0''' e8y ù proirja ù jnakperlyoduku ùgsuk ve sokrj jnak etnlk deerlkl fesjpgjsk. (Kot= Jna djgij engl slkzoldjg ogajn

zorukdj delgdlr).(:5 pujk)

Xkjv sresl


30/52

U.Jrlcogu 5/26/022 % C(x) lk nngerl ùgukuyorcturev1pogyder(^8 2 -0 :2Q)> % C(x)'lk trevl ugukuyorcturevnon1roots(cturev)> % trev lk nngerl ugukuyorcektei1pogylkt(^8 2 -0 :2Q)> % C(x) lk ekteirjgl ùgukuyorcekteinon1roots(cektei)> % ekteirjg lk nnùgukuyorAnon1^Cnon' cturevnon' cekteinon'Q> % A(x) lk nngerlJ1pogy(Anon)> % A(x) dekngealklk njtsjygjr ugukuyorsyas xhozua1vpj(pogy0sya(J,x),8) % A(x) dekngeal seaògln ogjrjn egde edlgdl

Hevjp 0)hgh

J1 ^-:2 0 8 5> 8.4 :.6 9 6> -:.4 4.0 -4 3> 8 -5 9 0Q>topgja12>k12>ekùy12>corn1:=4

cora1:=4k1k+:>lcJ(a,n) ; ekùyekùy1J(a,n)>

egseekdtopgja1topgja+J(a,n)>ortjgjaj1topgja/k>

lc9*ortjgjaj ; ekùydlsp('jrjkjk sjy')dlsp(J(a,n))dlsp('jrjkjk jdres')dlsp(a),dlsp('lkhl sjtr')dlsp(n),dlsp('kh stuk')

`rejnegseekd

ekdekd

Hevjp 8)hgejr jgg

hghx1 2=pl/522=0*pl>y1 0*slk(x)>z1 (hos(x)).[8+slk(8*x)>pgot8(x,y,z)J1ajx(z)>H1clkd(z11J)dlsp('x lk deerl')V1x(H)dlsp('y klk deerl')\1y(H)dlsp('z lk deerl')]1J

irldxgjèg('x'),ygjèg('y'),zgjèg('z')


31/52

U.Jrlcogu 5/26/022

0'

x)t(y

t0:0

'0

t0:0

'0 ex8x0xe8x8 -0Q>

`1^2>:Q>

h1^-0 8Q>d1^0Q>

t12=2.2:=3>

x21^0 :Q>

u1exp(-0*t)>

^y xQ1gsla(j,`,h,d,u,t,x2)

pgot(t,y) ,xgjèg('t'),ygjèg('z(t)'),irld

Hevjp 5) )3/plwthos(*w*:22*0Pll)H/:(Gl '''

':0

'

:

xxl

xl

)3/plwthos(*w*:22*0x*Px*)H/:(Gx 0:'0

)3/plwthos(*G

w*:22*0x*)G/P(x*)GH/:(x 0:

'0

)3/plwthos(*w*:22*0G/:

2

x

x

G/P)H*G/(:

:2

x

x

dt

d

0

:

0

:

)3/plwthos(*w*:22*0*2x

x2:xl)t(y

0

::

Rroirja jjdj verlgaltlr=

t12=2.22:=2.20>

T12.20>

G1 8/(0*pl*52)>H1:/(pl*522)>P10>w10*pl*52>J1sqrt(0)*:22*w>u1J*hos(w*t-pl/3)>

x21^2 2Q>

j1^2 :>-:/(G*H) P/GQ> `1^2>:/GQ> h1^: 2Q> d12>

^jnla xQ1gsla(j,`,h,d,u,t,x2)> % jnla lcjdesl

Ssegcjkl1G*dlcc(jnla)./dlcc(t)> % òyut t dek : dt

Sdlrekhjkl1P*jnla> % òyut t lge jyk

Snjpjsltejkl1(:/H)*huatrjpz(t,jnla)> % òyut t lge jykSsegcier1sqrt((:/T)*trjpz(t(0=ekd),Ssegcjkl.[0))

Sdlrekhier1sqrt((:/T)*trjpz(t,Sdlrekhjkl.[0))


32/52

U.Jrlcogu 5/26/022

uzuk1gekitf(t)>J1:2*rjkd(:,uzuk)> % jnla ljretl retlglyor

usttepe1ajx(J)> % jnlalk ajx.tepe kontjs fesjpgjkyorjgttepe1 alk(J)> % jnlalk alk.tepe kontjs fesjpgjkyor

tusttepe1clkd(J11usttepe)> % jnak ajx.tepe kontjsk t klk njkh

% egeajkkdj jgd fesjpgjkyor

tjgttepe1clkd(J11jgttepe)> % jnak alk. kontjsk t klk njkh

% egeajkkdj jgd fesjpgjkyor

t:1t(tusttepe)> % jnak tepe kontjskj ugjgjk t jk ùgukuyor

t01t(tjgttepe)> % jnak alk kontjskj ugjgjk t jk ùgukuyor

T10*(t0-t:)> % jnak perlyodu ùgukuyor

H1T/2.222:> % jna perlyodukj nj ga sjyskdj ugjgd

% ùgukuyor

@1J(:=H)> % jnak ga jrjg njdjr ogjk egjajkgjr yekl ventre

% jtjkyor.Jnsl njgde ekteirjgde òyut fjtjs ogur.

jnlaetnlk1sqrt((:/T)*trjpz(t(:=H),@.[0)) % jnak etnlk deerl ùgukuyor


33/52

U.Jrlcogu :9/25/02:2

XJ AF.CJN. EGN-EGK. [email protected]\LXJG JKJG] CKJGXLKJS XOPUGJPL

Xoru :) 0 0 0 0 0 0 0X R Z D X R Z D jntlc ogajyjk i . (D= Dlstorslyok ih). enlg :de `lrnjykjj lglnlk ierlgla ve jna dellagerl verlgaltlr. @u njykjntjk enlgek jntlc ogajyjk i deerlklùgukuz. Kot= Jntlc i, jklihk `lr perlyot òyukhj ortjgjajsdr. X1S*L deerlkdedlr (etnlk deerger).\jtjy ensek (wt) jrt deerlkl 2.22: jgkz.

enlg : enlg 0

Xoru 0)Eglklzde Devget Aeteorogoml Ekstltskdek jgkjk lnl stukgu `lr ajtrls ogsuk (r= J(5222,0). gn stk rjnjaogjrjn rzijr lddetlkl, lnlkhl stuk lse derehe ogjrjn rzijrk ykk isterslk. yge `lr proirja yjzk nl,jgtkdj enrjkdj ek `yn rzijr lddetl ve ykk isterek (enlg 0denlke èkzeyek) ventr ogsuk.

Xoru 8)c(z)1u(x,y)+lv(x,y) ogajn zere yx vu ve xy vu jrtgjrkk jyk jkdj sjgjkajs duruaukdj c(z)

coknslyokukukHjuhfy- Pleajkk nouguku sjgjd sygeklr. c(z)1 0 0 0x 0xy l(y x ) lcjdeslklk Hjuhfy-

Pleajkk nougukuk sjgjyp sjgjajdk test edek `lr proirja yjzkz. Test sokuhuku dlspnoautu lgeverlklz. Egge lgea yjpgajyjhjntr.

Xoru 4) enlg 8denl devreye lglnlkdurua dekngeagerl jjdj verlgaltlr.gn nougjr scr jgkjhjntr. t12=2.2:=0 llkfer lnl dlrek jnakk dellal jgt jgtj lzdlrek ajtgj`dosyjs oguturukuz. Rro`gea llkde gslanoautudj ùgukjk noaut sjtrgjr lge zgehentlr.

enlg 8

Xoru 5 )j), orlmlk aernezgl ve yjrjp :dek `yn ogjk ferfjkil `lr eaèr ogsuk. Jjdjnl ekteirjg sokuhukuùgukuz. (Noapgens delnekgl coknslyokgjr teorlsl sorusu)

5z 0dz ?

z(z :)

`)0

2

hos5xL dx

x :

Kot `jreal= :/:2 0/ 82 8/:2 4/ 82 5/ 02

Xkjv sresl


34/52


35/52


Rroirja jjdj verlgaltlr=

j1^-4 -4 >: 2Q>`1^0>2Q>h1^: 2Q>d1^2Q>t12=2.2:=0>uzuk1gekitf(t)>x21^2 2Q>u102*okes(uzuk,:)>^y xQ1gsla(j,`,h,d,u,t,x2)>P010>% x(=,:) njpjslte ierlglal1dlrekh ierlglaldlr (devreye `jnkz)P0_dlrekh_jnlal1x(=,:)/P0>% njpjslte dlrekhe pjrjgeg `jg ogduukdjk dogjy ierlglagerl elttlr% P: dlrekhlklk ierlglal1 E-ò`lk ierlglal dlrG1:>ò`lk_ierlglal1G*(:*x(=,:)+2*x(=,0)+2*u)>P:10>P:_dlrekh_jnlal1(u-ò`lk_ierlglal)/P:>su`pgot(0::),pgot(t,P0_dlrekh_jnlal),xgjèg( 't'),ygjèg('lP0'),irldsu`pgot(0:0),pgot(t,P:_dlrekh_jnlal),xgjèg( 't'),ygjèg('lP:'),irld

Hevjp 5)j)i(z)15z-0, f(z)1z(z-:) ogajn zere clk jynrgngjr f(z) 1 z-:1 2 dekngealklk nngerl ogjk z12 ve z1:dlr. @u kontjgjr c llk `lrlkhl aerteèdek nutuptur. Fer lnl nutup dj erlslklk llkdedlr. Jyrhj>

: 0L( , z ) L( , z ) : ogduukdjk>

: '

0 '

5z 0 i(2)Pes(c ,z ) Pes( ,2) 0

z(z :) f (2)

5z 0 i(:)Pes(c ,z ) Pes( ,:) 8

z(z :) f (:)

5z 0 dz 0 l(0 8) :2 lz(z :)

egde edlglr.

`)5

5x 5

0 0

: hos5x : : : eL dx Pe^ e dxQ Pe^0 l Q e

0 0 0 0l 0x : x :

1 2.2:25


36/52


XJ AF.CJN. EGN-EGK. [email protected]\LXJG JKJG] CKJGXLKJS XOPUGJPL

Xoru :) 0 0 0 0 0 0 0X R Z D X R Z D jntlc ogajyjk i . (D= Dlstorslyok ih). enlg :de `lrnjykjj lglnlk ierlgla ve jna dellagerl verlgaltlr. @u njykjntjk enlgek jntlc ogajyjk i deerlklùgukuz. Kot= Jntlc i, jklihk `lr perlyot òyukhj ortjgjajsdr. X1S*L deerlkdedlr (etnlk deerger).\jtjy ensek (wt) jrt deerlkl 2.22: jgkz.

enlg : enlg 0

Xoru 0)Eglklzde Devget Aeteorogoml Ekstltskdek jgkjk lnl stukgu `lr ajtrls ogsuk (r= J(5222,0). gn stk rjnjaogjrjn rzijr lddetlkl, lnlkhl stuk lse derehe ogjrjn rzijrk ykk isterslk. yge `lr proirja yjzk nl,jgtkdj enrjkdj ek `yn rzijr lddetl ve ykk isterek (enlg 0denlke èkzeyek) ventr ogsuk.

Xoru 8)c(z)1u(x,y)+lv(x,y) ogajn zere yx vu ve xy vu jrtgjrkk jyk jkdj sjgjkajs duruaukdj c(z)

coknslyokukukHjuhfy- Pleajkk nouguku sjgjd sygeklr. c(z)1 0 0 0x 0xy l(y x ) lcjdeslklk Hjuhfy-

Pleajkk nougukuk sjgjyp sjgjajdk test edek `lr proirja yjzkz. Test sokuhuku dlspnoautu lgeverlklz. Egge lgea yjpgajyjhjntr.

Xoru 4) enlg 8denl devreye lglnlkdurua dekngeagerl jjdj verlgaltlr.gn nougjr scr jgkjhjntr. t12=2.2:=0 llkfer lnl dlrek jnakk dellal jgt jgtj lzdlrek ajtgj`dosyjs oguturukuz. Rro`gea llkde gslanoautudj ùgukjk noaut sjtrgjr lge zgehentlr.

enlg 8

Xoru 5 )j), orlmlk aernezgl ve yjrjp :dek `yn ogjk ferfjkil `lr eaèr ogsuk. Jjdjnl ekteirjg sokuhukuùgukuz. (Noapgens delnekgl coknslyokgjr teorlsl sorusu)

5z 0dz ?

z(z :)

`)0

2

hos5xL dx

x :

Kot `jreal= :/:2 0/ 82 8/:2 4/ 82 5/ 02

Xkjv sresl


37/52


]AGEPHevjp :) wt12=2.22:=0*pl>

S_t1002*sqrt(0)*slk(wt)>uzuk1gekitf(wt)>jnla1zeros(:,uzuk)>corn1:=uzuk>

lcwt(n)71 pl/8>

jnla(n)1-52>egselcpl/87 wt(n) & wt(n)7 pl+pl/8>jnla(n)1:22>

egselcpl+pl/871 wt(n) & wt(n)710*pl>jnla(n)1-52>

ekdekdpgot(wt,S_t),fogd ok,pgot(wt,jnla),irldjkliuh1S_t.*jnla>R1aejk(jkliuh)S1sqrt(aejk(S_t.[0))L1sqrt(aejk(jnla.[0))X1S*L

Jrjkjk_iuh1sqrt(X[0-R[0)

Hevjp 0)hgejr jgghghJ1^ 5.


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39/52

U. Jrlcogu - I. etlkeg ::.25.02::XJ. AF.CJN. EGN-EGK. AF.@G.

XJ\LXJG JKJG] CKJGXLKJS XOPUGJPL

X:) j)\unjrdj verlgek dlcerjkslyeg dekngea slstealke lglnlk lgn nouggjr> x(2)10, y(2)12, , ogduukj ire x(t) ve y(t)

dellalkl t 1 ^2 = 2.2:= :2Q jrjgkdj lzdlrek ajtgj`proirjak gslanoautuku nuggjkjrjn yjzkz.Trevger t'ye iredlr.`)Jyk pro`geal ode45noautu lge zkz.

X0) (Gjurekt jga)x1 2.: llk, deerl lge ) jga jrjskdjnl "cjrnk autgjn deerlklk" :e-:5dek nn njgdksjyskùgjk ajtgj`proirjak yjzkz.

X8)enlg :'de verlgek ORJAR ekteirjg jghdr. @u tr `lr opjapk tjka `jkts, englk feaek jgtkdj verlgaltlr. Ps18 ofa, H12.5C, P1 4ofa, G10 F, dr. t12 jkkdj njpjslte ierlglal scr vogt, ò`lk jna scr japer ogduukj ire, ò`lk jnakk ^2 = 5Q sk jrjgkdj dellalkllzekve ò`lk jnakk ortjgjaj deerlkl ùgjk`lr ajtgj`proirja yjzkz. vogt jgkjhjntr.

enlg :

X4)X4) j) 508 08

xxx dekngealklk x21:.0 lgn nougu jgtkdj nnk Kewtok ykteal lge egge jda jda ùgajyj jgrnek, `lrlkhl ve(ieren ogursj) lnlkhl lterjsyok sokukdj epslgok12.22: nougu jgtkdj yjnksjaj sjgjkp sjgjkajdk noktrog edlklz, dekngealk nnkùgup ùgajdkz yjzkz, ùgausjkz nn yjzkz (fesjp egge yjpgjhjn, ajtgj`noautu nuggjkgajyjhjntr).(:5 p)`) j nnkdj verlgek dekngeal uyiuk `lr ajtgj`noautgjr nuggjkjrjn zek `lr proirja yjzkz. (5p)

X5) Jjdj `lr prlzdek cjrng djnlnjgjrdj ggek vogtjm deergerl verlgaentedlr.j)

Serlgek senlz jdet kontjdjk ieaeye jgjk 5. ve 9. aerteèdek lnl cjrng erlklk njtsjygjrk ùgukuz. (5p)`)

j nnkdj egde ettllklz erlger jrjskdjnl ortjgjaj njreseg fjtjy fesjpgjyjk ve lnl erl jrjskdjnl cjrnk autgjn deerlkl lzdlrek `lrajtgj`proirja yjzkz. (5p)

h) 6. ve 00. djnlnjgjrj njrgn iegek vogtjm deergerlkl ajtgj`noautgjr yjrdaygj fesjpgjtkz.(5p)

(t:,v:)1(:,002) (t0,v0)1(8,0:


40/52

U. Jrlcogu - I. etlkeg ::.25.02::

]AGEP:) j) x

y

\unjrdjnl eltglnger pro`geade verlgek dekngeagerde nuggjkgrsj>

(8t) \unjrdjnl eltglnger ajtrlseg coradj yjzgrsj>

^

Q

^ Q egde edlglr.j1^2 : 2 2> -5 2 4 2> 2 2 2 :>: 2 -0 2Q>

`1^2 2>2 :>2 2>: 2Q>h1^: 2 2 2>2 2 : 2Q>d1^2 2>2 2Q>x21^0 2 2 :Q> t12=2.2:=:2>

u1^8+hos(8*t)> -0+slk(0*t)Q>^y xQ1gsla(j,`,h,d,u,t,x2)>

su`pgot(0::),pgot(t,x(=,:))> xgjèg('t'), ygjèg('x(t)'),irldsu`pgot(0:0),pgot(t,x(=,8))> xgjèg('t'), ygjèg('y(t)'),irld

`) ^t xQ1ode45('ihozturev:',^2 :2Q, ^0 2 2 :Q')>

cliure(0), su`pgot(0::),pgot(t,x(=,:))> xgjèg('t'), ygjèg('x(t)'),irldsu`pgot(0:0),pgot(t,x(=,8))> xgjèg('t'), ygjèg('y(t)'),irld

cukhtlokturev1ihozturev:(t,x)

turev1^x(0)> >x(4)> (8*t)Q>0) J1exp(2.:)>

top1:>

x12.:>

cor n1:=:2222

top1top+(x[n)/cjhtorljg(n)>

lc j`s(top-J)7:e-:5

dlsp('jrjkgjk k sjys')

dlsp(n)

`rejn

egse

ekd

ekd

8) ORJAR'k nkj NI\ uyiugjkdkdj>

t12=2.22:=5> x212>Ss14*exp(t)> G10> P14> Ps18> H12.5>So1-(:/(Ps*H))*huatrjpz(t,Ss)> % ORJAR n ierlgla dellalj1^-P/GQ> `1^:/GQ> h1^:Q> d1^2Q>^y xQ1gsla(j,`,h,d,So,t,x2)>pgot(t,x)> xgjèg('t'), ygjèg('L:'),irld % jna lzlglyoraejk(x) % jnak ortjgjaj deerl ùgukuyor


41/52


4)j) , Ql1:. lterjsyok llk j`s(x(2)-x(:))12.08:;2.22: ogduukdjk lterjsyok devja eder.l10. lterjsyok llk j`s(x(:)-x(0))12.246;2.22: ogduukdjknn `ugukjajatr.

`) x1czero(csogve)(8*x[8+x[0+0*x-5,:.0)

5) j) t1:=0=:5S1^002 0:

dekngealklk nngerldlr. @u nnger dlr.

llk>

8/6@jslt nutup ogduukdjk glalt jyk zjajkdj rezld deerldlr. llk> egde edlglr. O fjgde,

Pez(c, `ugukur.


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43/52


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45/52

U. Jrlcogu - I. etlkeg 09.25.02::

`) ^t xQ1ode08('ihozturev',^2 5Q, ^2 2 2Q')>

pgot(t,x(=,:)),pgot(t,x(=,:)), xgj`eg('t'), ygj`eg('l(t)')tltge(ode noautu lge dlcerjkslyeg dekngea za)>

cukhtlokturev1ihozturev(t,x)

corn1:=gekitf(t)lct(n)70

vh(:,n)152*t(n)>egselct(n);10 && t(n)718vh(:,n)1-52*(t(n)-4)>

egselct(n);8vh(:,n)152>

ekd

turev1^-8*x(:)+x(0)+0*x(8)+vh(:,n)>(:/8)*x(:)-(6/8)*x(0)+(4/8)*x(8)>x(:)+0*x(0)-8*x(8)Q>ekd

8) hgejr jgghghS`1002> Sj1002*exp(m*pl/8)> Sh1002*exp(-m*pl/8)>]fjt1:+0*m> ]j10-8*m> ]`1:+8*m> ]h10+0*m> ]kotr10>

syas xyzukrejg% x1Lj, y1L`, z1Lh jgkatrJ1Sj-x*(]fjt+]j)-(x+y+z)*]kotr>@1S`-y*(]fjt+]`)-(x+y+z)*]kotr>H1Sh-z*(]fjt+]h)-(x+y+z)*]kotr>X1sogve(J,@,H)Lj1dou`ge(X.x) % j cjz jna (cjzr)L`1dou`ge(X.y) % j cjz jna (cjzr)Lh1dou`ge(X.z) % j cjz jna (cjzr)Lkotr1Lj+L`+Lh % ktr jnaJ:1j`s(Lj) % J: japeraetreslJ01j`s(Lh) % J0 japeraetreslJ81j`s(Lkotr) % J8 japeraetreslfogd ok

hoapjss(L`),hoapjss(Lkotr)text(rejg(L`)+2.:,laji(L`),'J0')text(rejg(Lkotr)+2.:,laji(Lkotr), 'J8'),irld

4)hgejr jgghghx1^2.2z1^44.28 4


46/52

U. Jrlcogu - I. etlkeg 09.25.02::gla () gla () gla ( ) @jslt nutup ogduukdjk glalt jyk zjajkdj rezld deerldlr. llk>gla( ) () gla( ) () gla ogur. @ygehe

llk >

() ( ) () , egde edlglr. O fjgde,

Pez(c,) ()() ()() `ugukur.


47/52

U.Jrlcogu (:2/26/02::)

XJ. AF. CJN. EGN. EGK. AF. @G.XJ\LXJG JKJG] \J] ONUGU CKJGXLKJS XOPUGJPL

Xoru :) )t(zy3

dt

dy::

dt

yd3

dt

yd

0

0

8

8

''

8'

0: yx>yx>yx ve ta lgn nouggjr : jgjrjn, t12=2.2:=5 llk, z(t)13*u(t-8)

y(t) 10:

xx deerlklgslanoautu nuggjkjrjn lzdlrek ajtgj`proirjak yjzkz.

( Kot=u(t)=`lrla `jsjajn coknslyokudur) (05 pujk)

Xoru 0) t8

0

:

'0

': e

0

2

x

x

53

:2

x

x

> t8

0

:e8

x

x2:y

(02 pujk)

y(t) erlslkl ^2 3Q sjklye jrjgkdj ode45noautu lge lzdlrlklz. :)2(x: > 0)2(x0

Xoru 8) C C:01:2>C:21C e tj`jkkdj goijrltajdr). C:2deerlklk jrjkajsk %

2.:fjssjslyetge yjptrkz. Dki durduraj togerjksk 2.22: jgkz. (02 pujk)

Xoru 4)t8t02

e:6e4)5.2tslk(8)t(c

yunjrdj verlgek c(t) coknslyokuk trevlklk, c(t)klk nagjtlc ekteirjglke orjkkk ek

ynsen ogduu t jkk tesplt edek `lr jgiorltaj yjzkz. t12=2.:=:2 sk. sjysjgnoaut ve ykteagerl nekdlklz selklz. (:2 pujk)

Xoru 5) t5e0

:

y

x

08

80

y

x

dt

d

> x(2)1:, y(2)1 -: (05 pujk)

x(t) deerlklk tja zak s doaekl yjrda lge`ugjk ajtgj`proirjakyjzkz

Xkjv sresl


48/52

U.Jrlcogu (:2/26/02::)

HESJRGJP

:)

)t(zy3dt

dy::

dt

yd3

dt

yd0

0

8

8

)t(zx3x::x3x :08'8

'08

':0

:

xx

xx

yx

)t(z

:

2

2

x

x

x

3::3

:22

2:2

x

x

x

dt

d

8

0

:

8

0

:

J1

3::3

:22

2:2

> @1

:

2

2

> H1 2::

j1^2 : 2> 2 2 :> -3 -:: -3Q>`1^2>2>:Q> h1^: : 2Q> d12>x21^: : :Q>t12=2.2:=5>uzuk1gekitf(t)>cor s1:=uzuk

lc t(s);12 & t(s)718z(s)1 2> % `lr ilrlgl slstea

egse

z(s)1 3>ekd

ekdy1gsla(j,`,h,d,z,t,x2)pgot(t,y)ygjèg(y),xgjèg(t)tltge(gsla noautu lge dlcerjkslyeg dekngea za)>

0)t8

0

:

'0

': e

0

2

x

x

53

:2

x

x

> t8

0

:e8

x

x2:y

:)2(x: > 0)2(x0

0': xx

t80:

'0 e0x5x3x

;; x21^: 0Q>;; ^t xQ1ode45(hevjp02::,2,3, x2)

;; pgot(t,x(=,:)+8*exp(-8*t)) % y(t) dellalkl lzdlrlglyor;; xgjèg(t),ygjèg(y),irld ok

Jgt proirja dosyjs jjdj verlgaltlr>cukhtlok dx1hevjp02::(t,x)r1exp(-8*t)>dx1^x(:)>3*x(:)-5*x(0)-0*rQ>


49/52

U.Jrlcogu (:2/26/02::)

8) C C:01:2>cor x1:=2.22:=822

C::1((0:622/C:0)*goi:2((085+x)/(085+C C:21Ca1-4*exp(-02*t)+:6*exp(-8*t)+8*slk(t-2.5)> tt1dlcc(a)./dlcc(t)>ss:1ss(:=ekd-:)>

ss01tt./ss:>^j `Q1ajx(ss0)sokuh1t(`)

5) hgejr jgg, hghJ1sogve('s*x-:10*x+8*y+(:/(s-5))','s*y+:18*x+0*y+0*(:/(s-5))','x','y')> x_s1lgjpgjhe(J.x)


50/52

U.Jrlcogu (05/29/02:0)

XJ. AF. CJN. EGN. EGK. AF. @G.XJ\LXJG JKJG] \J] ONUGU CKJGXLKJS XOPUGJPL

Xoru :) )(04

8

8

tr

dt

dx

dt

xd

dlcerjkslyeg dekngealkde> ''8

'0: x>x> xxxx jgkjhjnve ta lgn nouggjr: jgkjhjntr.

t12=2.2:=5 llk, r(t)18*u(t-0)+ te 4*5 ogjrjn verlgaentedlr. Xlstealk y(t) n deerl>y(t) 1 )(40 0: trxx ogjrjn verlgdllke ire,gsla noautuku nuggjkjrjn, su`pgot

noautu yjrda lge ste dx/dt dellalkl, jgtj lse y(t) dellalkl (verlgek t jrjgkdj)lzdlrek ajtgj`proirjak yjzkz. ( Kot=u(t)=`lrla `jsjajn coknslyokudur) (05 pujk)

Xoru 0)

)(382

0:

90

:8 8

0

:

'0

':

tu

e

x

x

x

x t

>

)(3082:

8

0

:

tu

e

x

xy

t

(02 pujk)

y(t) erlslkl ^2 8Q sjklye jrjgkdj ode45noautu lge lzdlrlklz. :)2(: x > 2)2(0 x

Xoru 8) t12=2.:=:>

R1^5 4,6 4,3 4,409 4,034 4,:05 4,2:3 8,


51/52

U.Jrlcogu (05/29/02:0)

HESJRGJP

:)

)(048

8

trdt

dy

dt

yd

)(040

'

8 trxx

)(*)4/:(5.2 0

'8 trxx

'08

':0

:

x

x

x

x

xx

)(42:0y)(

05.2

2

2

25.22

:22

2:2

8

0

:

8

0

:

8

0

:

tr

x

x

x

tr

x

x

x

x

x

x

dt

d

J1

25.22

:22

2:2

> @1

05.2

2

2

> H1 2:0

j1^2 : 2> 2 2 :> 2 2.5 2Q>`1^2>2>2.05Q> h1^0 -: 2Q> d14>x21^: : :Q>t12=2.2:=5>uzuk1gekitf(t)>cor n1:=uzuk

lc t(n);12 & t(n)710r(n)1 5*exp(-4*t(n))> % `lr ilrlgl slstea

egser(n)1 8+5*exp(-4*t(n))>

ekdekd^y xQ 1 gsla(j,`,h,d,r,t,x2)su`pgot(0,:,:), pgot(x(=,0),t), xgjèg('t'), ygjèg('dx/dt'), irldsu`pgot(0,:,0),pgot(y,t), xgjèg('t'), ygjèg('y(t)'), irld

0)

)(382

0:

90

:8 8

0

:

'0

':

tu

e

x

x

x

x t

>

)(3082:

8

0

:

tu

e

x

xy

t

:08 80:':

texxx

:690 0:'0 xxx

;; x21^-: 2Q>;; ^t xQ1ode45(hevjp02:0,^2 8Q, x2)

;; pgot(t,x(=,:)+8*exp(-8*t)-:0) % y(t) dellalkl lzdlrlyor;; xgjèg(t),ygjèg(y),irld ok

Jgt proirja dosyjs jjdj verlgaltlr>cukhtlok dx1hevjp02:0(t,x)r1exp(-8*t)>dx1^8*x(:)-x(0)+r-:0> 0*x(:)+9*x(0)+:6Q>


52/52

U.Jrlcogu (05/29/02:0)

8) hght12=2.:=:>R1^5 4.6 4.3 4.409 4.034 4.:05 4.2:3 8.deier01spglke(t,R,2.35)>lcj`s(deier:-deier0)72.:2

dlsp('jrjkgjk deer'),dlsp((deier:+deier0)*2.5)egse

dlsp('jrjkgjk deere ugjgjajatr')ekd

4) hght1:=2.2:=0>a1-4*exp(-02*t)+:6*exp(-8*t)+8*slk(t-2.5)> tt1huatrjpz(a)> % c(t) ùgukuyor

cturev_01dlcc(a)./dlcc(t)> % c(t)'klk lnlkhl trevl ùgukuyort:1t(:=ekd-:)>su`pgot(0,:,:), pgot(t,tt), xgjèg('t'), ygjèg('c(t)')su`pgot(0,:,0), pgot(t:,cturev_0), xgjèg('t'), ygjèg('dc0(t)/dt0')

5) dxx

xxL

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)slk(*0

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0 x

xzc

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mlaz

zzcesladxxcax:

)),((Pe0)()slk( > k=styjr dzgeadenl nutup sjys

@jslt nutupgjrk rezld fesj`kdj nuggjkgjk>

Pes(c,o

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zeggll ùrjdj nuggjkgjhjn ogursj>

Pes(c, l)1llxlx

lxlz 0

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))((

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:

*:*)),((Pe0)()slk(

elzcesladxxcxm

ll

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