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Ordinary Differential Equations Exam

Date post: 16-Dec-2015
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Laplace Transform
7
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(Prs)

(10) 1. Find r[/(r)1, rvhere i(r) = {1,' ?:i."'{ 1t) = t U(t- -) = tn 7(t) , n; tt

Yo{+ 3t'1. f ,)" G,nrr) = (- ,), q,n Gr= f\"+t)t

(10) 2. I)etermine Llt' - fer + e4t cootl.

NAI,IE ,K!t.,nttlo * '- "ru *r *: ..

ql+\=t1(t-z) / ', G(s)= {'': (- rr J'n f a t\ [ t-,' ^-',1- r'' .r.,F)=(-') i*;t- i

.L-1)

I

sb\)

>,1

/-'IroL

s- !---[s-,,)t t I

(10) g. Detelnrine f-' I u ltr* | .=lLs'+4s+13J(StS= sL+ks+rr -) st+ks+t3: (s*.f tq

t-'f 3srs l= t^'L' L *-t*'l I o(J 3s+ 6 L-LQtr) tq(s+r)z tt:I F)t (+)StZ

I ss*-s IL (stz)"+q J--1

I2l l_. ? 2. I\s+z\ f3 \_l(si-|'*r'

\t3-t.*, {*rt} =

• (ro) 4,Firrri /t'l#A]

(1u) o.[=o]

p,s'+ trA 1- rls'+ css(s'rt)

A + Bs*C Is s."t lt-'l'lA Ipl

L

C= o, 6=-*-t'E l-)4'/r - (v,). Is F;.]-,/-l*;. /;'iLg/1!r.l= _L-\, S \l:\*,rutlirrl= +.j --s1* ,t ,/\-b-- c

**, ( t'* ,l(:'+r) -(10) s '-' ['. (H)] [k,.d\ l{t +rt)i= F'i r) = -+f,"= i- "fr 'L'lt'{t+ro}]='{*.#i

-> t {f0 = *z- sss[rt) a- 'i-r's (t)

*) Zi+tt-oo5t"'! =[*\ r(")a"Js

*c,cr[A -l F(*)= *

nr;rt = tilIJ = -rfnt,7L

,\ .b1 + ,1(-, t\i

L t t-t"'-rJ

f i, -g*rd,I \o ,t^L+r \,/s- J1y= ..'+ t

4u "- L" da

1.N **=E

r\\ .

t*r,a- + t

Ot t {(r)J =i+ "[(r) = r

"url"l : ul ,;r,laL-t-l- L"J t/,V nle/\n l@-) j, U *"Jr\$"bl,^\ - +t

I

S,ffi-!,n*rplfrl +r) re- ,e" [t)

• {2A} 7. Solre t1' the method of Laplace T\'anslbru the iuitia} r,alue probler rr;,1" 4-Ztyt - ,ly:1y(o) : tl; y'{tl) = o.

&1r"\ -z {ris'Y(i -sylo)- v,lo) t"\ v\'- {r,)

, ,lty'j ,, /r\v] , t"\t]

^-r,L .

LtPt*a,E Dop.,inr

It* y'l = Gr)"+o)= *(civ')) = -*(,vr,l-rr,,: -Yl'l-'Yh)

e st [,) ,A W *.(-, y,(,) -y,r) - { /r,r =

(i) + yr(s) =o3t^('\ - s7v\-_a -J+c

\s

21s) =

("*) r' +c

\5

LI-t*

7z -s/ue-lL,I t -5/EI se

I

- z rlr(s) =[s"-i yn>(+ .*)Y

y(,\. uJt+- i)d' _YG) = J ,nt ?() d,

L-S stn

-))\

g s'1,se

Lt -s /,,:5eal

-s /q,l-se ds +c-;

Itn ordut (sot'ilir'og)orTe

-JZs' ,1trs\

j,/ (tZt

-5 /yoL C

&*\

\ - 5ltlSE

wlr.t f[..coru,yn I

4s- -/

zts /q

= 4 +ce-s, -F,. -s"/qs'e

Y (t) (sor",rncd ilr LNtaLft o"^*,;rr)

C v-rrsf \",

+4*! -t-t^t ont rr*ws'.

!/r1 =6 , ,.!n90r9@

zt:

J.5i

s' J z

I .7,)',lL+qe* ) +\r3 -sl Ito

ob

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