Question

Some the following DE's using Laplace Transform. 1) <3> y - 2 y = a* (0) = | (<4> Y" + 4y' + 4y = 0; Y(0) = 2 ; Y6u =-| - <s> Y" + 4y = f coszt, y(o)= 0; y(o)=4

Answer 1

(3) Given differential equation y'- 2y = ebt ė yro) = 1 First we take the Laplace thansform of all the terms in the diffenential equation. We will plug the initial conditions. into it. SY (8) – YO) 2 resj = 1(854) = 3-6 » (3-2) res) -1 =3-5 = Y(s)-65+)- = 55-555-27 Now we do panttal fraction, - Let, yes) = A + B : A CS-2) +BCS-5)=3-4 => SCATB) = (2015.B) = s-Y .. we get, A+B= 1 2A+ 5 B = 4 Solving we get A= 15 , B= 2 33 ::Y) = . 3-5 + og stz Now taking inverse taplace transformation 9029 = 17 (706) - 5) + 1962) WS weget, :. Solutim of the IVP is y = 1/3 est + 2 eet CSScanned with CamScanner

23+7 47 Giren differential equation .. '+491+4y=0, yco)=2, gro)- Laking Laplace transform of all the terms and plugging the initial conditions g meget, st 4659 –: syco- '(0) + 4(876)-710))+4(900)) = 0 => stY() - 28 +1+45 7(s)-8 +4 405) = 0 => YS) (84745+49 = 23+7. van => Y63) = - 28+7 7 phucth - TS+22 Now we do partial fraction. tet, Yes) = 1.25 + 1 : A+28+ SB = 25+7 in B=Q2, A=3 . 2. Y()= 842 924 ista Now taking inverse laplare transformation weget Y(A) = L*(Yl) 3176623)+22160+2) 1 = 3te 2t+2e-2t. - Solution of the FVpis . y (t) = 36 62ft eeze Scanned with CamScanner be

57 y" + 4y = 800198 , 910) = 0, y' (O=4. Taking Laplace transform of all the terms and plugging into the initial conditions, we get. SRY(s) – syu) – y'(0) +4 ( Y(:)) ..=82*2 → sty() -4+4 Y(5) - → Y 883 (344) = y + y = 45*.8416 → 768) ingenting = 4 y + 8 Torty) 34-344 + 8.787yja Now taking in vense Laplace transform weget. Y(A) = (Y(3) = 4 (844)+847 (594)) we have to man) = 4stat. Let' araya tom gjor we see als) = -5 5474 - =-4 ts saty :: 966) =14&Q (5) --4 as Situs sų tsin27 :. We have the solution of the IV p. 7(4) = 414 (344)+ 8.14 (topky)) 4.sin2t +86 toe sinat y(t) = 2 3in2+ + 2t sinat. this is the solution.. CS Scanned with CamScanner