Question

La placE TRANSFORM To solve y"-24'ty = f(t), yco)=7 fit) = {e-s 125 Show all STEPS yicos=1

Answer 1

Elixen . y"_24'+y: f1t), y(o): 1, 4'lo). I ,125 SI : 0 { t-5, 17,5 y" - 2y'+y = f(t) Applying Laplace transforms on both sides 214") -2 214') + 214) = 26914)) Lly") •2(y) s y(s) -5-1 > Lly) - SY(S) - YO) Lly'): syis) - 1 >3 C s? Y(S) - S410) - Y'(o) 2 Lly) = y(s) s09 26f1tjj ft dit D goed .st oedt + s (4. sjest de 5 D Ot 1(4-5) est dit S It est dt - 5 let st 5 5 ust (er you] + 5 Leary S S 5 B CS Scanned with CamScannen

-55 - 55 ୧ s -5S 11 eiss 5 e ( 5+Ys] + (-56 [5+s] sef sepas S S - 55 s2 ale ripity 47(4)) -58 e alo Substitaling 26 ③.0.0 in 0, we get ${y(s) - S=1 7 SY(S) – $7415) ='é -55 52 -55 Y(s) ( 5+25+1] = e t St3 -55 e s 3 t at Y(s) Ś (5-1)? (5-1)} (5-2) Applying invenise laplace transform on be side CS Scanned with CamScanner

2 -55 e -55 25+1 3-25 2 + $(5-1)3 s² 2 (5-1) 5 -55 е 20 st 네 -55 5 e + (5-1)" B S(+-5) ult) + + s(+-5) ult) + -2 et 514-5) ult) + 5tet slt-5)uct l'less (2+t-zel +5ett) 814-5)ult) s(5-1) I'l S ellt+1) celt) (5-117 t L"? ostetult) y(t) = (2+2-zeltstet) $12-5)4%e°(2+1) (14) + 34 et ult) t :-. ylt) - (2 +7-2ett Stel) 812-5) 417) tetult) + 4t et ult) CS Scanned with CamScanner
Given differential equation is solved by using Laplace and inverse Laplace transform.