Question

Use the convolution theorem to find the inverse Laplace transform of the given function. 1 اولیا اور = (4) (2 + 1) دی 1 (t) =

Answer 1

eriven that I' { 50 (541) 8 (1) use convelution theorem a { st(st) ) T'{ Filsbu facs)} here first at 33 & Fa (s) = 1 52+) now iff, } - Ustab=1120 Ilffals)} = { sh? sinto (silber 8422}= sinat now according to convelution theorem ("If ni sati ht] =t") Tlff,(s) falsip filt) * Fall) S'ficu) afzle-u) ty that sint Stu2 sin lt-w)dy + न 1 S 4² sin (4-8 du 6 now apply Integral by parts ſfg' = $9-fflg f = 4² . Izau g'a Sin 14-t) C8 (4-t) ga T Tlns u cop (4-4) de1] y t [ - 4cos(4-t) -S-a4 a (-4G$C 4-7)+2 f4 ca$ (4+) dk) → From © con Sider Sucose-t) dy (fgl= fg- Iflg f-4 g=COBI 4-t) f'=1 sur scurtset = usin(4+) -sciment du y sin(4-1) 4 colety@ Integrate by parts ya Similunt)

기 SubstilureGoinD ㅋ --cRlqrt) + a q sin(x-t) + 2 C8(t) DIE G.Sin(t) - CB(4t) tgtt 2t Sijolt-t) - -Bt-t) - o to + cs8(t) 8t - 고 U 볼 -0-148 여 +8 1 ) (Sastata sint) Os sint dt = cost) { testant) t2 a tt 1