Question

Find the Laplace transform F(s) L(f(t)) given f(t) = 5e-4 sin(5t) + 2e cos(6t). F(8) =

Answer 1

Here basically we use the shifting property and linearity property of the Laplace transform.

Guven function f(t)= 5ē at Run (5t) + 2ězt comC6t). Want to find the Laplace transform of f(t). le F(s)= L (160). Now the shifting theorem for Lablace transfosuonation is: ។ L486113= F(s), then L{ eetg (1)} = F(3-9), where a is ) real number 07 Complex Now led 9.(+) = Reist. then Laplace transform of gi L( 9.(1)}= len 5t? 5 FG, (S) 5?+ 8? As L{ len at} = a 4 st

Thus Lleut regt}= G, (S- (-41) = G, (S+4) 5+ (S+4) S Now let 92 (1) = (ol 67. then Laplace trane formation of 92(1); L {92(1)} = 1 { 108 613 Go(s)(Ray) 6²+5² the above shifting theorem Ljett go(t)}= G ( S+2) = + (stije then using

2824 Cox (61). Since f(t)=584t ein est + 2e then using the linearity of Laplace transformation F(S) = l{}(1} = 5.44eal Rai 5t} +24fēt com bf} s - 5 +2 52+ (5+4)? 6²+ (5+2)2 25 25 1 + 25+ (5+4)? 6+ (*+2)

Thus we are done!