Question

Use the Laplace transform to solve the following initial value problem: 44" + 2y + 18y = 3 cos(3t), y(0) = 0, y(0) = 0. a. First, take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation and then solve for L{y(t)}. Do not perform partial fraction decomposition since we will write the solution in terms of a convolution integral. 3s L{y(t)}(s) = (452 + 25 +2s + 18)(52+9) b. Express the solution y(t) in terms of a convolution integral. 9 arctan 4s + 1 V71 s(t) = 0 So ap log(32 +9) – log(22++9) + Zo arctan( 1 ) 1 dw 3 20171

Answer 1

3 "}dolly = 11 3 (25 + 1½)2 + 7/4 Do hacia (15+1/4)² + 7/16 2.) 3 4 (s +1/4)² + 7/16 12 s² + 71 ( By first 16 stifting property n/t 3 4 ext sin (171 t) (3774) 3 14t X sin (1571 4 JFI t 3 - 1/4 t 12 pis} = sin 71 571 374 ) = f(t) (say) By convolution Theeren, 242 +15).40)} = f(+) * git) t - Star f(W get-uldu t ie 3 S (45+25+18) (s?+9) sin (IH u). 171 Cos 3(tu) du. t - 34 sinnu.com - cos 3 ct-uldu C45+25+18) (s'+9) t 3 ie. yit) = se JFT Cos 3 (t-u) du sin (hu).