Answer to Question #343118 in Differential Equations for Tobias Felix

Question

Find the Laplace transforms of the following function:

1. L{t^2sinkt}

Answer 1

We have

$L(sin kt)=\frac{k}{s^2+k^2}$

$L(t^2sin(kt))=(-1)^2\frac{d^2}{s^2}(\frac{k}{k^2+s^2})=\frac{d}{ds}\frac{-2ks}{(k^2+s^2)^2}=\frac{-2k(k^2+s^2)^2-8ks^2(k^2+s^2)}{(k^2+s^2)^4}=\frac{6ks^2-2k^3}{(s^2+k^2)^3}$