Hey this problem is making me insane so have at it and let me know what I keep screwing up.
Express the solution of the initial value problem in terms of a convolution integral:
$$y''''+37y''+36y=g(t),\\ y(0)=4,\ y'(0)=y''(0)=y'''(0)=0.$$
I keep getting stuck after I apply Laplace transform to both sides and apply the initial conditions. Here's what I got so far:
\begin{align}s^4L(y)-4s^3+37s^2L(y)-4s+36L(y)&=L(g)\\ L(y)&=\frac{L(g)+4s^3-4s}{(s^2+36)(s^2+1)}\end{align}
and this is where I'm confused on how/what to do. Any help is appreciated!