Convolution applied to Inverse Transfrom Problem

I have a inverse transfrom function which need to be separated into two equations and then apply the convolution property. Included is an example that is supposedly worked, but is missing a couple of key steps. Hopefully BM can assist in filling in the blanks.

Laplace Transform to solve IVP

Use Laplace transform to solve an IVP style problem. Where y^n(t)+a^2y(t)=0 and a is not equal to zero.

Finding Inverse Transforms f(t) from F(s). REPOST

I had a question posted about Inverse transforms and received a reply which had a something in one line that I didn't quite pick up. Problem requires that the inverse transform be taken of F(s) equation.

Using definition of Laplace transform, find transform of function f(t)

Simple Laplace transform of a product of two trig identities of which I cannot remember how to integrate.

What is the largest interval on which the solution can be defined

Show how you would have done things by hand. Find the solution y(t) to the initial value problem attached. What is the largest interval on which the solution can be defined?