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?

Find the general solution to the driven differential equation

Show how you would have done things by hand. Find the general solution to the driven differential equation attached.

Find the solution to the attached

Show how you would have done things by hand. Find the solution to the attached.

Consider the differential equation attached

Show how you would have done things by hand. Consider the differential equation attached. The graph is shown. a) Is this differential equation linear or nonlinear? Is it autonomous or nonautonomous? b) Without solving, use the graph to determine the limiting value... (see attached for rest).

Use variation of parameters

Show how you would have done things by hand. One solution of the equation attached is y(t) = t. Find the general solution. Use variation of parameters to find a particular solution of the equation attached.

Differential Equation: Continuous Functions; Fundamental Set of Solutions; Coefficient Functions

Consider the attached differential equation where I = (a,b) and p,q are continuous functions on I. (a) Prove that if y1 and y2 both have a maximum at the same point in I, then they can not be a fundamental set of solutions for the attached equation. (b) Let I = {see attachment}. Is {cos t, cos 2t} a fundamental set of solu ...continues