I say ugly because it involves omegas and too much messy algebra for my liking.

Yeah, here it is:

I need to solve for

using Laplace transforms for each of the following cases: when

and when

Both

and

are constants here. So basically in short, I'll need to solve the following two equations with the initial conditions I listed already:

1.

2.

I'll start with Case #1 (

). Here's what I've done so far:

Taking the Laplace transform of both sides gives me:

Using the facts that

,

, as well as

, I rewrote the above expression as:

Once I apply the intial conditions and factor out

, the 2nd and 3rd terms on the left side disappear and I am left with:

Dividing both sides by

gives me:

It looks like finding the inverse transform is going to involve partial fraction decomposition and will be in the form

. I'll probably have to solve for A, B, C and D. The next step after that, as far as I've gone, is:

And that's all I feel like doing for now, because proceeding is going to be ugly...

...bleh, anyone else want to give it a go?