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:
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?