I thought I had these problems down, but I found one for which I am having trouble.

My attempt:

Let where satisfies the BCs

.

So, I picked .

Now the new BCs and IC for are

.

Now assume then

Case I:

Let

But this doesn't seem to help me solve for . Normally I would solve for one constant and then solve for \omega, plug this into then use the IC to solve for the last constant.