Can you show me how you got that the DE for was
? Because I'm getting
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.
Okay, so I figured this problem out a while back but I just didn't post back. However, for completeness I think I should post the procedure to solve this non-homogeneous PDE.
First we solve the DEQ
Once we have a solution to this DEQ, we need to normalize the eigenfunctions. Now that we have the normalized eigenfunctions, we take the innerproduct
.
Here we must remember that . Thus we have
.
Then of course, we sum up all .