You will need to transform your PDE to one with BCs fixed to zero. Try and choosing and such that
I have the question
Use eigenfunction expansion to solve
with IC: and BC's: and where and are given constants.
With this method, I am supposed to start with a trial solution based on
which based on the boundary conditions I would guess to be
but when checking this against the boundary conditions I find that and do not meet the boundary conditions. What am I missing here?
Let me show you a bit more. First off,
, and so the PDE becomes
Now for the BC's. You want so
. These you solve for a and b.
Next, the IC
Now you have a new problem
This you can solve by the separation of variables. Once you have the solution, then use