Consider the following non-homogenous heat equation on
Find a solution of the form
where are the eigenfunctions of an appropriate homogenous problem, and find explicit expressions for
Sub these back into the PDE to obtain
Also since we have
which is homogeneous.
Then we have
Since this is homogenous, we can solve by letting
but now I get a little lost.....I think I know where to go, just not exactly how to get there. Already have calculated that , which will lead to what I need to prove (I think).