If you want steay state (i.e. ) then solve
subject to your BC's.
What would be my next step?
Separation of variables?
Case 1 -- K=0
using boundary condtions I got
so there exist no nontrivial solution
Case 2 --- K > 0
using boundary conditions also produces nontrivial solutions.
Case 3 ---- K < 0
and this is where I run into problems again
which just leaves me with
so my question is if
produces a nontirvial answer
for n = 0,1,2,3,...
my question is what does
Can I just consider that a nontrivial solution with ?
And am I approaching the problem correctly so far? Thanks in advance
Your problem is
The first boundary condition is a problem so we'll change this problem into a new (do-able) problem.
Let and choose and such that the new problem is
Without the source term ( ), separation of a variables would lead to what you have
Hence we look for a solution of the form
noting that we'll need a Fourier series of the form
for the source term
PS. Yes on