Particular solution to the Non-Homogeneous Poisson

Question

I am stuck with the following Non-Homogeneous Poisson equation:

The boundary conditions are:

My effort so far is that I got the homogeneous solution (with some BCs applied):

How do I find the particular solution?

Answer 1

If anyone is interested, I found out why the system was un-solvable!

This is essentially an un-damped resonantly forced system, and it blows up.

The rhs of the equation has the same kx and ky as those in the homogeneous solution. If the homogeneous solution is given, say kx1 and ky1, the particular solution becomes: Now it is clear, that at kx=kx1 and ky=ky1 the solution is infinite.