Non-homogeneous PDE with Non-homogeneous BC's - Eigenfunction Expansion

I have an exam tomorrow, and I came across one question that I don't really know how to start and I was really hoping someone could help me out with it.

Use the method of eigenfunction expansion to obtain a solution to

with initial condition:

BC's:

where are given constants.

So I need to start with a trial solution based on the homogeneous part of the problem i.e.

which if the boundary conditions were something like I would start with a solution

but I don't know how to determine a trial solution if the BC's are non-homogeneous.