Trivial Solutions of the modified Wave Equation

I recently found the solution to a modified wave equation where u=0 at the x boundaries, . I used separation of variables, and found that I had to have a positive separation constant (eigenvalue?) in order to get non-trivial solutions, which is the case for the original wave equation. I strongly suspect that this is the case when other terms are added to the wave equation (non-homogeneous, u sub x added, etc.). Is there an easy way to see that this must be true without solving each case for all possible variations?