You probably have to use Separation of Variables.
I'm not sure I think your last steps there are valid. In pde's, separation of variables doesn't look like it does in ode's. You would assume that in which case and Finally, Plugging all this into the pde yields
I'm not sure exactly what this means. Does it mean that any solution that is a product of a function of and a function of is a solution to the pde? I'm inclined to think so.
There are a few other's. I really like the separation of variables of the Sine-Gordon equation as presented in Debnath's book (rather clever). A very good question is - when does a nonlinear PDE (in variables ) admit (functional) separable solutions in the form