I'm having trouble figuring out where to go with this PDE. We've done equations that can be reduced to have homogeneous boundary conditions, but we haven't covered much where they can't (or at least not in the way I've tried). In the question below I tried the method I've learned so far, letting u(x,t) = X(x)T(t), but that got me nowhere because the factor of gamma*u threw things off a bit.
Anyways, my question is: What should I substitute for u(x,t) so that I can use separation of variables? I'm just looking for a starting point for this question, and then I should be able to take it from there.