I have been dealing with the following PDE, where $f:\mathbb{R}^2\rightarrow\mathbb{C}$:
$$ \frac{\partial^2 f}{\partial x^2}-A \frac{\partial^2 f}{\partial y^2}-Be^{4x}f+Ce^{6x}g(y)f=0 , $$ where $A$, $B$, and $C$ are positive real constants. I want an analytical solution, but the textbooks and papers usually do not consider hyperbolic equations like this one, at least they don't show any method for solving it analitically, presenting an explicit solution $f(x,y)$.
The equation becomes trivial if $B=C=0$, or if $g$ is either a constant or an exponential of the form $e^{\lambda y}$, for some $\lambda\neq 0$. I would like to find some kind of analytical method to give at least an "algorithm" that makes it possible to solve this equation for a function $g$ as general as possible.
This PDE comes from a physics problem where there is an enormous ambiguity concerning boundary conditions (b.c.), so any b.c. "good enough" can be used. That is why I cannot present this problem in a very precise way as an Initial Value Problem.
I have verified that the equation is not separable, and Fourier and Laplace Transforms do not seem to help, as far as I could try those methods. I have also tried some usual change of variables with no success.
If anyone could come up with an idea, that would help me a lot. Thanks in advance.