I'd like to solve a PDE of the form $$ \frac{\partial^2 z(x,y)}{\partial y^2} = \frac{\partial}{\partial x}\left(g(x)\frac{\partial z(x,y)}{\partial x}\right) $$ where $g$ is known and $z$ is the funciton to be solved for. This looks like a modified version of the wave equation; if $g$ were a constant it would reduce to the wave equation. It's linear and homogeneous, so it has some nice properties, but I've yet to find a method to solve it analytically.
A back-up option is to use separation of variables and pursue a series solution. Writing $z(x,y)=X(x)Y(y)$, $$ \frac{1}{Y}\frac{\partial^2 Y}{\partial y^2} = -\omega^2 = \frac{1}{X}\frac{\partial}{\partial x}\left(g(x)\frac{\partial X}{\partial x}\right) \, . $$ The $X$ part becomes $$ 0 = g(x)\frac{\partial^2 X}{\partial x^2} + \frac{\partial g}{\partial x}\frac{\partial X}{\partial x} + \omega^2 X $$ which I still do not know yet know how to solve.
Can anyone offer some suggestions for how to attack this problem, or do I appear to be out of luck?
EDIT: I did not specify $g$ because I'm interested in the case where it is any monotonically increasing function - essentially, it's a control parameter. Although I'd like to keep it general, any case that makes this problem solvable analytically would be still be useful to know. The case $g(x)=ax^2$ for some $a$ yields a Cauchy-Euler ODE which I can solve analytically. I'm not aware of any others.