The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed constants.


I'm not very experienced at solving pde-s, so before I immerse myself into the subject I would like to ask some experts if they know a simple way to solve it, or know about its solution in the literature. 



Wave equation with linear coefficients
Perform a Fourier transform on both variables and turn this second order PDE into a first order one. Solve it then perform an inverse Fourier transform.


