I am stuck with the separation of variables for the following PDE:
$ -Ay^{2}\partial _{y}^{2}f(x,y)-y^{2}\partial _{x}^{2}f(x,y)+iBy \partial _{x} f(x,y)+C= \lambda _{n}f(x,y)$
Here, $A, B, C$ are constants and $ i = \sqrt{-1} $.
I believe that the solution to the equation of $y$ should be a Bessel function but I don't know how to split this into two linear equations in the variables $x$ and $y$.
To eliminate the dependence on $x$, I could take the Laplace transform so I get a function of $ f(x,s)$, but I do not know what else to do.
Any hints?