I am interested in solving a differential equation of the form
$$a\,\frac{\partial^2 f}{\partial x^2} + by\,\frac{\partial f}{\partial x} + cy\,\frac{\partial f}{\partial y}+itf = 0$$
for $f=f(x,y)$, with $a$, $b$, $c$ and $t$ constant, and boundary condition $f(x_0,y) = 1$, and the other boundary term left general. If a second boundary condition is required, I am interested in conditions of the form $\dfrac{\partial f}{\partial x} (x_1,y) = 0$.
Since there is only a second derivative in one of the variables, I thought it might be tractable, but I have tried the method of characteristics and couldn't find get the equation into a form I could solve.
Are there some clever tricks to solve differential equations of this form?
Many thanks in advance!