I am trying to solve the following, parabolic homogenous PDE:
$$\partial_y\left[(x^2-y^2)\partial_yf\right]+x\partial_xf+y\partial_yf=0$$
Equivalently, one can write it in the following canonical form:
$$\partial^2_yf+\frac{x}{(x^2-y^2)}\partial_xf-\frac{y}{(x^2-y^2)}\partial_yf=0$$
which also leads to the following form:
$$\partial^2_yf+\partial_x g\partial_xf+\partial_y g\partial_yf=0$$
with $g(x,y)=\log(\sqrt{x^2-y^2})$. I have attempted the method of separation of variables and consulted the Handbook of linear partial differential equations for engineers and scientists by Polyanin et al, but couldn't proceed. I am looking for a general solution that I can match with a boundary condition at $f(x, \pm 1)=e^{-\lambda x^2}$. The problem is on the domain $(x,y) \in [1,\infty)\times[-1,1]$