Let $d_1$ and $d_2$ be positive constants. I'm considering a 2D Poisson-like problem of the form

$$ d_1\frac{\partial^2 u}{\partial x_1^2} + d_2\frac{\partial^2 u}{\partial x_2^2} = f$$ in the domain $\Omega = B(0,1)$ the unit ball. Assume also the Dirichlet boundary condition $u = 0$ on $\partial \Omega$.

I'm interested in finding an analytical solution. My only idea was to rescale the problem so that it becomes a "standard" Poisson problem on an ellipse with axes $\sqrt{d_1}$ and $\sqrt{d_2}$ and then find the Green's function for an elliptical domain, but it isn't clear for me how to do this.

Is there some other, more straightforward, approach to this?