I want to find a solution to the PDE $$ (1-x^2)\frac{\partial T}{\partial y} = \frac{\partial^2 T}{\partial x^2} $$ With boundary conditions $$ T(x=a) = T(x=-a) = 0\\ T'(x=0) = 0\\ T(y = 0) = 1 $$
I think this is solved by separation of variables like $$ T(x,y) = X(x) \cdot Y(y) $$ which leads to the Sturm Liouville Problem $$ d/dx( X') + (\lambda (1-x^2))X = 0 $$
I found a source that claims the solution is of the form $$ T(x,y) = \sum_{m=0}^{\infty} (A_m e^{-\tilde\lambda_m^2 y} \cdot \sum_{n=0}^{\infty}a_{nm}x^n) $$
Ideally someone could provide me a way to find this solution. But I'd be also happy to just find a simple way to proove correctness and calculate the coefficients.
Edit: I tried it on my own, but do not get the correct results. Here is what I tried: Using a wolfram alpha solver, I got the solution, which is some parabolic cylinder function. I then expressed this solution as taylor series like $$ X(x) = \sum_{n=0}^{\infty} a_n x^n $$ To calculate the coefficients $A_m$ I applied the BC $T(y=0)=1$ which gives $$ \sum_{m=0}^\infty A_m \cdot a_{nm} = \begin{cases}0 & k>0 \\ 1 & k = 0\end{cases} $$
I tried to approximate the values by using a finite sum. But the solution seems quite unstable (the values $A_m$ for larger $m$ are very large with alternating sign).
Can someone tell me what's wrong with this solution and how to solve the problem?