I solved my PDE $\partial_X\theta - \frac{2}{Y(1-Y)}\partial_{YY}\theta = 0$ by using separation of variables (cf. this post). I got two solutions $F(X)$ and $G(Y)$ where $\theta(X,Y) = F(X)\, G(Y)$. Now, I want to find the eigenvalues and eigenfunctions of this Sturm-Liouville problem. How can i get the general solution for this?
$F(X) = a_1 e^{-\lambda X}$
$G(Y)=c_1\, D_{(\sqrt{2\lambda}−8)/16}\left(\frac{(2Y−1)\lambda^{1/4}}{2^{3/4}}\right) + c_2\, D_{−(\sqrt{2\lambda}+8)/16}\left(\text{i}\frac{(2Y−1)\lambda^{1/4}}{2^{3/4}}\right)$
Here, $D_n(z)$ is the parabolic cylinder function. The boundary conditions are $\theta(X,0) = 0$ and $\partial_Y\theta(X,1) = 0$ for $X\geq 0$. The initial condition is $\theta(0,Y)=1$.