After spending a couple of weeks, I was able to find the solution to a certain differential equation, given below (Well they are the eigenfunctions to be exact):
$$y_n(x) = \frac{e^{ax^2}}{\sqrt{x}}\cdot \prod_{i=1}^n \left( \frac{1}{\sqrt{x}}-t_i \right) $$
where $a\in \mathbb{R}^-$ and the $t_i$'s are constants.
I wish to ask the question whether anyone knows a way to find $$\int_{\alpha}^{\infty}\left|y_n(x)\right|^2$$
Where $\alpha>0$
As it would be nice if I could find a normalization constant for this function. Is there a way that this can be done with contour integration or Feynman's parametrization trick?