Consider the singular Sturm-Liouville problem
$$xy'' + (1 − x)y' + \lambda y = 0 = (e^{-x^2}y')' + \lambda e^{-x} y$$
$$\lim_{x \to 0}|y(x)|< \infty, \hspace{0.4cm} \lim_{x \to \infty}\frac{y(x)}{x^k} < \infty \hspace{0.5cm} \text{for some positive integer k}$$
Show that the eigenvalues of this problem are $\lambda_n = n$, $n=0,1,2,...$ And the corresponding eigenfunctions are the Laguerre polynomials $L_n (x)$.
Definition:
The Laguerre polynomial of order $n$ can be defined as follows:
$$L_n (x) = \frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x})$$
That after developing is of the form:
$$L_n (x) = \sum_{k=0}^{n}{(-1)^{k} \binom{n}{k}\frac{1}{k!}x^k} = \sum_{k=0}^{n}{(-1)^{k} \frac{n!}{(n-k)!k!k!}x^k}$$
More information about Laguerre polynomials here
I understand and know that one of the main conditions for posting a question to this community is to attach at least one attempted solution or proof of the problem raised. However, I have spent hours trying to solve this problem, but all my attempts have been unsuccessful, so, feeling frustrated I decided to share this problem with this community with the purpose of knowing a possible solution on this exercise instead of posting alongside this problem futile attempts.
I apologize for my inability to attach any coherent attempt on this Sturm-Liouville problem and welcome any help.