I consider this problem.
Suppose $y=y(x)$ on $0<x<\infty$ with boundary conditions: $y(0) = y(\infty) = 0$ satisfying
$$x^2 \frac{d^2y}{dx^2}+x \frac{dy}{dx}+(ax-x^2)y=\lambda^2y$$
where $a \ge 0$ and $\lambda > 0$
This ODE has a trivial solution: $y = 0$.
I want to know when it has a non-trivial solution. Some calculations gave $a=0,\lambda \in \Bbb N$.
But I can't prove $a = 0$ is necessary to have a non-trivial solution. Please tell me your idea.