I have a differential equation of the form a y'' + b y/x = E y (The origin is a 1D Schrödinger equation for a potential of the form $-1/x$). I am only interested in the ground state energy, i.e. the lowest order solution.
Is there a good, systematic way to tackle this? I used a lot of hand waving:
I said that for $x \rightarrow \infty$, the potential term is negligible and the equation is a simple homogeneous 2nd order ODE with constant coefficients, which has solution $e^{-kx}$ for some $k$. So as an overall ansatz I choose $f(x)e^{-kx}$, which yields a (f'' - 2k f' + k^2 f) + b f/x = E f.
I then argue -- that is where the hand-waving occurs -- that the ground state would have a polynomial of the lowest possible order for $f$. A constant (order $0$) is not possible, since then nothing cancels the $1/x$ in the equation, so I try the ansatz $f(x) = x$. With that, I can indeed solve the equation and obtain conditions for $k$ and $E$:
$-2ka + b = 0$ $ak^2 = E$
This allows me to solve for $k$ and $E$.
But is there a better, more rigorous way?