How to arrive at Bessel function solution to 1D polynomial potential

1. Nov 15, 2011

Peeter

My quantum text, leading up to the connection formulas for WKB and the Bohr-Sommerfeld quantization condition states that for

\begin{align}u'' + c x^n u = 0 \end{align}

one finds that one solution is

\begin{align}u &= A \sqrt{\eta k} J_{\pm m}(\eta) \\ m &= \frac{1}{{n + 2}} \\ k^2 &= c x^n \\ \eta &= \int_0^x dx' k(x')\end{align}

I'd like to know how this was arrived at. Could somebody outline what set of change of variables would one make to put the differential equation above into the standard Bessel differential form?

2. Nov 15, 2011

dextercioby

Try first a change of variables xn/2 = t. Compute the ODE in terms of u(t) then.