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[tex]\begin{align}u'' + c x^n u = 0 \end{align} [/tex]

one finds that one solution is

[tex]\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} [/tex]

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?