For a problem solving class I need to find the general solution of ODE $y''+(e^{-2x}-\frac{1}{16})y=0$ in terms of $J_{\nu}$ and $J_{-\nu}$, if possible. $\nu$ represents the Bessel parameter.
A hint is given, namely that a useful substitution would be $e^{-x}=z$; this should lead to the ODE being reduced to a Bessel equation. Substituting this value leads to the ODE $z^2y''+(z^2-\frac{1}{16})y=0$, which has some form of a Bessel equation with $\nu=\frac{1}{4}$, apart from missing a first order derivative $y'$.
What is the proper way of solving this equation by reduction to a Bessel equation?
Can this be done?
Best Answer
Letting $t = \mathrm{e}^{-x}$ we have $$ \dfrac{d}{dx} = \frac{dt}{dx}\frac{d}{dt} = -\mathrm{e}^{-x}\frac{d}{dt} = -t\frac{d}{dt}\\ \dfrac{d^2}{dx^2} = -t\frac{d}{dt}-t\frac{d}{dt} = t^2\frac{d^2}{dt^2}-t\frac{d}{dt} $$
To tidy up this post with an answer (As I see that you already found your way).