Second order differential equation with multiple bessel functions

Question

I have an differential equation which is $af(R)=\frac{1}{R}\frac{\partial}{\partial R} \sqrt{R}\frac{\partial}{\partial R}\left(f(R) 3\nu\sqrt{R}+g(R)cR^2\right)$ where $c,\nu, a$ are all constants. It is known that $g(R)=R^{-1/4}J_{1/4}(R)$, but where to proceed from here? A test solution $f=R^bJ_{1/4}(R)$ doesn't seem to work.

Answer 1

Below, there is a calculus which is more a guideline to solve this hard problem, than a complete solving. Theoretically the most difficult part could be overcome, but with big equations, which is not carried out here.

Nevertheless, the incomplete result suggests the general form of the function $f(R)$, in which the argument of some Bessel functions is not always $(R)$, but also $\left(\sqrt{\frac{\alpha}{3\nu}}R\right)$