Is there a better particular solution ansatz to this ODE?

Question

I am trying to find the particular solution, $y(x)$, for (spherical Bessel ) ODE, where $$y''+\frac{2}{x} y' +\frac{A^2 x^2 - B}{x^2}y = C $$ Here $A,B,C\in \mathbb {R} _{>0}$.

I know what the homogeneous solution is and I tried the usual polynomial ansatz for the particular solution $y(x) = \sum a_n x^n$ but I get an infinite series.

Could there possibly be a better ansatz? Anyhow, thought I should check. Thank you.

Answer 1

It is possible to find a particular solution and thus analytically solve the ODE. But this involves complicated integrals where Bessel functions are involves (see below).

This seems not simpler that solving with infinite series.