Solution to an ODE in the form $y'' + \frac{m}{mx+b} y' + k(mx + b)y = 0$

Question

I am trying to find a solution to a second order ODE in the form $y'' + \frac{m}{mx+b} y' + k(mx + b)y = 0$ where $m,a,b$ are constants. I cannot find a method to solve such form. Any tips would be greatly appreciated.

Answer 1

The point $x=-b/m$ is a singular point for this DE. Near any other point $x \in \mathbb C$, this DE has an analytic solution. And near $x=-b/m$ there are series solutions (described by Frobenius, and included in undergraduate ODE texts). These series solutions possibly involve fractional powers of $(x+\frac{b}{m})$ or logarithms.

In this case, the solution is in terms of Bessel functions: $$ y(x) = C_{{1}}{{\rm J}_{0}\left({\frac {2}{3\,m}\sqrt {k} \left( mx+b \right) ^{{\frac{3}{2}}}}\right)}+C_{{2}}{{\rm Y}_{0}\left({\frac {2 }{3\,m}\sqrt {k} \left( mx+b \right) ^{{\frac{3}{2}}}}\right)} $$ The $Y_0$ term behaves logarithmically near $-b/m$.