I'd just like to ask about any non-trivial solutions or methods for finding such about IVPs of the form below? $$y''+ay'+bf(x)y=0, \quad y(0)=y_0, y'(0)=m$$ with $a,b =$ constants. Or perhaps the general solution (or method for finding so) for just the DE might do.
For the most part, I am only really mostly taught in the case of constant coefficients, but I reckoned it shouldn't be too difficult. It's an IVP of a linear homogeneous differential equation, even if of the second order, so I was hoping there having been already stuff about it posted online. Seemed like a Sturm-Liouville problem, but I'm having a hard time finding solutions or even methods of solving this particular form.
I tried looking up reduction of order, but that kinda needs a given solution to solve it (or at least what I've found does so), and the series solutions are more focused on solving polynomial coefficients, but that's kinda limited.
So with that, maybe there's something I'm missing or the like, so I was hoping asking for help here. Thanks in advance.