I am a novice at differential equations, so apologies if this question is naive. I am looking for solution methods for the following ODE: \begin{equation} A s^2 f''(s) + B s f'(s) + C f'(s)^{\frac{\gamma }{\gamma -1}} + D f(s) = 0\ , \end{equation} where the upper-case letters are constants, and $0 < \gamma < 1$. This is almost a Cauchy-Euler equation, except for the non-linear term $ C f'(s)^{\frac{\gamma }{\gamma -1}}$. I have the following question:
Is is possible to say anything about the general solution to this equation?
I ask this because the Cauchy-Euler equation has a known explicit solution; I wondered if something similar might be possible here. But perhaps the non-linearity makes this impossible?
If the answer to the above question is negative, or if the question itself if too vague, here is some additional information regarding my specific application. Consider that $f^0(s) = \phi s^\gamma$, with $\phi = -(-1)^{\gamma } \gamma ^{-\gamma } C^{1-\gamma } (\gamma (A (\gamma -1)+B)+D)^{\gamma -1}$ is a solution to this equation. As $s \to \infty$, I would like $f(s) \to f^0 (s)$; is this constraint enough to nail down $f$, up to constants of integration? I appreciate that this constraint might be ill-defined since $f^0$ grows without bound, but hopefully the idea is clear.