I am trying to solve the following nonlinear ODE for a function $P(x)$:
Here, $k$ and $c$ are arbitrary parameters. By rescaling $P(x)$, one can without loss of generality set $c=1$, so really this equation is only parameterized by $k$.
I wonder if there exists a closed form solution for this equation, at least for integer $k$? Although it looks very simple, my efforts to solve it have so far been unsuccessful. Nonetheless, I have been able to find some very simple solutions: for instance,
- when $k=1$, the equation can be integrated, resulting in
- when $k=-2$, by inspection I found
- when $k=-3$, by inspection I found
These solutions look simple enough that I'm hopeful that a general solution exists. Perhaps there even exists some standard transformation to linearize such equations?