I am trying to solve the following nonlinear ODE for a function $P(x)$:

$$\left(1-x^2\right)P''(x)+k(k+1)P(x)=cP(x)^\frac{k-1}{k+1}.$$

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

$$P(x)=\frac{1}{2}+a\left(1-x^2\right)+b\left[x+\left(1-x^2\right)\mathrm{arctanh}\,x\right],$$

- when $k=-2$, by inspection I found

$$P(x)=\sqrt{2}\frac{4x}{3+x^2},$$

- when $k=-3$, by inspection I found

$$P(x)=120\frac{1-x^2}{\left(5-x^2\right)^2}.$$

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?