Simple-looking nonlinear ODE with fractional power 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?
 A: Using contact transformations, Lie showed that second order equations of this type can be transformed to the trivial equation $\frac{d^2 P^\prime}{dx^{\prime 2}} = 0$.
Edit: an algorithm for this is described in Kumei and Bluman When Nonlinear Differential Equations are Equivalent to Linear Differential Equations, SIAM J. Math. 42(5) (1982) 1157-1173.
Alternatively, a more general technique is described in Bluman and Dridi New Solutions for ordinary differential equations, J. Symb. Comp. 47 (2012) pp. 76-88. It uses a combination of invertible and non-invertible mappings. They treat several examples in detail.
Edit(2): The paper by N. Euler, Transformation Properties of $\ddot{x}+f_1(t)\dot{x}+f_2(t)x+f_3(t)x^n=0$ J. Nonlin. Math. Phys. 4(3-4) (1997) pp. 310-337 provides the conditions for an equation of this type to be integrable, as well as the conditions for it to have Lie symmetries.
He also considers non-point transformations $X=F(x,t)$, $dT = G(x,t)dt$ to transform the equation to $\frac{d^2X}{dT^2}+k_1\frac{dX}{dT}+k_2X^p=0$ for constants $k_1$, $k_2$ and $p\in \mathscr{Q}$. I've tried substituting the corresponding values for your equation into his conditions, but the results underline your point about the lack of symmetries.
A: For $k=0$, according to Maple the general solution is
$$ \text{arctanh}(x) \pm \int^{P(x)/\sqrt{x^2-1}} \dfrac{du}{\sqrt{u^2 - 2 c \ln(u) + b}} = a $$
Another family of solutions (for all $k$) is 
$$ P(x) = \left( {\frac {k \left( k+1 \right) }{c}} \right) ^{-(k+1)/2}{x}^{k+1
}
$$
For $k = -3$, there are some other closed-form solutions, e.g.
$$ P(x) =  \frac{8}{c(1-x^2)} $$
and
$$ P(x) = \frac{24 (21 x^2-5)}{c (3x^2+5)^2}$$
