Finding extrema of non-linear second-order ODE I'm dealing with a non-linear ODE of the form 
$$a\frac{d^{2}y}{dx^{2}}+bA(x)\frac{dy}{dx}+cy^{3}+y=[A(x)]^{2}$$
where a,b, c are positive constants and the function A(x) is a real valued and continuously differentiable (except, maybe, at x=0). The only unchangeable boundary value is that $y\rightarrow 0$ when $x\rightarrow \infty $.  At $x=0$ y might be required to be a given real number or it might be allowed to diverge, both cases are of interest.
Unsurprisingly, I can't solve the equation. I only want to study its qualitative behaviour so
(a) Is there a way to find if the solutions of the equations have extrema in the interval $[0,\infty )$?
If "yes", then
(b)  Is there a way to approximately know where the extrema are located?

Numerically computing the solution to the differential equation using most approaches would rewrite the problem as
$$\begin{cases}z'=\frac1a\left(A^2-bAz-cy^3-y\right)\\y'=z\end{cases}$$
In which case one would be computing $y$ and $y'$ simultaneously, and the extrema can be numerically deduced based on $y'$, which you can search for being zero with some tolerance, or very large in magnitude. Determining if it actually results in a relative extrema can't be done in general of course, but you can gain some insight on it based on whether or not $z'=y''$ is positive or negative near the point of interest, or simply by looking at how $y'$ behaves around the points of interest.
