Analytical solution to non-linear 2nd order ODE Is there an analytical solution to the following non-linear ODE?
$$y''(x) = A \frac{y}{B+\alpha y} ... (1)$$ where $A,B,\alpha$ are constants.
The boundary conditions are:
$$y'(x = 0) = y'(x = L) = 0$$

Let us assume that $A = B = \alpha = 1$ for the sake of simplicity.
Then we can multiply both sides by $y'$, whence we get
\begin{align*}
y'' = \frac{y}{1 + y} & \Rightarrow y''y' = \left(\frac{y}{1+y}\right)y' = \left(1 - \frac{1}{1+y}\right)y'
\end{align*}
Then we can integrate both sides in order to obtain
\begin{align*}
\frac{(y')^{2}}{2} = y - \ln|1 + y| + c
\end{align*}
which is not solvable in terms of elementary functions.
Similar procedure can be applied to the general case.
Hopefully this helps!
