Analytical solution to non-linear 2nd order ODE

Question

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:

$$y'(x = 0) = y'(x = L) = 0$$

Answer 1

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!

Answer 2

$y(x)=0$ is a solution to the problem.