A particular nonlinear second-order ODE with parameter

Question

I am trying to solve the ODE:
$$ y'' = \frac{ay(y')^2}{1+ay^2}$$
where $a \geq 0$ is a given parameter, and suppose we have initial conditions $y(0) = \mu, y'(0) = \eta$. WolframAlpha tells me that when $a = 1$ we have a general solution of the form $y = \sinh(c_1(c_2+x))$. Of course if $a = 0$ we get a solution of the form $y = c_1 x + c_2$. Is there anything that can be said about the case of general $a$?

Answer 1

For $a>0$, let $z:=\sqrt{a}y$. Rewriting the ODE in terms of $z$, it becomes $$ z''=\frac{z(z')^2}{1+z^2}, \tag{1} $$ which is the original ODE in the case $a=1$. Therefore, you can use the solution provided by WolframAlpha in this case to write the solution for $a>0$ as $$ y(x)=\frac{\sinh(c_1(c_2+x))}{\sqrt{a}}. \tag{2} $$