How to obtain an exact solution to nonlinear second order ODE

Question

I need help in analytically solving this nonlinear second order ODE,

$A y(x) + y'(x) \Bigg( B + \frac{C y'(x)}{D y'(x) - y''(x)} \Bigg) = 0$.

Any help is appreciated.

Answer 1

The equation can be written as $$ y''=D\,y'+\frac{C\,y'^2}{A\,y+B\,y'}. $$ The independent variable does not appear explicitly. Let $y'=p$ and consider $p$ as a function of $y$. Then $$ y''=\frac{dp}{dx}=\frac{dp}{dy}\,\frac{dy}{dx}=p\,\frac{dy}{dp}. $$ The equation becomes $$ p\,\frac{dy}{dp}=D\,p+\frac{C\,p^2}{A\,y+B\,p}. $$ One solution is $p=0$, which implies $y$ is constant. If $p\ne0$ we get $$ \frac{dy}{dp}=D+\frac{C\,p}{A\,y+B\,p}, $$ which is an homogeneous equation. You can solve for $p$ as a function of $y$ and then for $y$ as a function of $x$.