TITLE: Bernoulli Differential Equation of Second Order
QUESTION [7 upvotes]: How one can solve a Bernoulli differential equation of second order?
i.e., solve the DE
\begin{align}
\frac{{d^2 y}}{{dx^2 }} + p\left( x \right)\frac{{dy}}{{dx}} 
+ q \left( x \right)y = g\left( x \right)y^n 
\end{align}
where $p$, $q$ and $g$ are continuous functions in an interval $(a,b)$ and $n$ is a real number.

REPLY [5 votes]: Suppose that  someone could answer to your question, i.e. give an analytic formula for $y(x)$ as general solution of :
\begin{align}
\frac{{d^2 y}}{{dx^2 }} + p\left( x \right)\frac{{dy}}{{dx}} 
+ q \left( x \right)y = g\left( x \right)y^n 
\end{align}
Then, this result would apply in case of $g(x)=0$ :
\begin{align}
\frac{{d^2 y}}{{dx^2 }} + p\left( x \right)\frac{{dy}}{{dx}} 
+ q \left( x \right)y = 0 
\end{align}
So, the general solution of the homogeneous second order ODE would be discovered. This would be a great discovery !.
As a consequence, you will have to wait probably un long time until a genious gives the general solution of your problem.