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.
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$\begingroup$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.
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