Inhomogenous NON-linear differential equation

[Thoughtknot]

I'm having some trouble solving an equation that is similar to a Bernoulli equation. It is of the form

\begin{equation}
\ddot{x}+f(x)\dot{x}^2 = g(x)
\end{equation}

Where x is a function of time, perhaps. I feel moderately certain that there should exist an exact solution, but I've so far been unable to find it, and I have not run into any great amount of non-linear ODEs before.

Does anyone have any idea if it can be solved? Could it be solved by some clever substitution?

 

[Eynstone]

The substitution x'' = x'(dx'/dx) would reduce this to a Bernoulli differential equation.