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\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?