I've been doing some work that involves particle's motion subjected to central forces. I was trying to compute the perihelion advance of the particle's path. For that I've been following this reference, where found the equation of motion, I considered a small perturbation to circular orbit. The problem was that instead of the typical harmonic oscillator ODE I ended up with an equation of the form: $$\ddot{x}+A~\dot{x}^2 + B~x=0,$$ where $A$ and $B$ are constants. Can we retrieve any information from this equation without actually solving it, I mean, in the sense that we can find an expression for the apsidal angle if the equation was of an harmonic oscillator?
Note: In case someone is wondering why the appearance of the $\dot{x}^2$ term, it is because the considered space is curved.
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