I came across this differential equation in a problem I'm working on:
$$m \ddot{r} - \frac{a}{r^3} + b =0 \, ,$$
where $m$, $a$ and $b$ are positive constants and $r=r(t)$. Furthermore, since this problem arises in a certain physical system, I'm only interested in functions such that $r(t) \geq 0$ at all times $t$.
I would like to obtain the first integral of this equation, but I don't know how to proceed. I only know how to solve ordinary linear differential equations and some specific non-linear ones, but not this one.
Can you give me any hints as to how I might solve this?