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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?

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Hint: Multiply your equation by $2\dot{r}$, then it can be rewritten as $$\frac{d}{dt}\left(m\dot{r}^2+\frac{a}{r^2}+2br\right)=0.$$

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    $\begingroup$ @SchlomoSteinbergerstein Actually your differential equation describes the motion of a particle in an external potential independent of time. Therefore one should certainly have a first integral corresponding to the particle energy (hence of the form "squared velocity plus some function of $r$"). $\endgroup$
    – Start wearing purple
    Apr 5 '14 at 14:24

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