2
$\begingroup$

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?

Improve this question
$\endgroup$
5
$\begingroup$

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

Improve this answer
$\endgroup$
  • 1
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.