I have found the following ODE in the context of a mechanics problem and am now asked to find a first integral of this equation.(adsbygoogle = window.adsbygoogle || []).push({});

[tex](m_1+m_2)\ddot{r}-m_1Cr^{-3}+m_2g=0[/tex]

I know this means that I'm supposed to find an equation of the form [itex]F(\dot{r},r)=\mbox{const.}[/itex] but I don't know how to achieve that.

Am I expected to guess a coordinate transformation whose associated constant of the motion (in the sense of Noether's theorem) is of the form [itex]F(\dot{r},r)=\mbox{const.}[/itex]? Or is there a more direct approach? Certainly the equation cannot be integrated directly because what's [itex]\int r^{-3}dt[/itex]??

Thanks for the help!

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# First integral of the motion

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