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I am trying to find a solution for this equation in terms of r or t:

[tex] \frac{d^2r}{dt^2} = \frac{m}{r(r-2m)}\left( \left( \frac{dr}{dt}\right)^2-\left(1-\frac{2m}{R}\right)\right)[/tex]

It seems I should integrate it twice with respect t, but I have no idea how to do that with a dr/dt term in the middle of the right hand side. I need to get rid of all dr and dt variables and obtain an answer that contains just r,t and R variables, if that is possible.

One possible (given) solution that I am not convinced is correct is:

[tex]t=\frac{R}{2}\sqrt{\frac{R}{2m}}\left(cos^{-1}\left(\frac{2r}{R}-1\right)+2\sqrt{\frac{r(R-r)}{R^2}}\right)[/tex]

but I have no idea how the solution was obtained.

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# How the heck do I integrate this?

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