Short Question: if a question says to 'integrate the equation of motion', what does it mean?
Long Question:
Question: Take a planet of mass $M$ and place a satellite at rest at a distance $R$ from the planet, where $R$ is much greater than the planet's radius. How long does the satellite take to hit the surface of the planet?
Part 1 of the question asks the reader to perform dimensional analysis. This yields
$\textrm{Time taken }T=C\sqrt{\frac{R^3}{GM}}$
Part 2 - integrate the equation of motion of the satellite to show that $C=\pi /2\sqrt{2}$.
As far as I'm aware, the equation of motion for the satellite is
$\ddot{r}=-\frac{GM}{r^2}.$
I've tried solving this differential equation, but to no avail ($r=\frac{9}{2}GMt^{\frac{2}{3}}$ is a particular solution, but I have no idea how to find the more general case; substituting the dimensionless quantity $\kappa=\frac{1}{GM}r^3t^{-2}$ almost worked but not quite). I also tried using the potential $V=\frac{-GMm}{r}$ to form the equation
$T=\int_R^0{\frac{dr}{2\sqrt{\frac{GM}{r}-\frac{GM}{R}}}}.$
However, this integral doesn't look as if it's going to give the right answer. The $\pi$ in the given expression for $C$ seems to suggest that we're going to get an integral involving a $\sin$ substitution.
So - what does 'integrate the equation of motion' mean? Integrate which equation? And with respect to what?