I have this equation:
$$\ddot{r}=-\frac{G M}{r^{2}}$$
I have to show that the solution of $r(t)$ can be parametrized by $\theta$ like this:
$$ r=A(1-\cos \theta), \quad t=B(\theta-\sin \theta), \quad A^{3}=G M B^{2} $$
I could probably show by direct substitution that this parametrization is correct using double chain rule. Assuming I don't know what the solution is going to be, how could I go about obtaining this parametrization?
I can multiply both sides by $\dot{R}$:
$$\dot{R} \ddot{R}=-\frac{G M\dot{R}}{R^{2}}$$
which leads me to:
$$\frac{d}{d t}\left[\frac{1}{2} \dot{R}^{2}-\frac{G M}{R}\right]=0$$
which gives me:
$$\frac{d r}{d t}=\frac{2 G M}{r}+C.$$
I don't know how to continue to get to the parametrization.