I'm just reading over some Cosmology notes and there is a little ODE solve that I am not quite understanding.
I have an equation of the form:
$$ \ddot{R}=-\frac{GM}{R^{2}} $$
Integrating gives:
$$ \dot{R}^{2}=+\frac{2GM}{R}+\mathcal{E} $$
Where $\mathcal{E}$ is just a constant of integration. The notes are essentially saying that this can be solved with a parameter $\theta$ such that r is given by:
$$ r(\theta)=R_{0}(1-\cos\theta) $$
In terms of density this can be written as:
$$ \dot{R}^{2}=\frac{4\pi{G}}{3}\rho_{0}R^{2} $$
There is also a solution of the form:
$$ \rho(\theta)=\rho_{0}\frac{9(\theta-\sin\theta)^{2}}{2(1-\cos\theta)^{3}} $$
Could anyone run through exactly how this is come about?