Imagine you have Newton's Law of gravitation. How would you solve this differential equation. $$ \frac{\mathrm d^2 r(t)}{\mathrm d t^2 } =-\frac{GM}{r(t)^2} $$
I know the solution of this equation is a conic section, but how would you arrive to that? Also, I know that the final result is a function $r(\theta)$ not $r(t)$, but why? I've tried to search on the internet but I feel like they don't explain too much the process. I would like, if possible, a way of understanding each of the steps of the derivation.
