Two Body Problem with differential equations?

Question

I have been learning up on the two body problem that uses differential equations to solve for a equation of a satellite relative to the planet. The two sources (linked below) both use different techniques, but I seem to be getting lost somewhere in a few places.

1. In the video, around time $12$:$30$, the accelerations of the satellite minus the acceleration of the planet is reduced to just acceleration. Why is this?

2. The other issue I am having is with the general solution of the differential equation found $\left( a+ \frac{G(m_p)}{r^3} \vec{r}=0\right)$. This is because the two sources go different routes.

Can anyone help to explain and clarify it? What is the general process for solving the above differential equation?

Sources:

• Orbital Motion: Two-Body Problem and Solution of Two-Body Differential Equation

• Astromechanics: The Two Body Problem

• Astromechanics: The Two Body Problem (Continued)