newtonian mechanics - How to prove that a motion is Simple Harmonic Motion (SHM)?

I would like to know how one could show and prove that a given motion is simple harmonic motion.

Once given an answer, I'll apply that technique to an example I am trying to figure out.

Thank you in advance!

I believe a motion can be proved simple harmonic, if the relation between its is as such: $$ a_x = - \omega^2\cdot x $$

And as such the period time is: $$ T =\frac{2\pi}{\omega} $$

Question-so-far: How do you prove such for a given force $F = \frac{G\cdot m_e \cdot M}{R_E} \cdot r$ ? Or any force that has non-trivial constants?