Basically equation of motion is solving Newton's 2nd Law.
Unless we know the nature of F, we just solve for
Otherwise, to get the proper solution, you need to solve
Perform integration twice and applying proper initial conditions
If start from rest,
Of course, the exact form will depend on the nature of force is being applied too. For instant, if F is caused by a spring pulling on the mass, then solution involves a sinusoidal displacement.