What Exactly Does Equation (2) Mean? (Equations of Motion from PE function)

[humancentered666]

Homework Statement

(There isn't really one, I'm just confused about the interpretation of the equation.)

Relevant Equations

F=ma (1)
Fᵢ({x})=-∂V({x})/∂xᵢ (2)
V({x}) is a potental in a system.

What exactly is this equation telling me? How can I use it to work out the Equations of Motion given an equation of potential energy? Won't I have to solve a PDE? I'm extremely sorry if this question comes off ignorant.

 

[archaic]

https://en.wikipedia.org/wiki/Scalar_potential

 

[haruspex]

How can I use it to work out the Equations of Motion given an equation of potential energy? Won't I have to solve a PDE?

If you are given the potential then, as (2) shows, you can differentiate it with respect to position to find the force vector.
If your aim is to find position as a function of time, that is not always solvable analytically. Even the simple case of an inverse square law is nontrivial.