What does it mean to "Evaluate the potential of leading order?"

Question

Given the potential $V(r)$ of some particle $m$ that is affected by it, how does one use Newton's Second Law to evaluate the potential of leading order?

I got the DE:

$$mr''r^3=2b-ar,$$ where $a$,$b$ are constant and $'$ is time differentiation.

Answer 1

You have skimped on context here. Multiplying by $r'$ and integrating yields the conserved energy equation, $$ mr' ^2 /2= a/r -b/r^2 -c , $$ suggesting your potential is $$ V= c-a/r+ b/r^2 ~. $$ One normally ignores constants (zero point energy), so the leading term at large distances is -a/r, a Coulomb-like term. Is this what you have in mind?