Using changing acceleration due to gravity?

[Wooh]

We know that, with gravity for example, that ma = GmM/r^2. For simplicity's sake and the sake of my question, let us say that a=c/r^2, where c is GM. Basically, I am wondering how I can use this to create the most accurate displacement equations possible. My problem, however, is that a is dv/dt...or dx^2/dt^2, but I doubt you can do dx^2/r=cdt^2 and integrate twice or whatnot.

Does anyone have some clarity? I have though to do a=c/(r-x)^2, where r is the initial distance and x is the distance traveled, but that still yields nothing helpful.

 

[Tide]

I recommend studying a good mechanics book that discusses planetary motion. Exactly those equations solved to find planetary orbits and the like. However, it's not for the faint of heart if you're just starting out!