We have two particles m1 and m2, at positions x1 and x2. I am trying to come up with the equation of motion of particle m1 due to the gravitational atraction of particle m2, that is an equation that will give x1 as a function of time. (No other force acts on the particles.)
So we have F=m1*a and F=Gm1m2/(x2-x1)^2, from which we get a=-Gm2/(x2-x1)^2. Since the distance between the particles changes, the acceleration also changes, and so this must be solved analytically.
d^2x1/dt2 = -Gm2/(x2-x1)^2
Now, I know I have to integrate this once with respect to time to get velocity, and integrate velocity with respect to time to find displacement, but my calculus is quite rusty and I am not sure how to do this.
Any help would be appreciated.