I'm trying to work a differential-equation-type homework problem and haven't yet cracked it. I'd be grateful of any help toward solving it. The problem is:

"A particle at rest is attracted toward a center of force according to the relation F = -mk^2 / x^3. Show that the time required for the particle to reach the force center from a distance d is d^2 / k."

My intuition is to treat this as a 2nd-order separable ODE, x'' = -k^2 / x^3 with initial values x = d and v = 0 and final values x = 0 and v = u and solve for the function x(t).
Yeah, I used the chain rule to get x" = dv/dt = (dv/dx)(dx/dt) = v(dv/dx). Using that I integrated both sides and whittled it down to dx/dt = k/x + C, but I'm not totally sure my reckoning is correct, and not confident that I can separate and integrate dx/dt = k/x + C to solve for x(t).