oscillating particle problem

A particle has mass m and moves in the region x > 0 under the force , where and are constants. The particle starts from the position of equilibirum with speed v. Find the limiting values of x in the subsequent motion. Show also that the period of oscillation is independent of v.

so far i have done:

*for the position of equilibrium this is where F=0

as x>0.

*for the period

I tried to use

at x=a V=0 so

as max, min x T=0

so

however i get lost with constants of integration