Period function monotonicity of planar vector fields

In this paper, we study planar differential systems possessing a center at the origin. We introduce functional conditions for centers and obtain sufficient conditions for monotonicity of period function and isochronicity of the center of systems of ODE reducible to the equation x⋅⋅=f(x,x⋅). These criteria are applied to obtain sufficient conditions for isochronicity of some special important systems.