On the period function of x"+f(x)x^2+g(x)=0

We study the period function T of a center O of the titleʹs equation. A sufficient condition for the monotonicity of T, or for the isochronicity of O, is given. Such a condition is also necessary, when f and g are odd and analytic. In this case a characterization of isochronous centers is given. Some classes of plane systems equivalent to such equation are considered, including some Kuklesʹ systems