On the period function of \(x^{\prime\prime}+f(x)x^{\prime2}+g(x)=0\) (Q1419820)

The paper is devoted to a study of the period function \(T(x, y)\), which associates to every point \((x,y)\) from a neighborhood of the center \(O\) of the equation \(x''+f(x){x'}^2+g(x)=0\) the corresponding period \(T\). The function \(T\) has a strong relationship to the existence and uniqueness of the solutions of some boundary value problem. The author considers some classes of planar systems equivalent to such equation. The article contains a sufficient condition for the monotonicity of \(T\), or for the isochronicity of \(O\), which is also necessary, when \(f\) and \(g\) are odd and analytic.