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On the period function of ${x}^{\text{'}\text{'}}+f\left(x\right){x}^{\text{'}2}+g\left(x\right)=0$. (English) Zbl 1048.34068
The paper is devoted to a study of the period function $T\left(x,y\right)$, which associates to every point $\left(x,y\right)$ from a neighborhood of the center $O$ of the equation ${x}^{\text{'}\text{'}}+f\left(x\right){{x}^{\text{'}}}^{2}+g\left(x\right)=0$ 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.

##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE) 34C25 Periodic solutions of ODE 34C07 Theory of limit cycles of polynomial and analytic vector fields
##### Keywords:
center; period function; monotonicity; polynomial systems