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Zbl 1048.34068
Sabatini, M.
On the period function of $x^{\prime\prime}+f(x)x^{\prime2}+g(x)=0$. (English)
[J] J. Differ. Equations 196, No. 1, 151-168 (2004). ISSN 0022-0396

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.
[Alexander Grin (Grodno)]

MSC 2000:: *34C05 Qualitative theory of some special solutions of ODE
34C25 Periodic solutions of ODE
34C07 Theory of limit cycles of polynomial and analytic vector fields

Keywords: center; period function; monotonicity; polynomial systems

Cited in: Zbl 1184.34048

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Zentralblatt MATH Berlin [Germany]