IJPAM: Volume 32, No. 2 (2006)

THE MONOTONICITY OF THE PERIOD FUNCTION FOR
$x^{\prime \prime }+f_{2}(x)x^{\prime 2}+f_{1}(x)x^{\prime }+g(x)=0$

Khalil I.T. Al-Dosary
College of Arts and Sciences
Univesity of Sharjah
P.O. Box 27272, Sharjah, UNITED ARAB EMIRATES
e-mail: dosary@sharjah.ac.ae


Abstract.We consider the second-order differential equations of the form\begin{equation*}
x^{\prime \prime }+f_{2}(x)x^{\prime 2}+f_{1}(x)x^{\prime }g(x)=0\,.
\end{equation*}In this paper we deal with the so-called monotonicity and isochronicity problem. We study this problem by means of a new appropriate equivalent differential system of the form\begin{equation*}
\left\{
\begin{array}{l}
x^{\prime }=y-xB_{1}(x)-xyB_{2}(x)\,...
...^{\prime }=-C(x)-yB_{1}(x)-y^{2}B_{2}(x)\,.
\end{array}%
\right.
\end{equation*} We use this system to determine the monotonicity charactor of the period function. Then we give necessary and sufficient condition for monotonicty and isochronicty when $f_{1}$, $f_{2}$ and $g$ are analytic functions. We apply this chracterization to estimate the region of monotonicty of the period function of a particular case of the considered equation.

Received: August 20, 2006

AMS Subject Classification: 34C25

Key Words and Phrases: center, period function, monotonicity, isochronous center

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 32
Issue: 2