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This article is cited in 1 scientific paper (total in 1 paper)
Differentical equations, dynamical systems and optimal control
Explicit expression for a first integral for some classes of two-dimensional differential systems
R. Boukoucha Department of Technology, Faculty of Technology,
University of Bejaia,
06000 Bejaia, Algeria
Abstract:
In this paper we are interested in studying
the existence of first integrals and then the trajectories for
classes of two-dimensional differential
systems of the forms
\begin{equation*}
\left\{
\begin{array}{l}
x^{\prime }=\frac{P\left( x,y\right) ^{\alpha }}{T\left( x,y\right) ^{\beta }
}+x\frac{R\left( x,y\right) ^{\gamma }}{S\left( x,y\right) ^{\delta }}, \\
y^{\prime }=\frac{Q\left( x,y\right) ^{\alpha }}{K\left( x,y\right) ^{\beta }
}+y\frac{R\left( x,y\right) ^{\gamma }}{S\left( x,y\right) ^{\delta }},
\end{array}
\right.
\end{equation*}
and
\begin{equation*}
\left\{
\begin{array}{l}
x^{\prime }=x\left( \frac{P\left( x,y\right) ^{\alpha }}{T\left(
x,y\right)
^{\beta }}+\frac{R\left( x,y\right) ^{\gamma }}{S\left( x,y\right) ^{\delta }
}\right) , \\
y^{\prime }=y\left( \frac{Q\left( x,y\right) ^{\alpha }}{K\left(
x,y\right)
^{\beta }}+\frac{R\left( x,y\right) ^{\gamma }}{S\left( x,y\right) ^{\delta }
}\right) ,
\end{array}
\right.
\end{equation*}
where $a,$ $b,$ $n,$ $m$ are positive integers, $\alpha ,$ $\beta ,$
$\gamma ,$ $\delta \in
\mathbb{Q}
$ and $P\left( x,y\right) ,$ $Q\left( x,y\right) ,$ $R\left( x,y\right) ,$ $
T\left( x,y\right) ,$ $K\left( x,y\right) ,$ $S\left( x,y\right) $
are homogeneous polynomials of degree $n,$ $n,$ $m,$ $a,$ $a,$ $b$
respectively. Concrete examples exhibiting the applicability of our
result are introduced.
Keywords:
autonomous differential system, Kolmogorov system, first integral, trajectories, Hilbert 16th problem.
Received October 21, 2016, published September 14, 2017
Citation:
R. Boukoucha, “Explicit expression for a first integral for some classes of two-dimensional differential systems”, Sib. Èlektron. Mat. Izv., 14 (2017), 903–913
Linking options:
https://www.mathnet.ru/eng/semr833 https://www.mathnet.ru/eng/semr/v14/p903
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