Explicit expression for a first integral for some classes of two-dimensional differential systems

R. Boukoucha

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 903-913

Résumé

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.

Détail

Keywords: autonomous differential system, Kolmogorov system, first integral, trajectories, Hilbert 16th problem.