Qualitative Study of a Class of Quartic Differential System with an Unstable Node

In this paper, we introduce a class of quartic differential system from the integrability and the existence of limit cycle. We show that this class is integrable and we give the explicit expression of first integral. After this, we prove, under suitable conditions on the parameters, that this class admits a non-algebraic limit cycle surrounding an unstable node. Concerning the singular points at infinity, we show that there is only one singular point. As an illustration, a phase portrait is drawn at the end of this paper.Mathematics Subject Classification: 34A05, 34C05, 34C07, 34C25.