Explicit expression for a first integral and phase portrait for a class of planar differential system

where A1(x,y), A2(x,y), B1(x,y), B2(x,y), P(x,y), Q(x,y), R(x,y), S(x,y) are homogeneous polynomials of degree a, a, b, b, n, n, m, m respectively. Writing this system in polar coordinates, from which we can compute a formal First integral for the system. Finally, we give the curves which are formed by the trajectories for the system.

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Introduction

We consider two-dimensional autonomous systems of differential equations of the form

where F(x,y) and G(x,y) are reals functions. In the qualitative theory of planar dynamical systems [1,8,9] , one of the most important topics is related to the second part of the unsolved Hilbert 16th problem [18] . There is a huge literature about limit cycles, most of them deal essentially with their detection, their number and their stability and rare are papers concerned by giving them explicitly [2,3,6,11,13] . There exist three main open problems in the qualitative theory of real planar differential systems, the distinction between a centre and a focus, the determination of the number of limit cycles and their distribution, and the determination of its integrability. The importance for searching first integrals of a given system was already noted by Poincaré in his discussion on a method to obtain polynomial or rational first integrals. One of the classical tools in the classification of all trajectories of a dynamical system is to find first integrals. Giné and Llibre characterized a large classes of polynomial differential systems in terms of the existence of first integrals [4,7,9,12,16,17,21] . For more details about first integral see for instance [10,14,15,19,20] , see the references quoted in those articles. We recall that in the phase plane, a limit cycle of system ( 1) is an isolated periodic orbit in the set of all periodic orbits of system (1). System (1) is integrable on an open set Ω of ℝ 2 if there exists a non-constant C 1 function H: Ω→ℝ, called a first integral of the system on Ω, which is constant on the trajectories of the system (1) contained in Ω i.e. if

in the points of Ω. Moreover, H=h is the general solution of this equation, where h is an arbitrary constant. It is well known that for differential systems defined on the plane ℝ 2 the existence of a first integral determines their phase portrait see [5] .

In this paper we study the integrability of two-dimensional autonomous systems in the plane of the form

where A1(x,y), A2(x,y), B1(x,y), B2(x,y), P(x,y), Q(x,y), R(x,y), S(x,y) are homogeneous polynomials of degree a, a, b, b, n, n, m, m respectively. Writing this system in polar coordinates, from which we can compute a formal First integral for the system. Finally, we give curves which are formed by the trajectories for the system. We define the trigonometric functions

(cos ,sin ) (cos ,sin ) ( ) (cos ,sin )(cos )sin (cos ,sin )(sin )sin , (cos ,sin ) (cos ,sin )

(cos ,sin ) (cos ,sin ) ( ) (cos ,sin )(cos )sin (cos ,sin )(sin )sin . (cos ,sin ) (cos ,sin )

Main result

Our main result on the existence of a First integral and the curves which are formed by the trajectories of the 2dimensional differential systems (2) is the following.

Moreover the phase portrait of the differential system (2), in Cartesian coordinates is given by 4)

The general solution of linear equation ( 5) is

where 𝛼∈ℝ., which has the first integral

Hence statement (a) of Theorem



The general solution of equation ( 6) is

where 𝛼∈ℝ, which has the first integral

Hence statement (b) of Theorem 1 is proved.

Assume now that f3 (𝜃) =0 for all 𝜃∈ℝ, then from (3) it follows that 𝜃' =0 So the straight lines through the origin of coordinates of the differential system (2) are invariant by the flow of this system. Hence, y

x is a first integral of the system. The curves H=h with h∈ℝ, which are formed by trajectories of the differential system (2), are written in Cartesian coordinates as y=hx where h∈ℝ. This completes the proof of statement (c) of Theorem 1.∎

Conclusion

The elementary method used in this paper seems to be fruitful to investigate planar differential systems of ODEs in order to obtain explicit expression for a first integral and characterizes its trajectories; this is a one of the classical tools in the classification of all trajectories of dynamical systems.

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