The Analysis Of The Integrability Of A Class Of Second Order Autonomous Systems And Applications

In this paper,the integrability of a class of second order autonomous systems is studied.First of all,the existence of algebraic curve solutions of the equation is considered.Three conditions for the equation having no algebraic curve solutions are given by using the Division theorem.In addition the equation has algebraic curve solutions if the equation is Liouville integrability is proved,which provides a theoretical basis for finding algebraic curve solutions.Afterwards two conditions that the equation can be linearized into Laguerre form by using the method of Lie point transformation are given.Two conditions are given and proved when the integrating factor ?(x,y)exists.At the same time the exact solutions and infinitesimal symmetries of the equation are obtained.Finally,the relevant conclusions are applied to two partial differential equations.