Solutions And Integrability Analysis Of Two Classes Of Nonlinear Differential Equations

In this paper,we extend the first integral method which is mainly suitable for second-order differential equations.Combined with Lie group theory and linearization of third-order equations,the integrability of two classes of third-order nonlinear differential equations is analyzed and solved.At first,the first integral method in the literature is extended,and the method is extended to the third-order differential equation for the first time.Secondly,for a class of third-order nonlinear differential equations,two independent first integrals of the equation are obtained by using the extended first integral method,and the two first integrals are used to reduce the order;two exact solutions of the equation are obtained by using the function transformations;using Lie group theory to obtain the infinitesimal generating elements of the Lie group accepted by the equation,invariant solution and two exact solutions are obtained;it is proved that the differential equation can not be linearized.Then,for another class of third-order nonlinear differential equations,using Lie group theory to find the infinitesimal generating elements of the Lie group accepted by the equation,and its invariant solutions are obtained;according to the third-order equation linearization method,the conclusion is obtained whether the equation can be linearized.Finally,the application of the relevant theory and method to the solution of a class of Kuramoto-Sivashinsky type equation is considered.