Some Attempt And Application Of Lie Group Theory To Differential Equation Theory

In this paper, we use different means of the single parameter Lie group,study the integrability of the some nonlinearity ordinary differential equations,and prove the unaccepted any Lie group for the four classes system Painlevé-equation, making use of traditional theory of the Lie group,accordingly which cannot reduce order. We make use of method of Lie parenthesis, for definition quasi-homogenous,we investigate the first integral of peg-top system in the special instance.Of which:The first chapter briefly introduces the development of the Lie group theory and the background of selecting this question, presents the involved concepts and theorems in this lecture.In the second chapter, we shall study the accepted single parameter Lie group for Painlevé-equation.The third chapter, we shall discuss the first integral of the analogous peg-top system.The forth chapter,we mainly carries on summary to induce to the research result.