Poincar¨¦ Differential Equation Qualitative Theory Study

Mathematical analysis began with searching for the solutions to differential equations. At that time, however, nonlinear differential equations had no universal solutions and some problems of celestial mechanics were still required to be resolved. It inspired Poincare to introduce the idea of qualitative in solving differential equations and initiate a new branch of the qualitative theory of ordinary differential equations in real field. This theory broke through the obstacles in the traditional thoughts of differential equations' solution. It was an important advancement in the history of differential equations' research.This paper briefly recounts the background and origin of Poincare's qualitative theory in the 19th century, and emphasizes the innovations in his thoughts. It also analyses the difference between his thoughts and Enriques', and discusses the relations between his theory and Liapunov's stability theory of motion. It not only has some value of reference to understand and grasp qualitative theory of differential equations comprehensively, but also inspires us to ponder over the track of the emergency and development of new thoughts and theories in the course of science.