The Formation Of Ordinary Differential Equation Theory

On the basis of historical analysis and comparative study, this thesis focuses on the history of the formation of ODE theory. The main results are as follows:(1) The background of the origin of ODEs is reviewed in an all round way. The impact of the integration of mathematics with other subjects such as mechanics, physics, acoustics and geometry on germinating ODE is carefully analyzed.(2) The primitive formation of ODE in the process of founding calculus are studied. By paying more attention to Newton's method of series and his idea of variation of constants applied to the problem of three bodies as well as Leibniz's ingenious linking differential triangles with differential equations in his solution of problems related to curves, the thesis displays the way how differential equations stuck out through calculus as an object of study.(3) The significance to the origin of ODE theory of the five challenging problems in the 17^th and 18^th century is respectively discussed. The author pointed out that the appearance isochrone problem made the integral calculus first to be used in solving differential equations, that the catenary problem is a starting point to develop techniques for solving differential equations, that the successful quadrature of a hyperbola showed the essential advance of solving techniques, that the brachistochrone problem offered the most convincing example of applying idea of differential equations and that the orthogonal trajectory problem increased much theoretical color in researches on differential equations(4) The process that the ODE was gradually separated from the calculus as an independent object of study from the time of Bernoulli is systematically observed.(5) By probing into the four key facts—Euler's method of reduction of order in 1728, the problem of singular solution that Clairaut researched deeply in 1734, the adjoint equation found by Lagrange and Lagrange's breakthrough in solving linear equation with constant coefficients in 1743 and by observing the turn to paying more attention to the "definite problem"in the end of 18th.century and elaborating on three approaches to prove existence theorem, the thesis analyzes deeply the profound change of the research pattern of ODE in the 18th-century and reconstructs historically the course of the formation of the ODE theory since 1740 with considering the birth of existence theorem as a symbol of establishment of ODE theory.(6) The further development of ODE theory after forming the existence theorem is studied. The transmission of the ODE theory in China from the Opium War to 1959 is reviewed by five periods.