常微分方程解的存在唯一性定理的教学探索

Explore on the teaching of existence and uniqueness theorem for the solution of ordinary differential equation

研究了一阶常微分方程的初值问题,通过构造上、下控制函数结合上、下解方法及不动点理论,证明了当非线性项连续时解的存在性,当非线性项Lipschitz连续时解的唯一性.该方法也适用于其它类型的微分方程研究.结合多年的教学与科研经验对"常微分方程解的存在唯一性定理"的课堂教学进行了分析与探讨.

The existence and uniqueness of solution for the ordinary differential equation was investigated by constructing the upper and lower control function and using the method of upper and lower solutions as well as fixed point theorem.It was shown that the initial value problem has a solution when the nonlinear term is continuous and has a unique solution when the nonlinear term is Lipschitz continuous.The method is the same with other type of differential equation.Moreover,the classroom teaching of the existence and uniqueness theorem of the ordinary differential equations was discussed and analyzed with years of experience in teaching and scientific research.