常微分方程A(H)

理论课，4学分，3学时/每周，习题课2学时/周。先修课程：MA203a数学分析III或者MA213-16数学分析精讲，MA104b线性代数II。本课将理论和应用相互穿插，也将用软件模拟理论结果和帮助解决应用问题，培养用常微分方程解决数学问题和应用问题的意识和能力。理论部分包括：一阶线性方程，分离变量法，方向向量场，欧拉方法，存在和唯一性定理，相线分析，二阶线性方程，常数变易法，解的渐近行为，级数解法，拉普拉斯变换法，一阶线性方程组，一阶非线性自治系统，驻点的线性稳定性和分类，局部和整体相平面分析，零值线，不变区域，Lyapunov 函数，极限环, Poincare-Bendixson定理。Lecture, 4 credits. Prerequisites: Mathematical Analysis III (MA203a) or Real Analysis (MA213-16) and Linear Algebra II (MA104b). This course mixes theory with applications, and uses softwares to aid the understanding on theoretical results, and to help to solve application problems. The course aims to nurture the ability of the student to use ODE to solve problems arising in applications and other branches of mathematics. The theoretical part of the course covers: first order linear equations, separation of variables, direction fields, Euler’s method, existence and uniqueness theorem, phase line analysis, second order linear equations, variation of constants, asymptotic behavior of solutions, first order linear systems, first order nonlinear autonomous systems, linear stability and types of equilibria, local and global phase plane analysis, nullclines, invariant regions, Lyapunov function, limit cycles, Poincare-Bendixson Theorem, Sturm-Liouville theory.