### Ordinary Differential Equations

Course Code: B13006H

Course Name: Ordinary Differential Equations

Credits: 3.0

Pre-requisite: Calculus, Linear Algebra

Lecture Time: 60 periods

Course Description

This course is a basic course for undergraduates with major in mathematics. After studying, students should master the elementary knowledge about ordinary differential equations. It is aimed at providing prerequisite knowledge for future courses such as partial differential equations, numerical analysis and etc. It is also a fundamental course for the students who are engaged in the study of power system, symplectic geometric algorithm, perturbation theory, celestial mechanics, and other related fields.

Through this course, students should meet the following basic requirements: to master the introductory knowledge of the ordinary differential equations, and the basic concepts and methods related to the differential equations, to get a preliminary understanding of the basic idea of ordinary differential equations, and the way of thinking and to prepare for the study of future courses.

Topics and Schedule

1. Introduction (8 periods)

1.1.   Brief introduction of ordinary differential equations and development (1 period)

1.2.   First-order differential equations; Some basic concepts (1 period)

1.3.   Some elementary integration methods (1 period)

1.4. Formulation of the existence and uniqueness theorem (1 period)

1.5. Reduction of a general system of differential equations to a normal system (2 period)

1.6. Complex differential equations (1 period)

1.7. Some properties of linear differential equations(1 period)

2.    Linear Differential Equations with Constant Coefficients (16 periods)

2.1. Linear homogeneous equation with constant coefficients. The case of simple roots (3 periods)

2.2.   Linear homogeneous equation with constant coefficients. The case of multiple roots (2 periods)

2.3.   Stable polynomials (3 periods)

2.4.   Linear non-homogeneous equation with constant coefficients (2 periods)

2.5.   Method of elimination; Method complex amplitudes; (3 periods)

2.6. Autonomous systems of differential equations and their phase spaces (3 periods)

3.    Linear Differential Equations with Variable Coefficients (8 periods)

3.1.   The normal system of linear differential equations (3 periods)

3.2.   The linear differential equations of nth order(2 periods)

3.3.   The normal linear homogeneous system with periodic coefficients (3 periods)

4.    Existence Theorems (12 periods)

4.1.   Proof of the existence and uniqueness theorem for one equation (3 periods)

4.2.   Proof of the existence and uniqueness theorem for differential equations (2 periods)

4.3.Local theorems of continuity and differentiability of solutions (2 periods)

4.4.   First integrals (2 periods)

4.5. Global theorems of continuity and differentiability of solutions (3 periods)

5.    Stability (16 periods)

5.1.   Lyapunov’s theorem (4 periods)

5.2.   Limit cycles (4 periods)

5.3.   The states of equilibrium of a second-order autonomous system (4 periods)

5.4.   Stability of periodic solutions (4 periods)

Textbook

L.S.Pontryagin, Ordinary Differential Equations, Higher Education Press, 2006(Chinese Version)

References

V.I.Arnold, Ordinary Differential Equations, Third Edition, Science Press, 2001(Chinese Version)