Topics:
Second order linear equations - Chapter 3, 3.1-3.7
High order linear equations - Chapter 4, 4.1-4.4
Systems of first order linear equations - Chapter 7, 7.1-7.9
Nonlinear differential equations and stability - Chapter 9.1-9.3
You are advised to look again at every definition, theorem, example
and try to solve the following (not necessarily all) problems:
Chapter 3:
Sec. 3.1: 21-24, 27-28
Sec. 3.2: 7-10, 13-16
Sec. 3.3: 15-20, 24-26
Sec. 3.4: 28-31
Sec. 3.5: 18-24
Sec. 3.6: 13-18, 31-32
Sec. 3.7: 13-18
Chapter 4:
Sec. 4.1: 18-20, 21-24
Sec. 4.3: 1-4
Sec. 4.4: 1-4
Chapter 7:
Sec. 7.4: 1-6
Sec. 7.5: 1-4
Sec. 7.6: 1-4, 13-16
Sec. 7.7: 1-4, 13-16, 18
Sec. 7.8: 1-4, 19
Sec. 7.9: 1-6, 14-17
Chapter 9
Sec. 9.1: 1-6, 13-14
Sec. 9.2: 1-4
Sec. 9.3: 1-8, 17-18, 25-26
Wish you a good preparation!
_______________________________________________________________
INTRODUCTION TO DIFFERENTIAL EQUATIONS
Instructor: 1. Dr. VU HOANG LINH, Associate Professor
Faculty of Mathematics, Mechanics, and Informatics
Vietnam National University, Hanoi
Office: 3rd floor, Building T3, 334 Nguyen Trai, Thanh Xuan, Hanoi
Phone: (04)38581135 (office)
E-mail: linhvh@vnu.edu.vn (official),
vuhoanglinh@hus.edu.vn (preferred for this course)
Course topic: This course is intended for mathematician students who require a basic knowledge of differential equations; topics included are basic mathematical concepts, existence and uniqueness theorems, solving techniques, applications of ordinary differential equations and an introduction to stability theory.
Grading:The final grade will be given based on Homework Assignments & Class Activity (20%), Midterm Tests (20%), Final Exam (60%)
Course outline:
Introduction (4 hours)
Some basic mathematical models and notions
Solutions of some differential equations
Classification of differential equations
First order differential equations (12 hours)
Linear equations with variable coefficients
Separable equations
Exact equations and integrating factors
Some applications
Numerical approximations: Euler’s method (optional)
The existence and uniqueness theorem
Second order differential equations (6 hours)
Homogeneous equations with constant coefficients
Fundamental solutions of linear homogeneous equations
Linear independence and the Wronskian
Complex roots and repeated roots of the characteristic equations
Nonhomogeneous equations: method of undetermined coefficients
Variation of parameters
Applications: vibrations
Higher order differential equations (3 hours)
General theory
Homogeneous equations with constant coefficients
The method of undetermined coefficients
The method of variation of parameters
The Laplace transform (optional) (4 hours)
Definition
Solution of initial value problems
Step functions
Differential equations with discontinuous forcing functions
Systems of first order linear equations (10 hours)
Introduction
Review of matrix algebra
Basic theory
Homogeneous linear systems with constant coefficients
Stability theory (6 hours)
Linear systems
Autonomous systems and stability
Almost linear systems
Some model problems
Lyapunov’s second method
Periodic solutions and limit cycles (optional)
Chaos and strange attractors (optional)
Remark: The planned topics and the course schedule may be subjected to changes. We offer more than those in the syllabus of MATH 307, Department of Mathematics, UW.
Textbooks:
1. Elementary Differential Equations and Boundary Value Problems, W.E. Boyce and R.C. DiPrima, John Wiley & Sons, 8th edition, 2001.
2. Basics of Differential Equations and Stability Theory, N.T. Hoan and P. Phu, Education Publishing House, 2000. (in Vietnamese: Cơ sở phương trình vi phân và lý thuyết ổn định, NXB Giáo dục)
Advanced readings:
3. An Introduction to Ordinary Differential Equations, E.A. Coddington, Dover, New York, 1989.
4. Differential Equations and Dynamical Systems, L. Perko, Springer, 1991.
Further on-line readings and useful links:
Differential Equation from Wikipedia, the free online encyclopedia. See also my article on differential-algebraic equations at Scholarpedia, the free peer-reviewed encyclopedia.
Differential Equations, MIT OpenCourseWare, 2006.
Online Notes / Differential Equations, by Paul Dawkins, Lamar University
Notes on Differential Equation, by Bob Terrell, Cornell University
Differential equations and modeling: some demos and video clips from Department of Physics, University of Southern California. http://physics.usc.edu/demolab/videos.html
The midterm exam is expected in the last week of March. Two or three quick tests (15-30 minutes) may be held during the semester without preannouncement.