MAT3270B Ordinary Differential Equations, 2008-09

General Information

Course Description

Objectives:

(1) To acquaint with standard techniques in solving linear or nonlinear ordinary differential equations;

(2) To understand the basic theory of linear ODES and the stability theory of nonlinear ODES;

(3) To solve basic boundary value problems.

Syllabus and Teaching Scheme:

Part 1: Introduction and First Order Differential Equations (2.5 weeks)

(a) Mathematical models leading to ODEs: falling subject; Compound interest. Solutions of ODE.

(b) Explicitly sovable equations: linear, separable, exact, homogeneous equations. Techniques in solving nonlinear equations.

(c) Blow up and nonuniqueness phenomena; fundamental existence and uniqueness for Initial Value Problems -a sketch of proof.

We will cover: 1.1 - 1.3, 2.1 - 2.8

Quiz 1

Part 2: Linear Theory (2.5 weeks)

(a) Linear second order equations with constant coefficients.

(b) General structure for the solution space for linear equations; variations of parameters.

(c) Free and forced vibrations; resonance.

(d) Higher order linear equations: general theory, homogeneous equations.

We will cover: Chapters 3, 4

Midterm Examination

Part 3: Systems of ODEs (1.5 weeks)

First order systems; the exponential of a matrix; Fundamental solutions; solving systems using linear algebra.

We will cover: Chapter 7

Quiz 2

Part 4: Stability (2.5 weeks)

(a) Phase portraits for 2 $\times$ 2 linear autonomous systems; concept of stability and asymptotic stability.

(b) The method of linearisation.

(c) Liapunov's function.

(d) Applications to population models.

(e) Liapunov's second method, periodic solutions, limit cycles and chaos.

We will cover: 9.1 - 9.8

Quiz 3

Part 5: Boundary Value Problems (2.5 weeks)

(a) Linear Homogeneous Boundary Value Problems.

(b) Sturm-Liouville two-point boundary value problems and eigenvalue probllems.

(c) Properties of the eigenvalues and eigenfunctions.

(d) Nonhomogeneous BVPs.

We will cover: 11.1 - 11.4

Quiz 4

Final Examination: December 3rd - December 22nd, 2008

Assignments:

Each week I will give homework assignment. You do not need to turn in the assignment I STRONGLY suggest you do ALL of them. Your TA will answer questions from the homework. About every 2.5 weeks there will be a quiz. Midterm: around 7th Week. One final examination is scheduled.

Textbooks

• Boyce and DiPrima, "Elementary Differential Equations and Boundary Value Problems", 8th ed., J. Wiley and Sons, Singapore (Please buy this book at the University Bookstore on campus)

References

• Coddington, "An introduction to Ordinary Differential Equations", Prentice-Hall, N.J.
• Brauer, "Differential Equations and Applications", Springer-Verlag, New York.

Assignments

• Assignment 1 (4-9-08)
• Assignment 2 (11-9-08)
• Assignment 3 (18-9-08)
• Assignment 4 (25-9-08)

Solutions

• Solution for Assignment 1 (revised)
• Solution for Assignment 2 (figures inserted)
• Solution for Assignment 3
• Solution for Assignment 4

Assessment Scheme