Unit information: Ordinary Differential Equations 2 in 2013/14

Unit name	Ordinary Differential Equations 2
Unit code	MATH20101
Credit points	20
Level of study	I/5
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Slastikov
Open unit status	Not open
Pre-requisites	MATH11002 and MATH11003
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Differential equations are a natural means to express the laws that govern a wide variety of systems: mechanical systems, systems of chemical reactants, of animal populations, wave phenomena, and many more. This course discusses analytical techniques for finding solutions and for understanding their behaviour. Topics include linear differential equations as well as methods for nonlinear differential equations including an introduction to chaotic systems. Aspects of the general theory of differential equations will be discussed, the emphasis, however, will be on practical methods rather than mathematical rigour.

Aims

The aim of this unit is to introduce the students to the basic theory of ordinary differential equations and give a competence in solving ordinary differential equations by using analytical or numerical methods.

Syllabus

Here is a brief syllabus of the course:

1. What is ODE? Existence and uniqueness of solutions. Simple examples from physics.
2. First order ODE's. Linear equations, autonomous equations, separable equations, exact equations. Methods of finding approximate solutions.
3. Linear systems of ODE's. Homogeneous systems, linear systems with constant coefficients. Stability analysis. Non-homogeneous systems.
4. Power series solutions of ODE's. Taylor series method. Existence of analytic solutions. Frobenius method.
5. Nonlinear systems of ODE's. Linearization about critical point. Stability analysis. Examples from physics.
6. Variational problems and boundary value problems.

There may be minor changes to this syllabus, or to the order of presentation.

Relation to Other Units

This unit develops the ordinary differential equations material in Core Mathematics. Partial differential equations are treated in a separate unit, Applied Differential Equations 2. Together with Calculus 2, these courses provide essential tools for mathematical methods and applied mathematics units at Levels 3 and 4. Calculus 2 is recommended but not required as a corequisite.

Intended Learning Outcomes

• The student will learn to formulate ordinary differential equations (ODEs) and seek understanding of their solutions, either obtained exactly or approximately by analytic or numerical methods.

• Students should understand the concept of a solution to an initial value problem, and the guarantee of its existence and uniqueness under specific conditions.

• The student will recognize basic types of differential equations which are solvable, and will understand the features of linear equations in particular.

• Students will learn to use different approaches to investigate equations which are not easily solvable. In particular, the student will be familiar with phase plane analysis.

• Students will become proficient with the notions of linearization, equilibrium, stability. They will learn to use the eigenvalue method for autonomous systems on the plane.

Transferable Skills:

• Increased understanding of the relationship between mathematics and the “real world” (meaning the physical, biological, economic, etc. systems).

• Development of problem-solving and analytical skills.

Teaching Information

The final mark for Ordinary Differential Equations is calculated as follows:

• 100% from a 2½-hour written examination in May/June

More information is given below.

Summer Examination

The examination in May/June consists of a 2 ½-hour paper consisting of FIVE questions; you should attempt FOUR. If you attempt more than four, your best four answers will be used for assessment. Calculators may NOT be used.

Assessment Information

The final mark for Ordinary Differential Equations is calculated as follows:

• 100% from a 2½-hour written examination in May/June

More information is given below.

Summer Examination

The examination in May/June consists of a 2 ½-hour paper consisting of FIVE questions; you should attempt FOUR. If you attempt more than four, your best four answers will be used for assessment. Calculators may NOT be used.

Reading and References

This year, the preferred text for the course is:

Differential Equations: An Applied Approach, by J. M. Cushing, Pearson (Prentice Hall) 1 st ed. (2004)

The course syllabus includes (but not restricted to): Chapters 1, 2, 3, 4, 5, 6, section 7.3 and Chapter 8. There will be some additional material given in lectures and handouts.

You may also use the following text:

Elementary Differential Equations and Boundary Value Problems, by William E. Boyce, Richard C. Diprima, John Wiley & Sons, 8 th ed. (2005),

however some topics are not covered here. Additional literature will be given during the lectures.