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Ordinary Differential Equations 2 (MATH 20101)

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Administrative Information

  1. Unit number and title: MATH 20101 Ordinary Differential Equations 2
  2. Level: I/5
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: In similar form: 09/10
  6. Unit Organiser: Valeriy Slastikov
  7. Lecturer: Dr V. Slastikov
  8. Teaching block: 2
  9. Prerequisites: MATH 11007 Calculus 1 and MATH 11005 Linear Algebra & Geometry; Calculus 2 recommended but not required.

Unit 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.

General Description of the Unit

The subject of differential equations is a very important branch of applied mathematics. Many phenomena from physics, biology and engineering may be described using ordinary differential equations. In order to understand the underlying processes we have to find and interpretate the solutions of these equations. This unit explains different methods of solution of ordinary differential equations: from analytical to numerical.

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.


Teaching Methods

Lectures - 33 sessions in which the lecturer will present the course material on the blackboard. Students are expected to attend all lectures, and to prepare for them by reading notes, handouts or texts, as indicated by the lecturer. The lectures are 3 per week, on weeks 1 to 11 - no class on week 12 .

Problems classes - 10 sessions with the lecturer, in which problems will be worked through as a demonstration, on the blackboard. Students are strongly encouraged to attend all problems classes.

Homework assignments - 10 problem sheets will be given out, one per week. Students will be required to turn in selected problems from the sheet, which will be marked by the postgraduate teaching assistants. 

Learning Objectives

  1. 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.
  2. Students should understand the concept of a solution to an initial value problem, and the guarantee of its existence and uniqueness under specific conditions.
  3. The student will recognize basic types of differential equations which are solvable, and will understand the features of linear equations in particular.
  4. 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.
  5. 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.

Assessment Methods

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.


Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

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.

Texts

There is no single book that covers all the material. Online lecture notes is the best approximation of the course.

Syllabus

Here is a brief syllabus of the course:

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

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