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Next: MC130 Mathematical Modelling Up: Year 1 Previous: MC126 Multivariate Calculus

MC127 Vectors and ODEs


MC127 Vectors and ODEs

Credits: 10 Convenor: Dr. M. Marletta Semester: 1 (weeks 1 to 6)


Prerequisites: essential: A-level Mathematics
Assessment: Coursework and/or tests: 20% One and a half hour exam: 80%

Lectures: 18 Classes: none
Tutorials: 5 Private Study: 47
Labs: 5 Seminars: none
Project: none Other: none
Total: 75

Explanation of Pre-requisites

This modules will assume that the student has a firm grasp of all the material in the core A-level syllabus. In particular, a thorough understanding of A-level calculus will be required: differentiation and integration of standard functions (polynomials, radicals, trigonometric, exponential and logarithmic functions) together with rules for dealing with products and compositions of functions. The module will also assume a knowledge of trigonometry, including standard trigonometric identities.

Course Description

In this course we present a number of standard mathematical methods which can be used for solving simple differantial equations. At the same time we shall also illustrate the fact that the theoretical study of differential equations - which usually do not fall into any of the categories for which explicit solution is possible - is an interesting and challenging subject. Many seemingly simple differential equations have solutions which exhibit unexpected behaviour, such as blowup in finite time. Other seemingly complicated equations may be much more tame because they are expressing an underlying energy principle.

The formal lectures in this module will be supplemented by computer labs in which students will acquire skills in the use of mathematical software packages. In addition to the IT skills which this will provide, it will also give students some experience of independent study. The topic covered will be vectors, which have many important applications in the study of ordinary differential equations.

Aims

This module has three aims:
1.
To give students an appreciation of the fact that there is sophisticated mathematics behind the study of differential equations;
2.
To equip students with the basic skills required to deal with differential equations which can be solved in closed form by standard techniques, thus preparing them for later modules in their degrees;
3.
To familiarize students with vectors through the use of an appropriate mathematical software package.

Objectives

1.
Students should know how to define an initial value problem for a first order equation, and give examples of initial value problems which do not have unique solutions.
2.
Students should know basic techniques for solving first order ODEs of separable and linear types.
3.
Students should know how to solve second-order equations with constant coefficients.
4.
Students should be able to identify conservative second-order equations and use energy methods to analyze them.
5.
Students should know how to use the TransMath package to study vectors.

Transferable Skills

The skills in elementary analysis of differential equations which a diligent student should acquire in this module are transferable to any situation where differential equations are encountered, including economics, financial modelling and investment analysis, engineering and most of the physical sciences. Students should also acquire some familiarity with the use of PCs.

Syllabus

1. First Order Scalar Equations
Definition of first order ODE y' = f(x,y) and of solutions of first order ODEs. Examples. Statement of Peano's Theorem (no proof). [1 lecture]

Solution techniques for simple first order ODEs, covering

Remark: We shall not cover so-called EXACT equations as these require a knowledge of partial derivatives. Also, exact equations usually only arise in mathematics exams. [4 lectures]

The need for initial conditions: the concept of an *initial value problem*. Examples. Statement of Picard's Theorem (no proof). Examples of failure of uniqueness in absence of Lipschitz continuity.
[2 lectures]

2. Second Order Scalar Equations

Definition of second order ODE y'' = f(x,y,y'). Examples. [1 lecture]

Second order linear equations with constant coefficients.


Second order conservative equations. Energy methods for reduction to a separable first order equation. Examples, including bungee jumping without injury. [3 lectures]

3. Vectors

There will be six lab classes during which you will use the TransMath package to explore the topic of vectors. Worksheets with questions will enable you to work out whether or not you have understood the material.

TOTAL: 18 lectures.

Reading list

Background:

R. Bronson, Schaum's Outline of Theory and Problems of Differential Equations, McGraw-Hill, 1994.

F. R. Giordano and M. R. Weir, Differential Equations, A Modelling Approach Addison Wesley, 1991.

A. Jeffrey, Linear Algebra and Ordinary Differential Equations

D. Pearson, Calculus and ODE's, Arnold, 1996.

J. Gilbert, Guide to Mathematical Methods, Macmillan, 1991.

Details of Assessment

The assessment will consist of a one-and-a-half hour exam in January, which will count for 80% of the assessment; assignments based on the computer labs, which will count for 10% of the assessment; and assessment based on the coursework problems, totalling 10%. The examination paper will contain 4 questions with full marks on the paper obtainable from 4 complete answers.


next up previous
Next: MC130 Mathematical Modelling Up: Year 1 Previous: MC126 Multivariate Calculus
S. J. Ambler
11/20/1999