	Department of Mathematics & Computer Science

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MA1002 Vectors and ODEs

MA1002(=MC127) Vectors and ODEs

Credits: 10

Convenor: Dr. P. Houston

Semester: 1 (weeks 1 to 6)

Prerequisites:	essential: A-level Mathematics
Assessment:	Continual assessment: 20%	One and a half hour exam: 80%

Lectures:	18	Problem Classes:	none
Tutorials:	5	Private Study:	52
Labs:	none	Seminars:	none
Project:	none	Other:	none
Surgeries:	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 differential 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.

Aims

The aims of this course are: (i) to develop and improve student's calculus skills (ii) to give students an appreciation of the fact that there is sophisticated mathematics behind the study of differential equations; (iii) 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; (iv) to familiarise students with vectors.

Objectives

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.
Students should know basic techniques for solving first order ODEs of separable and linear types.
Students should know how to solve second-order equations with constant coefficients.
Students should know how to compute Taylor's series in one dimension.
Students should know how to manipulate vectors.

Syllabus

Revision of basic calculus skills. Taylor's series. First order scalar equations: definition of a first order ODE; examples. Solution techniques for simple first order ODEs: separable equations; integrating factors. Statement of Picard's Theorem (no proof): examples of failure of uniqueness in absence of Lipschitz continuity. Second Order Scalar Equations: definition of a second order ODE; examples. Second order linear equations with constant coefficients: solution of homogeneous second order ODEs by reference to the roots of the auxiliary quadratic; solution of inhomogeneous equations using the variation-of-parameters method. Introduction to 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.

Reading list

Essential:

G. B. Thomas, R. Finney, M. D. Weir and F. R. Giordano, Thomas' Calculus, 10th Edition, Pearson Education, 2001.

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 final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination during the January exam period. The 20% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.