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Next: MC130 Mathematical Modelling
Up: Year 1
Previous: MC126 Multivariate Calculus
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 |
Problem Classes: |
none |
Tutorials: |
5 |
Private Study: |
47 |
Labs: |
5 |
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 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
- separable equations;
- linear equations (including integrating factors);
- nonlinear equations of the form y' = f(y/x) (if time permits);
- Bernoulli equations (if time permits).
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.
- Homogeneous equations. How to solve a homogeneous second order
equation with constant coefficients by reference to the roots
of the auxiliary quadratic. What to do in the case of coincident
roots. What to do in the case of complex roots, including the
case of purely imaginary roots. (Note: this can be done without
complex exponentials.) Examples, including damped oscillations
of a spring.[4 lectures]
- Inhomogeneous equations. How to find particular integrals for
various standard forms of inhomogeneity. The variation-of-parameters
method. Examples, including forced damped oscillation of a spring.
[3 lectures]
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: MC130 Mathematical Modelling
Up: Year 1
Previous: MC126 Multivariate Calculus
Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 10/4/2000
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This document has been approved by the Head of Department.
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