MA13312 Differential Equations
Introductory description
How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.
However, this theory is not only of interest to the applied mathematician: indeed, it is an integral part of any rigorous mathematical training, and is developed here in a systematic way. Just as a pure' subject like group theory can be part of the daily armoury of the
applied' mathematician , so ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology.
Module aims
To introduce simple differential and difference equations and methods for their solution, to illustrate the importance of a qualitative understanding of these solutions and to understand the techniques of phaseplane analysis.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
In this module we will cover relatively simple examples, first order equations dy/dx=f(x,y), linear second order equations
\ddot{x}(t)\dot{x}+q(t)x=g(t) and coupled first order linear systems with constant coefficients, for most of which we can find an explicit solution. However, even when we can write the solution down it is important to understand what the solution means, i.e. its `qualitative' properties. This approach is invaluable for equations for which we cannot find an explicit solution.
We also show how the techniques we learned for second order differential equations have natural analogues that can be used to solve difference equations.
The course looks at solutions to differential equations in the cases where we are concerned with one and twodimensional systems, where the increase in complexity will be followed during the lectures. At the end of the module, in preparation for more advanced modules in this subject, we will discuss why in threedimensions we see new phenomena, and have a first glimpse of chaotic solutions.
Learning outcomes
By the end of the module, students should be able to:
 You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations) and to interpret their qualitative behaviour;
 and to do the same for simple difference equations.
Indicative reading list
The primary text will be:
J. C. Robinson An Introduction to Ordinary Differential Equations, Cambridge University Press 2003.
Additional references are:
W. Boyce and R. Di Prima, Elementary Differential Equations and Boundary Value Problems, Wiley 1997.
C. H. Edwards and D. E. Penney, Differential Equations and Boundary Value Problems, Prentice Hall 2000.
K. R. Nagle, E. Saff, and D. A. Snider, Fundamentals of Differential Equations and Boundary Value Problems, Addison Wesley 1999.
Subject specific skills
See learning outcomes.
Transferable skills
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (25%) 
Tutorials  8 sessions of 30 minutes (3%) 
Private study  86 hours (72%) 
Total  120 hours 
Private study description
Review lectured material and work on set exercises.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D2
Weighting  Study time  

Assignments  15%  
2 hour online examination (Summer)  85%  

Assessment group R
Weighting  Study time  

2 hour examination  100%  

Feedback on assessment
Marked assignments and exam feedback.
Courses
This module is Core for:
 Year 1 of UMAAG100 Undergraduate Mathematics (BSc)
 Year 1 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 1 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 1 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 1 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 1 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 1 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)
 Year 1 of UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 1 of UMAAGV18 Undergraduate Mathematics and Philosophy with Intercalated Year
 Year 1 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 1 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 1 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 1 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Optional for:
 Year 1 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 1 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)