The University of Southampton
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# MATH1052 Differential Equations

## Module Overview

The module begins by revising and extending methods of integration from A-level. The module then goes on to look at the role of differential equations in modelling real problems. It then introduces the basic methods used for solving ordinary differential equations (ODEs) and showing how ODEs can be used to model a wide variety of real situations. Techniques for solving various classes of first order differential equations are considered and a number of practical applications will be investigated including applications to finance, biological systems and dynamics. Linear second order ODEs will then be studied and applications to oscillatory motion considered. It is important to note that many differential equations cannot be solved using analytical methods so attention will also be paid to the qualitative behaviour of solutions to ODEs by looking at the phase plane

### Aims and Objectives

#### Module Aims

The module begins by revising and extending methods of integration from A-level. The module then goes on to look at the role of differential equations in modelling real problems. It then introduces the basic methods used for solving ordinary differential equations (ODEs) and showing how ODEs can be used to model a wide variety of real situations.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Recall from school and perform accurately definite and indefinite integration for powers, exponentials logarithms, rational functions, trigonometric functions, using parts, substitution and reduction formulae
• Sketch the general behaviour of the solution of a first order ODE
• Identify and solve first order ODEs that are, separable, linear, exact, homogeneous and other special types of nonlinear equations
• Derive first order ODEs from physical situations by applying balance laws
• Find complementary solutions, particular integrals and general solutions of linear second order equations and interpret the results
• Derive second order ODEs from physical problems, solve and interpret the results
• Sketch the phase plane of a system of two first order ODEs and interpret the results.

### Syllabus

• Integration including: area definition; Fundamental Theorem of Calculus (statement only); definite and indefinite integration for powers, exponentials logarithms, rational functions, trigonometric functions, using parts, substitution and reduction formulae; • Introduction to Differential Equations; • Introduction: order, degree, general solution. • First order Differential Equations: • Vector fields, isoclines and integral curves. • Analytical techniques for the solution of classes of odes including: variables separable, linear, exact, homogeneous, Bernoulli and Clairaut equations. • Applications of Differential Equations to: e.g. economics, radiocarbon dating, pollution control, population dynamics, biological and physical situations. • Second order Differential Equations: linear with constant coefficients, Euler equations. • Undetermined coefficients, reduction of order, variation of parameters. • Simple harmonic Motion, Resonance. • Newton's Law of motion, kinetic energy, conservation of energy as first integrals of the motion; Solutions of systems of linear equations through matrix techniques; • Phase plane analysis of solutions of systems of equations, and their interpretation. • Applications of phase plane analysis to, e.g., physical, social, economic, political situations.

### Learning and Teaching

#### Teaching and learning methods

Lectures, workshops, exercise classes, private study

TypeHours
Teaching60
Independent Study90
Total study time150

Ayres, F. Mendleson, E., Calculus. Schaum’s outline series.

Bronson R, and Costa G. Differential Equations, Schaum’s outline series.

Robinson, J.C.,. An Introduction to Ordinary Differential Equations.

### Assessment

#### Assessment Strategy

Each of the weekly homeworks set will count towards final mark

#### Summative

MethodPercentage contribution
Class Test 10%
Coursework 20%
Exam  (2 hours) 70%

#### Repeat Information

Repeat type: Internal & External

One of the pre-requisites for MATH2011, MATH2038, MATH2039 AND MATH 2045

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Full printed skeletal lecture notes are provided for the class. Electronic copies of the notes are posted to Blackboard. Course texts may be provided by the library. There are no additional compulsory costs associated with the module

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.

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