MATH10232 -2008/2009

 

Specification

Aims

This course unit aims to introduce students to ordinary differential equations, primarily covering methods of solution and applications to physical situations.

Brief Description of the unit

The unit will cover first and second order ordinary differential equations including classification and standard solution methods. Applications will be drawn from the field of classical mechanics, but no prior experience in mechanics is expected or required. Matlab will be used to illustrate some of the ideas and methods.

Learning Outcomes

On completion of this unit successful students will be able to solve first order and second order linear problems and first order separable equations analytically. Use substitution methods and power series methods to find solutions. Be able to investigate solutions using direction fields and Euler's method. Have used Matlab as a mathematical tool and used differential equations to solve problems in mechanics and other applications.

Future topics requiring this course unit

Almost every applied mathematics course unit and many in pure mathematics and statistics.

Syllabus

1. Ordinary differential equations: order; linear and autonomous equations; solutions by integration; initial and boundary conditions; existence and uniqueness of solution.
2. Applications: motion in a straight line; population modelling including predator prey equations.
3. Computer methods: direction fields; Taylor series approximation; Taylor's method; Euler's method; some topics from Matlab.
4. Analytic methods for first order equations: separable equations; autonomous equations; linear equations; standard substitution methods.
5. Phase diagrams for two dimensional autonomous systems.
6. Motion in space: vector functions and space curves; arc length and curvature; velocity and acceleration; Kepler's law of planetary motion.
7. Second order equations: Linear homogeneous equations; power series; special problems {y″=f(x,y′); y″=f(y,y′); y″=y(y)}; application to springs including resonance.

Textbooks

• James Stewart, Calculus, Early Transcendentals, Thomson, 5th Edition, International Student Edition, 2003.
• C. H Edwards, Elementary differential equations with boundary value problems, Pearson Prentice Hall, 2004.

Teaching and learning methods

Assessment

Coursework; Weighting within unit 15%

Supervision; Weighting within unit 10%

Two hours and half end of semester examination; Weighting within unit 75%

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Arrangements