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School of Mathematics

# MATH11222 - 2007/2008

General Information
• Title: Calculus and Applications
• Unit code: MATH10222
• Credits: 10
• Ths course unit consists of the first half of MATH10222. It may not be taken with that course unit.
• Prerequisites: MATH10121
• Co-requisite units: None
• School responsible: Mathematics
• Members of staff responsible: Prof. Matthias Heil
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## Specification

Many

### Brief Description of the unit

The course is devoted to an introduction to the study of ordinary differential equations (ODEs).

### Future topics requiring this course unit

A large number of course units in applied mathematics, pure mathematics and statistics.

### Syllabus

1. General introduction. Notation. What are ODEs? Implicit versus explicit form. Classification: order, linearity, autonomous ODEs. Boundary and initial conditions. Boundary and initial value problems. Existence and uniqueness for linear and nonlinear ODEs. [3]
2. First-order ODEs. Graphical methods; separable ODEs, ODEs of homogeneous type; integrating factor. [4]
3. Second-order ODEs. Existence and uniqueness. Linear ODEs: superposition of solutions, fundamental solutions and the general solution for homogeneous ODEs. The general solution of constant-coefficient ODEs; particular solutions for specific RHS; the method of undetermined coefficients. [If time permits (probably not): Power series exapansions about regular points.] Some nonlinear ODEs with special properties (autonomous ODEs and ODEs that do not contain the dependent variable). [8]
4. Mechanics applications of second-order ODEs Damped harmonic motions of mechanical oscillators; harmonic forcing and resonance. [3]
5. Non-dimensionalisation and scaling. Exploiting small parameters in an ODE: perturbation methods. Motivation via the roots of quadratic polynomials (singular perturbations only mentioned); applications to selected (regularly perturbed) ODEs. [4]

### Teaching and learning methods

Four lectures and one supervision class each week. In addition, students should expect to spend at least six hours each week on private study. The course unit only lasts for the first six weeks of the semester.

### Assessment

Attendance at supervisions: weighting 5%
Submission of coursework at supervisions: weighting 5%
In-class test: weighting 15%
Three hours end of semester examination: weighting 75%

## Arrangements

Online course materials are available for this unit.