## MATH1400 Modelling with Differential Equations

### 10 creditsClass Size: 250

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

A-level Mathematics, or equivalent.

This module is approved as an Elective

### Objectives

To introduce the concept of mathematical modelling. To illustrate its application in various areas and to develop relevant methods for the solution of first and second order ODEs. On completion of this module, students should be able to: (a) set up simple first order differential equations to model processes such as radioactive decay and Newton cooling; (b) solve first order differential equations of various types such as separable, homogenous, linear, and to apply initial conditions to the general solution; (c) solve second order linear differential equations with constant coefficients by finding complementary functions and particular integrals, and to apply either initial or boundary conditions; (d) linearise systems of first order differential equations, find their equilibrium points, and classify the equilibrium points of systems of two variables; (e) apply the phase plane method to physical systems of two variables, such as the predator-prey model.

### Syllabus

The applied mathematician attempts to give a mathematical description (a mathematical model) of things in the real world. In the real world most things change with time. Mathematically a rate of change is expressed as a derivative so the applied mathematician deals mostly with equations involving derivatives - so called differential equations. This module develops the theory of differential equations and applies it to produce mathematical models describing e.g. the way in which the population of the world varies with time, and the way in which an influenza virus propagates through a university campus. Topics covered include: 1. The modelling process via simple examples: exponential growth and decay etc. 2. Solution of first order ODEs: linear via integrating factor, nonlinear via substitutions. 3. Application of first order ODEs to modelling population growth, etc. 4. Solution of second order ODEs (linear with constant coefficients) and simultaneous ODEs. 5. Application of second order ODEs to interacting population models etc. 6. Introduction to phase plane methods: critical points, node, saddle, focus, centre.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures (22 hours) and tutorials (11 hours).

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

2 hour written examination at end of semester (85%), coursework (15%).