## MATH1400 Modelling with Differential Equations

### 10 creditsClass Size: 250

Module manager: Professor A. Rucklidge
Email: a.m.rucklidge@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisite qualifications

A good A-level Mathematics grade or equivalent.

### This module is mutually exclusive with

 MATH1932 Calculus, ODEs and Several-Variable Calculus MATH1970 Differential Equations

This module is approved as an Elective

### Module summary

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.

### 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

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. Partial differentiation, classification of critical points of two-variable functions. 7. 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

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Tutorial 11 1.00 11.00 Private study hours 67.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

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

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated