Module Information

Module Identifier

PH16210

Module Title

Algebra and Differential Equations

Academic Year

2019/2020

Co-ordinator

Professor John Gough

Semester

Semester 1

Co-Requisite

MP10610 or MT10610

Mutually Exclusive

MT11210

Mutually Exclusive

MT10510

Mutually Exclusive

FG16210

Mutually Exclusive

MA10510

Mutually Exclusive

MA11210

Pre-Requisite

A Level Mathematics or equivalent

Other Staff

Course Delivery

Delivery Type	Delivery length / details
Workshop	5 x 2 Hour Workshops
Lecture	22 x 1 Hour Lectures

Assessment

Assessment Type	Assessment length / details	Proportion
Semester Assessment	Hand-ins from Workshops	30%
Semester Exam	2 Hours Semester Exam	70%
Supplementary Exam	2 Hours Written Examination	100%

Learning Outcomes

On successful completion of this module students should be able to:

1. Manipulate complex numbers and use DeMoivre’s theorem.
2. Employ the division algorithm for polynomials.
3. Derive identities involving the roots of a polynomial and its coefficients.
4. Determine the order, homogeneity, linearity and ordinary/partial character of differential equations.
5. Identify suitable solution strategies for common types of ordinary differential equation.
6. Solve separable and linear-homogeneous ODE and linear ODE with constant coefficients.

Aims

To equip students with concepts of algebra such as complex numbers, polynomials and functions, which are needed to understand physical concepts and solve physical problems.
To introduce the concept of ordinary differential equations (ODE) and fundamental solution strategies for ODE used in various physical contexts.

Brief description

This module covers the basic algebra needed to study physical concepts and processes quantitatively. It also introduces ordinary differential equations, underpinning topics such as acoustics and quantum mechanics.

Content

Complex numbers: Geometric representation, DeMoivre's theorem.
Polynomials: Polynomial division, symmetric functions, relations between roots of a polynomial and its coefficients
Functions of a real variable: Graphs of elementary functions (polynomia, exponential, logarithmic, trigonometric, hyperbolic etc.), periodic functions, even and odd functions. Operations on functions: addition, multiplication, division, composition. Asymptotes. Inverse functions.

Series: Convergence of series. Power Series.
Classifying differential equations: Order, ordinary vs. partial, homogeneity, linearity.
First-order equations with separable variables. Radioactive decay. Boundary conditions (e.g. initial values).
Homogeneous linear first-order equations. Integrating factor method. Higher orders. Free fall.
Non-homogeneous equations. Particular function. Driven oscillations. Special cases: Heterogeneous part solves homogeneous equation.
Linear ODE with constant coefficients. Characteristic polynomial. Special cases: Degenerate roots. Standing waves.

Module Skills

Skills Type	Skills details
Application of Number	Application of numbers occurs in examples.
Communication	Students will have to state definitions of mathematical terms concisely.
Improving own Learning and Performance	There is opportunity to learn from feedback in the workshops and so to improve perfromance.
Problem solving	Mathematical problems to be solved in each of the workshops.
Research skills	Research skills are developed through background reading on the module topics
Subject Specific Skills	Translating physical problems into mathematical equations and models.
Team work	There is opportunity for group work in the workshops where students are encouraged to work together to solve problems and learn from each other.

Notes

This module is at CQFW Level 4