# Gwybodaeth Modiwlau

Module Identifier
PH16210
Module Title
Algebra and Differential Equations
2015/2016
Co-ordinator
Semester
Semester 1
Mutually Exclusive
Mutually Exclusive
Mutually Exclusive
Mutually Exclusive
Mutually Exclusive
Other Staff

#### Course Delivery

Delivery Type Delivery length / details
Lecture 11 x 1 Hour Lectures
Workshop 11 x 2 Hour Workshops

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Assessment Hand-ins from Workshops 1-4  30%
Semester Exam 2 Hours   Semester Exam  70%
Supplementary Exam 2 Hours   Written Examination  100%

### Learning Outcomes

On completion of this module, students should be able to:
1. manipulate complex numbers and use DeMoivre’s theorem
2. use the division algorithm for polynomials
3. derive identities involving the roots of a polynomial and its coefficients
4. sketch the graphs of simple functions
5. explain the notion of inverse function
6. express functions in terms of power series
7. classify differential equations in terms of order, homogeneity, linearity and ordinary/partial character
8. identify suitable solution strategies for common types of ordinary differential equation
9. determine the number of boundary conditions needed to solve a particular differential equation
10. solve separable and linear-homogeneous ODE and linear ODE with constant coefficients
11. phrase simple physical problems such as radioactive decay or free fall in terms of an ODE, irrespective of the variable names used in the particular physical context

### 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 The focus of the module is on algebraic rather than numerical work, but application of number will feature in the physical examples.
Communication Students will have to state definitions of mathematical terms concisely in their own words.
Improving own Learning and Performance With four workshop and coursework cycles, students have ample opportunity to engage with feedback given during the semester to improve their own performance.
Information Technology Not specifically targeted or assessed beyond coursework submission through Blackboard.
Personal Development and Career planning Not specifically targeted.
Problem solving Mathematical problems to be solved in each of the four workshops.
Research skills To the extent that background reading is required.
Subject Specific Skills Translating physical problems into mathematical equations and models.
Team work The workshops will be run in small groups, and students are encouraged to solve problems together and learn from each other.

### Notes

This module is at CQFW Level 4