# Module Information

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

Lecture | 22 x 1 Hour Lectures |

Workshop | 5 x 2 Hour Workshops |

#### 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 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 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

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