# Module Information

- Professor Pete Vukusic (Professor - Exeter University)

#### Course Delivery

#### 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 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