|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%|
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.
To introduce the concept of ordinary differential equations (ODE) and fundamental solution strategies for ODE used in various physical contexts.
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.
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.
|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.|
This module is at CQFW Level 4