Algebra and Differential Equations

Cod y Modiwl: PH16210
Teitl y Modiwl: Algebra and Differential Equations
Blwyddyn Academaidd: 2025/2026
Semester: Semester 1
Cyd-gysylltydd y Modiwl: Professor John Gough
Rhestr Ddarllen: Gweld ar Aspire
Rhagofynion: A Level Mathematics or equivalent
Cyd-Ofynion: MP10610 neu MT10610
Anghymharus (Unrhyw Flwyddyn Acad): MT10510
Anghymharus (Unrhyw Flwyddyn Acad): MT11210
Anghymharus (Unrhyw Flwyddyn Acad): MA10510
Anghymharus (Unrhyw Flwyddyn Acad): FG16210
Anghymharus (Unrhyw Flwyddyn Acad): MA11210
Staff Eraill sy'n Cyfrannu: Professor John Gough

Dulliau Asesu

Math o Asesiad	Manylion Asesiad	Cyfran
Asesiad Semester	Online Test 2:	15%
Asesiad Semester	Online Test 1:	15%
Arholiad Semester	Semester Exam: 2 Awr	70%
Arholiad Ailsefyll	Written Examination: 2 Awr	100%

Canlyniadau Dysgu

Wedi cwblhau'r modiwl dylai'r myfyrwyr fedru:

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.

Disgrifiad cryno

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.

Nod

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.

Cynnwys

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.

Sgiliau Modiwl

Math o Sgiliau	Manylion Sgiliau
Cyfathrebu	Students will have to state definitions of mathematical terms concisely.
Datrys Problemau	Mathematical problems to be solved in each of the workshops.
Gwaith Tim	There is opportunity for group work in the workshops where students are encouraged to work together to solve problems and learn from each other.
Gwella dysgu a pherfformiad ei hun	There is opportunity to learn from feedback in the workshops and so to improve perfromance.
Rhifedd	Application of numbers occurs in examples.
Sgiliau pwnc penodol	Translating physical problems into mathematical equations and models.
Sgiliau ymchwil	Research skills are developed through background reading on the module topics

Nodau

Mae'r modiwl hwn yn cydymffurfio a FfCChC Lefel 4