Differential Equations module (MA31002)

Differential equations are one of the main ways mathematics describes change. They appear when we model motion, heat flow, waves, population growth, chemical reactions, and many other processes that evolve over time or space.

In this module, you will learn how to recognise different types of differential equation and choose analytical methods for solving them.

You will begin with ordinary differential equations (ODEs), i.e. equations for a function of one variable. Special emphasis is placed on special classes, such as separable and linear problems.

You will also study partial differential equations (PDEs), which describe quantities that change in both space and time. You will meet linear and nonlinear examples, learn how initial and boundary conditions complete the statement of a problem, and classify second order linear PDEs as elliptic, parabolic or hyperbolic. Using separation of variables, you will solve the heat (diffusion) equation, Schrödinger and wave equation. This builds directly on eigenvalue problems that are also studied in the module.

By the end of the module, you will have a strong analytical toolkit for solving ordinary and partial differential equations and for understanding how they model real processes.

What you will learn

In this module, you will:

• solve ODEs using analytical methods
• study second order linear equations and their applications
• solve boundary value problems and eigenvalue problems for second order equations
• study partial differential equations, their types, and their initial and boundary conditions
• use separation of variables to solve PDEs

By the end of this module, you will be able to:

• classify differential equations and choose suitable solution methods
• solve first and second order ODEs
• rewrite higher order equations as systems
• use linear algebra to solve linear systems of ODEs
• classify second order linear PDEs as elliptic, parabolic or hyperbolic
• solve initial-boundary value problems for the heat, Schrödinger and wave equations by separation of variables

Assignments / assessments

• Coursework (20%)
• Written exam (80%)

Teaching methods / timetable

• lectures, introducing the main ideas, methods, and worked examples in differential equations
• lecture notes will be available before class, helping you prepare and focus on understanding during sessions
• interactive discussion, giving you opportunities to ask questions and connect methods to model problems
• tutorials, where you will practise solving differential equations individually and in groups
• support from lecturers and peers, helping you build confidence with analytical methods and systems of equations