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 first order ordinary differential equations, including separable, linear, homogeneous, exact, and Bernoulli equations. These methods help you move from a model involving rates of change to an explicit or implicit solution.

You will then study second order linear differential equations, which are central to modelling oscillations, forces, and systems with acceleration. Techniques such as reduction of order, undetermined coefficients, and variation of parameters will help you solve both homogeneous and nonhomogeneous equations.

The module also introduces systems of first order differential equations. You will see how higher order equations can be rewritten as systems, how linear algebra helps solve them, and how phase planes can be used to understand the behaviour of solutions.

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

What you will learn

In this module, you will:

• solve first order ordinary differential equations using analytical methods
• study second order linear equations and their applications
• use undetermined coefficients and variation of parameters
• analyse systems of first order differential equations
• use phase planes to understand solution behaviour

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 systems of ODEs

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