Differential Equations module (MA31002)

This module explores an area of calculus involving derivatives of functions. Derivatives give the rate of change of a function. For example, speed is the rate of change of distance, while acceleration is the rate of change of speed. Differential equations are equations that involve functions and their derivatives.

Therefore a differential equation can be used to describe a relationship between distance, speed, and acceleration. These equations have many other applications. These include modelling systems in engineering, economics, and biology. Differential equations are an essential modelling tool in science and engineering. They describe how a particular quantity changes over time or space.

Studying differential equations is useful. They provide the tools to solve real-world problems. By learning how to formulate and solve these equations, you will gain insights into the behaviour of complex systems. This knowledge is highly practical. It equips you with analytical skills that are valuable in research and industry. Moreover, mastering differential equations enhances problem-solving abilities. This can prepare you for advanced studies in mathematics and related disciplines.

What you will learn

In this module, you will:

• study techniques to solve single differential equations with first and second derivative terms. These are called first and second order equations
• learn about systems of differential equations. You will also learn how you can visualise solutions by plotting solution trajectories in something called a phase plane
• be introduced to how functions can be written using a series expansion called a Fourier series. You will learn how this structure is useful within certain differential equations

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

• solve examples of first order and second order differential equations. You will be able to apply results from linear algebra to the study of systems of differential equations. This will allow you to derive general solutions
• solve systems of differential equations, plus plot phase plane trajectories to visualise results
• represent arbitrary functions using Fourier series decomposition. This will allow you to construct solutions to partial differential equations
• use the software package Maple to do simple calculations and to present solutions in graphical formats

Assignments / assessment

• coursework (20%)
• final exam (80%)

Teaching methods / timetable

• one two-hour lecture weekly
  • key points of the week's content will be discussed
  • lecture notes covering the full module content will be available before classes
  • in-class time will be prioritised for interactive discussion
• one two-hour tutorial weekly
  • solve problems individually and in groups
  • support with difficulties will be provided by your lecturers and peers