### Introduction

In this Section we start to learn how to solve second order differential equations of a particular type: those that are linear and have constant coefficients. Such equations are used widely in the modelling of physical phenomena, for example, in the analysis of vibrating systems and the analysis of electrical circuits.

The solution of these equations is achieved in stages. The first stage is to find what is called a ‘complementary function’. The second stage is to find a ‘particular integral’. Finally, the complementary function and the particular integral are combined to form the general solution.

#### Prerequisites

• understand what is meant by a differential equation
• understand complex numbers ( HELM booklet 10)

#### Learning Outcomes

• recognise a linear, constant coefficient equation
• understand what is meant by the terms ‘auxiliary equation’ and ‘complementary function’
• find the complementary function when the auxiliary equation has real, equal or complex roots