### Introduction

Sections 19.2 and 19.3 have introduced several techniques for solving commonly occurring first-order and second-order ordinary differential equations. In this Section we solve a number of these equations which model engineering systems.

#### Prerequisites

- understand what is meant by a differential equation
- be familiar with the terminology associated with differential equations: order, dependent variable and independent variable
- be able to integrate standard functions

#### Learning Outcomes

- recognise and solve first-order ordinary differential equations, modelling simple electrical circuits, projectile motion and Newton’s law of cooling
- recognise and solve second-order ordinary differential equations with constant coefficients modelling free electrical and mechanical oscillations
- recognise and solve second-order ordinary differential equations with constant coefficients modelling forced electrical and mechanical oscillations

#### Contents

1 Modelling with first-order equations1.1 Applying Newton’s law of cooling

1.2 Modelling electrical circuits

2 Modelling free mechanical oscillations

2.1 Case 1: No damping

2.2 Case 2: Light damping

2.3 Case 3: Heavy damping

3 Modelling forced mechanical oscillations

4 Modelling forces on beams

5 Engineering Example 3

5.1 Shear force and bending moment of a beam

6 Engineering Example 4

6.1 Deflection of a uniformly loaded beam