### Introduction

equations. In particular we shall consider initial value problems. We shall find that the initial conditions are automatically included as part of the solution process. The idea is simple; the Laplace transform of each term in the differential equation is taken. If the unknown function is $y\left(t\right)$ then, on taking the transform, an algebraic equation involving $Y\left(s\right)=\mathcal{L}\left\{y\left(t\right)\right\}$ is obtained. This equation is solved for $Y\left(s\right)$ which is then inverted to produce the required solution $y\left(t\right)={\mathcal{L}}^{-1}\left\{Y\left(s\right)\right\}$ .

#### Prerequisites

- understand how to find Laplace transforms of simple functions and of their derivatives
- be able to find inverse Laplace transforms using a variety of techniques
- know what an initial-value problem is

#### Learning Outcomes

- solve initial-value problems using the Laplace transform method

#### Contents

1 Solving ODEs using Laplace transforms2 Solving systems of differential equations

3 Applications of systems of differential equations

3.1 Electrical circuits

3.2 Two masses on springs

4 Engineering Example 1

4.1 Charge on a capacitor

5 Engineering Example 2

5.1 Deflection of a uniformly loaded beam