### 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 $Y\left(s\right)=\mathsc{L}\left\{y\left(t\right)\right\}$ is obtained. This equation is solved for $Y\left(s\right)$ which is then inverted to produce the required solution $y\left(t\right)={\mathsc{L}}^{-1}\left\{Y\left(s\right)\right\}$ .

#### 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