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