Introduction & Chapter Objectives

Introduction

The Laplace transform is a powerful technique used to convert differential equations into simpler algebraic equations. By transforming a function of time into a function of a complex variable, the Laplace transform allows us to analyze and solve problems that arise in mathematics, physics, and engineering. In this chapter, we introduce the definition of the Laplace transform, study its main properties, and learn how to compute inverse transforms. We then apply these ideas to the solution of ordinary differential equations, particularly those involving initial conditions. This method provides a systematic and efficient approach to solving problems where traditional techniques may be difficult or time-consuming.

Chapter Objectives

At the end of this chapter, students should be able to:

1.  Understand the definition of the Laplace transform and its fundamental properties.

 2.  Compute Laplace transforms and inverse Laplace transforms for common functions.

3.  Use operational properties such as linearity, shifting, and convolution.

4.  Solve first- and second-order differential equations using the Laplace transform.

 5. Apply the Laplace transform to initial value problems arising in physics and engineering.