Chapter 1 - Introduction to differential equations

Abstract

The purpose of Differential Equations with Mathematica, fifth edition, is twofold. First, we introduce and discuss the topics covered in typical undergraduate and beginning graduate courses in ordinary and partial differential equations including topics such as Laplace transforms, Fourier series, eigenvalue problems, and boundary-value problems. Second, we illustrate how Mathematica is used to enhance the study of differential equations not only by eliminating the computational difficulties but also by overcoming the visual limitations associated with the explicit solutions to differential equations. In each chapter, we first present the material in a manner similar to most differential equations texts and then illustrate how Mathematica can be used to solve some typical problems. For example, in Chapter 2, we introduce the topic of first-order equations. First we show how to solve certain types of problems by hand, and then show how Mathematica can be used to assist in the same solution procedures. Finally, we illustrate how Mathematica commands like DSolve and NDSolve can be used to solve some frequently encountered equations exactly and/or numerically. In Chapter 3, we discuss some applications of first-order equations. Since we are experienced and understand the methods of solution covered in Chapter 2, we make use of DSolve and similar commands to obtain solutions. In doing so we are able to emphasize the applications themselves as opposed to becoming bogged down in calculations.