Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition

Introduction

n this introductory chapter we try to give perspective to your study of differential equations. We first formulate several problems to introduce terminology and to illustrate some of the basic ideas that we will return to frequently in the remainder of the book. We introduce three major methods—geometrical, analytical, and numerical—for investigating the solutions of these problems. We then discuss several ways of classifying differential equations in order to provide organizational structure for the book.

1.1 Mathematical Models, Solutions, and Direction Fields

Many of the principles, or laws, underlying the behavior of the natural world are statements, or relations, involving rates at which one variable, say y, changes with respect to another variable, t, for example. Most often, these relations take the form of equations containing y and certain of the derivatives y′, y″, . . . , y⁽ⁿ⁾ of y with respect to t. The resulting equations are then referred to as differential equations. Some examples of differential equations that will be studied in detail later on in the text, are

The subject of differential equations was motivated by problems in mechanics, elasticity, astronomy, ...