chapter 1, Ordinary Differential Equations

Excerpt

Historical Comments: Historically, the study of differential equations originated with the introduction of the calculus by Sir Isaac Newton(1642–1727) and Gottfried Wilhelm von Leibniz (1646–1716). Although mathematics began as a recreation for Newton, he became known as a great mathematician by the age of 24 after his invention of the calculus, discovery of the law of universal gravitation, and experimental proof that white light is composed of all colors. Leibniz completed his doctorate in philosophy by the age of 20 at the University of Altdorf Afterward he studied mathematics under the supervision of Christian Huygens (1629–1695) and, independently of Newton, helped develop the calculus. Leibniz corresponded regularly with other mathematicians concerning differential equations, and he developed several methods for solving first-order equations.

Other prominent mathematicians who contributed to the development of differential equations and their applications were members of the famous Bernoulli family of Switzerland, the most famous of which are James (1654–1705) and John (1667–1748). Over the years there have been a host of mathematicians who contributed to the general development of differential equations.

The objective of this chapter is to review the basic ideas found in a first course in ordinary differential equations (ODEs). In doing so, we will concentrate primarily on those we deem most important in engineering applications. Because DEs are considered the most fundamental models that are used in a wide variety of physical phenomena, they play a central role in many of the following chapters of this text.

1.1 Introduction

Differential equations (DEs) play a fundamental role in engineering and science because they can be used in the formulation of many physical laws and relations. The development of the theory of DEs is closely interlaced with the development of mathematics in general, and it is indeed difficult to separate the two. In fact, most of the famous mathematicians from the time of Newton and Leibniz had a part in the cultivation of this fascinating subject. The first problems studied that involved the notion of DE came from the field of mechanics. Consequently, some of the terminology that persists today (like “forcing function”) had its beginning in these early mechanics problems.

At its most basic level, Newton's second law of motion is commonly expressed by the simple algebraic formulation