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Unformatted text preview: 1 The Starting Point: Basic Concepts and Terminology Let us begin our study of differential equations with a few basic questions questions that any beginner should ask: What are differential equations? What can we do with them? Solve them? If so, what do we solve for? And how? and, of course, What good are they, anyway? In this chapter, we will try to answer these questions (along with a few you would not yet think to ask), at least well enough to begin our studies. With luck we will even raise a few questions that cannot be answered now, but which will justify continuing our study. In the process, we will also introduce and examine some of the basic concepts, terminology and notation that will be used throughout this book. 1.1 Differential Equations: Basic Definitions and Classifications A differential equation is an equation involving some function of interest along with a few of its derivatives. Typically, the function is unknown, and the challenge is to determine what that function could possibly be. Differential equations can be classified either as ordinary or as partial. An ordinary differential equation is a differential equation in which the function in question is a function of only one variable. Hence, its derivatives are the ordinary derivatives encountered early in calculus. For the most part, these will be the sort of equations well be examining in this text. Here are a few examples: dy dx = 8 dy dx + 4 y = x 2 3 4 The Starting Point: Basic Concepts and Terminology d 2 y dx 2 8 dy dx = 9 . 8 d 3 y dx 3 + 5 x d 2 y dx 2 7 dy dx + 832 y = sin ( x 2 ) d 42 y dx 42 = parenleftbigg d 3 y dx 3 parenrightbigg 2 In each of these equations, y denotes a function that is given by some, yet unknown, formula of x . 1 A partial differential equation is a differential equation in which the function of interest depends on two or more variables. Consequently, the derivatives of this function are the partial derivatives developed in the later part of most calculus courses. 2 Because the methods for studying partial differential equations often involve solving ordinary differential equations, it is wise to first become reasonably adept at dealing with ordinary differential equations before tackling partial differential equations. As already noted, this text is mainly concerned with ordinary differential equations. So let us agree that, unless otherwise indicated, the phrase differential equation in this text means ordinary differential equation. If you wish to further simplify the phrasing to DE or even to something like Diffy-Q, go ahead. This author, however, will not be so informal....
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