Chapter 1 Differential Equations You probably have seen differential equations before, and may even have taken a course on them. However, both to help you review and to set up some terminology peculiar to this book, we will begin with some elementary discussion of differential equations. As we learn in calculus, the derivative df dx measures how much the value of the function f will change if the value of the variable x is changed by a tiny amount. In particular, like an exchange rate for converting one currency to another, the derivative is the factor by which you would need to multiply an infinitesimal change in x to get the corresponding change in f. A differential equation, which is nothing other than an equation involving derivatives of some unknown function, represents an exact relationship between such rates. For example, the simple differential equation dP dt = kP (1.1) poses the problem of finding a function P (t) whose rate of change with respect to t is exactly the number k multiplied by the value of the function P itself. Anyone who has taken a course in calculus can come up with a solution to this abstract puzzle1 for any constant C the function P (t) = Cekt (1.2) 1 In addition, there are explicit techniques that one learns in a differential equa- tions course (e.g. “separation of variables”) which can be utilized here. However, most of those analytic techniques will not be needed for the subject studied in this textbook. Consequently, we may intentionally avoid mentioning them so that the book is equally useful to those who are familiar with them and those who are not. 1 http://dx.doi.org/10.1090/stml/054/01
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