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)

1In

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