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
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