Chapter
6
Differential Equations
and Mathematical
Modeling
O
ne way to measure how light in the ocean di-
minishes as water depth increases involves
using a Secchi disk. This white disk is 30
centimeters in diameter, and is lowered into the
ocean until it disappears from view. The depth of this
point (in meters), divided into 1.7, yields the coeffi-
cient
k
used in the equation
l
x
l
0
e
kx
. This equation
estimates the intensity
l
x
of light at depth
x
using
l
0
,
the intensity of light at the surface.
In an ocean experiment, if the Secchi disk disap-
pears at 55 meters, at what depth will only 1% of
surface radiation remain? Section 6.4 will help you
answer this question.
320

Section 6.1
Slope Fields and Euler’s Method
321
Chapter 6 Overview
One of the early accomplishments of calculus was predicting the future position of a
planet from its present position and velocity. Today this is just one of a number of occa-
sions on which we deduce everything we need to know about a function from one of its
known values and its rate of change. From this kind of information, we can tell how long a
sample of radioactive polonium will last; whether, given current trends, a population will
grow or become extinct; and how large major league baseball salaries are likely to be in
the year 2010. In this chapter, we examine the analytic, graphical, and numerical tech-
niques on which such predictions are based.
Slope Fields and Euler’s Method
Differential Equations
We have already seen how the discovery of calculus enabled mathematicians to solve
problems that had befuddled them for centuries because the problems involved moving
objects. Leibniz and Newton were able to model these problems of motion by using equa-
tions involving derivatives—what we call
differential equations
today, after the notation of
Leibniz. Much energy and creativity has been spent over the years on techniques for solv-
ing such equations, which continue to arise in all areas of applied mathematics.
6.1
What you’ll learn about
• Differential Equations
• Slope Fields
• Euler’s Method
. . . and why
Differential equations have always
been a prime motivation for the
study of calculus and remain so
to this day.
EXAMPLE 1
Solving a Differential Equation
Find all functions
y
that satisfy
dy dx
sec
2
x
2
x
5.
SOLUTION
We first encountered this sort of differential equation (called
exact
because it gives the
derivative exactly) in Chapter 4. The solution can be any antiderivative of sec
2
x
2
x
5,
which can be any function of the form
y
tan
x
x
2
5
x
C
. That family of func-
tions is the
general
solution to the differential equation.
Now try Exercise 1.
Notice that we cannot find a unique solution to a differential equation unless we are given
further information. If the general solution to a first-order differential equation is continuous,
the only additional information needed is the value of the function at a single point, called an
initial condition
. A differential equation with an initial condition is called an
initial value
problem
. It has a unique solution, called the