Differential Equations
By analyzing pairs of differ
ential equations we gain
insight into population
cycles of predators and
prey, such as the Canada
lynx and snowshoe hare.
C
H A P T E R
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R
W
1000
150
100
50
2000
3000
0
t
R
t¡
t£
W
120
80
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t™
2000
1000
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5E10(pp 622631)
1/18/06
9:18 AM
Page 622
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Perhaps the most important of all the applications of cal
culus is to differential equations. When physical scientists
or social scientists use calculus, more often than not it is to
analyze a differential equation that has arisen in the pro
cess of modeling some phenomenon that they are studying.
Although it is often impossible to find an explicit formula for the solution of a dif
ferential equation, we will see that graphical and numerical approaches provide the
needed information.

10.1
Modeling with Differential Equations
In describing the process of modeling in Section 1.2, we talked about formulating a math
ematical model of a realworld problem either through intuitive reasoning about the phe
nomenon or from a physical law based on evidence from experiments. The mathematical
model often takes the form of a
differential equation,
that is, an equation that contains an
unknown function and some of its derivatives. This is not surprising because in a real
world problem we often notice that changes occur and we want to predict future behavior
on the basis of how current values change. Let’s begin by examining several examples of
how differential equations arise when we model physical phenomena.
Models of Population Growth
One model for the growth of a population is based on the assumption that the population
grows at a rate proportional to the size of the population. That is a reasonable assumption
for a population of bacteria or animals under ideal conditions (unlimited environment, ade
quate nutrition, absence of predators, immunity from disease).
Let’s identify and name the variables in this model:
The rate of growth of the population is the derivative
. So our assumption that the
rate of growth of the population is proportional to the population size is written as the
equation
where
k
is the proportionality constant. Equation 1 is our first model for population
growth; it is a differential equation because it contains an unknown function
P
and its
derivative
.
Having formulated a model, let’s look at its consequences. If we rule out a population
of 0, then
for all
t
. So, if
, then Equation 1 shows that
for all
t
.
This means that the population is always increasing. In fact, as
increases, Equation 1
shows that
becomes larger. In other words, the growth rate increases as the popula
tion increases.
dP dt
P t
P t
0
k
0
P t
0
dP dt
dP
dt
kP
1
dP dt
P
the number of individuals in the population
the dependent variable
t
time
the independent variable
623

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 Fall '09
 hamrick
 Derivative, DI, Euler

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