Chap9_Sec1 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

DIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS
9

Perhaps the most important of
all the applications of calculus is
to differential equations.
DIFFERENTIAL EQUATIONS

When physical or social scientists use
calculus, more often than not, it is to analyze
a differential equation that has arisen in
the process of modeling some phenomenon
they are studying.
DIFFERENTIAL EQUATIONS

It is often impossible to find an explicit
formula for the solution of a differential
equation.

Nevertheless, we will see that graphical and numerical
approaches provide the needed information.
DIFFERENTIAL EQUATIONS

9.1
Modeling with
Differential Equations
In this section, we will learn:
How to represent some mathematical models
in the form of differential equations.
DIFFERENTIAL EQUATIONS

In describing the process of modeling in
Section 1.2, we talked about formulating
a mathematical model of a real-world problem
through either:

Intuitive reasoning about the phenomenon

A physical law based on evidence from experiments
MODELING WITH DIFFERENTIAL EQUATIONS

The model often takes the form of
a differential equation.

This is an equation that contains an unknown
function and some of its derivatives.
DIFFERENTIAL EQUATION

This is not surprising.

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.
MODELING WITH DIFFERENTIAL EQUATIONS

Let’s begin by examining several
examples of how differential equations
arise when we model physical
phenomena.
MODELING WITH DIFFERENTIAL EQUATIONS

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.
MODELS OF POPULATION GROWTH

That is a reasonable assumption for
a population of bacteria or animals under
ideal conditions, such as:

Unlimited environment

Adequate nutrition

Absence of predators

Immunity from disease
MODELS OF POPULATION GROWTH

Let’s identify and name the variables
in this model:

t = time (independent variable)

P = the number of individuals in the population
(dependent variable)
MODELS OF POPULATION GROWTH

The rate of growth of
the population is the derivative
dP / dt .
MODELS OF POPULATION GROWTH

Hence, 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.
dP
kP
dt
=
POPULATION GROWTH MODELS
Equation 1

Equation 1 is our first model for
population growth.

It is a differential equation because it contains
an unknown function P and its derivative dP / dt.
POPULATION GROWTH MODELS

Having formulated
a model, let’s look at its
consequences.
POPULATION GROWTH MODELS

If we rule out a population of 0, then
P ( t ) > 0 for all t
So, if k > 0, then Equation 1 shows that:
P’ ( t ) > 0 for all t
POPULATION GROWTH MODELS

This means that the population is
always increasing.