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In this section, we will learn: How to represent some mathematical models in the form of differential equations.

1.
9

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS

2.
DIFFERENTIAL EQUATIONS

Perhaps the most important of

all the applications of calculus is

to differential equations.

Perhaps the most important of

all the applications of calculus is

to differential equations.

3.
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.

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.

4.
DIFFERENTIAL EQUATIONS

It is often impossible to find an explicit

formula for the solution of a differential

Nevertheless, we will see that graphical and numerical

approaches provide the needed information.

It is often impossible to find an explicit

formula for the solution of a differential

Nevertheless, we will see that graphical and numerical

approaches provide the needed information.

5.
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.

9.1

Modeling with

Differential Equations

In this section, we will learn:

How to represent some mathematical models

in the form of differential equations.

6.
MODELING WITH 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

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

7.
DIFFERENTIAL EQUATION

The model often takes the form of

a differential equation.

This is an equation that contains an unknown

function and some of its derivatives.

The model often takes the form of

a differential equation.

This is an equation that contains an unknown

function and some of its derivatives.

8.
MODELING WITH DIFFERENTIAL EQUATIONS

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.

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.

9.
MODELING WITH DIFFERENTIAL EQUATIONS

Let’s begin by examining several

examples of how differential equations

arise when we model physical

Let’s begin by examining several

examples of how differential equations

arise when we model physical

10.
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.

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.

11.
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

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

12.
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)

Let’s identify and name the variables

in this model:

t = time (independent variable)

P = the number of individuals in the population

(dependent variable)

13.
MODELS OF POPULATION GROWTH

The rate of growth of

the population is the derivative

The rate of growth of

the population is the derivative

14.
POPULATION GROWTH MODELS Equation 1

Hence, our assumption that the rate of

growth of the population is proportional to

the population size is written as the equation

dP

kP

dt

where k is the proportionality constant.

Hence, our assumption that the rate of

growth of the population is proportional to

the population size is written as the equation

dP

kP

dt

where k is the proportionality constant.

15.
POPULATION GROWTH MODELS

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.

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.

16.
POPULATION GROWTH MODELS

Having formulated

a model, let’s look at its

Having formulated

a model, let’s look at its

17.
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

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

18.
POPULATION GROWTH MODELS

This means that the population is

always increasing.

In fact, as P(t) increases, Equation 1 shows that

dP/dt becomes larger.

In other words, the growth rate increases as

the population increases.

This means that the population is

always increasing.

In fact, as P(t) increases, Equation 1 shows that

dP/dt becomes larger.

In other words, the growth rate increases as

the population increases.

19.
POPULATION GROWTH MODELS

Equation 1 asks us to find a function whose

derivative is a constant multiple of itself.

We know from Chapter 3 that exponential functions

have that property.

In fact, if we let P(t) = Cekt, then

P’(t) = C(kekt) = k(Cekt) = kP(t)

Equation 1 asks us to find a function whose

derivative is a constant multiple of itself.

We know from Chapter 3 that exponential functions

have that property.

In fact, if we let P(t) = Cekt, then

P’(t) = C(kekt) = k(Cekt) = kP(t)

20.
POPULATION GROWTH MODELS

Thus, any exponential function

of the form P(t) = Cekt

is a solution of Equation 1.

In Section 9.4, we will see that there is

no other solution.

Thus, any exponential function

of the form P(t) = Cekt

is a solution of Equation 1.

In Section 9.4, we will see that there is

no other solution.

21.
POPULATION GROWTH MODELS

Allowing C to vary through all the real

numbers, we get the family of solutions

P(t) = Cekt, whose graphs are shown.

Allowing C to vary through all the real

numbers, we get the family of solutions

P(t) = Cekt, whose graphs are shown.

22.
POPULATION GROWTH MODELS

However, populations have only

positive values.

So, we are interested only in the solutions with C > 0.

Also, we are probably concerned only with values of t

greater than the initial time t = 0.

However, populations have only

positive values.

So, we are interested only in the solutions with C > 0.

Also, we are probably concerned only with values of t

greater than the initial time t = 0.

23.
POPULATION GROWTH MODELS

The figure shows the physically

meaningful solutions.

The figure shows the physically

meaningful solutions.

24.
POPULATION GROWTH MODELS

Putting t = 0, we get:

P(0) = Cek(0) = C

The constant C turns out to be

the initial population, P(0).

Putting t = 0, we get:

P(0) = Cek(0) = C

The constant C turns out to be

the initial population, P(0).

25.
POPULATION GROWTH MODELS

Equation 1 is appropriate for modeling

population growth under ideal conditions.

However, we have to recognize that a more

realistic model must reflect the fact that

a given environment has limited resources.

Equation 1 is appropriate for modeling

population growth under ideal conditions.

However, we have to recognize that a more

realistic model must reflect the fact that

a given environment has limited resources.

26.
POPULATION GROWTH MODELS

Many populations start by increasing in

an exponential manner.

However, the population levels off when

it approaches its carrying capacity K

(or decreases toward K if it ever exceeds K.)

Many populations start by increasing in

an exponential manner.

However, the population levels off when

it approaches its carrying capacity K

(or decreases toward K if it ever exceeds K.)

27.
POPULATION GROWTH MODELS

For a model to take into account both

trends, we make two assumptions:

dP

kP

1. dt if P is small.

(Initially, the growth rate is proportional to P.)

dP

0

2. dt if P > K.

(P decreases if it ever exceeds K.)

For a model to take into account both

trends, we make two assumptions:

dP

kP

1. dt if P is small.

(Initially, the growth rate is proportional to P.)

dP

0

2. dt if P > K.

(P decreases if it ever exceeds K.)

28.
POPULATION GROWTH MODELS Equation 2

A simple expression that incorporates both

assumptions is given by the equation

dP P

kP 1

dt K

If P is small compared with K, then P/K is close to 0.

So, dP/dt ≈ kP

If P > K, then 1 – P/K is negative. So, dP/dt < 0

A simple expression that incorporates both

assumptions is given by the equation

dP P

kP 1

dt K

If P is small compared with K, then P/K is close to 0.

So, dP/dt ≈ kP

If P > K, then 1 – P/K is negative. So, dP/dt < 0

29.
LOGISTIC DIFFERENTIAL EQUATION

Equation 2 is called the logistic

differential equation.

It was proposed by the Dutch mathematical biologist

Pierre-François Verhulst in the 1840s—as a model

for world population growth.

Equation 2 is called the logistic

differential equation.

It was proposed by the Dutch mathematical biologist

Pierre-François Verhulst in the 1840s—as a model

for world population growth.

30.
LOGISTIC DIFFERENTIAL EQUATIONS

In Section 9.4, we will develop techniques

that enable us to find explicit solutions of

the logistic equation.

For now, we can deduce qualitative characteristics

of the solutions directly from Equation 2.

In Section 9.4, we will develop techniques

that enable us to find explicit solutions of

the logistic equation.

For now, we can deduce qualitative characteristics

of the solutions directly from Equation 2.

31.
POPULATION GROWTH MODELS

We first observe that the constant

functions P(t) = 0 and P(t) = K are

This is because, in either case, one of the factors

on the right side of Equation 2 is zero.

We first observe that the constant

functions P(t) = 0 and P(t) = K are

This is because, in either case, one of the factors

on the right side of Equation 2 is zero.

32.
EQUILIBRIUM SOLUTIONS

This certainly makes physical sense.

If the population is ever either 0 or at

the carrying capacity, it stays that way.

These two constant solutions are called

equilibrium solutions.

This certainly makes physical sense.

If the population is ever either 0 or at

the carrying capacity, it stays that way.

These two constant solutions are called

equilibrium solutions.

33.
POPULATION GROWTH MODELS

If the initial population P(0) lies between

0 and K, then the right side of Equation 2

is positive.

So, dP/dt > 0 and the population increases.

If the initial population P(0) lies between

0 and K, then the right side of Equation 2

is positive.

So, dP/dt > 0 and the population increases.

34.
POPULATION GROWTH MODELS

However, if the population exceeds

the carrying capacity (P > K), then 1 – P/K

is negative.

So, dP/dt < 0 and the population decreases.

However, if the population exceeds

the carrying capacity (P > K), then 1 – P/K

is negative.

So, dP/dt < 0 and the population decreases.

35.
POPULATION GROWTH MODELS

Notice that, in either case, if the population

approaches the carrying capacity (P → K),

then dP/dt → 0.

This means the population levels off.

Notice that, in either case, if the population

approaches the carrying capacity (P → K),

then dP/dt → 0.

This means the population levels off.

36.
POPULATION GROWTH MODELS

So, we expect that the solutions of the logistic

differential equation have graphs that look

something like these.

So, we expect that the solutions of the logistic

differential equation have graphs that look

something like these.

37.
POPULATION GROWTH MODELS

Notice that the graphs move away from

the equilibrium solution P = 0 and move

toward the equilibrium solution P = K.

Notice that the graphs move away from

the equilibrium solution P = 0 and move

toward the equilibrium solution P = K.

38.
MODELING WITH DIFFERENTIAL EQUATIONS

Let’s now look at an example

of a model from the physical

Let’s now look at an example

of a model from the physical

39.
MODEL FOR MOTION OF A SPRING

We consider the motion of an object

with mass m at the end of a vertical

We consider the motion of an object

with mass m at the end of a vertical

40.
MODEL FOR MOTION OF A SPRING

In Section 6.4, we discussed

Hooke’s Law.

If the spring is stretched (or compressed) x units from

its natural length, it exerts a force proportional to x:

restoring force = -kx

where k is a positive constant (the spring constant).

In Section 6.4, we discussed

Hooke’s Law.

If the spring is stretched (or compressed) x units from

its natural length, it exerts a force proportional to x:

restoring force = -kx

where k is a positive constant (the spring constant).

41.
SPRING MOTION MODEL Equation 3

If we ignore any external resisting forces

(due to air resistance or friction) then,

by Newton’s Second Law, we have:

2

d x

m 2 kx

dt

If we ignore any external resisting forces

(due to air resistance or friction) then,

by Newton’s Second Law, we have:

2

d x

m 2 kx

dt

42.
SECOND-ORDER DIFFERENTIAL EQUATION

This is an example of a second-order

differential equation.

It involves second derivatives.

This is an example of a second-order

differential equation.

It involves second derivatives.

43.
SPRING MOTION MODEL

Let’s see what we can guess about

the form of the solution directly from

the equation.

Let’s see what we can guess about

the form of the solution directly from

the equation.

44.
SPRING MOTION MODEL

We can rewrite Equation 3 in the form

2

d x k

2

x

dt m

This says that the second derivative of x

is proportional to x but has the opposite sign.

We can rewrite Equation 3 in the form

2

d x k

2

x

dt m

This says that the second derivative of x

is proportional to x but has the opposite sign.

45.
SPRING MOTION MODEL

We know two functions with this property,

the sine and cosine functions.

It turns out that all solutions of Equation 3

can be written as combinations of certain

sine and cosine functions.

We know two functions with this property,

the sine and cosine functions.

It turns out that all solutions of Equation 3

can be written as combinations of certain

sine and cosine functions.

46.
SPRING MOTION MODEL

This is not surprising.

We expect the spring to oscillate about

its equilibrium position.

So, it is natural to think that trigonometric

functions are involved.

This is not surprising.

We expect the spring to oscillate about

its equilibrium position.

So, it is natural to think that trigonometric

functions are involved.

47.
GENERAL DIFFERENTIAL EQUATIONS

In general, a differential equation is

an equation that contains an unknown

function and one or more of its derivatives.

In general, a differential equation is

an equation that contains an unknown

function and one or more of its derivatives.

48.
The order of a differential equation is

the order of the highest derivative that

occurs in the equation.

Equations 1 and 2 are first-order equations.

Equation 3 is a second-order equation.

the order of the highest derivative that

occurs in the equation.

Equations 1 and 2 are first-order equations.

Equation 3 is a second-order equation.

49.
INDEPENDENT VARIABLE

In all three equations, the independent

variable is called t and represents time.

However, in general, it doesn’t have to

represent time.

In all three equations, the independent

variable is called t and represents time.

However, in general, it doesn’t have to

represent time.

50.
INDEPENDENT VARIABLE Equation 4

For example, when we consider

the differential equation

y’ = xy

it is understood that y is an unknown

function of x.

For example, when we consider

the differential equation

y’ = xy

it is understood that y is an unknown

function of x.

51.
A function f is called a solution of a differential

equation if the equation is satisfied when

y = f(x) and its derivatives are substituted

into the equation.

Thus, f is a solution of Equation 4 if

f’(x) = xf(x)

for all values of x in some interval.

equation if the equation is satisfied when

y = f(x) and its derivatives are substituted

into the equation.

Thus, f is a solution of Equation 4 if

f’(x) = xf(x)

for all values of x in some interval.

52.
SOLVING DIFFERENTIAL EQUATIONS

When we are asked to solve a differential

equation, we are expected to find all possible

solutions of the equation.

We have already solved some particularly simple

differential equations—namely, those of the form

y’ = f(x)

When we are asked to solve a differential

equation, we are expected to find all possible

solutions of the equation.

We have already solved some particularly simple

differential equations—namely, those of the form

y’ = f(x)

53.
SOLVING DIFFERENTIAL EQUATIONS

For instance, we know that the general

solution of the differential equation y’ = x3

4

is given by x

y C

4

where C is an arbitrary constant.

For instance, we know that the general

solution of the differential equation y’ = x3

4

is given by x

y C

4

where C is an arbitrary constant.

54.
SOLVING DIFFERENTIAL EQUATIONS

However, in general, solving

a differential equation is not an easy

There is no systematic technique that enables

us to solve all differential equations.

However, in general, solving

a differential equation is not an easy

There is no systematic technique that enables

us to solve all differential equations.

55.
SOLVING DIFFERENTIAL EQUATIONS

In Section 9.2, though, we will see how to

draw rough graphs of solutions even when

we have no explicit formula.

We will also learn how to find numerical

approximations to solutions.

In Section 9.2, though, we will see how to

draw rough graphs of solutions even when

we have no explicit formula.

We will also learn how to find numerical

approximations to solutions.

56.
SOLVING DIFFERENTIAL EQNS. Example 1

Show that every member of the family

of functions 1 ce t

y t

1 ce

is a solution of the differential equation

1

y ' 2 y 1

2

Show that every member of the family

of functions 1 ce t

y t

1 ce

is a solution of the differential equation

1

y ' 2 y 1

2

57.
SOLVING DIFFERENTIAL EQNS. Example 1

We use the Quotient Rule to differentiate

the expression for y:

y'

1 ce t

ce t

1 ce t

ce t

t 2

1 ce

cet c 2e 2t cet c 2 e 2t 2cet

t 2 t 2

1 ce 1 ce

We use the Quotient Rule to differentiate

the expression for y:

y'

1 ce t

ce t

1 ce t

ce t

t 2

1 ce

cet c 2e 2t cet c 2 e 2t 2cet

t 2 t 2

1 ce 1 ce

58.
SOLVING DIFFERENTIAL EQNS. Example 1

The right side of the differential equation

t 2

1 1 1 ce

2

y 2

1 t

1

2 1 ce

1 cet 2 1 cet 2

1

2

2 1 ce t

t t

1 4ce 2ce

2

2

2 1 ce

t

1 ce

t

The right side of the differential equation

t 2

1 1 1 ce

2

y 2

1 t

1

2 1 ce

1 cet 2 1 cet 2

1

2

2 1 ce t

t t

1 4ce 2ce

2

2

2 1 ce

t

1 ce

t

59.
SOLVING DIFFERENTIAL EQNS. Example 1

Therefore, for every value of c,

the given function is a solution of

the differential equation.

Therefore, for every value of c,

the given function is a solution of

the differential equation.

60.
SOLVING DIFFERENTIAL EQNS.

The figure shows graphs of seven members

of the family in Example 1.

The differential equation shows that,

if y ≈ ±1, then y’ ≈ 0.

This is borne out

by the flatness of

the graphs near

y = 1 and y = -1.

The figure shows graphs of seven members

of the family in Example 1.

The differential equation shows that,

if y ≈ ±1, then y’ ≈ 0.

This is borne out

by the flatness of

the graphs near

y = 1 and y = -1.

61.
SOLVING DIFFERENTIAL EQNS.

When applying differential equations, we are

usually not as interested in finding a family

of solutions (the general solution) as we are

in finding a solution that satisfies some

additional requirement.

In many physical problems, we need to find

the particular solution that satisfies a condition

of the form y(t0) = y0

When applying differential equations, we are

usually not as interested in finding a family

of solutions (the general solution) as we are

in finding a solution that satisfies some

additional requirement.

In many physical problems, we need to find

the particular solution that satisfies a condition

of the form y(t0) = y0

62.
INITIAL CONDITION & INITIAL-VALUE PROBLEM

This is called an initial condition.

The problem of finding a solution of

the differential equation that satisfies

the initial condition is called an initial-value

This is called an initial condition.

The problem of finding a solution of

the differential equation that satisfies

the initial condition is called an initial-value

63.
INITIAL CONDITION

Geometrically, when we impose an initial

condition, we look at the family of solution

curves and pick the one that passes through

the point (t0, y0).

Geometrically, when we impose an initial

condition, we look at the family of solution

curves and pick the one that passes through

the point (t0, y0).

64.
INITIAL CONDITION

Physically, this corresponds to measuring

the state of a system at time t0 and using

the solution of the initial-value problem

to predict the future behavior of the system.

Physically, this corresponds to measuring

the state of a system at time t0 and using

the solution of the initial-value problem

to predict the future behavior of the system.

65.
INITIAL CONDITION Example 2

Find a solution of the differential equation

1

y' 2 y 2

1

that satisfies the initial condition y(0) = 2.

Find a solution of the differential equation

1

y' 2 y 2

1

that satisfies the initial condition y(0) = 2.

66.
INITIAL CONDITION Example 2

Substituting the values t = 0 and y = 2

into the formula from Example 1,

t

1 ce

y

1 cet

0

we get: 1 ce 1 c

2 0

1 ce 1 c

Substituting the values t = 0 and y = 2

into the formula from Example 1,

t

1 ce

y

1 cet

0

we get: 1 ce 1 c

2 0

1 ce 1 c

67.
INITIAL CONDITION Example 2

Solving this equation for c,

we get:

2 – 2c = 1 + c

This gives c = ⅓.

Solving this equation for c,

we get:

2 – 2c = 1 + c

This gives c = ⅓.

68.
INITIAL CONDITION Example 2

So, the solution of the initial-value

problem is:

1 t

1 e 3 e

3

t

y t

1 t

1 e 3 e

3

So, the solution of the initial-value

problem is:

1 t

1 e 3 e

3

t

y t

1 t

1 e 3 e

3