Elementary_Differential_Equations-Boyce-8e-solutions.pdf

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CHAPTER 1.

——

Chapter One

Section 1.1

For

C  "Þ&

, the slopes are

negative , and hence the solutions decrease. For

C  "Þ&

, the

slopes are

positive

, and hence the solutions increase. The equilibrium solution appears to

C> œ"Þ&

, to which all other solutions converge.

For

C "Þ&

, the slopes are

:9=3 tive , and hence the solutions increase. For C "Þ&

negative

diverge away from the equilibrium solution

, and hence the solutions decrease. All solutions appear to

C > œ  "Þ&

For

C   "Î#

, the slopes are

:9=3 tive , and hence the solutions increase. For

C   "Î# , the slopes are

negative

, and hence the solutions decrease. All solutions

diverge away from

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page 1

, the slopes are

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CHAPTER 1.

——

the equilibrium solution

C > œ  "Î#

For

C #

, the slopes are

:9=3 tive , and hence the solutions increase. For

C #

the slopes are

negative

, and hence the solutions decrease. All solutions diverge away

from

the equilibrium solution

C> œ#

8. For solutions to approach the equilibrium solution

all

C > œ #Î$

, we must have

C  ! C  #Î$ C  ! C  #Î$

for

, and

for

. The required rates are satisfied by the

differential equation

Cœ#$C

9. For solutions

other

than

C> œ#

to diverge from

Cœ# C>

must be an

increasing

function for

C#

, and a

decreasing

function for

C#

. The simplest differential

equation

whose solutions satisfy these criteria is

CœC#

10. For solutions

other C > œ "Î$

than

to diverge from

C œ "Î$

, we must have

C  !

for

C  "Î$ C  ! C  "Î$

, and

for

. The required rates are satisfied by the differential

equation

Cœ$C"

12.

Note that

Cœ! Cœ! Cœ&

for

and

. The two equilibrium solutions are

C>œ!

and

C> œ&

. Based on the direction field,

C ! C&

for

; thus solutions with initial

values

greater &

than diverge from the solution

C> œ& !C&

. For

, the slopes are

negative

, and hence solutions with initial values

between

and all decrease toward the

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page 2

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CHAPTER 1.

——

solution

C> œ! C!

. For

, the slopes are all

positive

; thus solutions with initial

values

less than approach the solution

C > œ !

14.

Observe that

Cœ! Cœ! Cœ#

for

and

. The two equilibrium solutions are

C>œ!

and

C> œ#

. Based on the direction field,

C ! C#

for

; thus solutions with initial

values

greater #

than diverge from

C> œ# !C#

. For

, the slopes are also

positive

, and hence solutions with initial values

between

and all increase toward the

solution

C> œ# C!

. For

, the slopes are all

negative

; thus solutions with initial

values

less !

than diverge from the solution

C > œ !

16. Let

ab ab

+ Q>

be the total amount of the drug

in milligrams

in the patient's body at

any

given time

> 2<=

. The drug is administered into the body at a

constant

rate of

&!!

71Î2<Þ

The rate at which the drug

accumulation rate of the drug is described by the differential equation

leaves

the bloodstream is given by

!Þ%Q > Þ

Hence the

œ &!!  !Þ% Q 71Î2< Þ

Based on the direction field, the amount of drug in the bloodstream approaches the

equilibrium level of

"#&! 71 A3>238 + 0/A 29?<= Þ

18. Following the discussion in the text, the differential equation is

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page 3

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CHAPTER 1.

——

7œ71@

# #

or equivalently,

œ1 @ Þ

# #

After a long time,

¸ ! Þ

Hence the object attains a

terminal velocity

given by

@œ Þ

Using the relation

@ œ 71

, the required

drag coefficient

œ !Þ!%!) 51Î=/- Þ

19.

All solutions appear to approach a linear asymptote

A3>2 =69:/ /;?+6 >9 "

. It is easy to

verify that

C> œ>$

is a solution.

20.

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CHAPTER 1.

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All solutions approach the equilibrium solution C> œ!Þ

23.

All solutions appear to

diverge

from the sinusoid

C > œ  =38Ð>  Ñ  "

which is also a solution corresponding to the initial value

C ! œ  &Î#

25.

All solutions appear to converge to

C> œ!

. First, the rate of change is small. The

slopes

eventually increase very rapidly in

magnitude

26.

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page 5

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