9781133105060 App F1 · C 10. 0 10 Ce k(0) x 0.25 t 1 x 0 t 0 x 0 t 0. t x 10 Ce kt 0 x 10. dx dt k...

Appendix F.1 ■ Solutions of Differential Equations F1

■ Find general solutions of differential equations.

■ Find particular solutions of differential equations.

General Solution of a Differential Equation

A differential equation is an equation involving a differentiable function and one ormore of its derivatives. For instance,

Differential equation

is a differential equation. A function is a solution of a differential equation if the equation is satisfied when and its derivatives are replaced by and its derivatives. For example,

Solution of differential equation

is a solution of the differential equation shown above. To see this, substitute for andin the original equation.

Substitute for y and

In the same way, you can show that and are alsosolutions of the differential equation. In fact, each function given by

General solution

where is a real number, is a solution of the equation. This family of solutions is calledthe general solution of the differential equation.

Example 1 Verifying Solutions

Determine whether each function is a solution of the differential equation

a. b.

SOLUTION

a. Because and it follows that

So, is a solution.

b. Because and it follows that

So, is also a solution.

Checkpoint 1

Determine whether is a solution of the differential equation ■y� � y.y � Ce4x

y � Ce�x

y� � y � Ce�x � Ce�x � 0.

y� � Ce�x,y� � �Ce�x

y � Cex

y� � y � Cex � Cex � 0.

y� � Cex,y� � Cex

y � Ce�xy � Cex

y� � y � 0.

C

y � Ce�2x

y �12e�2xy � 2e�2x, y � �3e�2x,

� 0

y�.y� � 2y � �2e�2x � 2�e�2x�

y� � �2e�2xy

y � e�2x

f�x�yy � f�x�

y� � 2y � 0

F Differential Equations

F.1 Solutions of Differential Equations

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Particular Solutions and Initial Conditions

A particular solution of a differential equation is any solution that is obtained byassigning specific values to the arbitrary constant(s) in the general solution.*

Geometrically, the general solution of a differential equation represents a family ofcurves known as solution curves. For instance, the general solution of the differentialequation is

General solution

Figure F.1 shows several solution curves corresponding to different values of Particular solutions of a differential equation are obtained from initial conditions

placed on the unknown function and its derivatives. For instance, in Figure F.1, supposeyou want to find the particular solution whose graph passes through the point This initial condition can be written as

when Initial condition

Substituting these values into the general solution produces which impliesthat So, the particular solution is

Particular solution

Example 2 Finding a Particular Solution

For the differential equation

verify that is a solution. Then find the particular solution determined by theinitial condition when

SOLUTION You know that is a solution because and

Furthermore, the initial condition when yields

General solution

Substitute initial condition.

Solve for C.

and you can conclude that the particular solution is

Particular solution

Try checking this solution by substituting for and in the original differential equation.

Checkpoint 2

For the differential equation verify that is a solution. Thenfind the particular solution determined by the initial condition when ■

*Some differential equations have solutions other than those given by their general solutions. These are calledsingular solutions. In this brief discussion of differential equations, singular solutions will not be discussed.

x � 4.y � 1y � Cx2xy� � 2y � 0,

y�y

y � �2x3

27.

�2

27� C

2 � C��3�3

y � Cx3

x � �3y � 2

� 0.

� 3Cx3 � 3Cx3

xy� � 3y � x�3Cx2� � 3�Cx3�

y� � 3Cx2y � Cx3

x � �3.y � 2y � Cx3

xy� � 3y � 0

y � 3x2.

C � 3.3 � C�1�2,

x � 1.y � 3

�1, 3�.

C.

y � Cx2.

xy� � 2y � 0

F2 Appendix F ■ Differential Equations

STUDY TIP

To determine a particular solution, the number of initial conditions must matchthe number of constants inthe general solution.

−3

−2

3

2

1

3 2 −2 −3

y

x

General solution:y = Cx2

(1, 3)

Solution Curves for FIGURE F.1

xy� � 2y � 0

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You are working in the marketing department of a company that is producing a newcereal product to be sold nationally. You determine that a maximum of 10 million unitsof the product could be sold in a year. You hypothesize that the rate of growth of thesales (in millions of units) is proportional to the difference between the maximumsales and the current sales. As a differential equation, this hypothesis can be written as

The general solution of this differential equation is

General solution

where is the time in years. After 1 year, 250,000 units have been sold. Sketch the graphof the sales function over a 10-year period.

SOLUTION Because the product is new, you can assume that when So,you have two initial conditions.

when First initial condition

when Second initial condition

Substituting the first initial condition into the general solution produces

which implies that

Substituting the second initial condition into the general solution produces

which implies that

So, the particular solution is

Particular solution

The table shows the annual sales during the first 10 years, and the graph of the solutionis shown in Figure F.2.

Checkpoint 3

Repeat Example 3 using the initial conditions when and when ■t � 1.

x � 0.3t � 0x � 0

t 1 2 3 4 5 6 7 8 9 10

x 0.25 0.49 0.73 0.96 1.19 1.41 1.62 1.83 2.04 2.24

x � 10 � 10e�0.0253t.

k � �ln 0.975 � 0.0253.

�ln 0.975 � k

0.975 � e�k

�9.75 � �10e�k

0.25 � 10 � 10e�k(1)

C � 10.

0 � 10 � Ce�k(0)

t � 1x � 0.25

t � 0x � 0

t � 0.x � 0

t

x � 10 � Ce�kt

0 � x � 10.dxdt

� k�10 � x�,

x

Rate ofchange

of x

is propor-tional to

the differencebetween 10 and x.

3

2

1

x

t

x = 10 − 10e−0.0253t

Sales Projection

Time (in years)

Sale

s (i

n m

illio

ns o

f un

its)

1 2 3 4 5 6 7 8 9 10

FIGURE F.2

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In the first three examples in this section, each solution was given in explicit form, such as Sometimes you will encounter solutions for which it is moreconvenient to write the solution in implicit form, as shown in Example 4.

Example 4 Sketching Graphs of Solutions

Given that

General solution

is the general solution of the differential equation

sketch the particular solutions represented by and

SOLUTION The particular solutions represented by and are shown inFigure F.3.

Graphs of Five Particular SolutionsFIGURE F.3

Checkpoint 4

Given that

is the general solution of

sketch the particular solutions represented by and ■C � 4.C � 1, C � 2,

xy� � 2y � 0

y � Cx2

2 2

2

3

2

2

2

2

3

1

y

x

x

y

x

y

x

x

C = 0

C = 1

C = −1 C = −4

y y

C = 4

± 4±1,C � 0,

±4.C � 0, ±1,

2yy� � x � 0

2y2 � x2 � C

y � f�x�.

SUMMARIZE (Section F.1)

1. Explain how to verify a solution of a differential equation (page F1). For anexample of verifying a solution, see Example 1.

2. Describe the difference between a general solution of a differential equationand a particular solution (pages F1 and F2). For an example of a generalsolution of a differential equation and a particular solution, see Example 2.

3. Describe a real-life example of how a differential equation can be used tomodel the sales of a company’s product (page F3, Example 3).

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Exercises F.1

Verifying Solutions In Exercises 1–12, verify that thefunction is a solution of the differential equation. SeeExample 1.

Solution Differential Equation

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Determining Solutions In Exercises 13–16, determinewhether the function is a solution of the differential equation

13. 14.

15. 16.

Determining Solutions In Exercises 17–20, determinewhether the function is a solution of the differential equation

17.

18.

19.

20.

Finding a Particular Solution In Exercises 21–24,verify that the general solution satisfies the differentialequation. Then find the particular solution that satisfiesthe initial condition. See Example 2.

21. General solution:Differential equation:Initial condition: when

22. General solution:Differential equation:Initial condition: when

23. General solution:Differential equation:Initial condition: and when

24. General solution:Differential equation:Initial condition: and when

Sketching Graphs of Solutions In Exercises 25 and26, the general solution of the differential equation isgiven. Sketch the particular solutions that correspond tothe indicated values of C. See Example 4.

General Solution Differential Equation C-Values

25.

26.

Finding General Solutions In Exercises 27–34, useintegration to find the general solution of the differentialequation.

27. 28.

29. 30.

31. 32.

33. 34.dydx

� 4 sin xdydx

� cos 4x

dydx

�x

1 � x2

dydx

� x�x2 � 6

dydx

�x � 2

xdydx

�1

1 � x

dydx

� 2x3 � 3xdydx

� 3x2

0, ±1, ±4y� � y � 0y � Ce�x

0, ±1, ±2�x � 2�y� � 2y � 0y � C�x � 2�2

x � 0y� � 6y � 5y� � y� � 12y � 0

y � C1e4x � C2e

�3x

x � 1y� � 0.5y � 5xy� � y� � 0

y � C1 � C2 ln x

x � 1y � 22x � 3yy� � 0

2x2 � 3y2 � C

x � 0y � 3y� � 2y � 0

y � Ce�2x

y � x ln x

y � xex

y � cos x

y �29 xe�2x

y�� 3y� � 2y � 0.

y � 3 sin 2xy �4x

y � 5 ln xy � e�2x

y �4� � 16y � 0.

y� � 3y� � 4y � 0y � C1e4x � C2e

�x

y� � y � 0y � C1 sin x � C2 cos x

y� � 3x2y� � 6xy � 0y � ex3

x2y� � 2y � 0y � x2

y� � �2x � 1�y � 0y � Cex�x2

x�y� � 1� � �y � 4� � 0y � x ln x � Cx � 4

xy� � y � x�3x � 4�y � x2 � 2x �Cx

xy� � 3x � 2y � 0y � Cx2 � 3x

y� �2x

y � 0y � 4x2

y� �3x

y � 0y � 2x3

y� � 2y � 0y � e�2x

y� � 4yy � Ce4x

The following warm-up exercises involve skills that were covered in earlier sections. You will usethese skills in the exercise set for this section. For additional help, review Sections 2.2, 2.6, 4.3, and 4.4.

In Exercises 1–4, find the first and second derivatives of the function.

1. 2. 3. 4.

In Exercises 5 and 6, solve for k.

5. 6. 14.75 � 25 � 25e�2k0.5 � 9 � 9e�k

y � �3ex2y � �3e2xy � �2x3 � 8x � 4y � 3x2 � 2x � 1

SKILLS WARM UP F.1

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Finding a Particular Solution In Exercises 35–38,some of the curves corresponding to different values ofC in the general solution of the differential equation areshown in the figure. Find the particular solution thatpasses through the point plotted on the graph.

35. 36.

37. 38.

39. Biology The limiting capacity of the habitat of awildlife herd is 750. The growth rate of the herdis proportional to the unutilized opportunity for growth,as described by the differential equation


When the population of the herd is 100. After 2 years, the population has grown to 160.

(a) Write the population as a function of

(b) What is the population of the herd after 4 years?

40. Investment The rate of growth of an investment isproportional to the amount in the investment at any time

That is,


The initial investment is $1000, and after 10 years thebalance is $3320.12. What is the particular solution?

41. Safety Assume that the rate of change per hour in thenumber of miles of road cleared by a snowplow isinversely proportional to the depth of the snow. Thisrate of change is described by the differential equation

Show that

is a solution of this differential equation.

43. Verifying a Solution Show that is a solution of the differential equation

where is a constant.

44. Using a Solution The function is a solution of the differential equation

Is it possible to determine or from the informationgiven? If so, find its value.

True or False? In Exercises 45 and 46, determinewhether the statement is true or false. If it is false, explainwhy or give an example that shows it is false.

45. A differential equation can have more than one solution.

46. If is a solution of a differential equation, thenis also a solution.y � f�x� � C

y � f�x�

kC

dydx

� 0.07y.

y � Cekx

k

y � a � b� y � a� � �1k �dy

dt

y � a � Cek�1�b�t

s � 25 �13

ln 3 ln

h2

dsdh

�kh

.

hs

A � Cekt.

dAdt

� kA.

t.

t.N

t � 0,

N � 750 � Ce�kt.

dNdt

� k�750 � N�.

dNdt

1

2

−1

−2

x

(2, 1) y

1 2 3 −1 −2 −3

4 5 6

x

(0, 3)

y

2xy� � y � 0y� � y � 0

y2 � 2Cxy � Cex

4 3 2

−4 −3

4 3 −3 x

(3, 4)

y

7

−4 −3 −2 −1

4 3 2 1

6 5 4 x

(4, 4)

y

yy� � 2x � 02xy� � 3y � 0

2x2 � y2 � Cy2 � Cx3

42. HOW DO YOU SEE IT? The graph shows a solution of one of the following differential equations. Determine the correct equation.Explain your reasoning.

(i) (ii)

(iii) (iv) y� � 4 � xyy� � �4xy

y� �4xy

y� � xy

x

y

9781133105060_App_F1.qxp 12/27/11 1:36 PM Page F6

Appendix F.2 ■ Separation of Variables F7

F.2 Separation of Variables■ Use separation of variables to solve differential equations.

■ Use differential equations to model and solve real-life problems.

Separation of Variables

The simplest type of differential equation is one of the form You know thatthis type of equation can be solved by integration to obtain

In this section, you will learn how to use integration to solve another important familyof differential equations—those in which the variables can be separated. This techniqueis called separation of variables.

Essentially, the technique of separation of variables is just what its name implies.For a differential equation involving and you separate the variables by grouping the variables on one side and the variables on the other. After separating variables,integrate each side to obtain the general solution.

Example 1 Solving a Differential Equation

Find the general solution of

SOLUTION Begin by separating variables, then integrate each side.


Separate variables.

Integrate each side.

General solution

Checkpoint 1

Find the general solution of ■dydx

�x2

y.

y3

3� y �

x2

2� C

��y2 � 1� dy � �x dx

(y2 � 1� dy � x dx

dydx

�x

y2 � 1

dydx

�x

y2 � 1.

yxy,x

y � � f �x� dx.

y� � f �x�.

Separation of Variables

If and are continuous functions, then the differential equation

has a general solution of

� 1

g�y� dy � � f �x� dx � C.

dydx

� f �x�g�y�

gf

TECH TUTOR

You can use a symbolic integration utility to solve adifferential equation that hasseparable variables. Use a symbolic integration utilityto solve the differential equation in Example 1.

9781133105060_App_F2.qxp 12/27/11 1:36 PM Page F7






Separate variables.


Find antiderivative of each side.

Multiply each side by 2.

So, the general solution is Note that is used as a temporary constantof integration in anticipation of multiplying each side of the equation by 2 to producethe constant C.

Checkpoint 2


■


Find the general solution of Use a graphing utility to graph several solutions.



Separate variables.



By taking the natural logarithm of each side, you can write the general solution as

General solution

The graphs of the particular solutions given by 5, 10, and 15 are shown in Figure F.4.

Checkpoint 3


Use a graphing utility to graph the particular solutions given by 2, and 4. ■C � 1,

2y dydx

� �2x.

C � 0,

y � ln�x2 � C�.

ey � x2 � C

�ey dy � � 2x dx

ey dy � 2x dx

ey dydx

� 2x

ey dydx

� 2x.

dydx

�x � 1

y.

C1y2 � x2 � C.

y2 � x2 � C

y2

2�

x2

2� C1

�y dy � �x dx

y dy � x dx

dydx

�xy

dydx

�xy

.

−5

6 −6

5

C = 0

C = 5 C = 10

C = 15

FIGURE F.4

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Solve the differential equation subject to the initial condition when

SOLUTION


Subtract from each side.

Separate variables.



Multiply each side by 2.

To find the particular solution, substitute the initial condition values to obtain

This implies that or So, the particular solution that satisfies theinitial condition is

Particular solution

Checkpoint 4

Solve the differential equation subject to the initial condition when ■


Example 3 in Section F.1 uses the differential equation

to model the sales of a new product. Solve this differential equation.

SOLUTION


Separate variables.


Find antiderivatives.

Multiply each side by

Exponentiate each side.

Solve for x.

Checkpoint 5

Solve the differential equation for ■0 � y � 65.dydx

� k�65 � y�

x � 10 � Ce�kt

10 � x � e�kt�C1

�1. ln�10 � x� � �kt � C1

�ln�10 � x� � kt � C1

� 110 � x

dx � �k dt

1

10 � x dx � k dt

dxdt

� k�10 � x�

0 � x � 10dxdt

� k�10 � x�,

x � 0.y � 2ex � yy� � 0

y2 � �ex2� 2.

C � 2.1 � �1 � C,

�1�2 � �e�0�2� C.

y2 � �ex2� C

y2

2� �

12

ex2� C1

�y dy � ��xex2 dx

y dy � �xex2 dx

xex2 y dydx

� �xex2

xex2� yy� � 0

x � 0.y � 1xex2

� yy� � 0

STUDY TIP

In Example 5, the context ofthe original model indicatesthat is positive. So, when you integrate

you can writerather than

Also note in Example 5that the solution agrees withthe one that was given inExample 3 in Section F.1.

�ln�10 � x�.�ln�10 � x�,1��10 � x�,

�10 � x�

9781133105060_App_F2.qxp 12/27/11 1:36 PM Page F9


Application

Example 6 Corporate Investing

A corporation invests part of its receipts at a rate of dollars per year in a fund forfuture corporate expansion. The fund earns percent interest per year compounded continuously. The rate of growth of the amount in the fund is

where is the time (in years). Solve the differential equation for as a function of where when

SOLUTION You can solve the differential equation using separation of variables.


Differential form

Separate variables.

Integrate.

Assume and multiply each side by r.

Exponentiate each side.

Solve for A.

General solution

Using when you find the value of

So, the differential equation for as a function of can be written as

Checkpoint 6

Use the result of Example 6 to find when and years. ■t � 25

r � 5.9%,P � $550,000,A

A �Pr

�ert � 1�.

tA

C �Pr

0 � Cer�0� �Pr

C.t � 0,A � 0

A � Cert �Pr

A �C3e

rt � Pr

rA � P � ert�C2

rA � P > 0 ln�rA � P� � rt � C2

1r ln�rA � P� � t � C1

dA

rA � P� dt

dA � �rA � P� dt

dAdt

� rA � P

t � 0.A � 0t,At

dAdt

� rA � P

Ar

P


1. Explain how to use separation of variables to solve a differential equation(page F7). For examples of solving a differential equation using separationof variables, see Examples 1, 2, 3, 4, and 5.

2. Describe a real-life example of how separation of variables can be used to solve a differential equation that models corporate investing (page F10,Example 6).

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Exercises F.2

Separation of Variables In Exercises 1–6, decidewhether the variables in the differential equation can beseparated.

1. 2.

3. 4.

5. 6.

Solving a Differential Equation In Exercises 7–26,use separation of variables to find the general solution ofthe differential equation. See Examples 1 and 2.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

Solving a Differential Equation In Exercises27–30, (a) find the general solution of the differentialequation and (b) use a graphing utility to graph theparticular solutions given by 2, and 4. SeeExample 3.

27. 28.

29. 30.

Finding a Particular Solution In Exercises 31–38, usethe initial condition to find the particular solution of thedifferential equation. See Example 4.

Differential Equation Initial Condition

31.

32.

33.

34.

35.

36.

37.

38.

Finding an Equation In Exercises 39 and 40, find anequation of the graph that passes through the point andhas the specified slope. Then graph the equation.

39. Point: 40. Point:

Slope: Slope: y� �2y3x

y� � �9x

16y

�8, 2��1, 1�

y � 1 when x � 0dydx

� 2xy sin x2


� y cos x

y � 0 when x � 0y� � ex�2y

y � �2 when x � 5�x2 � 16y� � 5x


� x2�1 � y�

y � �5 when x � 0x�y � 4� � y� � 0

y � 4 when x � 1�x � �y y� � 0

y � 4 when x � 0yy� � ex � 0

dydx

� 0.25x�4 � y�dydx

� y � 3

dydx

� �2xy

dydx

� x

C � 1,

y dydx

� 6 cos��x�y dydx

� sin x

y� � y�x � 1� � 0�2 � x�y� � 2y

y� �xy

�x

1 � ydydx

� �1 � y

ex�y� � 1� � 1ey dydt

� 3t2 � 1

y� � y � 5y� � xy � 0

�1 � y� dydx

� 4x � 0x2 � 4ydydx

� 0

dydx

� x2y3y2 dydx

� 1

dydx

�x2 � 3

y2

dydx

�x � 1

y3

drds

� 0.05sdrds

� 0.05r

dydx

�1x

dydx

� 2x

x dydx

�1y

dydx

� x � y

dydx

�x

x � ydydx

�1x

� 1

dydx

�x � 1

xdydx

�x

y � 3

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 4.4, 5.2, and 5.3.

In Exercises 1–6, find the indefinite integral and check your result by differentiating.

1. 2. 3.

4. 5. 6.

In Exercises 7–10, solve the equation for C or k.

7. 8.

9. 10. �6�2 � 3�6� � e�k10 � 2e2k

��1�2 � ��2�2 � C�3�2 � 6�3� � 1 � C

�xe1�x2 dx�e2y dy� y2y2 � 1

dy

� 2x � 5

dx��t3 � t1�3� dt� x3�2 dx

SKILLS WARM UP F.2

9781133105060_App_F2.qxp 12/27/11 1:36 PM Page F11


Velocity In Exercises 41 and 42, solve the differentialequation to find velocity v as a function of time t if when The differential equation models the motionof two people on a toboggan after consideration of theforces of gravity, friction, and air resistance.

41.

42.

43. Biology: Cell Growth The growth rate of a sphericalcell with volume is proportional to its surface area For a sphere, the surface area and volume are related by

So, a model for the cell’s growth is

Solve this differential equation.

45. Radioactive Decay The rate of decomposition ofradioactive radium is proportional to the amount present at any time. The half-life of radioactive radium is 1599years. What percent of a present amount will remainafter 25 years?

46. Radioactive Decay The rate of decomposition ofradioactive einsteinium is proportional to the amount present at any time. The half-life of radioactive einsteinium is 276 days. After 100 days, 0.5 gramremains. What was the initial amount?

47. Weight Gain A calf that weighed 60 pounds at birthgains weight at the rate

where is the weight (in pounds) and is the time (inyears).

(a) Solve this differential equation.

(b) Use a graphing utility to graph the particular solutions for

0.9, and 1.

(c) The animal is sold when its weight reaches 800 pounds. Find the time of sale for each of themodels in part (b).

(d) What is the maximum weight of the animal for eachof the models in part (b)?

k � 0.8,

tw

dwdt

� k�1200 � w�

dVdt

� kV 2�3.

S � kV 2�3.

S.V

12.5 dvdt

� 43.2 � 1.75v

12.5 dvdt

� 43.2 � 1.25v

t � 0.v � 0

44. HOW DO YOU SEE IT? The differentialequation

represents the rate of change of the number of unitsproduced per day by a new employee, where is thenumber of units and is the time (in days). The general solution of this differential equation is

The graphs below show the particular solutions forand 1. Match the value of with each

graph. Explain your reasoning.

1 2 3 4 5

5

10

15

20

25

30

Days

Uni

ts

N

t

A

B

C

k0.6,k � 0.3,

N � 30 � 30e�kt.

tN

dNdt

� k�30 � N�

Business Capsule

After finding that the camera he wanted was soldout at a local store, Jack Abraham was inspired

to start Milo.com. Named after his dog, the siteshows buyers which nearby stores currently have a product in stock. This benefits not only shoppersbut also retailers, as Milo drives foot traffic into theirstores. In just one year, Milo.com grew to covermore than 140 retailers in 50,000 locations acrossthe United States. In 2010, the company wasbought by eBay, where Abraham now leads thelocal division.

48. Research Project Use your school’s library,the Internet, or some other reference source togather information about a company that offersinnovative products or services. Collect dataabout the revenue that the company has generatedand find a mathematical model of the data. Writea short paper that summarizes your findings.

9781133105060_App_F2.qxp 12/27/11 1:36 PM Page F12

■ Quiz Yourself F13

QUIZ YOURSELF

Take this quiz as you would take a quiz in class. When you are done, check yourwork against the answers given in the back of the book.

In Exercises 1–4, verify that the function is a solution of the differential equation.

Solution Differential Equation

1.

2.

3.

4.

In Exercises 5 and 6, verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition.

5. General solution:

Differential equation:

Initial condition: and when

6. General solution:

Differential equation:

Initial condition: and when

In Exercises 7–10, use separation of variables to find the general solution of the differential equation.

7. 8.

9. 10.

In Exercises 11 and 12, (a) find the general solution of the differential equation and(b) use a graphing utility to graph the particular solutions given by and

11. 12.

In Exercises 13 and 14, use the initial condition to find the particular solution of thedifferential equation.


13.

14.

15. Find an equation of the graph that passes through the point and has a slopeof Then graph the equation.

16. Ignoring resistance, a sailboat starting from rest accelerates at a rate proportionalto the difference between the velocities of the wind and the boat. With a 20-knotwind, this acceleration is described by the differential equation where is the velocity of the boat (in knots) and is the time (in hours). After halfan hour, the boat is moving at 10 knots. Write the velocity as a function of time.

tvdv�dt � k�20 � v�,

y� � 3x2y.�0, 2�

y � �3 when x �12

dydx

� y sin � x

y � 1 when x � 0y� � 2y � 1 � 0

dydx

�y

x � 3dydx

�x2 � 1

2y

C � ± 1.C � 0

dydx

�x

3y2 � 1y

dydx

�1

2x � 1

y� � �x � 2��y � 1�dydx

� �4x � 4

x � 2y� � 4y � 0

x2y� � 3xy� � 3y � 0

y � C1x � C2x3

x � ��6y� � 1y � 2

y� � 9y � 0

y � C1 sin 3x � C2 cos 3x

2xy� � y � x3 � xy �x3

5� x � C�x

xy� � 2y� � 0y �1x

y� � y � 0y � C1 cos x � C2 sin x

2y� � y � 0y � Ce�x� 2

9781133105060_App_F2.qxp 12/27/11 1:36 PM Page F13


F.3 First-Order Linear Differential Equations■ Solve first-order linear differential equations.

■ Use first-order linear differential equations to model and solve real-life problems.

First-Order Linear Differential Equations

In this section, you will see how to solve a very important class of differential equations—first-order linear differential equations. The term “first-order” refers to thefact that the highest-order derivative of in the equation is the first derivative.

To solve a linear differential equation, write it in standard form to identify the functions and Then integrate and form the expression

Integrating factor

which is called an integrating factor. The general solution of the equation is

General solution

Example 1 Solving a Linear Differential Equation


SOLUTION For this equation, and So, the integrating factor is

Integrating factor

This implies that the general solution is

General solution

Checkpoint 1


■y� � y � 10.

�12

ex � Ce�x.

� e�x�12

e2x � C� �

1ex�ex�ex� dx

y �1

u�x��Q�x�u�x� dx

� ex.

� e�dx

u�x� � e�P�x�dx

Q�x� � ex.P�x� � 1

y� � y � ex.

y �1

u�x��Q�x�u�x� dx.

u�x� � e�P�x� dx

P�x�Q�x�.P�x�

y

Definition of a First-Order Linear Differential Equation

A first-order linear differential equation is an equation of the form

where and are functions of An equation that is written in this form is said to be in standard form.

x.QP

y� � P�x�y � Q�x�

9781133105060_App_F3.qxp 12/27/11 1:41 PM Page F14

Appendix F.3 ■ First-Order Linear Differential Equations F15

In Example 1, the differential equation was given in standard form. For equationsthat are not written in standard form, you should first convert to standard form so thatyou can identify the functions and

Example 2 Solving a Linear Differential Equation

Find the general solution of Assume

SOLUTION Begin by writing the equation in standard form.

Standard form,

In this form, you can see that and So,

which implies that the integrating factor is

Integrating factor

This implies that the general solution is

Form of general solution

Substitute.

Simplify.

General solution

Checkpoint 2

Find the general solution of Assume ■x > 0.xy� � y � x.

� x2�ln x � C�.

� x2� 1x dx

�1

1�x2� x� 1x2� dx

y �1

u�x�� Q�x�u�x� dx

�1x2.

�1

eln x2

� e�ln x2


� �ln x2

� �2 ln x

� P�x� dx � �� 2x dx

Q�x� � x.P�x� � �2�x

y� � P�x�y � Q�x�y� � ��2x�y � x

x > 0.xy� � 2y � x2.

Q�x�.P�x�

TECH TUTOR

From Example 2, you can see that it can be difficultto solve a linear differentialequation. Fortunately, the task is greatly simplified bysymbolic integration utilities.Use a symbolic integration utility to find the particularsolution of the differentialequation in Example 2, giventhe initial condition when x � 1.

y � 1

Guidelines for Solving a Linear Differential Equation

1. Write the equation in standard form.

2. Find the integrating factor.

3. Evaluate the integral below to find the general solution.

y �1

u�x��Q�x�u�x� dx


y� � P�x�y � Q�x�

9781133105060_App_F3.qxp 12/27/11 1:41 PM Page F15


Application

Example 3 Intravenous Feeding

Glucose is added intravenously to the bloodstream at the rate of units per minute, andthe body removes glucose from the bloodstream at a rate proportional to the amountpresent. Assume that is the amount of glucose in the bloodstream at time and thatthe rate of change of the amount of glucose is

where is a constant. Find the general solution of the differential equation.

SOLUTION In standard form, this linear differential equation is

Standard form

which implies that and So, the integrating factor is

Integrating factor

and the general solution is

General solution

Checkpoint 3

Use the general solution

from Example 3 to find the particular solution determined by the initial conditionwhen (Assume and ) ■q � 0.05.k � 0.05t � 0.A � 0

A �qk

� Ce�kt

�qk

� Ce�kt.

� e�kt�qk

ekt � C� � e� kt�qektdt

A �1

u�t� �Q�t�u�t� dt

� ekt

� e�k dt

u�t� � e�P�t� dt

Q�t� � q.P�t� � k

dAdt

� kA � q

k

dAdt

� q � kA

tA

q


1. State the definition of a first-order linear differential equation (page F14).For examples of solving a first-order linear differential equation, seeExamples 1 and 2.

2. State the guidelines for solving a first-order linear differential equation (page F15). For examples of solving a first-order linear differential equation,see Examples 1 and 2.

3. Describe a real-life example of how a first-order linear differential equationcan be used to analyze intravenous feeding (page F16, Example 3).

9781133105060_App_F3.qxp 12/27/11 1:41 PM Page F16

Appendix F.3 ■ First-Order Linear Differential Equations F17

Exercises F.3

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 4.2, 4.4, and 5.1–5.3.

In Exercises 1–4, simplify the expression.

1. 2. 3. 4.

In Exercises 5–10, find the indefinite integral.

5. 6. 7.

8. 9. 10. � x�1 � x2�2 dx� �4x � 3�2 dx� x � 1

x2 � 2x � 3 dx

� 1

2x � 5 dx� xe3x2 dx� 4e2x dx

e2 ln x�xe�ln x31e�x �e�x � e2x�e�x�e2x � ex�

SKILLS WARM UP F.3

Writing in Standard Form In Exercises 1–6, write thefirst-order linear differential equation in standard form.

1. 2.

3. 4.

5. 6.

Solving a Linear Differential Equation In Exercises7–18, find the general solution of the first-order linear differential equation. See Examples 1 and 2.

7. 8.

9. 10.

11.

12.

13.

14.

15.

16.

17.

18.

Using Two Methods In Exercises 19–22, solve for y intwo ways.

19.

20.

21.

22.

Matching In Exercises 23–26, match the differentialequation with its solution without solving the differentialequation. Explain your reasoning.

Differential Equation Solution

23. (a)

24. (b)

25. (c)

26. (d)

Finding a Particular Solution In Exercises 27–34, findthe particular solution that satisfies the initial condition.


27.

28.

29.

30.

31.

32.

33.

34.

35. Sales The rate of change (in thousands of units) insales of a biomedical syringe is modeled by

where is the time (in years). Solve this differential equation and use the result to complete the table.

t 0 1 2 3 4 5 6 7 8 9 10

S 0

t

dSdt

� 0.2�100 � S� � 0.2t

S

y � 2 when x � 42xy� � y � x3 � x

y � 3 when x � �1xy� � 2y � 3x2 � 5x

y � 2 when x � 1y� � �2x � 1�y � 0

y � 6 when x � 0y� � 3x2y � 3x2

y � 4 when x � 0y� � y � x

y � 2 when x � 2xy� � y � 0

y � 4 when x � 1y� � 2y � e�2x

y � 3 when x � 0y� � y � 6ex

y � Ce2xy� � 2xy � x

y � x2 � Cy� � 2xy � 0

y � �12 � Cex2y� � 2y � 0

y � Cex2y� � 2x � 0

y� � 4xy � x

y� � 2xy � 2x

y� � 3y � �2

y� � y � 4

xy� � y � x2 ln x

x3y� � 2y � e1�x2

xy� � y � x2 � 1

�x � 1�y� � y � x2 � 1

y� � 5y � e5x

y� � 2xy � 10x

dydx

�e�2x

1 � e�2x

dydx

�x2 � 3

x

dydx

� 3y � e�3xdydx

� y � e4x

dydx

� 5y � 15dydx

� 3y � 6

x � x2�y� � y�y � 1 � �x � 1�y�

xy� � y � x3yxy� � y � xex

y� � 5�2x � y� � 0x3 � 2x2y� � 3y � 0

9781133105060_App_F3.qxp 12/27/11 1:41 PM Page F17


37. Vertical Motion A falling object encounters airresistance that is proportional to its velocity Theacceleration due to gravity is meters per secondper second. The rate of change in velocity is

Solve this differential equation to find as a function oftime

38. Velocity A booster rocket carrying an observation satellite is launched into space. The rocket and satellitehave mass and are subject to air resistance proportionalto the velocity at any time A differential equation thatmodels the velocity of the rocket and satellite is

where is the acceleration due to gravity. Solve the differential equation for as a function of

39. Learning Curve The management at a medical supplyfactory has found that the maximum number of units anemployee can produce in a day is 40. The rate ofincrease in the number of units produced with respectto time (in days) by a new employee is proportional to

This rate of change of performance withrespect to time can be modeled by

(a) Solve this differential equation.

(b) Find the particular solution for a new employeewho produced 10 units on the first day at the factoryand 19 units on the twentieth day.

40. Investment Let be the amount in a fund earninginterest for years at the annual rate of (in decimalform), compounded continuously. If a continuous cashflow of dollars per year is withdrawn from the fund,then the rate of decrease of is given by the differentialequation

where when

(a) Solve this equation for as a function of

(b) Use the result from part (a) to find whenand

years.t � 5P � $250,000,r � 0.07,A0 � $2,000,000,

A

t.A

t � 0.A � A0

dAdt

� rA � P

AP

rtA

dNdt

� k�40 � N�.

40 � N.t

N

t.vg

mdvdt

� �mg � kv

t.vm

t.v

dvdt

� kv � 9.8.

�9.8v.

36. HOW DO YOU SEE IT? The rate of changein the spread of a rumor in a school is modeled by

where is the percent (in decimal form) of the students who have heard the rumor and is the time(in hours), with corresponding to 8:00 A.M.The graph shows the particular solution for this differential equation.

(a) What percent of the students have heard therumor by 8:00 A.M.?

(b) At what time have 50% of the students heard therumor?

(c) What percent of the students have heard therumor by 3:00 P.M.?

1 2 3 4 5 6 7

0.2

0.4

0.6

0.8

1.0

Time (0 ↔ 8:00 A.M.)

Perc

ent

(in

deci

mal

for

m)

P

t

t � 0t

P

dPdt

� k�1 � P�

41. Project: Weight Loss For a project analyzinga person’s weight loss, visit this text’s website atwww.cengagebrain.com. (Data Source: TheCollege Mathematics Journal)

9781133105060_App_F3.qxp 12/27/11 1:41 PM Page F18

Appendix F.4 ■ Applications of Differential Equations F19

F.4 Applications of Differential Equations■ Use differential equations to model and solve real-life problems.

Applications of Differential Equations

Example 1 Modeling a Chemical Reaction

During a chemical reaction, substance A is converted into substance B at a rate that isproportional to the square of the amount of substance A. When 60 grams of Aare present, and after 1 hour only 10 grams of A remain unconverted. Howmuch of A is present after 2 hours?

SOLUTION Let be the amount of unconverted substance A at any time From thegiven assumption about the conversion rate, you can write the differential equation asshown.

Using separation of variables or a symbolic integration utility, you can find the generalsolution to be

General solution

To solve for the constants and use the initial conditions. That is, because when you can determine that Similarly, because when it follows that

which implies that So, the particular solution is

Substitute for k and C.

Particular solution

Using the model, you can determine that the amount of unconverted substance A after2 hours is

In Figure F.5, note that the chemical conversion is occurring rapidly during the firsthour. Then, as more and more of substance A is converted, the conversion rate slowsdown.

Checkpoint 1

Use the chemical reaction model in Example 1 to find the amount of substance A(in grams) as a function of (in hours) given that grams when and

grams when ■t � 2.y � 5t � 0y � 40t

y

� 5.45 grams.

y �60

5�2� � 1

�60

5t � 1.

y ��1

��1�12�t � �1�60�

k � �112.

10 ��1

k � �1�60�

t � 1,y � 10C � �160.t � 0,

y � 60k,C

y ��1

kt � C.

dydt

� ky2

t.y

�t � 1�,t � 0,

Rate ofchange

of y

is propor-tional to

the squareof y.

y

t

60

50

40

30

20

10

(0, 60)

(1, 10) (2, 5.45)

y = 605t + 1

1 2 3

Time (in hours)

Am

ount

(in

gra

ms)

Chemical Reaction

FIGURE F.5

9781133105060_App_F4.qxp 12/27/11 1:44 PM Page F19


Example 2 Modeling Advertising Awareness

The new cereal product from Example 3 in Section F.1 is introduced through an advertising campaign to a population of 1 million potential customers. The rate atwhich the population hears about the product is assumed to be proportional to the number of people who are not yet aware of the product. By the end of 1 year, half ofthe population has heard of the product. How many will have heard of it by the end of2 years?

SOLUTION Let be the number (in millions) of people at time who have heard ofthe product. This means that is the number (in millions) of people who have notheard of it, and is the rate at which the population hears about the product. Fromthe given assumption, you can write the differential equation as shown.


General solution

To solve for the constants and use the initial conditions. That is, because when you can determine that Similarly, because when itfollows that which implies that


Particular solution

This model is shown graphically in Figure F.6. Using the model, you can determine thatthe number of people who have heard of the product after 2 years is

FIGURE F.6

Checkpoint 2

Repeat Example 2 given that by the end of 1 year, only one-fourth of the populationhave heard of the product. ■

y

t

1.25

1.00

0.75

0.50

0.25

(2, 0.75)

(1, 0.50)

(0, 0)

1 2 3 4 5

Time (in years)

Pote

ntia

l cus

tom

ers

(in

mill

ions

)

Advertising Awareness

y = 1 − e−0.693t

� 0.75 or 750,000 people.

y � 1 � e�0.693�2�

y � 1 � e�0.693t.

k � �ln 0.5 � 0.693.

0.5 � 1 � e�k,t � 1,y � 0.5C � 1.t � 0,

y � 0k,C

y � 1 � Ce�kt.

dydt

� k�1 � y�

dy�dt�1 � y�

ty

Rate ofchange

of y

is propor-tional to

the differencebetween 1

and y.

9781133105060_App_F4.qxp 12/27/11 1:44 PM Page F20


Earlier in the text, you studied two models for population growth: exponentialgrowth, which assumes that the rate of change of is proportional to and logisticgrowth, which assumes that the rate of change of is proportional to and where is the population limit.

The next example describes a third type of growth model called a Gompertzgrowth model. This model assumes that the rate of change of is proportional to andthe natural log of where is the population limit.

Example 3 Modeling Population Growth

A population of 20 wolves has been introduced into a national park. The forest serviceestimates that the maximum population the park can sustain is 200 wolves. After 3 years, the population is estimated to be 40 wolves. If the population follows a Gompertzgrowth model, how many wolves will there be 10 years after their introduction?

SOLUTION Let be the number of wolves at any time From the given assumptionabout the rate of growth of the population, you can write the differential equation asshown.


General solution

To solve for the constants and use the initial conditions. That is, because when you can determine that

Similarly, because when it follows that

which implies that So, the particular solution is

Particular solution

Using the model, you can estimate the wolf population after 10 years to be

In Figure F.7, note that after 10 years the population has reached about half of the estimated maximum population. Try checking the growth model to see that it yields

when and when

Checkpoint 3

A population of 10 wolves has been introduced into a national park. The forest service estimates that the maximum population the park can sustain is 150 wolves.After 3 years, the population is estimated to be 25 wolves. If the population follows a Gompertz growth model, how many wolves will there be 10 years after their introduction? ■

t � 3.y � 40t � 0y � 20

� 100 wolves.

y � 200e�2.3026e�0.1194�10�

y � 200e�2.3026e�0.1194t.

k � 0.1194.

40 � 200e�2.3026e�k�3�

t � 3,y � 40

� 2.3026.

C � ln 10

t � 0,y � 20k,C

y � 200e�Ce�kt.

dydt

� ky ln 200

y

t.y

LL�y,yy

L1 � y�L,yy

y,y

y

t

20018016014012010080604020 (0, 20)

(3, 40)

(10, 100)

y = 200e−2.3026e−0.1194t

Num

ber

of w

olve

s

2 4 6 8 10 12 14

Time (in years)

Population Growth

FIGURE F.7

Rate ofchange

of y

is propor-tional to

the productof y and

the log ofthe ratio of200 and y.

9781133105060_App_F4.qxp 12/27/11 1:44 PM Page F21


In genetics, a commonly used hybrid selection model is based on the differentialequation

In this model, represents the portion of the population that has a certain characteristicand represents the time (measured in generations). The numbers and are constants that depend on the genetic characteristic that is being studied.

Example 4 Modeling Hybrid Selection

You are studying a population of beetles to determine how quickly characteristic D willpass from one generation to the next. At the beginning of your study you findthat half the population has characteristic D. After four generations you findthat 80% of the population has characteristic D. Use the hybrid selection model abovewith and to find the percent of the population that will have characteristicD after 10 generations.

SOLUTION Using and the differential equation for the hybrid selectionmodel is


General solution

To solve for the constants and use the initial conditions. That is, because when you can determine that Similarly, because when itfollows that

which implies that


Particular solution

Using the model, you can estimate the percent of the population that will have characteristic D after 10 generations to be given by

Using a symbolic algebra utility, you can solve this equation for to obtain The graph of the model is shown in Figure F.8.

Checkpoint 4

Repeat Example 4 given that only 25% of the population has characteristic D whenand 50% of the population has characteristic D when ■t � 4.t � 0

y � 0.96.y

y�2 � y��1 � y�2 � 3e0.5199�10�.

y�2 � y��1 � y�2 � 3e0.5199t.

k �18

ln 8 � 0.2599.

0.8�1.2��0.2�2 � 3e8k

t � 4,y � 0.8C � 3.t � 0,y � 0.5k,C

y�2 � y��1 � y�2 � Ce2kt.

dydt

� ky�1 � y��2 � y�.

b � 1,a � 2

b � 1a � 2

�t � 4�,�t � 0�,

kb,a,ty

dydt

� ky�1 � y��a � by�.

y

t

1.00.90.80.70.60.50.40.30.20.1

(0, 0.5)

(4, 0.8)

Perc

ent o

f po

pula

tion

2 4 6 8 10 12

Time (in generations)

Hybrid Selection

y(2 − y)(1 − y)2 = 3e0.5199t

FIGURE F.8

9781133105060_App_F4.qxp 12/27/11 1:44 PM Page F22


Example 5 Modeling a Chemical Mixture

A tank contains 40 gallons of a solution composed of 90% water and 10% alcohol. Asecond solution containing half water and half alcohol is added to the tank at the rate of 4 gallons per minute. At the same time, the tank is being drained at the rateof 4 gallons per minute, as shown in Figure F.9. Assuming that the solution is stirredconstantly, will there be at least 14 gallons of alcohol in the tank after 10 minutes?

SOLUTION Let be the number of gallons of alcohol in the tank at any time Thepercent of alcohol in the 40-gallon tank at any time is Moreover, because 4 gallons of solution are being drained each minute, the rate of change of is

where 2 represents the number of gallons of alcohol entering each minute in the 50%solution. In standard form, this linear differential equation is

Standard form

Using an integrating factor or a symbolic integration utility, you can find the generalsolution to be

General solution

Because when you can conclude that So, the particular solution is

Particular solution

Using this model, you can determine that the amount of alcohol in the tank when is

Yes, there will be at least 14 gallons of alcohol in the tank after 10 minutes.

Checkpoint 5

A tank contains 50 gallons of a solution composed of 90% water and 10% alcohol. A second solution containing half water and half alcohol is added to the tank at the rate of 5 gallons per minute. At the same time, the tank is being drained at the rate of 5 gallons per minute. Assuming that the solution is stirred constantly, howmuch alcohol will be in the tank after 10 minutes? ■

y � 20 � 16e��10��10 � 14.1 gallons.

t � 10

y � 20 � 16e�t�10.

C � �16.t � 0,y � 4

y � 20 � Ce�t�10.

y� �1

10y � 2.

dydt

� �4� y40� � 2

yy�40.

t.y4 gal/min

4 gal/min

FIGURE F.9


1. Describe a real-life example of how a differential equation can be used tomodel a chemical reaction (page F19, Example 1).

2. Describe a real-life example of how a differential equation can be used tomodel population growth (page F21, Example 3).

3. Describe a real-life example of how a differential equation can be used tomodel hybrid selection (page F22, Example 4).

Rate ofchange

of y

is equal to theamount of alcohol

draining out

plus the amountof alcoholentering.

9781133105060_App_F4.qxp 12/27/11 1:44 PM Page F23


Exercises F.4

Chemical Reaction In Exercises 1 and 2, use thechemical reaction model described in Example 1to find the amount y (in grams) as a function oftime t (in hours). Then use a graphing utility tograph the function.

1. grams when grams when

2. grams when grams when

Advertising Awareness In Exercises 3 and 4, use theadvertising awareness model described in Example 2 tofind the number of people y (in millions) aware of theproduct as a function of time t (in years).

3. when when

4. when when

Population Growth In Exercises 5 and 6, use theGompertz growth model described in Example 3 to findthe population y as a function of time t (in years).

5. when when

6. when when

Hybrid Selection In Exercises 7 and 8, use the hybridselection model described in Example 4 to find the percenty (in decimal form) of the population that has the indicatedcharacteristics as a function of time t (in generations).

7. when when

8. when when

Finding a Particular Solution In Exercises 9–14,assume that the rate of change in y is proportional to y.Solve the resulting differential equation andfind the particular solution that passes through the points.

9.

10.

11.

12.

13.

14.

15. Chemical Reaction During a chemical reaction, acompound changes into another compound at a rate proportional to the unchanged amount Write the differential equation for the chemical reaction model.Find the particular solution when the initial amount ofthe original compound is 20 grams and the amountremaining after 1 hour is 16 grams.

16. Chemical Reaction Using the result of Exercise 15,when will 75% of the compound have been changed?When will 95% of the compound have been changed?

17. Population Growth The rate of change of thepopulation of a city is proportional to the population at any time (in years). In 2000, the population was200,000, and the constant of proportionality was 0.015.Estimate the population of the city in the year 2020.

tP

y.

�1, 4�, �2, 1��2, 2�, �3, 4��0, 60�, �5, 30��0, 4�, �4, 1��0, 4�, �1, 6��0, 1�, �3, 2�

dy/dx � ky

t � 2t � 0; y � 0.75y � 0.6

t � 4t � 0; y � 0.4y � 0.1

t � 4t � 0; y � 60y � 30

t � 2t � 0; y � 150y � 100

t � 2y � 0.9t � 0;y � 0

t � 1y � 0.75t � 0;y � 0

t � 1y � 12t � 0;y � 75

t � 2y � 4t � 0;y � 45

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 4.6, F.2, and F.3.

In Exercises 1–4, use separation of variables to find the general solution of thedifferential equation.

1. 2.

3. 4.

In Exercises 5–8, use an integrating factor to solve the first-order linear differential equation.

5. 6.

7. 8.

In Exercises 9 and 10, write the equation that models the statement.

9. The rate of change of with respect to is proportional to the square of

10. The rate of change of with respect to is proportional to the difference of and t.xtx

x.xy

xy� � 2y � x2y� � xy � x

y� � 2y � e�2xy� � 2y � 4

dydx

�x � 4

4y3

dydx

� 2xy

2y dydx

� 3dydx

� 3x

SKILLS WARM UP F.4

9781133105060_App_F4.qxp 12/27/11 1:44 PM Page F24


18. Fruit Flies The rate of change of an experimental population of fruit flies is proportional to the population

at any time (in days). There were 100 flies after the second day of the experiment and 300 flies after thefourth day. Approximately how many flies were in theoriginal population?

19. Chemistry A wet towel hung from a clothesline to dryloses moisture through evaporation at a rate proportionalto its moisture content. After 1 hour, the towel has lost40% of its original moisture content. How long will it take the towel to lose 80% of its original moisturecontent?

20. Meteorology The barometric pressure (in inchesof mercury) at an altitude of miles above sea leveldecreases at a rate proportional to the current pressureaccording to the model

where inches when Find the barometricpressure (a) at the top of Mt. St. Helens (8364 feet) and(b) at the top of Mt. McKinley (20,320 feet).

21. Sales Growth The rate of change in sales (in thousands of units) of a new product is proportional tothe difference between and at any time (in years),where is the maximum number of units of the newproduct available. When Write and solvethe differential equation for this sales model.

22. Sales Growth Use the result of Exercise 21 to findthe particular solutions when (a) and when and (b) and when

23. Biology A population of eight beavers has been introduced into a new wetlands area. Biologists estimate that the maximum population the wetlands cansustain is 60 beavers. After 3 years, the population is 15 beavers. The population follows a Gompertz growthmodel. How many beavers will there be in the wetlandsafter 10 years?

24. Biology A population of 30 rabbits has been introduced into a new region. It is estimated that themaximum population the region can sustain is 400 rabbits. After 1 year, the population is estimated to be90 rabbits. The population follows a Gompertz growthmodel. How many rabbits will there be after 3 years?

25. Biology At any time (in years), the rate of growth ofthe population of deer in a state park is proportionalto the product of and where is themaximum number of deer the park can maintain.

(a) Use a symbolic integration utility to find the generalsolution.

(b) Find the particular solution given the conditionswhen and when

(c) Find when

(d) Find when

26. Biology At any time (in years), the rate of growth of the population of fish in a pond is proportional tothe product of and where is themaximum number of fish the pond can maintain.

(a) Use a symbolic integration utility to find the generalsolution.

(b) Find the particular solution given the conditionswhen and when

(c) Find when

(d) Find when

27. Chemical Mixture A 100-gallon tank is full of a solution containing 25 pounds of a concentrate. Startingat time distilled water is admitted to the tank atthe rate of 5 gallons per minute, and the well-stirredsolution is withdrawn at the same rate.

(a) Find the amount of the concentrate in the solutionas a function of by solving the differential equation

(b) Find the time required for the amount of concentratein the tank to reach 15 pounds.

28. Chemical Mixture A 200-gallon tank is half full of distilled water. At time a solution containing0.5 pound of concentrate per gallon enters the tank atthe rate of 5 gallons per minute, and the well-stirredmixture is withdrawn at the same rate.

(a) Find the amount of the concentrate in the solutionas a function of by solving the differential equation

(b) Find the amount of concentrate in the tank after 30 minutes.

29. Population Growth When predicting populationgrowth, demographers must consider birth and deathrates as well as the net change caused by the differencebetween the rates of immigration and emigration. Let be the population at time and let be the net increaseper unit time due to the difference between immigrationand emigration. So, the rate of growth of the populationis given by

is constant.

Solve the differential equation to find as a function of t.P

NdPdt

� kP � N,

NtP

Q� � �5� Q100� �

52

.

tQ

t � 0,

Q� � �5� Q100�.

tQ

t � 0,

N � 700.t

t � 1.N

t � 2.N � 500t � 0N � 200

L � 1000L � N,NN

t

N � 350.t

t � 1.N

t � 4.N � 200t � 0N � 100

L � 500L � N,NN

t

t � 1.S � 50L � 500t � 2,S � 25L � 100

S � 0.t � 0,L

tSL

S

x � 0.y � 29.92

dydx

� �0.2y

xy

tP

9781133105060_App_F4.qxp 12/27/11 1:44 PM Page F25


31. Investment A large corporation starts at time to invest part of its profit at a rate of dollars per year ina fund for future expansion. Assume that the fund earns

percent interest per year compounded continuously.The rate of growth of the amount in the fund is given by

where when and is in decimal form.Solve this differential equation for as a function of

Investment In Exercises 32–34, use the result ofExercise 31.

32. Find for each situation.

(a)

(b)

33. Find if the corporation needs $120,000,000 in 8 years and the fund earns 8% interest compounded continuously.

34. Find if the corporation needs $800,000 and it caninvest $75,000 per year in a fund earning 13% interestcompounded continuously.

Medical Science In Exercises 35–38, a medicalresearcher wants to determine the concentration C (inmoles per liter) of a tracer drug injected into a movingfluid with flow R (in liters per minute). Solve this problemby considering a single-compartment dilution model (seefigure). Assume that the fluid is continuously mixed andthat the volume V (in liters) of fluid in the compartment isconstant.

35. Mixture If the tracer is injected instantaneously attime then the concentration of the fluid in thecompartment begins diluting according to the differentialequation

(a) Solve this differential equation to find the concentration as a function of time.

(b) Find the limit of as

36. Mixture Use the solution of the differential equationin Exercise 35 to find the concentration as a function oftime. Then use a graphing utility to graph the function.

(a)

(b)

37. Mixture In Exercises 35 and 36, it was assumed thatthere was a single initial injection of the tracer drug intothe compartment. Now consider the case in which thetracer is continuously injected (beginning at ) at aconstant rate of mol/min. The concentration of thefluid in the compartment begins diluting according tothe differential equation

(a) Solve this differential equation to find the concentration as a function of time.

(b) Find the limit of as

38. Mixture Use the solution of the differential equationin Exercise 37 to find the concentration as a function oftime. Then use a graphing utility to graph the function.

(a) and

(b) and R � 1.0 L�minV � 2 liters,Q � 1 mol�min,

R � 0.5 L�minV � 2 liters,Q � 2 mol�min,

t →�.C

C � 0 when t � 0.dCdt

�QV

� �RV�C,

Qt � 0

V � 2 liters, R � 1.5 L�min, and C0 � 0.6 mol�L

V � 2 liters, R � 0.5 L�min, and C0 � 0.6 mol�L

t →�.C

C � C0 when t � 0.dCdt

� ��RV�C,

t � 0,

Flow R (pure)

Flow R (concentration C)

Volume V

Tracer injected

t

P

P � $250,000, r � 0.15, and t � 10 years

P � $100,000, r � 0.12, and t � 5 years

A

t.Art � 0,A � 0

dAdt

� rA � P

Ar

Pt � 0

30. HOW DO YOU SEE IT? In a learning theoryproject, the rate of change in the percent (indecimal form) of correct responses after trialscan be modeled by

The graph shows the particular solutions for two different groups.

(a) What was the percent of correct responsesbefore any trials for each group?

(b) What is the limit of as approaches infinityfor each group?

(c) After how many trials are 75% of the responsescorrect for each group?

tP

5 10 15 20 25 30 35 40

0.2

0.4

0.6

0.8

1.0

AB

Number of trials

Perc

ent

(in

deci

mal

for

m)

P

n

dPdn

� kP�1 � P�.

nP

9781133105060_App_F4.qxp 12/27/11 1:44 PM Page F26

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