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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Checkpoint

7.1
  1. Nonlinear
  2. Linear, nonhomogeneous
7.4

Linearly independent

7.5

y(x)=c1e3x+c2xe3xy(x)=c1e3x+c2xe3x

7.6
  1. y(x)=ex(c1cos3x+c2sin3x)y(x)=ex(c1cos3x+c2sin3x)
  2. y(x)=c1e−7x+c2xe−7xy(x)=c1e−7x+c2xe−7x
7.7

y(x)=e−2x+e5xy(x)=e−2x+e5x

7.8

y(x)=ex(2cos3xsin3x)y(x)=ex(2cos3xsin3x)

This figure is the graph of y(x) = e^x(2 cos 3x − sin 3x) It has the positive x axis scaled in increments of even tenths. The y axis is scaled in increments of twenty. The graph itself starts at the origin. Its amplitude increases as x increases.
7.9

y(t)=te−7ty(t)=te−7t

This figure is the graph of y(t) = te^−7t. The horizontal axis is labeled with t and is scaled in increments of tenths. The y axis is scaled in increments of 0.5. The graph passes through the origin and has a horizontal asymptote of the positive t axis.


At time t=0.3,t=0.3, y(0.3)=0.3e(−7*0.3)=0.3e−2.10.0367.y(0.3)=0.3e(−7*0.3)=0.3e−2.10.0367. The mass is 0.0367 ft below equilibrium. At time t=0.1,t=0.1, y(0.1)=0.3e−0.70.1490.y(0.1)=0.3e−0.70.1490. The mass is moving downward at a speed of 0.1490 ft/sec.

7.10

y(x)=c1ex+c2e4x2y(x)=c1ex+c2e4x2

7.11

y(t)=c1e2t+c2te2t+sint+costy(t)=c1e2t+c2te2t+sint+cost

7.12
  1. y(x)=c1e4x+c2exxexy(x)=c1e4x+c2exxex
  2. y(t)=c1e−3t+c2e2t5cos2t+sin2ty(t)=c1e−3t+c2e2t5cos2t+sin2t
7.13

z1=3x+311x2,z1=3x+311x2, z2=2x+211xz2=2x+211x

7.14
  1. y(x)=c1cosx+c2sinx+cosxln|cosx|+xsinxy(x)=c1cosx+c2sinx+cosxln|cosx|+xsinx
  2. x(t)=c1et+c2tet+tetln|t|x(t)=c1et+c2tet+tetln|t|
7.15

x(t)=0.1cos(14t)x(t)=0.1cos(14t) (in meters); frequency is 142π142π Hz.

7.16

x(t)=17sin(4t+0.245),x(t)=17sin(4t+0.245), frequency=42π0.637,frequency=42π0.637, A=17A=17

7.17

x(t)=0.6e−2t0.2e−6tx(t)=0.6e−2t0.2e−6t

7.18

x(t)=12e−8t+4te−8tx(t)=12e−8t+4te−8t

7.19

x(t)=−0.24e−2tcos(4t)0.12e−2tsin(4t)x(t)=−0.24e−2tcos(4t)0.12e−2tsin(4t)

7.20

x(t)=12cos(4t)+94sin(4t)+12e−2tcos(4t)2e−2tsin(4t)x(t)=12cos(4t)+94sin(4t)+12e−2tcos(4t)2e−2tsin(4t)
Transient solution:12e−2tcos(4t)2e−2tsin(4t)Transient solution:12e−2tcos(4t)2e−2tsin(4t)
Steady-state solution:12cos(4t)+94sin(4t)Steady-state solution:12cos(4t)+94sin(4t)

7.21

q(t)=−25etcos(3t)7etsin(3t)+25q(t)=−25etcos(3t)7etsin(3t)+25

7.22
  1. y(x)=a0n=0(−1)nn!x2n=a0ex2y(x)=a0n=0(−1)nn!x2n=a0ex2
  2. y(x)=a0(x+1)3y(x)=a0(x+1)3

Section 7.1 Exercises

1.

linear, homogenous

3.

nonlinear

5.

linear, homogeneous

11.

y=c1e5x+c2e−2xy=c1e5x+c2e−2x

13.

y=c1e−2x+c2xe−2xy=c1e−2x+c2xe−2x

15.

y=c1e5x/2+c2exy=c1e5x/2+c2ex

17.

y=ex/2(c1cos3x2+c2sin3x2)y=ex/2(c1cos3x2+c2sin3x2)

19.

y=c1e−11x+c2e11xy=c1e−11x+c2e11x

21.

y=c1cos9x+c2sin9xy=c1cos9x+c2sin9x

23.

y=c1+c2xy=c1+c2x

25.

y=c1e((1+22)/3)x+c2e((122)/3)xy=c1e((1+22)/3)x+c2e((122)/3)x

27.

y=c1ex/6+c2xex/6y=c1ex/6+c2xex/6

29.

y=c1+c2e9xy=c1+c2e9x

31.

y=−2e−2x+2e−3xy=−2e−2x+2e−3x

33.

y=3cos(2x)+5sin(2x)y=3cos(2x)+5sin(2x)

35.

y=e6x+2e−5xy=e6x+2e−5x

37.

y=2ex/5+75xex/5y=2ex/5+75xex/5

39.

y=(2e6e−7)e6x(2e6e−7)e−7xy=(2e6e−7)e6x(2e6e−7)e−7x

41.

No solutions exist.

43.

y=2e2x2e2+1e2xe2xy=2e2x2e2+1e2xe2x

45.

y=4cos3x+c2sin3x,infinitely many solutionsy=4cos3x+c2sin3x,infinitely many solutions

47.

5y+19y4y=05y+19y4y=0

49.

a. y=3cos(8x)+2sin(8x)y=3cos(8x)+2sin(8x)
b.

This figure is a periodic graph. It has an amplitude of 3.5. Both the x and y axes are scaled in increments of 1.
51.

a. y=e(−5/2)x[−2cos(352x)+43535sin(352x)]y=e(−5/2)x[−2cos(352x)+43535sin(352x)]
b.

This figure is a graph of an oscillating function. The x and y axes are scaled in increments of even numbers. The amplitude of the graph is decreasing as x increases.

Section 7.2 Exercises

55.

y=c1e−4x/3+c2ex2y=c1e−4x/3+c2ex2

57.

y=c1cos4x+c2sin4x+120e−2xy=c1cos4x+c2sin4x+120e−2x

59.

y=c1e2x+c2xe2x+2x2+5xy=c1e2x+c2xe2x+2x2+5x

61.

y=c1ex+c2xex+12sinx12cosxy=c1ex+c2xex+12sinx12cosx

63.

y=c1cosx+c2sinx13xcos2x59sin2xy=c1cosx+c2sinx13xcos2x59sin2x

65.

y=c1e−5x+c2xe−5x+16x3e−5x+425y=c1e−5x+c2xe−5x+16x3e−5x+425

67.

a. yp(x)=Ax2+Bx+Cyp(x)=Ax2+Bx+C
b. yp(x)=13x2+43x359yp(x)=13x2+43x359

69.

a. yp(x)=(Ax2+Bx+C)exyp(x)=(Ax2+Bx+C)ex
b. yp(x)=(14x258x3332)exyp(x)=(14x258x3332)ex

71.

a. yp(x)=(Ax2+Bx+C)excosxyp(x)=(Ax2+Bx+C)excosx +(Dx2+Ex+F)exsinx+(Dx2+Ex+F)exsinx
b. yp(x)=(110x21125x27250)excosxyp(x)=(110x21125x27250)excosx +(310x2+225x+39250)exsinx+(310x2+225x+39250)exsinx

73.

y=c1+c2e−2x+115e3xy=c1+c2e−2x+115e3x

75.

y=c1e2x+c2e−4x+xe2xy=c1e2x+c2e−4x+xe2x

77.

y=c1e3x+c2e−3x8x9y=c1e3x+c2e−3x8x9

79.

y=c1cos2x+c2sin2x32xcos2x+34sin2xln(sin2x)y=c1cos2x+c2sin2x32xcos2x+34sin2xln(sin2x)

81.

y=347343+4343e7x+27x2e7x449xe7xy=347343+4343e7x+27x2e7x449xe7x

83.

y=5725+325e5x+15xe5x+425e−5xy=5725+325e5x+15xe5x+425e−5x

85.

yp=12+103x2lnxyp=12+103x2lnx

Section 7.3 Exercises

87.

x+16x=0,x+16x=0, x(t)=16cos(4t)2sin(4t),x(t)=16cos(4t)2sin(4t), period =π2sec,=π2sec, frequency =2πHz=2πHz

89.

x+196x=0,x+196x=0, x(t)=0.15cos(14t),x(t)=0.15cos(14t), period =π7sec,=π7sec, frequency =7πHz=7πHz

91.

a. x(t)=5sin(2t)x(t)=5sin(2t)
b. period =πsec,=πsec, frequency =1πHz=1πHz
c.

This figure is the graph of a function. It is a periodic function with consistent amplitude. The horizontal axis is labeled in increments of 1. The vertical axis is labeled in increments of 1.5.


d. t=π2sect=π2sec

93.

a. x(t)=et/5(20cos(3t)+15sin(3t))x(t)=et/5(20cos(3t)+15sin(3t))
b. underdamped

95.

a. x(t)=5e−4t+10te−4tx(t)=5e−4t+10te−4t
b. critically damped

97.

x(π)=7eπ/46x(π)=7eπ/46 ft below

99.

x(t)=329sin(4t)+cos(128t)1692sin(128t)x(t)=329sin(4t)+cos(128t)1692sin(128t)

101.

q(t)=e−6t(0.051cos(8t)+0.03825sin(8t))120cos(10t)q(t)=e−6t(0.051cos(8t)+0.03825sin(8t))120cos(10t)

103.

q(t)=e−10t(−32t5)+5,I(t)=2e−10t(160t+9)q(t)=e−10t(−32t5)+5,I(t)=2e−10t(160t+9)

Section 7.4 Exercises

105.

y=c0+5c1n=1(x/5)nn!=c0+5c1ex/5y=c0+5c1n=1(x/5)nn!=c0+5c1ex/5

107.

y=c0n=0(x)2n(2n)!+c1n=0(x)2n+1(2n+1)!y=c0n=0(x)2n(2n)!+c1n=0(x)2n+1(2n+1)!

109.

y=c0n=0x2nn!=c0ex2y=c0n=0x2nn!=c0ex2

111.

y=c0n=0x2n2nn!+c1n=0x2n+11357(2n+1)y=c0n=0x2n2nn!+c1n=0x2n+11357(2n+1)

113.

y=c1x3+c2xy=c1x3+c2x

115.

y=13x+2x33!12x44!+16x66!120x77!+y=13x+2x33!12x44!+16x66!120x77!+

Chapter Review Exercises

117.

True

119.

False

121.

second order, linear, homogeneous, λ22=0λ22=0

123.

first order, nonlinear, nonhomogeneous

125.

y=c1sin(3x)+c2cos(3x)y=c1sin(3x)+c2cos(3x)

127.

y=c1exsin(3x)+c2excos(3x)+25x+225y=c1exsin(3x)+c2excos(3x)+25x+225

129.

y=c1ex+c2e−4x+x4+e2x18516y=c1ex+c2e−4x+x4+e2x18516

131.

y=c1e(−3/2)x+c2xe(−3/2)x+49x2+427x1627y=c1e(−3/2)x+c2xe(−3/2)x+49x2+427x1627

133.

y=e−2xsin(2x)y=e−2xsin(2x)

135.

y=e1xe41(e4x1)y=e1xe41(e4x1)

137.

θ(t)=θ0cos(glt)θ(t)=θ0cos(glt)

141.

b=ab=a

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