Calculus Volume 3

# Chapter 7

### 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

$y(x)=ex(2cos3x−sin3x)y(x)=ex(2cos3x−sin3x)$ 7.9

$y(t)=te−7ty(t)=te−7t$ At time $t=0.3,t=0.3,$ $y(0.3)=0.3e(−7*0.3)=0.3e−2.1≈0.0367.y(0.3)=0.3e(−7*0.3)=0.3e−2.1≈0.0367.$ The mass is 0.0367 ft below equilibrium. At time $t=0.1,t=0.1,$ $y′(0.1)=0.3e−0.7≈0.1490.y′(0.1)=0.3e−0.7≈0.1490.$ The mass is moving downward at a speed of 0.1490 ft/sec.

7.10

$y(x)=c1e−x+c2e4x−2y(x)=c1e−x+c2e4x−2$

7.11

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

7.12
1. $y(x)=c1e4x+c2ex−xexy(x)=c1e4x+c2ex−xex$
2. $y(t)=c1e−3t+c2e2t−5cos2t+sin2ty(t)=c1e−3t+c2e2t−5cos2t+sin2t$
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

$x(t)=17sin(4t+0.245),x(t)=17sin(4t+0.245),$ $frequency=42π≈0.637,frequency=42π≈0.637,$ $A=17A=17$

7.17

$x(t)=0.6e−2t−0.2e−6tx(t)=0.6e−2t−0.2e−6t$

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

$q(t)=−25e−tcos(3t)−7e−tsin(3t)+25q(t)=−25e−tcos(3t)−7e−tsin(3t)+25$

7.22
1. $y(x)=a0∑n=0∞(−1)nn!x2n=a0e−x2y(x)=a0∑n=0∞(−1)nn!x2n=a0e−x2$
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.

$y=c1e5x/2+c2e−xy=c1e5x/2+c2e−x$

17.

$y=e−x/2(c1cos3x2+c2sin3x2)y=e−x/2(c1cos3x2+c2sin3x2)$

19.

$y=c1e−11x+c2e11xy=c1e−11x+c2e11x$

21.

$y=c1cos9x+c2sin9xy=c1cos9x+c2sin9x$

23.

$y=c1+c2xy=c1+c2x$

25.

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

27.

$y=c1e−x/6+c2xe−x/6y=c1e−x/6+c2xe−x/6$

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.

$y=2e−x/5+75xe−x/5y=2e−x/5+75xe−x/5$

39.

$y=(2e6−e−7)e6x−(2e6−e−7)e−7xy=(2e6−e−7)e6x−(2e6−e−7)e−7x$

41.

No solutions exist.

43.

$y=2e2x−2e2+1e2xe2xy=2e2x−2e2+1e2xe2x$

45.

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

47.

$5y″+19y′−4y=05y″+19y′−4y=0$

49.

a. $y=3cos(8x)+2sin(8x)y=3cos(8x)+2sin(8x)$
b. 51.

a. $y=e(−5/2)x[−2cos(352x)+43535sin(352x)]y=e(−5/2)x[−2cos(352x)+43535sin(352x)]$
b. ### Section 7.2 Exercises

55.

$y=c1e−4x/3+c2ex−2y=c1e−4x/3+c2ex−2$

57.

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

59.

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

61.

$y=c1e−x+c2xe−x+12sinx−12cosxy=c1e−x+c2xe−x+12sinx−12cosx$

63.

$y=c1cosx+c2sinx−13xcos2x−59sin2xy=c1cosx+c2sinx−13xcos2x−59sin2x$

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. $yp(x)=−13x2+43x−359yp(x)=−13x2+43x−359$

69.

a. $yp(x)=(Ax2+Bx+C)e−xyp(x)=(Ax2+Bx+C)e−x$
b. $yp(x)=(14x2−58x−3332)e−xyp(x)=(14x2−58x−3332)e−x$

71.

a. $yp(x)=(Ax2+Bx+C)excosxyp(x)=(Ax2+Bx+C)excosx$ $+(Dx2+Ex+F)exsinx+(Dx2+Ex+F)exsinx$
b. $yp(x)=(−110x2−1125x−27250)excosxyp(x)=(−110x2−1125x−27250)excosx$ $+(−310x2+225x+39250)exsinx+(−310x2+225x+39250)exsinx$

73.

$y=c1+c2e−2x+115e3xy=c1+c2e−2x+115e3x$

75.

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

77.

$y=c1e3x+c2e−3x−8x9y=c1e3x+c2e−3x−8x9$

79.

$y=c1cos2x+c2sin2x−32xcos2x+34sin2xln(sin2x)y=c1cos2x+c2sin2x−32xcos2x+34sin2xln(sin2x)$

81.

$y=−347343+4343e7x+27x2e7x−449xe7xy=−347343+4343e7x+27x2e7x−449xe7x$

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. d. $t=π2sect=π2sec$

93.

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

95.

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

97.

$x(π)=7e−π/46x(π)=7e−π/46$ 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.

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

### Section 7.4 Exercises

105.

$y=c0+5c1∑n=1∞(−x/5)nn!=c0+5c1e−x/5y=c0+5c1∑n=1∞(−x/5)nn!=c0+5c1e−x/5$

107.

$y=c0∑n=0∞(x)2n(2n)!+c1∑n=0∞(x)2n+1(2n+1)!y=c0∑n=0∞(x)2n(2n)!+c1∑n=0∞(x)2n+1(2n+1)!$

109.

$y=c0∑n=0∞x2nn!=c0ex2y=c0∑n=0∞x2nn!=c0ex2$

111.

$y=c0∑n=0∞x2n2nn!+c1∑n=0∞x2n+11⋅3⋅5⋅7⋯(2n+1)y=c0∑n=0∞x2n2nn!+c1∑n=0∞x2n+11⋅3⋅5⋅7⋯(2n+1)$

113.

$y=c1x3+c2xy=c1x3+c2x$

115.

$y=1−3x+2x33!−12x44!+16x66!−120x77!+⋯y=1−3x+2x33!−12x44!+16x66!−120x77!+⋯$

### Chapter Review Exercises

117.

True

119.

False

121.

second order, linear, homogeneous, $λ2−2=0λ2−2=0$

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.

$y=c1e−x+c2e−4x+x4+e2x18−516y=c1e−x+c2e−4x+x4+e2x18−516$

131.

$y=c1e(−3/2)x+c2xe(−3/2)x+49x2+427x−1627y=c1e(−3/2)x+c2xe(−3/2)x+49x2+427x−1627$

133.

$y=e−2xsin(2x)y=e−2xsin(2x)$

135.

$y=e1−xe4−1(e4x−1)y=e1−xe4−1(e4x−1)$

137.

$θ(t)=θ0cos(glt)θ(t)=θ0cos(glt)$

141.

$b=ab=a$