Chapter 7

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Chapter 7

Calculus Volume 3Chapter 7

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Chapter 7

Checkpoint

7.1

1. Nonlinear

2. Linear, nonhomogeneous

7.4

Linearly independent

7.5

$y(x) = c_{1}e^{3x} + c_{2}xe^{3x}$

7.6

1. $y(x) = e^{x}\left( {c_{1}\text{cos}\mspace{2mu} 3x + c_{2}\text{sin}\mspace{2mu} 3x} \right)$

2. $y(x) = c_{1}e^{-7x} + c_{2}xe^{-7x}$

7.7

$y(x) = \text{−}e^{-2x} + e^{5x}$

7.8

$y(x) = e^{x}(2\mspace{2mu}\text{cos}\mspace{2mu} 3x - \text{sin}\mspace{2mu} 3x)$

7.9

$y(t) = te^{-7t}$

At time $t = 0.3,$ $y(0.3) = 0.3e^{(-7*0.3)} = 0.3e^{-2.1} \approx 0.0367.$ The mass is 0.0367 ft below equilibrium. At time $t = 0.1,$ $y^{\prime}(0.1) = 0.3e^{-0.7} \approx 0.1490.$ The mass is moving downward at a speed of 0.1490 ft/sec.

7.10

$y(x) = c_{1}e^{\text{−}x} + c_{2}e^{4x} - 2$

7.11

$y(t) = c_{1}e^{2t} + c_{2}te^{2t} + \text{sin}\mspace{2mu} t + \text{cos}\mspace{2mu} t$

7.12

1. $y(x) = c_{1}e^{4x} + c_{2}e^{x} - xe^{x}$

2. $y(t) = c_{1}e^{-3t} + c_{2}e^{2t} - 5\mspace{2mu}\text{cos}\mspace{2mu} 2t + \text{sin}\mspace{2mu} 2t$

7.13

$z_{1} = \frac{3x + 3}{11x^{2}},$ $z_{2} = \frac{2x + 2}{11x}$

7.14

1. $y(x) = c_{1}\text{cos}\mspace{2mu} x + c_{2}\text{sin}\mspace{2mu} x + \text{cos}\mspace{2mu} x\mspace{2mu}\text{ln}\left| {\text{cos}\mspace{2mu} x} \right| + x\mspace{2mu}\text{sin}\mspace{2mu} x$

2. $x(t) = c_{1}e^{t} + c_{2}te^{t} + te^{t}\text{ln}|t|$

7.15

$x(t) = 0.1\mspace{2mu}\text{cos}\mspace{2mu}\left( {14t} \right)$ (in meters); frequency is $\frac{14}{2\pi}$ Hz.

7.16

$x(t) = \sqrt{17}\text{sin}\mspace{2mu}\left( {4t + 0.245} \right),$ $\text{frequency} = \frac{4}{2\pi} \approx 0.637,$ $A = \sqrt{17}$

7.17

$x(t) = 0.6e^{-2t} - 0.2e^{-6t}$

7.18

$x(t) = \frac{1}{2}e^{-8t} + 4te^{-8t}$

7.19

$x(t) = -0.24e^{-2t}\text{cos}\mspace{2mu}\left( {4t} \right) - 0.12e^{-2t}\text{sin}\mspace{2mu}\left( {4t} \right)$

7.20

$x(t) = - \frac{1}{2}\mspace{2mu}\text{cos}\mspace{2mu}\left( {4t} \right) + \frac{9}{4}\mspace{2mu}\text{sin}\mspace{2mu}\left( {4t} \right) + \frac{1}{2}e^{-2t}\text{cos}\mspace{2mu}\left( {4t} \right) - 2e^{-2t}\text{sin}\mspace{2mu}\left( {4t} \right)$

$\text{Transient solution:}\ \frac{1}{2}e^{-2t}\text{cos}\mspace{2mu}\left( {4t} \right) - 2e^{-2t}\text{sin}\mspace{2mu}\left( {4t} \right)$

$\text{Steady-state solution:}\ - \frac{1}{2}\mspace{2mu}\text{cos}\mspace{2mu}\left( {4t} \right) + \frac{9}{4}\mspace{2mu}\text{sin}\mspace{2mu}\left( {4t} \right)$

7.21

$q(t) = -25e^{\text{−}t}\text{cos}\mspace{2mu}\left( {3t} \right) - 7e^{\text{−}t}\text{sin}\mspace{2mu}\left( {3t} \right) + 25$

7.22

1. $y(x) = a_{0}\sum\limits_{n = 0}^{\infty}\frac{{(-1)}^{n}}{n!}x^{2n} = a_{0}e^{\text{−}x^{2}}$

2. $y(x) = a_{0}\left( {x + 1} \right)^{3}$

Section 7.1 Exercises

1.

linear, homogenous

3.

nonlinear

5.

linear, homogeneous

11.

$y = c_{1}e^{5x} + c_{2}e^{-2x}$

13.

$y = c_{1}e^{-2x} + c_{2}xe^{-2x}$

15.

$y = c_{1}e^{5x\text{/}2} + c_{2}e^{\text{−}x}$

17.

$y = e^{\text{−}x\text{/}2}\left( {c_{1}\text{cos}\mspace{2mu}\frac{\sqrt{3}x}{2} + c_{2}\text{sin}\mspace{2mu}\frac{\sqrt{3}x}{2}} \right)$

19.

$y = c_{1}e^{-11x} + c_{2}e^{11x}$

21.

$y = c_{1}\text{cos}\mspace{2mu} 9x + c_{2}\text{sin}\mspace{2mu} 9x$

23.

$y = c_{1} + c_{2}x$

25.

$y = c_{1}e^{{({{({1 + \sqrt{22}})}\text{/}3})}x} + c_{2}e^{{({{({1 - \sqrt{22}})}\text{/}3})}x}$

27.

$y = c_{1}e^{\text{−}x\text{/}6} + c_{2}xe^{\text{−}x\text{/}6}$

29.

$y = c_{1} + c_{2}e^{9x}$

31.

$y = -2e^{-2x} + 2e^{-3x}$

33.

$y = 3\mspace{2mu}\text{cos}\mspace{2mu}\left( {2x} \right) + 5\mspace{2mu}\text{sin}\mspace{2mu}\left( {2x} \right)$

35.

$y = \text{−}e^{6x} + 2e^{-5x}$

37.

$y = 2e^{\text{−}x\text{/}5} + \frac{7}{5}xe^{\text{−}x\text{/}5}$

39.

$y = \left( \frac{2}{e^{6} - e^{-7}} \right)e^{6x} - \left( \frac{2}{e^{6} - e^{-7}} \right)e^{-7x}$

41.

No solutions exist.

43.

$y = 2e^{2x} - \frac{2e^{2} + 1}{e^{2}}xe^{2x}$

45.

$y = 4\mspace{2mu}\text{cos}\mspace{2mu} 3x + c_{2}\text{sin}\mspace{2mu} 3x,\ \text{infinitely many solutions}$

47.

$5y^{''} + 19y^{\prime} - 4y = 0$

49.

a\. $y = 3\mspace{2mu}\text{cos}(8x)$

b.

51.

a\. $y = e^{(-5\text{/}2)x}\left\lbrack {-2\mspace{2mu}\text{cos}\mspace{2mu}\left( {\frac{\sqrt{35}}{2}x} \right) + \frac{4\sqrt{35}}{35}\mspace{2mu}\text{sin}\mspace{2mu}\left( {\frac{\sqrt{35}}{2}x} \right)} \right\rbrack$

b.

Section 7.2 Exercises

55.

$y = c_{1}e^{-4x\text{/}3} + c_{2}e^{x} - 2$

57.

$y = c_{1}\text{cos}\mspace{2mu} 4x + c_{2}\text{sin}\mspace{2mu} 4x + \frac{1}{20}e^{-2x}$

59.

$y = c_{1}e^{2x} + c_{2}xe^{2x} + 2x^{2} + 5x + 4$

61.

$y = c_{1}e^{\text{−}x} + c_{2}xe^{\text{−}x} + \frac{1}{2}\mspace{2mu}\text{sin}\mspace{2mu} x - \frac{1}{2}\mspace{2mu}\text{cos}\mspace{2mu} x$

63.

$y = c_{1}\text{cos}\mspace{2mu} x + c_{2}\text{sin}\mspace{2mu} x - \frac{1}{3}x\mspace{2mu}\text{cos}\mspace{2mu} 2x - \frac{5}{9}\mspace{2mu}\text{sin}\mspace{2mu} 2x$

65.

$y = c_{1}e^{-5x} + c_{2}xe^{-5x} + \frac{1}{6}x^{3}e^{-5x} + \frac{4}{25}$

67.

a\. $y_{p}(x) = Ax^{2} + Bx + C$

b. $y_{p}(x) = - \frac{1}{3}x^{2} + \frac{4}{3}x - \frac{35}{9}$

69.

a\. $y_{p}(x) = \left( {Ax^{2} + Bx + C} \right)e^{\text{−}x}$

b. $y_{p}(x) = \left( {\frac{1}{4}x^{2} - \frac{5}{8}x - \frac{33}{32}} \right)e^{\text{−}x}$

71.

a\. $y_{p}(x) = \left( {Ax^{2} + Bx + C} \right)e^{x}\text{cos}\mspace{2mu} x$ $+ \left( {Dx^{2} + Ex + F} \right)e^{x}\text{sin}\mspace{2mu} x$

b. $y_{p}(x) = \left( {- \frac{1}{10}x^{2} - \frac{11}{25}x - \frac{27}{250}} \right)e^{x}\text{cos}\mspace{2mu} x$ $+ \left( {- \frac{3}{10}x^{2} + \frac{2}{25}x + \frac{39}{250}} \right)e^{x}\text{sin}\mspace{2mu} x$

73.

$y = c_{1} + c_{2}e^{-2x} + \frac{1}{15}e^{3x}$

75.

$y = c_{1}e^{2x} + c_{2}e^{-4x} + xe^{2x}$

77.

$y = c_{1}e^{3x} + c_{2}e^{-3x} - \frac{8x}{9}$

79.

$y = c_{1}\text{cos}\mspace{2mu} 2x + c_{2}\text{sin}\mspace{2mu} 2x - \frac{3}{2}x\mspace{2mu}\text{cos}\mspace{2mu} 2x + \frac{3}{4}\mspace{2mu}\text{sin}\mspace{2mu} 2x\mspace{2mu}\text{ln}\mspace{2mu}(\text{sin}\mspace{2mu} 2x)$

81.

$y = - \frac{347}{343} + \frac{4}{343}e^{7x} + \frac{2}{7}x^{2}e^{7x} - \frac{4}{49}xe^{7x}$

83.

$y = - \frac{57}{25} + \frac{3}{25}e^{5x} + \frac{1}{5}xe^{5x} + \frac{4}{25}e^{-5x}$

85.

$y_{p} = \frac{1}{2} + \frac{10}{3}x^{2}\text{ln}\mspace{2mu} x$

Section 7.3 Exercises

87.

$x^{''} + 16x = 0,$ $x(t) = \frac{1}{6}\mspace{2mu}\text{cos}\mspace{2mu}\left( {4t} \right) - 2\mspace{2mu}\text{sin}\mspace{2mu}\left( {4t} \right)\text{,}$ period $= \frac{\pi}{2}\mspace{2mu}\text{sec},$ frequency $= \frac{2}{\pi}\mspace{2mu}\text{Hz}$

89.

$x^{''} + 196x = 0,$ $x(t) = 0.15\mspace{2mu}\text{cos}\mspace{2mu}\left( {14t} \right)\text{,}$ period $= \frac{\pi}{7}\mspace{2mu}\text{sec},$ frequency $= \frac{7}{\pi}\mspace{2mu}\text{Hz}$

91.

a\. $x(t) = 5\mspace{2mu}\text{sin}\mspace{2mu}\left( {2t} \right)$

b. period $= \pi\ \text{sec},$ frequency $= \frac{1}{\pi}\mspace{2mu}\text{Hz}$

c.

d. $t = \frac{\pi}{2}\mspace{2mu}\text{sec}$

93.

a\. $x(t) = e^{\text{−}t\text{/}5}\left( {20\mspace{2mu}\text{cos}\mspace{2mu}\left( {3t} \right) + 15\mspace{2mu}\text{sin}\mspace{2mu}\left( {3t} \right)} \right)$

b. underdamped

95.

a\. $x(t) = e^{-4t}\left( –5\cos 11.866t~–~0.834\sin 11.866t \right)$

b. underdamped

97.

$x(\pi) = \frac{7e^{\text{−}\pi\text{/}4}}{6}$ ft below

99.

$x(t) = \frac{1}{2}\mspace{2mu}\text{cos}\mspace{2mu}\left( 8\sqrt{2}t \right)–\frac{4\sqrt{2}}{21}\text{sin}\left( {8\sqrt{2}t} \right) + \frac{8}{21}\text{sin}\left( {8\sqrt{2}t} \right)$

101.

$q(t) = e^{-6t}\left( {0.051\mspace{2mu}\text{cos}\mspace{2mu}\left( {8t} \right) + 0.03825\mspace{2mu}\text{sin}\mspace{2mu}\left( {8t} \right)} \right) - \frac{1}{20}\mspace{2mu}\text{cos}\mspace{2mu}\left( {10t} \right)$

103.

$q(t) = e^{-10t}\left( {-32t - 5} \right) + 5,\ \ I(t) = 2e^{-10t}\left( {160t + 9} \right)$

Section 7.4 Exercises

105.

$y = a_{0} + 5a_{1}{\sum\limits_{n = 1}^{\infty}{\frac{\left( {\text{−}x\text{/}5} \right)^{n}}{n!} = c_{0} + 5c_{1}e^{\text{−}x\text{/}5}}}$

107.

$y = a_{0}{\sum\limits_{n = 0}^{\infty}\frac{(x)^{2n}}{\left( {2n} \right)!}} + a_{1}{\sum\limits_{n = 0}^{\infty}\frac{(x)^{2n + 1}}{\left( {2n + 1} \right)!}}$

109.

$y = a_{0}{\sum\limits_{n = 0}^{\infty}{\frac{x^{2n}}{n!} = c_{0}e^{x^{2}}}}$

111.

$y = a_{0}{\sum\limits_{n = 0}^{\infty}{\frac{x^{2n}}{2^{n}n!} + a_{1}{\sum\limits_{n = 0}^{\infty}\frac{x^{2n + 1}}{1 \cdot 3 \cdot 5 \cdot 7\cdots(2n + 1)}}}}$

113.

$y = c_{0}x^{3} + \frac{c_{1}}{x}$

115.

$y = 1 - 3x + \frac{2x^{3}}{3!} - \frac{12x^{4}}{4!} + \frac{16x^{6}}{6!} - \frac{120x^{7}}{7!} + \text{⋯}$

Review Exercises

117.

True

119.

False

121.

second order, linear, homogeneous, $\lambda^{2} - 2 = 0$

123.

first order, nonlinear, nonhomogeneous

125.

$y = c_{1}\text{cos}\mspace{2mu}\left( {3x} \right) + c_{2}\text{sin}\mspace{2mu}\left( {3x} \right)$

127.

$y = c_{1}e^{x}\text{cos}\mspace{2mu}\left( {3x} \right) + c_{2}e^{x}\text{sin}\mspace{2mu}\left( {3x} \right) + \frac{2}{5}x + \frac{2}{25}$

129.

$y = c_{1}e^{\text{−}x} + c_{2}e^{-4x} + \frac{x}{4} + \frac{e^{2x}}{18} - \frac{5}{16}$

131.

$y = c_{1}e^{(-3\text{/}2)x} + c_{2}xe^{(-3\text{/}2)x} + \frac{4}{9}x^{2} + \frac{4}{27}x - \frac{16}{27}$

133.

$y = e^{-2x}\text{sin}\mspace{2mu}\left( {\sqrt{2}x} \right)$

135.

$y = \frac{e^{1 - x}}{e^{4} - 1}\left( {e^{4x} - 1} \right)$

137.

$\theta(t) = \theta_{0}\text{cos}\mspace{2mu}\left( {\sqrt{\frac{g}{l}}L} \right)$

141.

$b = \sqrt{a}$

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• Authors: Gilbert Strang, Edwin “Jed” Herman
• Publisher/website: OpenStax
• Book title: Calculus Volume 3
• Publication date: Mar 30, 2016
• Location: Houston, Texas
• Book URL: https://openstax.org/books/calculus-volume-3/pages/1-introduction
• Section URL: https://openstax.org/books/calculus-volume-3/pages/chapter-7

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