Chapter 7

Checkpoint

7.1

Nonlinear
Linear, nonhomogeneous

7.4

Linearly independent

7.5

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

7.6

$y (x) = e^{x} (c_{1} cos 3 x + c_{2} sin 3 x)$
$y (x) = c_{1} e^{−7 x} + c_{2} x e^{−7 x}$

7.7

$y (x) = − e^{−2 x} + e^{5 x}$

7.8

$y (x) = e^{x} (2 cos 3 x - sin 3 x)$

7.9

$y (t) = t e^{−7 t}$

At time $t = 0.3,$ $y (0.3) = 0.3 e^{(−7 * 0.3)} = 0.3 e^{−2.1} \approx 0.0367 .$ The mass is 0.0367 ft below equilibrium. At time $t = 0.1,$ $y^{'} (0.1) = 0.3 e^{−0.7} \approx 0.1490 .$ The mass is moving downward at a speed of 0.1490 ft/sec.

7.10

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

7.11

$y (t) = c_{1} e^{2 t} + c_{2} t e^{2 t} + sin t + cos t$

7.12

$y (x) = c_{1} e^{4 x} + c_{2} e^{x} - x e^{x}$
$y (t) = c_{1} e^{−3 t} + c_{2} e^{2 t} - 5 cos 2 t + sin 2 t$

7.13

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

7.14

$y (x) = c_{1} cos x + c_{2} sin x + cos x ln | cos x | + x sin x$
$x (t) = c_{1} e^{t} + c_{2} t e^{t} + t e^{t} ln | t |$

7.15

$x (t) = 0.1 cos (14 t)$ (in meters); frequency is $\frac{14}{2 π}$ Hz.

7.16

$x (t) = \sqrt{17} sin (4 t + 0.245),$ $frequency = \frac{4}{2 π} \approx 0.637,$ $A = \sqrt{17}$

7.17

$x (t) = 0.6 e^{−2 t} - 0.2 e^{−6 t}$

7.18

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

7.19

$x (t) = −0.24 e^{−2 t} cos (4 t) - 0.12 e^{−2 t} sin (4 t)$

7.20

$x (t) = - \frac{1}{2} cos (4 t) + \frac{9}{4} sin (4 t) + \frac{1}{2} e^{−2 t} cos (4 t) - 2 e^{−2 t} sin (4 t)$
$Transient solution: \frac{1}{2} e^{−2 t} cos (4 t) - 2 e^{−2 t} sin (4 t)$
$Steady-state solution: - \frac{1}{2} cos (4 t) + \frac{9}{4} sin (4 t)$

7.21

$q (t) = −25 e^{− t} cos (3 t) - 7 e^{− t} sin (3 t) + 25$

7.22

$y (x) = a_{0} \sum_{n = 0}^{\infty} \frac{{(−1)}^{n}}{n!} x^{2 n} = a_{0} e^{− x^{2}}$
$y (x) = a_{0} {(x + 1)}^{3}$

Section 7.1 Exercises

linear, homogenous

nonlinear

linear, homogeneous

11.

$y = c_{1} e^{5 x} + c_{2} e^{−2 x}$

13.

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

15.

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

17.

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

19.

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

21.

$y = c_{1} cos 9 x + c_{2} sin 9 x$

23.

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

25.

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

27.

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

29.

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

31.

$y = −2 e^{−2 x} + 2 e^{−3 x}$

33.

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

35.

$y = − e^{6 x} + 2 e^{−5 x}$

37.

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

39.

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

41.

No solutions exist.

43.

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

45.

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

47.

$5 y^{″} + 19 y^{'} - 4 y = 0$

49.

a. $y = 3 cos (8 x)$
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. $y = e^{(−5 / 2) x} [−2 cos (\frac{\sqrt{35}}{2} x) + \frac{4 \sqrt{35}}{35} sin (\frac{\sqrt{35}}{2} x)]$
b.

This figure is a graph of an oscillating function. The amplitude of the graph is decreasing as x increases.

Section 7.2 Exercises

55.

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

57.

$y = c_{1} cos 4 x + c_{2} sin 4 x + \frac{1}{20} e^{−2 x}$

59.

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

61.

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

63.

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

65.

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

67.

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

69.

a. $y_{p} (x) = (A x^{2} + B x + C) e^{− x}$
b. $y_{p} (x) = (\frac{1}{4} x^{2} - \frac{5}{8} x - \frac{33}{32}) e^{− x}$

71.

a. $y_{p} (x) = (A x^{2} + B x + C) e^{x} cos x$ $+ (D x^{2} + E x + F) e^{x} sin x$
b. $y_{p} (x) = (- \frac{1}{10} x^{2} - \frac{11}{25} x - \frac{27}{250}) e^{x} cos x$ $+ (- \frac{3}{10} x^{2} + \frac{2}{25} x + \frac{39}{250}) e^{x} sin x$

73.

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

75.

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

77.

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

79.

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

81.

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

83.

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

85.

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

Section 7.3 Exercises

87.

$x^{″} + 16 x = 0,$ $x (t) = \frac{1}{6} cos (4 t) - 2 sin (4 t),$ period $= \frac{π}{2} sec,$ frequency $= \frac{2}{π} Hz$

89.

$x^{″} + 196 x = 0,$ $x (t) = 0.15 cos (14 t),$ period $= \frac{π}{7} sec,$ frequency $= \frac{7}{π} Hz$

91.

a. $x (t) = 5 sin (2 t)$
b. period $= π sec,$ frequency $= \frac{1}{π} Hz$
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. $t = \frac{π}{2} sec$

93.

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

95.

a. $x (t) = e^{−4 t} (- 5 \cos 11.866 t - 0.834 \sin 11.866 t)$
b. underdamped

97.

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

99.

$x (t) = \frac{1}{2} cos (8 \sqrt{2} t) - \frac{4 \sqrt{2}}{21} sin (8 \sqrt{2} t) + \frac{8}{21} sin (8 \sqrt{2} t)$

101.

$q (t) = e^{−6 t} (0.051 cos (8 t) + 0.03825 sin (8 t)) - \frac{1}{20} cos (10 t)$

103.

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

Section 7.4 Exercises

105.

$y = a_{0} + 5 a_{1} \sum_{n = 1}^{\infty} \frac{{(− x / 5)}^{n}}{n!} = c_{0} + 5 c_{1} e^{− x / 5}$

107.

$y = a_{0} \sum_{n = 0}^{\infty} \frac{{(x)}^{2 n}}{(2 n)!} + a_{1} \sum_{n = 0}^{\infty} \frac{{(x)}^{2 n + 1}}{(2 n + 1)!}$

109.

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

111.

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