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ANSWERS TO SELECTED ODD-NUMBERED PROBLEMS

Exercises 1.1, Page 11

  1. linear, second order
  2. linear, fourth order
  3. nonlinear, second order
  4. linear, third order
  5. nonlinear, first order
  6. linear in x but nonlinear in y
  1. Domain of function is [−2, ∞); largest interval of definition for solution is (−2, ∞).
  2. Domain of function is the set of real numbers except x = 2 and x = −2; largest intervals of definition for solution are (−∞, −2) (−2, 2) or (2, ∞).
  3. defined on (−∞, ln 2) or on (ln 2, ∞)
  1. m = −2
  2. m = 2, m = 3
  3. m = 0, m = −1
  4. m = 3, m = 5
  5. y = 2
  6. no constant solutions
  7. y = 0
  8. y = −4, y = 4

Exercises 1.2, Page 17

  1. y = 1/(1 − 4e−x)
  2. y = 1/(x2 − 1); (1, ∞)
  3. y = 1/(x2 + 1); (−∞, ∞)
  4. x = −cos t + 8 sin t
  7. y = 5e−x − 1
  8. y = 0, y = x3
  9. half-planes defined by either y > 0 or y < 0
  10. half-planes defined by either x > 0 or x < 0
  11. the regions defined by y > 2, y < −2, or −2 < y < 2
  12. any region not containing (0, 0)
  13. yes
  14. no
  15. (a) y = cx

    (b) any rectangular region not touching the y-axis

    (c) No, the function is not differentiable at x = 0.

  16. (b) y = 1/(1 − x) on (−∞, 1);

           y = −1/(x + 1) on (−1, ∞)

  1. y = −sin 3x
  2. y = 0
  3. no solution

Exercises 1.3, Page 25

Chapter 1 in Review, Page 30

  4. (a), (d)
  5. (b)
  6. (b)
  7. y = c1 and y = c2ex, c1 and c2 constants
  8. y′ = x2 + y2
  9. (a) The domain is the set of all real numbers.

    (b) either (−∞, 0) or (0, ∞)

  10. For x0 = −1, the interval is (−∞, 0), and for x0 = 2, the interval is (0, ∞).
  11. (c)
  12. (−∞, ∞)
  13. (0, ∞)
  2. y′ + y = 4x − 2
  1. y0 = −3, y1 = 0

Exercises 2.1, Page 41

  1. 0 is asymptotically stable (attractor); 3 is unstable (repeller).
  2. 2 is semi-stable.
  3. −2 is unstable (repeller); 0 is semi-stable; 2 is asymptotically stable (attractor).
  4. −1 is asymptotically stable (attractor); 0 is unstable (repeller).

Exercises 2.2, Page 49

  1. y = − cos 5x + c
  2. y = e−3x + c
  3. y = cx4
  4. −3e−2y = 2e3x + c
  5. x3 ln x − x3 = y2 + 2y + ln | y| + c
  6. 4 cos y = 2x + sin 2x + c
  7. (ex + 1)−2 + 2(ey + 1)−1 = c
  8. S = cekr
  10. (y + 3)5 ex = c(x + 4)5 ey
  11. y = sin(x2 + c)
  12. x = tan(4t − π)
  20. (a) y = 2, y = −2, y = 2
  1. y = 1
  2. y = 1 + tan ( x)
  3. y(x) = (4h/L2)x2 + a

Exercises 2.3, Page 59

  1. y = ce5x, (−∞, ∞)
  2. y = e3x + ce−x, (−∞, ∞)
  3. y = + , (−∞, ∞)
  4. y = x−1 ln x + cx−1, (0, ∞)
  5. y = cx − x cos x, (0, ∞)
  6. y = x3 − x + cx−4 , (0, ∞)
  7. y = x−2 ex + cx−2 e−x, (0, ∞)
  8. x = 2y6 + cy4 , (0, ∞)
  9. y = sin x + c cos x, (−π/2, π/2)
  10. (x + 1)ex y = x2 + c, (−1, ∞)
  11. (sec θ + tan θ)r = θ − cos θ + c, (−π/2, π/2)
  12. y = e−3x + cx−1 e−3x, (0, ∞)
  13. y = x−1 ex + (2 − e)x−1 , (0, ∞)
  14. , (−∞, ∞)
  15. (x + 1)y = x ln x − x + 21, (0, ∞)
  16. , (0, ∞)
  20. x = −y2 − 2y − 2 + cey

Exercises 2.4, Page 66

  1. x2 − x + y2 + 7 y = c
  2. x2 + 4xy − 2y4 = c
  3. x2 y2 − 3x + 4y = c
  4. not exact
  5. xy3 + y2 cos x − x2 = c
  6. not exact
  7. xy − 2xex + 2ex − 2x3 = c
  8. x3 y3 − tan−1 3x = c
  9. −ln | cos x | + cos x sin y = c
  10. t4y − 5t3 − ty + y3 = c
  11. x3 + x2y + xy2 − y =
  12. 4ty + t2 − 5t + 3y2 − y = 8
  13. y2 sin x − x3y − x2 + y ln y − y = 0
  14. k = 10
  15. x2y2 cos x = c
  16. x2 − = c
  17. x2y2 + x3 = c
  18. 3x2y3 + y4 = c
  19. −2ye3x + e3x + x = c
  21. (c)
  1. (a)

    (b)

Exercises 2.5, Page 71

  1. y + x ln |x| = cx
  2. (x − y) ln |x − y| = y + c(x − y)
  3. x + y ln |x| = cy
  4. ln(x2 + y2) + 2 tan−1 (y/x) = c
  5. 4x = y(ln | y| − c)2
  6. y3 + 3x3 ln |x| = 8x3
  7. ln |x| = ey/x − 1
  8. y3 = 1 + cx−3
  9. y−3 = x + + ce3x
  10. et/y = ct
  11. y−3 = −x−1 + x−6
  12. y = −x − 1 + tan(x + c)
  13. 2y − 2x + sin 2(x + y) = c
  14. 4(y − 2x + 3) = (x + c)2
  15. −cot (x + y) + csc (x + y) = x + − 1
  1. (b) y = + (−x + cx−3)−1

Exercises 2.6, Page 76

  1. y2 = 2.9800, y4 = 3.1151
  2. y10 = 2.5937, y20 = 2.6533; y = ex
  3. y5 = 0.4198, y10 = 0.4124
  4. y5 = 0.5639, y10 = 0.5565
  5. y5 = 1.2194, y10 = 1.2696
  1. Euler: y10 = 3.8191, y20 = 5.9363

    RK4: y10 = 42.9931, y20 = 84.0132

Exercises 2.7, Page 83

  1. 7.9 years; 10 years
  2. 760; approximately 11 persons/yr
  3. 11 h
  4. 136.5 h
  1. I(15) = 0.00098I0 or approximately 0.1% of I0
  2. 15,963 years
  3. T(1) = 36.76°F; approximately 3.06 min
  4. approximately 82.1 s; approximately 145.7 s
  5. 390°F
  6. approximately 1.6 h
  7. A(t) = 200 − 170e−t/50
  8. A(t) = 1000 − 1000e−t/100
  10. 64.38 lb
  11. i(t) = − e−500t; i → as t → ∞
  12. q(t) = − e−50t; i(t) = e−50t
  14. (a)

    (b) v(t) → as t → ∞

    (c)

  1. (a)

    (b) 33 min

  2. (a) P(t) = P0
  3. (a) As t → ∞, x(t) → r/k.

    (b) x(t) = r/k − (r/k)e−kt; (ln 2)/k

  4. (a) tb = 50 s

    (b) 70 m/s

    (c) 1250 m

    (e)

  5. (a) v(0) = 5 m3 , v(t) = 0.8 + 4.2e−0.2t, approximately 0.117%

    (b) in approximately 11.757 min or at approximately 9:12 A.M.

    (c) approximately 829.114 m3/min

Exercises 2.8, Page 92

  1. (a) N = 2000

    (b) N(t) = ; N(10) = 1834

  2. 1,000,000; 52.9 mo
  1. For 0 < P0 < 1, time of extinction is

    t = − ln .

  2. time of extinction is

  3. where c = (a/b) − ln P0
  4. 29.3 g; X → 60 as t → ∞; 0 g of A and 30 g of B
  5. (a) h(t) = ;

    (b) 576 s or 30.36 min

  6. (a) approximately 858.65 s or 14.31 min

    (b) 243 s or 4.05 min

  7. (a)

    where

    (b)

    (c)

    where

  8. (a) where ρ is the weight density of water

    (b)

    (c)

  9. (a) W = 0 and W = 2

    (b) W(x) = 2 sech2 (x − c1)

    (c) W(x) = 2 sech2 x

Exercises 2.9, Page 101

  2. 5, 20, 147 days. The time when y(t) and z(t) are the same makes sense because most of A and half of B are gone, so half of C should have been formed.
  3. (a)

    (b) approximately 1.25 × 109 years

    (c)

    (d) 10.5% of P0, 89.5% of P0

  4. = 6 − x1 + x2

    = x1 − x2

  5. (a)

    (b) x1(t) + x2(t) = 150; x2(30) ≈ 47.4 lb

  1. L1 + (R1 + R2)i2 + R1i3 = E(t)

    L2 + R1i2 + (R1 + R3)i3 = E(t)

  2. i(0) = i0, s(0) = n − i0, r(0) = 0; because the population is assumed to be constant

Chapter 2 in Review, Page 104

  1. −A/k, a repeller for k > 0, an attractor for k < 0
  3. true
  5. false
  6. = (y − 1)2 (y − 3)2
  7. semi-stable for n even and unstable for n odd; semistable for n even and asymptotically stable for n odd
  1. 2x + sin 2x = 2 ln(y2 + 1) + c
  2. (6x + 1)y3 = −3x3 + c
  3. Q = ct−1 + t4 (−1 + 5 ln t)
  4. y = + c(x2 + 4)−4
  8. y = csc x, (π, 2π)
  9. (b)
  1. P(45) = 8.99 billion
  2. (b) approximately 3257 BC
  4. (a)

    (b)

  7. no
  11. x = −y + 1 + c2e−y
  13. (a) k = 0.083 seems to work well;

    k = 0.1063 and k = 0.0823

Exercises 3.1, Page 120

  1. y = ex − e−x
  2. y = 3x − 4x ln x
  1. (−∞, 2)
  2. (a)

    (b)

  3. (a) y = ex cos x − ex sin x

    (b) no solution

    (c) y = ex cos x + e−π/2ex sin x

    (d) y = c2ex sin x, where c2 is arbitrary

  4. dependent
  5. dependent
  6. dependent
  7. independent
  8. The functions satisfy the DE and are linearly independent on the interval since W(e−3x, e4x) = 7ex ≠ 0; y = c1e−3x + c2e4x.
  9. The functions satisfy the DE and are linearly independent on the interval since W(ex cos 2x, ex sin 2x) = 2e2x ≠ 0; y = c1ex cos 2x + c2ex sin 2x.
  10. The functions satisfy the DE and are linearly independent on the interval since W(x3 , x4) = x6 ≠ 0; y = c1x3 + c2x4 .
  11. The functions satisfy the DE and are linearly independent on the interval since W(x, x−2 , x−2 ln x) = 9x−6 ≠ 0; y = c1x + c2x−2 + c3x−2 ln x.
  1. (b) yp = x2 + 3x + 3e2x; yp = −2x2 − 6x − e2x

Exercises 3.2, Page 124

  1. y2 = xe2x
  2. y2 = sin 4x
  3. y2 = sinh x
  4. y2 = xe2x/3
  5. y2 = x4 ln | x |
  6. y2 = 1
  7. y2 = x cos(ln x)
  8. y2 = x2 + x + 2
  9. y2 = x2 ex
  10. y2 = e2x, yp = −
  11. y2 = e2x, yp = e3x

Exercises 3.3, Page 130

  1. y = c1 + c2e−x/4
  2. y = c1e3x + c2e−2x
  3. y = c1e−4x + c2xe−4x
  4. y = c1e2x/3 + c2e−x/4
  5. y = c1 cos 3x + c2 sin 3x
  6. y = e2x(c1 cos x + c2 sin x)
  8. y = c1 + c2e−x + c3e5x
  9. y = c1e−x + c2e3x + c3xe3x
  10. u = c1et + e−t(c2 cos t + c3 sin t)
  11. y = c1e−x + c2xe−x + c3x2 e−x
  14. u = c1er + c2rer + c3e−r + c4re−r + c5e−5r
  15. y = c1 + c2x + c3x2 + c4x3 + c5ex cos 2x + c6ex sin 2x
  16. y = 2 cos 4x − sin 4x
  17. y = −e−(t − 1) + e5(t − 1)
  18. y = 0
  19. y = − e−6x + xe−6x
  20. y = e5x − xe5x
  21. y = 0

Exercises 3.4, Page 139

  1. y = c1e−x + c2e−2x + 3
  2. y = c1e5x + c2xe5x + x +
  3. y = c1e−2x + c2xe−2x + x2 − 4x +
  5. y = c1 + c2ex + 3x
  6. y = c1ex/2 + c2xex/2 + 12 + x2 ex/2
  7. y = c1 cos 2x + c2 sin 2x − x cos 2x
  8. y = c1 cos x + c2 sin x − x2 cos x + x sin x
  9. y = c1ex cos 2x + c2ex sin 2x + xex sin 2x
  10. y = c1e−x + c2xe−x − cos x + sin 2x − cos 2x
  11. y = c1 + c2x + c3e6x − x2 − cos x + sin x
  12. y = c1ex + c2xex + c3x2 ex − x − 3 − x3 ex
  13. y = c1 cos x + c2 sin x + c3x cos x + c4x sin x + x2 − 2x − 3
  15. y = −200 + 200e−x/5 − 3x2 + 30x
  16. y = −10e−2x cos x + 9e−2x sin x + 7e−4x
  18. y = 11 − 11ex + 9xex + 2x − 12x2 ex + e5x
  19. y = 6 cos x − 6(cot 1) sin x + x2 − 1

Exercises 3.5, Page 144

  1. y = c1 cos x + c2 sin x + x sin x + cos x ln | cos x |
  2. y = c1 cos x + c2 sin x −x cos x
  3. y = c1 cos x + c2 sin x + − cos 2x
  4. y = c1ex + c2e−x + x sinh x
  6. y = c3 + c2e−x + 3x
  7. y = c1e−x + c2e−2x + (e−x + e−2x) ln (1 + ex)
  8. y = c1e−2x + c2e−x − e−2x sin ex
  9. y = c1e−t + c2te−t + t2 e−t ln t − t2 e−t
  10. y = c1ex sin x + c2ex cos x + xex sin x

    + ex cos x ln | cos x |

  11. y = e−x/2 + ex/2 + x2 ex/2 − xex/2
  13. y = e−4x + e2x − e−2x + e−x
  16. y = c1x−1/2 cos x + c2x−1/2 sin x + x−1/2
  17. y = c1 + c2 cos x + c3 sin x

    − ln | cos x | −sin x ln | sec x + tan x |

Exercises 3.6, Page 150

  1. y = c1x−1 + c2x2
  2. y = c1 + c2 ln x
  3. y = c1 cos(2 ln x) + c2 sin(2 ln x)
  5. y = c1 cos( ln x) + c2 sin( ln x)
  6. y = c1x−2 + c2x−2 ln x
  9. y = c1 + c2x + c3x2 + c4x−3
  10. y = c1 + c2x5 + x5 ln x
  11. y = c1x + c2x ln x + x(ln x)2
  12. y = c1x−1 + c2x − ln x
  13. y = 2 − 2x−2
  14. y = cos(ln x) + 2 sin(ln x)
  15. y = − ln x + x2
  16. any real constant
  22. y = c1x−10 + c2x2
  23. y = c1x−1 + c2x−8 + x2
  24. y = x2 [c1 cos(3 ln x) + c2 sin(3 ln x)] + + x
  25. y = 2(−x)1/2 − 5(−x)1/2 ln(−x), x < 0
  27. (a)

    (b)

Exercises 3.7, Page 155

  1. y = ln | cos(c1 − x) | + c2
  4. y3 − c1 y = x + c2
  6. (b)

    (c)

  8. y = 1 + x + x2 + x3 + x4 + x5 +
  9. y = 1 + x −x2 + x3 −x4 + x5 +

Exercises 3.8, Page 169

  2. x(t) = − cos 4t
  3. (a)

    (b) 4 ft/s; downward

    (c) t = , n = 0, 1, 2, …

  4. (a) the 20-kg mass

    (b) the 20-kg mass; the 50-kg mass

    (c) t = nπ, n = 0, 1, 2, …; at the equilibrium position; the 50-kg mass is moving upward whereas the 20-kg mass is moving upward when n is even and downward when n is odd.

  5. (a)

    (b)

    (c)

  6. (a)

    (b) ft;

    (c) 15 cycles

    (d) 0.721 s

    (e) + 0.0927, n = 0, 1, 2, …

    (f) x(3) = −0.597 ft

    (g) x′(3) = −5.814 ft/s

    (h) x″(3) = 59.702 ft/s2

    (i) ±8 ft/s

    (j) 0.1451 + ; 0.3545 + , n = 0, 1, 2, …

    (k) 0.3545 + , n = 0, 1, 2, …

  9. Compared to a single-spring system with spring constant k, the parallel-spring system is more stiff.
  1. (a) above

    (b) heading upward

  2. (a) below

    (b) heading upward

  3. s; s, x() = e−2 ; that is, the weight is approximately 0.14 ft below the equilibrium position.
  4. (a) x(t) = e−2t − e−8t

    (b) x(t) = −e−2t + e−8t

  5. (a) x(t) = e−2t(−cos 4t − sin 4t)

    (b) x(t) = e−2t sin(4t + 4.249)

    (c) t = 1.294 s

  6. (a) β >

    (b) β =

    (c) 0 < α <

  8. x(t) = e−4t + te−4t − cos 4t
  9. x(t) = − cos 4t + sin 4t + e−2t cos 4t − 2e−2t sin 4t
  10. (a) m = −k(x − h) 2 or

    where 2λ = β/m and ω2 = k/m

    (b) x(t) = e−2t(− cos 2t − sin 2t) + cos t + sin t

  1. x(t) = −cos 2t − sin 2t + t sin 2t + t cos 2t
  2. (b)
  1. 4.568 C; 0.0509 s
  2. q(t) = 10 − 10e−3t(cos 3t + sin 3t)

    i(t) = 60e−3t sin 3t; 10.432 C

  3. qp = sin t + cos t, ip = cos t − sin t
  1. q(t) = −e−10t(cos 10t + sin 10t) + ; C

Exercises 3.9, Page 178

  1. (a) (6L2 x2 − 4Lx3 + x4)
  2. (a) (3L2 x2 − 5Lx3 + 2x4)
  3. (a)

    (c)

  5. (a)

    (b)

    (c)

  6. λn = n2 , n = 1, 2, 3, …; yn = sin nx
  7. λn = , n = 1, 2, 3, …;

    y = cos

  8. λn = n2 , n = 0, 1, 2, …; yn = cos nx
  9. λn = , n = 1, 2, 3, …; yn = e−x sin
  10. λn = n2 , n = 1, 2, 3, …; yn = sin(n ln x)
  12. x = L/4, x = L/2, x = 3L/4
  1. ωn = , n = 1, 2, 3, …; yn = sin
  3. (a)

    (b)

Exercises 3.10, Page 192

  12. y = −x sin x − cos x ln
  13. y = (cos 1 − 2)e−x + (1 + sin 1 − cos 1)e−2x

    − e−2x sin ex

  14. y = 4x − 2x2 − x ln x
  16. ,

  17. ,

  21. yp(x) = −ex cos x − ex sin x + ex

Exercises 3.11, Page 199

  1. + x = 0
  1. (a) x(t) = 5

    (c)

  2. (a)
  3. (a) xy″ = r When t = 0, x = a, y = 0, dy/dx = 0.

    (b) When r ≠ 1,

    When r = 1,

    (c) The paths intersect when r < 1.

Exercises 3.12, Page 207

  1. x = c1et + c2tet

    y = (c1 − c2)et + c2tet

  2. x = c1 cos t + c2 sin t + t + 1

    y = c1 sin t − c2 cos t + t − 1

  3. x = c1 sin t + c2 cos t − 2c3 sin t − 2c4 cos t

    y = c1 sin t + c2 cos t + c3 sin t + c4 cos t

  4. x = c1e2t + c2e−2t + c3 sin 2t + c4 cos 2t + et

    y = c1e2t + c2e−2t − c3 sin 2t − c4 cos 2t − et

  5. x = c1 − c2 cos t + c3 sin t + e3t

    y = c1 + c2 sin t + c3 cos t − e3t

  6. x = c1et + c2e−t/2 cos t + c3e−t/2 sin t

    +

  7. x = c1e4t + et

    y = −c1e4t + c2 + 5et

  8. x = c1 + c2t + c3et + c4e−t − t2

    y = (c1 − c2 + 2) + (c2 + 1)t + c4e−t − t2

  9. x = c1et + c2e−t/2 sin t + c3e−t/2 cos t

    y = c1et + (−c2 − c3) e−t/2 sin t

    + (c2 − c3) e−t/2 cos t

    z = c1et + (−c2 + c3) e−t/2 sin t

    + (− c2 − c3) e−t/2 cos t

  10. x = −6c1e−t − 3c2e−2t + 2c3e3t

    y = c1e−t + c2e−2t + c3e3t

    z = 5c1e−t + c2e−2t + c3e3t

  11. x = e−3t+3 − te−3t+3

    y = −e−3t+3 + 2te−3t+3

  12. mx″ = 0

    my″ = −mg;

    x = c1t + c2

    y = −gt2 + c3t + c4

Chapter 3 in Review, Page 208

  1. y = 0
  2. false
  3. 8 ft
  5. (−∞, ∞); (0, ∞)
  6. y = c1e3x + c2e−5x + c3xe−5x + c4ex + c5 xex + c6 x2ex,

    y = c1x3 + c2x−5 + c3x−5 ln x + c4x + c5 x ln x

    + c6 x(ln x)2

  9. y = c1 + c2e−5x + c3 xe−5x
  11. y = e3x/2 (c2 cos x + c3 sin x) + x3

    + x2 + x −

  12. y = c1 + c2e2x + c3e3x + sin x − cos x + x
  13. y = ex (c1 cos x + c2 sin x) − ex cos x ln | sec x + tan x |
  14. y = c1x−1/3 + c2x1/2
  15. y = c1x2 + c2x3 + x4 − x2 ln x
  16. (a) y = c1 cos ωx + c2 sin ωx + A cos αx + B sin αx, ω ≠ α, y = c1 cos ωx + c2 sin ωx + Ax cos ωx + Bx sin ωx, ω = α

    (b) y = c1e−ωx + c2eωx + Aeαx, ω ≠ α,

         y = c1e−ωx + c2eωx + Axeωx, ω = α

  17. (a) y = c1 cosh x + c2 sinh x + c3 x cosh x + c4 x sinh x

    (b) yp = Ax2 cosh x + Bx2 sinh x

  18. y = ex−π cos x
  19. y = ex − e−x − x − sin x
  20. y = x2 + 4
  1. x = −c1et − c2e2t +

    y = c1et + c2e2t − 3

  2. x = c1et + c2e5t + tet

    y = −c1et + 3c2e5t − tet + 2et

  3. 14.4 lb
  4. 0 < m ≤ 2
  5. (a) q(t) = − sin 100t + sin 50t

    (b) i(t) = − cos 100t + cos 50t

    (c) t = , n = 0, 1, 2, …

  1. m + kx = 0
  3. (a)

    (b)

    (c) the amplitude and period of the shorter pendulum are half that of the longer pendulum

  1. (a)

         

Exercises 4.1, Page 223

  17. Use sinh kt = to show that

    {sinh kt} =

Exercises 4.2, Page 231

  1. t2
  2. t − 2t4
  3. 1 + 3t + t2 + t3
  4. t − 1 + e2t
  5. e−t/4
  6. sin 7t
  7. cos
  8. 2 cos 3t − 2 sin 3t
  9. − e−3t
  10. e−3t + et
  11. 0.3e0.1t + 0.6e−0.2t
  12. e2t − e3t + e6t
  13. − cos t
  14. −4 + 3e−t + cos t + 3 sin t
  15. sin t − sin 2t
  17. y = −1 + et
  18. y = e4t + e−6t
  19. y = e−t − e−4t
  20. y = 10 cos t + 2 sin t − sin t
  21. y = −e−t/2 + e−2t + et + e−t
  25. y = e−t − e−3t cos 2t + e−3t sin 2t

Exercises 4.3, Page 240

  7. e3t sin t
  8. e−2t cos t − 2e−2t sin t
  9. e−t − te−t
  10. 5 − t − 5e−t − 4te−t − t2 e−t
  13. y = te−4t + 2 e−4t
  14. y = e−t + 2te−t
  15. y = t + − e3t + te3t
  16. y = −e3t sin 2t
  17. y = − et cos t + et sin t
  18. y = (e + 1) te−t + (e − 1) e−t
  5. −sin t (t − π)
  6. (t − 1) − e−(t−1) (t − 1)
  7. (c)
  8. (f)
  9. (a)
  10. f(t) = 2 − 4 (t − 3); {f(t)} = e−3s
  11. f(t) = t2 (t − 1); {f(t)} =
  12. f(t) = t − t (t − 2); {f(t)} =
  14. f(t) = (t − a) − (t − b); {f(t)} =
  16. y = [5 − 5e−(t−1)] (t − 1)
  17. y = − + t + e−2t − (t − 1)

         − (t − 1) (t − 1) + e−2(t−1) (t − 1)

  18. y = cos 2t − sin 2(t − 2π) (t − 2π)

         + sin(t − 2π) (t − 2π)

  19. y = sin t + [1 − cos(t − π)] (t − π)

         −[1 − cos(t − 2π)] (t − 2π)

  20. x(t) = t − sin 4t − (t − 5) (t − 5)

       + sin 4(t − 5) (t − 5) − (t − 5)

       + cos 4(t − 5) (t − 5)

  21. q(t) = (t − 3) − e−5(t−3) (t − 3)
  22. (a) i(t) = e−10t − cos t + sin t

       − e−10(t−3π/2)

       + cos

       + sin

    (b) imax ≈ 0.1 at t ≈ 1.6, imin ≈ −0.1 at t ≈ 4.7

  23.       

  24.       

  25. (a) = k(T − 70 − 57.5t − (230 − 57.5t) (t − 4))

Exercises 4.4, Page 253

  5. y = −e−t + cos t − t cos t + t sin t
  6. y = 2 cos 3t + sin 3t + t sin 3t
  7. y = sin 4t + t sin 4t

       − (t − π) sin 4(t − π) (t − π)

  1. y = t3 + ct2
  12. et − 1
  13. et − t2 − t − 1
  1. f(t) = sin t
  2. f(t) = −e−t + et + tet + t2 et
  3. f(t) = e−t
  4. f(t) = e2t + e−2t + cos 2t + sin 2t
  5. y(t) = sin t − t sin t
  8. i(t) = 100[e−10(t−1) − e−20(t−1)] (t − 1)

         − 100[e−10(t−2) − e−20(t−2)] (t − 2)

  13. i(t) = (1 − e−Rt/L)

        + (1 − e−R(t−n)/L) (t − n)

  14. x(t) = 2(1 − e−t cos 3t − e−t sin 3t)

         + 4 [1 − e−(t−nπ) cos 3(t − nπ)

         − e−(t−nπ) sin 3(t − nπ)] (t − nπ)

Exercises 4.5, Page 259

  1. y = e3(t−2) (t − 2)
  2. y = sin t + sin t (t − 2π)
  3. y = −cos t + cos t
  4. y = − e−2t + [ − e−2(t−1)] (t − 1)
  5. y = e−2(t−2π) sin t (t − 2π)
  6. y = e−2t cos 3t + e−2t sin 3t

    + e−2(t−π) sin 3(t − π) (t − π)

    + e−2(t−3π) sin 3(t − 3π) (t − 3π)

Exercises 4.6, Page 263

  1. x = −e−2t + et

    y = e−2t + et

  2. x = −cos 3t − sin 3t

    y = 2 cos 3t − sin 3t

  3. x = −2e3t + e2t −

    y = e3t − e2t −

  4. x = −t − sin t

    y = −t + sin t

  5. x = 8 + t3 + t4

    y = − t3 + t4

  6. x = t2 + t + 1 − e−t

    y = − + e−t + te−t

  8. (b) i2 = − e−900 t

    i3 = − e−900t

    (c) i1 = 20 − 20e−900t

  9. i2 = −e−2t + e−15t + cos t + sin t

    i3 = e−2t + e−15t − cos t + sin t

  10. i1 = − e−100t cosh 50t − e−100t sinh 50t

    i2 = − e−100t cosh 50t − e−100t sinh 50t

  11. (a)

    (b) is a quadratic function, for a fixed value of θ its graph is a parabola.

    (c) Solve y(x) = 0 to find the range R. To prove the complementary-angle property, show that R(θ) = R(π/2 − θ).

    (d) Solve y′(x) = 0 and find the corresponding value of y(x).

    (e) For θ = 38°: range is 2728.96 ft, max. height is 533.02 ft

    For θ = 52°: range is 2728.96 ft, max. height is 873.23 ft

    (f) For θ = 38°: time to hit the ground is t ≈ 11.5437 s, max. height occurs at t ≈ 5.7718 s

    For θ = 52°: time to hit the ground is t ≈ 14.7752 s, max. height occurs at t ≈ 7.3876 s

    (g)

    A graph. 2 parabolas are graphed on an x y coordinate plane. The x axis ranges from 0 to 2500 in increments of 500. The y axis ranges from 0 to 800 in increments of 200. The first curve, labeled theta = 38 degrees, begins at the origin, goes up and to the right with decreasing steepness, reaches a high point at the approximate point (1365, 533), goes down to the right symmetrically, and ends on the x axis at the approximate point (2729, 0). The second curve, labeled theta = 52 degrees, begins at the origin, goes up and to the right with decreasing steepness above the first curve, reaches a high point at the approximate point (1365, 873), goes down to the right symmetrically, and ends on the x axis at the approximate point (2729, 0).
  12. (a)

Chapter 4 in Review, Page 266

  2. false
  3. true
  10. t5
  11. t2 e5t
  13. e5t cos 2t + e5t sin 2t
  15. cos π(t − 1) (t − 1) + sin π(t − 1) (t − 1)
  16. −5
  17. e−k(s−a) F(s − a)
  18. t
  20. f(t) (t − t0)
  21. f(t − t0) (t − t0)
  22. f(t) = t − (t − 1) (t − 1) − (t − 4);

    {f(t)} = e−s − e−4s;

    {e tf (t)} = e−(s−1)

    − e−4(s−1)

  23. f(t) = 2 + (t − 2) (t − 2);

    {f(t)} = e−2s;

    {e tf (t)} = e−2(s−1)

  25. y = 5te t + t2 et
  26. y = − + t + e−t − e−5t − (t − 2)

    − (t − 2) (t − 2) + e−(t−2) (t − 2)

    − e−5(t−2) (t − 2)

  28. y = 1 + t + t2
  30. x = − + e−2t + e2t

    y = t + e−2t − e2t

  31. i(t) = −9 + 2t + 9e−t/5

Exercises 5.1, Page 280

  1. R = , [−, )
  2. R = 10, (−5, 15)
  3. x − x3 + x5 − x7 +
  4. 1 + x2 + x4 + x6 + , (−π/2, π/2)
  6. 2c1 + (k + 1)ck11 + 6ck21]xk
  1. 5; 4

Exercises 5.2, Page 288

  1. x = 0, irregular singular point
  2. x = −3, regular singular point; x = 3, irregular singular point
  3. x = 0, 2i, −2i, regular singular points
  4. x = −3, 2, regular singular points
  5. x = 0, irregular singular point; x = −5, 5, 2, regular singular points
  6. for x = 1: p(x) = 5, q(x) =

    for x = −1: p(x) = , q(x) = x2 + x

  7. r1 = , r2 = −1
  8. r1 = , r2 = 0

  9. r1 = , r2 = 0

  10. r1 = , r2 = 0

  11. r1 = , r2 = 0

  12. r1 = , r2 =

  13. r1 = 0, r2 = −1

    = [C1 sinh x + C2 cosh x]

  14. r1 = 1, r2 = 0

    y(x) = C1x + C2 [x ln x − 1 + x2

    + x3 + x4 + ]

  15. r1 = r2 = 0

    where y1(x) = xn = ex

  1. (b)

    (c)

Exercises 5.3, Page 300

  1. y = c1 J1/3(x) + c2 J−1/3(x)
  2. y = c1 J5/2(x) + c2 J−5/2(x)
  3. y = c1 J0(x) + c2Y0(x)

  1. (b)

  1. (a)

Chapter 5 in Review, Page 304

  1. false
  2. r1 = , r2 = 0

    y1(x) = C1x1/2 [1 − x + x2 − x3 + ]

    y2(x) = C2 [1 − x + x2 − x3 + ]

  3. y1(x) = c0 [1 + x2 + x3 + x4 + ]

    y2(x) = c1 [x + x3 + x4 + ]

  4. r1 = 3, r2 = 0

  5. y(x) = 3[1 − x2 + x4 − x6 + ] − 2[x − x3

    + x5 − x7 + ]

  6. π
  7. x = 0 is an ordinary point

  9. (a) y = c1 J3/2(4x) + c2Y3/2(4x)

    (b) y = c1 I3(6x) + c2K3(6x)

  1. y(x) = c0y1(x) + c1y2(x), where

Exercises 6.1, Page 310

  1. for h = 0.1, y5 = 2.0801; for h = 0.05, y10 = 2.0592
  2. for h = 0.1, y5 = 0.5470; for h = 0.05, y10 = 0.5465
  3. for h = 0.1, y5 = 0.4053; for h = 0.05, y10 = 0.4054
  4. for h = 0.1, y5 = 0.5503; for h = 0.05, y10 = 0.5495
  5. for h = 0.1, y5 = 1.3260; for h = 0.05, y10 = 1.3315
  6. for h = 0.1, y5 = 3.8254; for h = 0.05, y10 = 3.8840;

    at x = 0.5 the actual value is y(0.5) = 3.9082

  7. (a) y1 = 1.2

    (b)

    (c) Actual value is y(0.1) = 1.2214. Error is 0.0214.

    (d) If h = 0.05, y2 = 1.21.

    (e) Error with h = 0.1 is 0.0214. Error with h = 0.05 is 0.0114.

  8. (a) y1 = 0.8

    (b) y″(c) = 5e–2c = 0.025e–2c ≤ 0.025 for 0 ≤ c ≤ 0.1

    (c) Actual value is y(0.1) = 0.8234. Error is 0.0234.

    (d) If h = 0.05, y2 = 0.8125.

    (e) Error with h = 0.1 is 0.0234. Error with h = 0.05 is 0.0109.

  9. (a) Error is 19h2 e–3(c–1).

    (b) y″(c) ≤ 19(0.1)2 (1) = 0.19

    (c) If h = 0.1, y5 = 1.8207. If h = 0.05, y10 = 1.9424.

    (d) Error with h = 0.1 is 0.2325. Error with h = 0.05 is 0.1109.

  10. (a) Error is .

    (b)

    (c) If h = 0.1, y5 = 0.4198. If h = 0.05, y10 = 0.4124.

    (d) Error with h = 0.1 is 0.0143. Error with h = 0.05 is 0.0069.

Exercises 6.2, Page 315

  1. y5 = 3.9078; actual value is y(0.5) = 3.9082
  2. y5 = 2.0533
  3. y5 = 0.5463
  4. y5 = 0.4055
  5. y5 = 0.5493
  6. y5 = 1.3333
  7. (a) 35.7678

    (c) v(t) = v(5) = 35.7678

  8. (a) h = 0.1, y4 = 903.0282;

    h = 0.05, y8 = 1.1 × 1015

  9. (a) y1 = 0.82341667

    (b) y(5)(c) = 40e–2c ≤ 40e2(0)

    = 3.333 × 10–6

    (c) Actual value is y(0.1) = 0.8234134413. Error is 3.225 × 10–6 ≤ 3.333 × 10–6.

    (d) If h = 0.05, y2 = 0.82341363.

    (e) Error with h = 0.1 is 3.225 × 10–6. Error with h = 0.05 is 1.854 × 10–7.

  10. (a) y(5)(c)

    (b) = 2.0000 × 10–6

    (c) From calculation with h = 0.1, y5 = 0.40546517.

    From calculation with h = 0.05, y10 = 0.40546511.

Exercises 6.3, Page 318

  1. y(x) = −x + ex; actual values are y(0.2) = 1.0214, y(0.4) = 1.0918, y(0.6) = 1.2221, y(0.8) = 1.4255; approximations are given in Example 1
  2. y4 = 0.7232
  3. for h = 0.2, y5 = 1.5569; for h = 0.1, y10 = 1.5576
  4. for h = 0.2, y5 = 0.2385; for h = 0.1, y10 = 0.2384

Exercises 6.4, Page 322

  1. y(x) = −2e2x + 5xe2x; y(0.2) = −1.4918, y2 = −1.6800
  2. y1 = −1.4928, y2 = −1.4919
  3. y1 = 1.4640, y2 = 1.4640
  4. x1 = 8.3055, y1 = 3.4199; x2 = 8.3055, y2 = 3.4199
  5. x1 = −3.9123, y1 = 4.2857; x2 = −3.9123, y2 = 4.2857
  6. x1 = 0.4179, y1 = −2.1824; x2 = 0.4173, y2 = −2.1821

Exercises 6.5, Page 325

  1. y1 = −5.6774, y2 = −2.5807, y3 = 6.3226
  2. y1 = −0.2259, y2 = −0.3356, y3 = −0.3308, y4 = −0.2167
  3. y1 = 3.3751, y2 = 3.6306, y3 = 3.6448, y4 = 3.2355, y5 = 2.1411
  4. y1 = 3.8842, y2 = 2.9640, y3 = 2.2064, y4 = 1.5826, y5 = 1.0681, y6 = 0.6430, y7 = 0.2913
  5. y1 = 0.2660, y2 = 0.5097, y3 = 0.7357, y4 = 0.9471, y5 = 1.1465, y6 = 1.3353, y7 = 1.5149, y8 = 1.6855, y9 = 1.8474
  6. y1 = 0.3492, y2 = 0.7202, y3 = 1.1363, y4 = 1.6233, y5 = 2.2118, y6 = 2.9386, y7 = 3.8490
  7. (c) y0 = −2.2755, y1 = −2.0755, y2 = −1.8589, y3 = −1.6126, y4 = −1.3275

Chapter 6 in Review, Page 326

  1. Comparison of Numerical Methods with h = 0.1

    A table consists of 5 rows and 4 columns. The column headers are as follows: x subscript n, Euler, Improved Euler, R K 4. The table entries are as follows. (row 1). x subscript n, 1.10. Euler, 2.1386. Improved Euler, 2.1549. R K 4, 2.1556. (row 2). x subscript n, 1.20. Euler, 2.3097. Improved Euler, 2.3439. R K 4, 2.3454. (row 3). x subscript n, 1.30. Euler, 2.5136. Improved Euler, 2.5672. R K 4, 2.5695. (row 4). x subscript n, 1.40. Euler, 2.7504. Improved Euler, 2.8246. R K 4, 2.8278. (row 5). x subscript n, 1.50. Euler, 3.0201. Improved Euler, 3.1157. R K 4, 3.1197.

    Comparison of Numerical Methods with h = 0.05

    A table consists of 5 rows and 4 columns. The column headers are as follows: x subscript n, Euler, Improved Euler, R K 4. The table entries are as follows. (row 1). x subscript n, 1.10. Euler, 2.1469. Improved Euler, 2.1554. R K 4, 2.1556. (row 2). x subscript n, 1.20. Euler, 2.3272. Improved Euler, 2.3450. R K 4, 2.3454. (row 3). x subscript n, 1.30. Euler, 2.5410. Improved Euler, 2.5689. R K 4, 2.5695. (row 4). x subscript n, 1.40. Euler, 2.7883. Improved Euler, 2.8269. R K 4, 2.8278. (row 5). x subscript n, 1.50. Euler, 3.0690. Improved Euler, 3.1187. R K 4, 3.1197.

  2. Comparison of Numerical Methods with h = 0.1

    A table consists of 5 rows and 4 columns. The column headers are as follows: x subscript n, Euler, Improved Euler, R K 4. The table entries are as follows. (row 1). x subscript n, 0.60. Euler, 0.6000, Improved Euler, 0.6048. R K 4, 0.6049. (row 2). x subscript n, 0.70. Euler, 0.7095. Improved Euler, 0.7191. R K 4, 0.7194. (row 3). x subscript n, 0.80. Euler, 0.8283. Improved Euler, 0.8427. R K 4, 0.8431. (row 4). x subscript n, 0.90. Euler, 0.9559. Improved Euler, 0.9752. R K 4, 0.9757. (row 5). x subscript n, 1.00. Euler, 1.0921. Improved Euler, 1.1163. R K 4, 1.1169.

    Comparison of Numerical Methods with h = 0.05

    A table consists of 5 rows and 4 columns. The column headers are as follows: x subscript n, Euler, Improved Euler, R K 4. The table entries are as follows. (row 1). x subscript n, 0.60. Euler, 0.6024, Improved Euler, 0.6049. R K 4, 0.6049. (row 2). x subscript n, 0.70. Euler, 0.7144. Improved Euler, 0.7194. R K 4, 0.7194. (row 3). x subscript n, 0.80. Euler, 0.8356. Improved Euler, 0.8431. R K 4, 0.8431. (row 4). x subscript n, 0.90. Euler, 0.9657. Improved Euler, 0.9757. R K 4, 0.9757. (row 5). x subscript n, 1.00. Euler, 1.1044. Improved Euler, 1.1170. R K 4, 1.1169.

  3. h = 0.2: y(0.2) ≈ 3.2; h = 0.1: y(0.2) ≈ 3.23
  4. x(0.2) ≈ 1.62, y(0.2) ≈ 1.84

Exercises 7.1, Page 332

  1. 6i + 12j; i + 8j; 3i;; 3
  2. ‹12, 0›; ‹4, −5›; ‹4, 5›;;
  3. −9i + 6j; − 3i + 9j; − 3i − 5j; 3;
  4. −6i + 27j; 0; −4i + 18j; 0;
  5. ‹6, −14›; ‹2, 4›
  6. 10i − 12j; 12i − 17j
  7. ‹20, 52›; ‹−2, 0›
  8. 2i + 5j
    A graph. 2 vectors are graphed on an x y coordinate plane. The first vector begins at the origin and ends at the point (2, 5). The second vector begins at the marked point (3, 2) and ends at the marked point (5, 7). A horizontal vector, labeled P subscript 1 P subscript 2 and pointing to the right, is placed to the right of the second vector.
  9. 2i + 2j
    A graph. 2 vectors are graphed on an x y coordinate plane. The first vector begins at the origin and ends at the marked point (2, 2). The second vector begins at the point (3, 3) and ends at the marked point (5, 5). A horizontal vector, labeled P subscript 1 P subscript 2 and pointing to the right, is placed to the right of the second vector.
  10. (1, 18)
  11. (a), (b), (c), (e), (f)
  12. ‹6, 15›
  14. ‹0, −1›; ‹0, 1›
  18. 3 vectors form a triangle. The first vector, labeled a, points up and to the left. The second vector, labeled 3 b, begins at the same point as the first vector, and points up to the right. The third vector, labeled 3 b minus a, begins at the point the first vector ends, points up to the right, and ends at the point the second vector ends.
  19. −(a + b)
  3. (b) approximately 31°

Exercises 7.2, Page 338

  1. −5
    A graph. 3 points are graphed on a three dimensional x y z coordinate plane. The coordinates of the 3 marked points are as follows: (3, 4, 0), (1, 1, 5), (6, negative 2, 0).
  1. The set {(x, y, 5)|x, y real numbers} is a plane perpendicular to the z-axis, 5 units above the xy-plane.
  2. The set {(2, 3, z)|z a real number} is a line perpendicular to the xy-plane at (2, 3, 0).
  3. (0, 0, 0), (2, 0, 0), (2, 5, 0), (0, 5, 0), (0, 0, 8), (2, 0, 8), (2, 5, 8), (0, 5, 8)
  4. (–2, 5, 0), (–2, 0, 4), (0, 5, 4); (–2, 5, −2); (3, 5, 4)
  5. the union of the coordinate planes
  6. the point (−1, 2, −3)
  7. the union of the planes z = −5 and z = 5
  9. 7; 5
  10. right triangle
  11. isosceles
  12. d(P1, P2) + d(P1, P3) = d(P2, P3)
  13. 6 or −2
  14. (4, , )
  15. P1(– 4, −11, 10)
  16. ‹– 3, −6, 1›
  17. ‹2, 1, 1›
  18. ‹2, 4, 12›
  19. ‹–11, −41, −49›
  21. 6
  22. ‹– , , − ›
  23. 4i − 4j + 4k
  24. A graph. 3 vectors are graphed on a three dimensional x y z coordinate plane. The first vector begins at the origin, goes up to the right and ends at a point labeled a. The second vector begins at the origin, goes up to the right under the first vector, and ends at a point labeled b. The third vector begins at the origin, goes up to the right between the first 2 vectors, and ends at a point that is aligned with the points a and b. A dashed downward sloping line begins at the point a, passes through the end point of the third vector, and ends at the point b. The length of the dashed line is labeled: 1 over 2 (a + b).

Exercises 7.3, Page 343

  1. 12
  2. −16
  3. 48
  4. 29
  5. 25
  6. ‹– , , 2›
  8. (a) and (f), (c) and (d), (b) and (e)
  9. ‹, − , 1›
  1. 1.11 radians or 63.43°
  2. 1.89 radians or 108.43°
  3. cos α = 1/, cos α = 2/, cos γ = 3/; α = 74.5°, α = 57.69°, γ = 36.7°
  4. cos α = , cos α = 0, cos γ = −/2; α = 60°, α = 90°, γ = 150°
  5. 0.955 radian or 54.74°; 0.616 radian or 35.26°
  6. α = 58.19°, α = 42.45°, γ = 65.06°
  8. −6/11
  9. 72/109
  10. ‹– , ›
  11. ‹– , , ›
  12. ‹, ›
  13. 1000 ft-lb
  14. 0; 150 N-m
  15. approximately 1.80 angstroms

Exercises 7.4, Page 350

  1. −5i − 5j + 3k
  2. ‹–12, −2, 6›
  3. −5i + 5k
  4. ‹–3, 2, 3›
  5. 0
  6. 6i + 14j + 4k
  7. −3i −2j − 5k
  1. −i + j + k
  2. 2k
  3. i + 2j
  4. −24k
  5. 5i − 5j − k
  6. 0
  8. −j
  9. 0
  10. 6
  11. 12i − 9j + 18k
  12. −4i + 3j − 6k
  13. −21i + 16j + 22k
  14. −10
  15. 14 square units
  16. square unit
  17. square units
  18. 10 cubic units
  19. coplanar
  20. 32; in the xy-plane, 30° from the positive x-axis in the direction of the negative y-axis; 16i − 16j
  21. A = i − k, B = j − k, C = 2k

Exercises 7.5, Page 357

  1. ‹x, y, z› = ‹1, 2, 1› + t ‹2, 3, −3›
  2. ‹x, y, z› = ‹, −, 1› + t ‹–2, 3, −›
  3. ‹x, y, z› = ‹1, 1, −1› + t ‹5, 0, 0›
  4. x = 2 + 4t, y = 3 − 4t, z = 5 + 3t
  5. x = 1 + 2t, y = −2t, z = −7t
  6. x = 4 + 10t, y = + t, z = + t
  10. x = 4 + 3t, y = 6 + t, z = −7 −t;

  11. x = 5t, y = 9t, z = 4t;
  12. x = 6 + 2t, y = 4 − 3t, z = −2 + 6t
  13. x = 2 + t, y = −2, z = 15
  14. Both lines pass through the origin and have parallel direction vectors.
  15. (0, 5, 15), (5, 0, ), (10, −5, 0)
  16. (2, 3, −5)
  17. Lines do not intersect.
  18. 40.37°
  19. x = 4 − 6t, y = 1 + 3t, z = 6 + 3t
  20. 2x − 3y + 4z = 19
  21. 5x − 3z = 51
  22. 6x + 8y − 4z = 11
  23. 5x − 3y + z = 2
  24. 3x − 4y + z = 0
  25. The points are collinear.
  26. x + y − 4z = 25
  27. z = 12
  28. −3x + y + 10z = 18
  29. 9x − 7y + 5z = 17
  30. 6x − 2y + z = 12
  31. orthogonal: (a) and (d), (b) and (c), (d) and (f), (b) and (e); parallel: (a) and (f), (c) and (e)
  32. (c) and (d)
  33. x = 2 + t, y = − t, z = t
  34. x = − t, y = − t, z = t
  35. (– 5, 5, 9)
  36. (1, 2, −5)
  37. x = 5 + t, y = 6 + 3t, z = −12 + t
  38. 3x − y − 2z = 10
  39. A graph. A shaded surface given by a triangular portion of a plane is graphed on a three dimensional x y z coordinate plane. The portion of the plane is formed by 3 lines. The first line, graphed on the x y plane, begins at the point (2, 0, 0), and ends at the point (0, 5, 0). The second line begins at the point (2, 0, 0) and ends at the point (0, 0, 10). The third line begins at the point (0, 5, 0) and ends at the point (0, 0, 10).
  40. A graph. A shaded surface given by a portion of a plane is graphed in the first octant of a three dimensional x y z coordinate plane. The portion of the plane is formed by 3 lines. The first line, graphed on the x y plane, begins at the point (0, 6, 0), and goes parallel to the x axis. The second line begins at the point (0, 6, 0) and ends at the point (0, 0, 2). The third line begins at the point (0, 0, 2) and goes parallel to the x axis.
  41. A graph. A shaded surface given by a triangular portion of a plane is graphed on a three dimensional x y z coordinate plane. The portion of the plane is formed by 3 lines. The first line, graphed on the x y plane, begins at the point (negative 4, 0, 0), and ends at the point (0, 2, 0). The second line begins at the point (negative 4, 0, 0) and ends at the point (0, 0, 4). The third line begins at the point (0, 2, 0) and ends at the point (0, 0, 4).

Exercises 7.6, Page 364

  1. not a vector space, axiom (vi) is not satisfied
  2. not a vector space, axiom (x) is not satisfied
  3. vector space
  4. not a vector space, axiom (ii) is not satisfied
  5. vector space
  6. a subspace
  7. not a subspace
  8. a subspace
  9. a subspace
  10. not a subspace
  1. (b) a = 7u1 − 12u2 + 8u3
  2. linearly dependent
  3. linearly independent
  4. f is discontinuous at x = −1 and at x = −3.

Exercises 7.7, Page 370

  1. where

  2. where

  3. (a)

    (b)

  4. (a)

    (b)

Chapter 7 in Review, Page 371

  1. true
  2. false
  3. true
  4. true
  5. true
  6. 9i + 2j + 2k
  7. 5i
  8. 14
  9. −6i + j − 7k
  10. (4, 7, 5)
  11. (5, 6, 3)
  13. 12, −8, and 6
  15. 2 units
  16. (i − j − 3k)/
  17. 2
  18. i + j + k
  19. sphere; plane
  21. The direction vectors are orthogonal and the point of intersection is (3, −3, 0).
  22. 14x − 5y − 3z = 0
  23. 30 N-m
  24. approximately 153 lb
  25. not a vector space
  26. a subspace; 1, x

Exercises 8.1, Page 381

  1. 2 × 4
  2. 3 × 3
  3. 3 × 4
  4. not equal
  5. not equal
  6. x = 2, y = 4
  7. c23 = 9, c12 = 12

  15. 4 × 5
  2. AB is not necessarily the same as BA.
  3. a11x1 + a12x2 = b1

    a21x1 + a22x2 = b2

  4. (b) MR =

    MP =

    (c) xS = ≈ 1.4072,

    yS = ≈ 0.2948,

    zS = ≈ 0.9659

Exercises 8.2, Page 394

  1. x1 = 4, x2 = −7
  2. x1 = − , x2 =
  3. x1 = 0, x2 = 4, x3 = −1
  4. x1 = −t, x2 = t, x3 = 0
  5. inconsistent
  6. x1 = 0, x2 = 0, x3 = 0
  7. x1 = −2, x2 = −2, x3 = 4
  8. x 1 = 1, x2 = 2 − t, x3 = t
  9. x1 = 0, x2 = 1, x3 = 1, x4 = 0
  10. inconsistent
  11. x1 = 0.3, x2 = −0.12, x3 = 4.1
  12. 2Na + 2H2O → 2NaOH + H2
  13. Fe3O4 + 4C → 3Fe + 4CO
  14. 3Cu + 8NHO3 → 3Cu(NO3)2 + 4H2O + 2NO
  15. i1 = , i2 = , i3 =
  18. Interchange row 1 and row 2 in I3.
  19. Multiply the second row of I3 by c and add to the third row.
  20. EA =
  21. EA =

Exercises 8.3, Page 399

  1. 2
  2. 1
  3. 3
  4. 2
  5. 3
  6. linearly independent
  7. linearly independent
  8. 5
  9. rank(A) = 2

Exercises 8.4, Page 405

  1. 9
  2. 1
  3. 2
  4. 10
  5. −7
  6. 17
  7. λ2 − 3λ − 4
  8. −48
  9. 62
  10. 0
  11. −85
  12. −x + 2y − z
  13. −104
  14. 48
  15. λ1 = −5, λ2 = 7

Exercises 8.5, Page 411

  1. Theorem 8.5.4
  2. Theorem 8.5.7
  3. Theorem 8.5.5
  4. Theorem 8.5.3
  5. Theorem 8.5.1
  6. −5
  7. −5
  8. 5
  9. 80
  10. −105
  1. 0
  2. −15
  3. −9
  4. 0
  5. 16

Exercises 8.6, Page 420

  9. singular matrix
  15. x = 5
  1. x1 = 6, x2 = −2
  2. x1 = , x2 = −
  3. x1 = 2, x2 = 4, x3 = −6
  4. x1 = 21, x2 = 1, x3 = −11
  5. x1 = , x2 = ; x1 = 6, x2 = 16; x1 = −2, x2 = −7
  6. System has only the trivial solution.
  7. System has nontrivial solutions.
  8. (c) i1 = ,

    i2 = ,

    i3 =

Exercises 8.7, Page 424

  1. x1 = −, x2 =
  2. x1 = 0.1, x2 = −0.3
  3. x = 4, y = −7
  4. x1 = −4, x2 = 4, x3 = −5
  5. u = 4, v = , w = 1
  6. k =
  7. T1 ≈ 450.8 lb, T2 ≈ 423 lb

Exercises 8.8, Page 432

  1. K3, λ = −1
  2. K3, λ = 0
  3. K2, λ = 3; K3, λ = 1
  4. λ1 = 6, λ2 = 1, K1 = , K2 = ; nonsingular
  5. λ1 = λ2 = −4, K1 = ; nonsingular
  6. λ1 = 3i, λ2 = −3i

    K1 = , K2 = ; nonsingular

  7. λ1 = 4, λ2 = −5,

    K1 = , K2 = ; nonsingular

  8. λ1 = 0, λ2 = 4, λ3 = −4,

    K1 = , K2 = , K3 = ; singular

  9. λ1 = λ2 = λ3 = −2,

    K1 = , K2 = ; nonsingular

  10. λ1 = −1, λ2 = i, λ3 = −i,

    K1 = , K2 = , K3 = ; nonsingular

  11. λ1 = 1, λ2 = 5, λ3 = −7,

    K1 = , K2 = , K3 = ; nonsingular

  12. Eigenvalues of A are λ1 = 6, λ2 = 4; eigenvalues of A−1 are corresponding eigenvectors for both A and A−1 are
  13. Eigenvalues of A are λ1 = 5, λ2 = 4, λ3 = 3, eigenvalues of A−1 are corresponding eigenvectors for both A and A−1 are

Exercises 8.9, Page 436

  6. (a) 4m, m > 1

    (b) Am = 0, m > 1

    (c)

  8. (b), (c), (d), (e), (f)

Exercises 8.10, Page 443

  1. (b) λ1 = −4, λ2 = −1, λ3 = 16
  2. (b) λ1 = 18, λ2 = λ3 = −8
  3. orthogonal
  4. orthogonal
  5. not orthogonal
  11. (b)

    (c)

Exercises 8.11, Page 450

  4. 7 and 2
  5. 4, 3, and 1
  6. approximately 0.2087
  7. (c)

    (d) 0.59

    (e) approximately 9.44EI/L2

Exercises 8.12, Page 458

  1. P = , D =
  2. not diagonalizable
  8. not diagonalizable

  16. Ellipse; using

    we get X2/4 + Y2/6 = 1.

    A graph. An inclined ellipse is graphed on a lowercase x lowercase y coordinate plane. The center of the ellipse is at the origin. A new uppercase X uppercase Y coordinate system is created by rotating the x and y axes 45 degree in a clockwise direction. The center of the ellipse is at the origin of the new uppercase X uppercase Y coordinate plane. The ellipse passes through the following points in the new uppercase X uppercase Y coordinate plane: (negative sqrt(6), 0), (0, 2), (sqrt(6), 0), (0, negative 2).
  17. Hyperbola; using

    we get X2/4 − Y2/4 = 1.

    A graph. An inclined hyperbola is graphed on a lowercase x lowercase y coordinate plane. The hyperbola consists of 2 symmetric and disconnected curves with respect to the origin. A new uppercase X uppercase Y coordinate system is created by rotating the x and y axes 60 degree in a counter clockwise direction. The hyperbola is now symmetric with respect to the uppercase Y axis in the new uppercase X uppercase Y coordinate plane. The first disconnected curve enters the top left of the viewing window in the second quadrant of the new uppercase X uppercase Y coordinate plane, passes through the approximate point (negative 3, 2.2), goes down and to the right with increasing steepness until the point (negative 2, 0), then continues going down to the left with decreasing steepness, passes through the approximate point (negative 3, negative 2.2), and exits the bottom left of the viewing window. The second disconnected curve enters the top right of the viewing window in the first quadrant of the new uppercase X uppercase Y coordinate plane, passes through the approximate point (3, 2.2), goes down and to the left with increasing steepness until the point (2, 0), then continues going down to the right with decreasing steepness, passes through the approximate point (3, negative 2.2), and exits the bottom right of the viewing window.

Exercises 8.13, Page 465

  13. x1 = 1, x2 = 1, x3 = 5
  14. x1 = 28, x2 = −5, x3 = 13
  15. x1 = 25, x2 = −4, x3 = −19
  18. −2
  19. 1600
  20. −78

Exercises 8.14, Page 469

  1. (a)
  2. (a)
  3. (a)
  4. STUDY_HARD
  5. MATH_IS_IMPORTANT_
  6. DAD_I_NEED_MONEY_TODAY
  7. (a)

Exercises 8.15, Page 474

  1. (0 1 1 0)
  2. (0 0 0 1 1)
  3. (1 0 1 0 1 0 0 1)
  4. (1 0 0)
  5. parity error
  6. (1 0 0 1 1)
  7. (0 0 1 0 1 1 0)
  8. (0 1 0 0 1 0 1)
  9. (1 1 0 0 1 1 0)
  10. code word; (0 0 0 0)
  11. (0 0 0 1)
  12. code word; (1 1 1 1)
  13. (1 0 0 1)
  14. (1 0 1 0)
  15. (a) 27 = 128

    (b) 24 = 16

    (c) (0 0 0 0 0 0 0), (0 1 0 0 1 0 1),

    (0 1 1 0 0 1 1), (0 1 0 1 0 1 0),

    (0 1 1 1 1 0 0), (0 0 1 0 1 1 0),

    (0 0 1 1 0 0 1), (0 0 0 1 1 1 1),

    (1 0 0 0 0 1 1), (1 1 0 0 1 1 0),

    (1 0 1 0 1 0 1), (1 0 0 1 1 0 0),

    (1 1 1 0 0 0 0), (1 1 0 1 0 0 1),

    (1 0 1 1 0 1 0), (1 1 1 1 1 1 1)

Exercises 8.16, Page 478

  1. y = 0.4x + 0.6
  2. y = 1.1x − 0.3
  3. y = 1.3571x + 1.9286
  4. v = −0.84T + 234, 116.4, 99.6

Exercises 8.17, Page 482

  1. (a) T = , X0 =

    (b)

    (c)

  2. (a) T = , X0 =

    (b)

    (c)

Chapter 8 in Review, Page 483

  3. false
  4. , −5
  5. 0
  6. false
  7. true
  8. false
  9. true
  10. false
  1. (a)
  2. x1 = −, x2 = 7, x3 =
  1. 240
  2. trivial solution only
  3. I2 + 10HNO3 → 2HIO3 + 10NO2 + 4H2O
  4. x1 = −, x2 = , x3 =
  5. x = X cos θ − Y sin θ,

    y = X sin θ + Y cos θ

  6. x1 = 7, x2 = 5, x3 = 23
  7. λ1 = 5, λ2 = −1, K1 = , K2 =
  8. λ1 = λ2 = −1, λ3 = 8,

  9. λ1 = λ2 = −3, λ3 = 5,

  11. hyperbola
  13. HELP_IS_ON_THE_WAY
  14. (a) (1 1 0 0 1)

    (b) parity error

Exercises 9.1, Page 492

  1. A graph. A helix is graphed in a three dimensional x y z coordinate plane. The helix begins at the point (0, 4, 0), and spirals upward in a clockwise direction around the vertical z axis maintaining the same horizontal distance to the z axis.
  2. A graph. A curve and dashed horizontal line is graphed on a three dimensional x y z coordinate plane. The dashed line begins at the origin, and goes along the x y plane. The curve begins just above the origin on the positive z axis, goes along the dashed line, oscillating equally above and under the line.
  3. A graph. An upward sloping curve is graphed on an x y coordinate plane. The curve begins at the origin, goes up and to the right with increasing steepness, and exits the top right of the viewing window.
  4. A graph. A curve and dashed horizontal line is graphed in the first octant of a three dimensional x y z coordinate plane. The dashed line begins at the origin, and goes along the x y plane. The curve begins at the point (0, 0, 2), goes down and to the right with increasing steepness, and ends at a point in the x y plane on the dashed horizontal line.
  5. A graph. A helix is graphed in a three dimensional x y z coordinate plane. The helix begins at the origin, and spirals upward in a counter clockwise direction around the vertical z axis. The horizontal distance of the curve to the z axis increases as the helix keeps climbing up.
  6. A graph. A paraboloid and plane is graphed on a three dimensional x y z coordinate plane. The paraboloid is graphed with its central axis passing through the vertical z axis. The low point of the paraboloid is at the origin. The plane is vertical, passes though the z axis, and intersects the paraboloid along a parabola labeled C. The low point of the parabola passes through the origin as well.
  7. A graph. A cylinder and surface shaped like a long curved roof is graphed on a three dimensional x y z coordinate plane. The cylinder is graphed with its central axis passing through the vertical z axis. The base of the cylinder is on the x y plane. The curved surface begins on the x y plane on the circular base of the cylinder and parallel to the y axis, curves upward toward the z axis, reaches a maximum height on the top circular base of the cylinder, curves down symmetrically and ends on the circular base at a point diametrically opposite to the starting point, remaining parallel to the y axis. The intersection of the cylinder and the curved surface is a closed curve labeled C and saddle shaped.
  8. 2i − 32j
  9. (1/t)i − (1/t2)j; −(1/t2)i + (2/t3)j
  10. ‹e2t(2t + 1), 3t2 , 8t − 1›; ‹4e2t(t + 1), 6t, 8›
  11. A graph. An ellipse and a vector is graphed on an x y coordinate plane. The center of the ellipse is at the origin. The ellipse passes through the following points: (negative 2, 0). (0, 6), (2, 3), (2, 0), (0, negative 6). The vector is tangent to the ellipse at the marked point (2, 3). It begins on the ellipse at the marked point (2, 3) and points up to the left.
  12. A graph. A curve, a dashed horizontal line, and a vector is graphed on a three dimensional x y z coordinate plane. The dashed horizontal line is graphed on the x y plane and is parallel to the y axis. The curve is graphed on a vertical plane that goes along the dashed horizontal line. The curve enters the bottom left of the viewing window, goes up perpendicular to the x z plane, reaches a high point on the x z plane, and then goes down, and exits the right bottom right of the viewing window. The vector begins on the curve at a marked point in the first octant, and points down the right, and ends on the dashed horizontal line.
  13. x = 2 + t, y = 2 + 2t, z = + 4t
  14. r(t) × r″(t)
  15. r(t) · [r′(t) × r′″(t)]
  16. 2 r′1(2t) − (1/t2)r′2(1/t)
  17. i + 9j + 15k
  18. et(t − 1)i + e−2tj + et2k + c
  19. (6t + 1)i + (3t2 − 2)j + (t3 + 1)k
  20. (2t3 − 6t + 6)i + (7t − 4t3/2 − 3)j + (t2 − 2t)k
  22. 6(e3π − 1)
  23. a cos(s/a)i + a sin(s/a)j
  24. Differentiate r (t) · r (t) = c2 .

Exercises 9.2, Page 495

  1. Speed is.
    A graph. An upward sloping curve and 2 vectors graphed on an x y coordinate plane. The curve begins at the origin, goes up and to the right with increasing steepness, and exits the top right of the viewing window. The first vector, labeled a(1), begins on the curve at a marked point at x = 1, and points up to the right above the curve. The second vector, labeled v(1), begins at the same marked point, and points up to the right under the curve.
  2. Speed is 2.
    A graph. A curve and 2 vectors are graphed on an x y coordinate plane. The curve enters the bottom left of the viewing window in the third quadrant, goes up and to the right with increasing steepness until the marked point (negative 1, 0), then goes up to the left symmetrically, and exits the top left of the viewing window. The first vector, labeled a(0), begins on the curve at a marked point (negative 1, 0), and points horizontally to the left along the x axis. The second vector, labeled v(0), begins at the same marked point, and points vertically up.
  3. Speed is.
    A graph. A curve and 2 vectors are graphed on a three dimensional x y z coordinate plane. The curve is graphed on the vertical plane passing through the dashed line x = 2. The curve enters the bottom of the viewing window, goes up to the left until a point, then goes up to the right, and exits the top right of the viewing window. The first vector, labeled a(2), begins on the curve at a marked point, and points horizontally to the right, under the curve. The second vector, labeled v(2), begins at the same marked point, and points up to the right, above the curve.
  4. Speed is.
    A graph. An upward sloping curve and 2 vectors are graphed on a three dimensional x y z coordinate plane. The curve is graphed on the vertical surface passing through a dashed curve on the x y plane. The dashed curve begins at the origin in the x y plane, and goes up to the right with increasing steepness. The curve begins at the origin, goes up with increasing steepness, and exits the top right of the viewing window. The first vector, labeled a(1), begins on the curve at a marked point at x = 1, and points up to the right, above the curve. The second vector, labeled v(1), begins at the same marked point, and points up to the right, under the curve.
  5. (0, 0, 0) and (25, 115, 0);

    v(0) = −2j − 5 k, a(0) = 2i + 2k,

    v(5) = 10i + 73j + 5k, a(5) = 2i + 30j + 2k

  6. (a) r(t) = (−16t2 + 240t)j + 240 ti and

    x(t) = 240 t, y(t) = −16t2 + 240t

    (b) 900 ft

    (c) 6235 ft

    (d) 480 ft/s

  7. 72.11 ft/s
  8. 97.98 ft/s
  9. (a) 4300 ft, approximately 7052.15 ft, approximately 576.89 ft/s

    (b)

    A graph. 2 curves are graphed on an x y coordinate plane. The x axis ranges from 0 to 8000 in increments of 2000. The y axis ranges from 0 to 4000 in increments of 1000. The first curve, labeled theta = 30 degrees, begins on the y axis at the approximate point (0, 1600), goes up and to the right with decreasing steepness, reaches a high point at the approximate point (3000, 2400), goes down to the right with increasing steepness, and ends on the x axis at the approximate point (8300, 0). The second curve, labeled theta = 60 degrees, begins at the same point as the first curve, goes up and to the right with decreasing steepness above the first curve, reaches a high point at the approximate point (3000, 4300), goes down to the right with increasing steepness, and ends on the x axis at the approximate point (7000, 0).
  10. approximately 175.62 ft/s
  11. Assume that (x0, y0) are the coordinates of the center of the target at t = 0. Then rp = rt when t = x0/(v0 cos θ) = y0/(v0 sin θ). This implies tan θ = y0/x0. In other words, aim directly at the target at t = 0.
  1. 191.33 lb
  3. (b) Since F is directed along r, we must have F = cr for some constant c. Hence τ = r × (cr) = c(r × r) = 0. If τ = 0, then dL/dt = 0. This implies that L a constant.

Exercises 9.3, Page 502

  1. T = (/5)(−sin ti + cos tj + 2k)
  2. T = (a2 + c2)−1/2(−a sin ti + a cos tj + ck),

    N = −cos ti − sin tj,

    B = (a2 + c2)−1/2(c sin ti − c cos tj + ak),

    ĸ = a/(a2 + c2)

  3. 3x − 3y + 4z = 3π
  7. 0, 5
  1. ĸ = 2, ρ = ; ĸ = 2/ ≈ 0.18,

    ρ = /2 ≈ 5.59; the curve is sharper at (0, 0).

Exercises 9.4, Page 507

  1. A graph. 3 parallel and downward sloping lines are graphed on an x y coordinate plane. The first line enters the left side of the viewing window in the second quadrant, goes down to the right, passes through the points (0, 1), (2, 0), and exits the bottom right of the viewing window. The second line enters the left side of the viewing window in the second quadrant under the first line, goes down to the right, passes through the points (0, 0), (2, negative 1), and exits the bottom right of the viewing window, under the first line. The third line enters the left side of the viewing window in the second quadrant under the second line, goes down to the right, passes through the points (negative 2, 0), (0, negative 1), and exits the bottom right of the viewing window, under the second line.
  2. A graph. 3 hyperbolas, each consisting of 2 disconnected curves are graphed on an x y coordinate plane. The curves are symmetric with respect to the y axis. The 3 disconnected curves on the left enter the top left of the viewing window in the second quadrant, go down to the right with increasing steepness until the x axis, then go down to the left symmetrically, and exit the bottom left of the viewing window. The 3 disconnected curves on the right enter the top right of the viewing window in the first quadrant, go down to the left with increasing steepness until the x axis, then go down to the right symmetrically, and exit the bottom right of the viewing window.
  3. A graph. 3 parabolas are graphed one under the other on an x y coordinate plane. The 3 parabolas enter the top left of the viewing window in the second quadrant, go down to the right with decreasing steepness, reach a low point on the y axis, go up to the right symmetrically, and exit the top right of the viewing window. The first curve reaches its low point above the origin, the second curve reaches its low point at the origin, and the third curve reaches its low point under the origin.
  4. elliptical cylinders
  5. ellipsoids
  6. 3 graphs. (graph 1). Caption. c = 0. Graph. A double cone is graphed on a three dimensional x y z coordinate plane. The central axis of the double cone is on the z axis. The common vertex of the cones is at the origin. (graph 2). Caption. c > 0. Graph. A hyperboloid is graphed on a three dimensional x y z coordinate plane. The central axis the hyperboloid is on the z axis. The hyperboloid is symmetric with respect to the origin. The horizontal circle that goes around the center of the hyperboloid is smaller than the bases, and is on the x y plane. (graph 3). Caption. c < 0. Graph. A hyperbola, consisting of 2 disconnected and symmetric paraboloids is graphed on a three dimensional x y z coordinate plane. The central axis of the hyperboloid is on the z axis. The hyperboloid is symmetric with respect to the origin.
  7. ∂z/∂x = 2x − y2 , ∂z/∂y = −2xy + 20y4
  8. ∂z/∂x = 20x3 y3 − 2xy6 + 30x4 ,

    ∂z/∂y = 15x4 y2 − 6x2 y5 − 4

  9. ∂z/∂x = 2x−1/2/(3y2 + 1),

    ∂z/∂y = −24/(3y2 + 1)2

  10. ∂z/∂x = −3x2 (x3 − y2)−2,

    ∂z/∂y = 2y(x3 − y2)−2

  11. ∂z/∂x = −10 cos 5x sin 5x,

    ∂z/∂y = 10 sin 5y cos 5y

  13. fx = 7y/(x + 2y)2, fy = −7x/(x + 2y)2
  14. gu = 8u/(4u2 + 5v3), gv = 15v2/(4u2 + 5v3)
  15. wx = x−1/2y, wy = 2 − (y/z)ey/z − ey/z, wz = (y2/z2)ey/z
  16. Fu = 2uw2 − v3 − vwt2 sin(ut2),

    Fv = −3uv2 + w cos(ut2), Fx = 128x7 t4 ,

    Ft = −2uvwt sin(ut2) + 64x8 t3

  1. ∂z/∂x = 3x2 v2 euv 2 + 2uveuv 2, ∂z/∂y = −4yuveuv 2
  2. ∂z/∂u = 16u3 − 40y(2u − v),

    ∂z/∂v = −96v2 + 20y(2u − v)

  3. ∂w/∂t = −3u(u2 + v2)1/2e−t sin θ

    −3v(u2 + v2)1/2e−t cos θ,

    ∂w/∂θ = 3u(u2 + v2)1/2e−t cos θ

    −3v(u2 + v2)1/2e−t sin θ

  4. ∂R/∂u = s2 t4 ev 2 − 4rst4 uve−u2 + 8rs2 t3 uv2 eu2v2,

    ∂R/∂v = 2s2 t4 uvev 2 + 2rst4 e−u2 + 8rs2 t3 u2 veu2v2

  6. dz/dt = (4ut − 4vt−3)/(u2 + v2)
  7. dw/dt|t=π = −2
  1. 5.31 cm2/s

Exercises 9.5, Page 512

  1. (2x − 3x2y2)i + (−2x3y + 4y3)j
  2. (y2/z3)i + (2xy/z3)j − (3xy2/z4)k
  3. 4i − 32j
  5. x + y
  6. (− 2)
  10. −1
  13. −2i + 2j − 4k,
  16. ±31
  17. u = i − j; u = i + j; u = −i − j
  18. Du f = (9x2 + 3y2 − 18xy2 − 6x2 y)/;

    Du F = (−6x2 − 54y2 + 54x + 6y − 72xy)/10

  19. (2, 5), (−2, 5)
  20. −16i − 4j
  21. x = 3e−4t, y = 4e−2t
  22. One possible function is f(x, y) = x3 − y3 + xy3 + exy.

Exercises 9.6, Page 517

  1. A graph. An upward sloping line and a vector is graphed on an x y coordinate plane. The line enters the bottom left of the viewing window in the third quadrant, goes up to the right, passes through the points (0, negative 2), (4, 0), and exits the top right of the viewing window. A vector begins on the line at a marked point in the first quadrant, is perpendicular to the line, and points down to the right.
  2. A graph. A parabola and a vector is graphed on an x y coordinate plane. The parabola enters the top left of the viewing window in the second quadrant, goes down to the right with decreasing steepness, reaches a low point at the point (0, 1), goes up to the right symmetrically, and exits the top right of the viewing window. A vector begins on the parabola at a marked point in the first quadrant, and points up to the left.
  3. A graph. An ellipse and a vector is graphed on an x y coordinate plane. The center of the ellipse is at the origin. The ellipse passes through the following approximate points: (negative 2.7, 0). (0, 4.3), (2.7, 0), (0, negative 4.3). The vector begins on the ellipse at a marked point in the third quadrant, and points down to the left.
  4. A graph. A hyperbola and a vector is graphed on an x y coordinate plane. The hyperbola consists of 2 disconnected and symmetric curves with respect to the x axis. The first disconnected curve enters the top left of the viewing window in the second quadrant, goes down to the right with decreasing steepness, reaches a low point at the point (1, 1), goes up to the right symmetrically, and exits the top right of the viewing window. The second disconnected curve enters the bottom left of the viewing window in the third quadrant, goes up to the right with decreasing steepness, reaches a high point at the point (1, negative 1), goes down to the right symmetrically, and exits the bottom right of the viewing window. The vector begins on the first disconnected curve at its low point (1, 1), and points vertically down.
  5. A graph. A shaded surface given by a portion of a plane is graphed in the first octant of a three dimensional x y z coordinate plane. The portion of the plane is formed by 3 lines. The first line, graphed on the x y plane, begins at the point (0, 2, 0), and goes parallel to the x axis. The second line begins at the point (0, 2, 0) and ends at the point (0, 0, 2). The third line begins at the point (0, 0, 2) and goes parallel to the x axis. A vector begins on the surface at a marked point, points outward and is normal to the surface.
  6. A graph. A hemisphere and a vector is graphed on a three dimensional x y z coordinate plane. The center of the hemisphere is at the origin. The circular base of the hemisphere passes through the following point: (5, 0, 0), (0, 5, 0), (negative 5, 0, 0), and (0, negative 5, 0). The high point of the hemisphere is at the point (0, 0, 5). The vector begins on the outer surface of the hemisphere at a marked point in the first octant, and is normal to the surface.
  7. (−4, −1, 17)
  8. −2x + 2y + z = 9
  9. 6x − 2y − 9z = 5
  10. 6x − 8y + z = 50
  11. 2x + y −z = (4 + 5π)/4
  12. x + y − z = 2
  13. (1/, , 3/), (−1/, −, −3/)
  14. (−2, 0, 5), (−2, 0, −3)
  1. x = 1 + 2t, y = −1 − 4t, z = 1 + 2t
  2. (x − )/4 = (y − )/6 = −(z − 3)

Exercises 9.7, Page 521

  1. A graph. 3 concentric circles and 8 vectors are graphed on an x y coordinate plane. The centers of the circles are at the origin. The first circle passes through the point (1, 0). The second circle passes through the approximate point (1.4, 0). The third circle passes through the point (2, 0). The first vector begins on the first circle at the point (0, 1), and ends on the third circle at the point (0, 2). The second vector begins on the first circle at the point (0, negative 1), and ends on the third circle at the point (0, negative 2). The third vector begins on the first circle at the point (1, 0), and ends on the third circle at the point (2, 0). The fourth vector begins on the first circle at the point (negative 1, 0), and ends on the third circle at the point (negative 2, 0). The fifth vector begins on the second circle at the midpoint of the arc in the first quadrant, and points up to the right. The sixth vector begins on the second circle at the midpoint of the arc in the third quadrant, and points down to the left in the opposite direction as the fifth vector. The seventh vector begins on the second circle at the midpoint of the arc in the second quadrant, and points up to the left. The eighth vector begins on the second circle at the midpoint of the arc in the fourth quadrant, and points down to the right in the opposite direction as the seventh vector.
  2. A graph. 3 concentric circles and 12 vectors are graphed on an x y coordinate plane. The centers of the circles are at the origin. The first circle passes through the point (1, 0). The second circle passes through the approximate point (1.4, 0). The third circle passes through the point (2, 0). The first vector begins on the first circle at the point (0, 1) points horizontally to the right and ends on the second circle. The second vector begins on the first circle at the point (1, 0) points vertically up and ends on the second circle at the same point the first vector ends. The third vector begins on the first circle at the point (0, negative 1) points horizontally to the left and ends on the second circle. The fourth vector begins on the first circle at the point (negative 1, 0) points vertically down and ends on the second circle at the same point the third vector ends. The fifth vector begins on the second circle at the point the first and second vectors end, and point up to the right. The sixth vector begins on the second circle at the point (0, 2) points horizontally to the right and ends at the same point the fifth vector ends. The seventh vector begins on the second circle at the point (2, 0) points vertically up and ends at the same point the fifth and sixth vectors end. The eighth vector begins on the second circle at the point the third and fourth vectors end, and point down to the left in the opposite direction as the fifth vector. The ninth vector begins on the second circle at the point (0, negative 2), points horizontally to the left and ends at the same point the eighth vector ends. The tenth vector begins on the second circle at the point (negative 2, 0) points vertically down and ends at the same point the eighth and ninth vectors end. The eleventh vector begins on the second circle at the midpoint of the arc in the second quadrant, and ends at the origin. The twelfth vector begins on the second circle at the midpoint of the arc in the fourth quadrant, and ends at the origin.
  3. A graph. 18 vectors are graphed on an x y coordinate plane. The first vector begins at the point (negative 3, 1), points vertically up, and ends at the point (negative 3, 2). The second vector begins at the point (negative 2, 1), points vertically up, and ends at the point (negative 2, 2). The third vector begins at the point (negative 1, 1), points vertically up, and ends at the point (negative 1, 2). The fourth vector begins at the point (1, 1), points vertically up, and ends at the point (1, 2). The fifth vector begins at the point (2, 1), points vertically up, and ends at the point (2, 2). The sixth vector begins at the point (3, 1), points vertically up, and ends at the point (3, 2). The seventh vector begins at the point (negative 3, negative 1), points vertically down, and ends at the point (negative 3, negative 2). The eighth vector begins at the point (negative 2, negative 1), points vertically down, and ends at the point (negative 2, negative 2). The ninth vector begins at the point (negative 1, negative 1), points vertically down, and ends at the point (negative 1, negative 2). The tenth vector begins at the point (1, negative 1), points vertically down, and ends at the point (1, negative 2). The eleventh vector begins at the point (2, negative 1), points vertically down, and ends at the point (2, negative 2). The twelfth vector begins at the point (3, negative 1), points vertically down, and ends at the point (3, negative 2). The thirteenth vector begins at the point (negative 3, negative 3), points vertically down, and ends at the point (negative 3, negative 6). The fourteenth vector begins at the point (negative 2, negative 3), points vertically down, and ends at the point (negative 2, negative 6). The fifteenth vector begins at the point (negative 1, negative 3), points vertically down, and ends at the point (negative 1, negative 6). The sixteenth vector begins at the point (1, negative 3), points vertically down, and ends at the point (1, negative 6). The seventeenth vector begins at the point (2, negative 3), points vertically down, and ends at the point (2, negative 6). The eighteenth vector begins at the point (3, negative 3), points vertically down, and ends at the point (3, negative 6).
  4. (x − y)i + (x − y)j; 2z
  5. 0; 4y + 8z
  6. (4y3 − 6xz2)i + (2z3 − 3x2)k; 6xy
  7. (3e−z − 8yz)i − xe−zj; e−z + 4z2 − 3ye−z
  8. (xy2 ey + 2xyey + x3 yzez + x3 yez)i − y2 eyj

    + (−3x2 yzez − xex)k; xyex + yex − x3 zez

  1. 2i + (1 − 8y)j + 8zk
  1. div F = 1 ≠ 0. If there existed a vector field G such that F = curl G, then necessarily div F = div (curl G) = 0.

Exercises 9.8, Page 530

  1. −125/3; −250(− 4)/12;
  2. 3; 6; 3
  3. −1; (π − 2)/2; π2/8; π2/8
  4. 21
  5. 30
  6. 1
  7. 1
  8. 460
  10. −
  11. −
  12. 0
  14. 70
  15. −
  16. e
  17. −4
  18. 0
  19. On each curve the line integral has the value .

Exercises 9.9, Page 540

  2. 14
  3. 3
  4. 330
  5. 1096
  6. ϕ = x4 y3 + 3x + y
  7. not a conservative field
  8. ϕ = x4 + xy + y4
  9. 3 + e−1
  10. 63
  11. 8 + 2e3
  12. 16
  13. π − 4
  14. ϕ = (Gm1m2)/|r|

Exercises 9.10, Page 547

  1. 24y − 20ey
  3. ln 5
  4. 2 − sin y
  5. A graph. 3 lines and a shaded region are graphed on an x y coordinate plane. The first line enters the bottom left of the viewing window, goes up to the right, passes through the points (0, 1), (2, 3), and exits the top right of the viewing window. The second line is horizontal, enters the bottom left of the viewing window, intersects the first line at the point (0, 1), and exits the bottom right of the viewing window. The third line is vertical, enters the bottom of the viewing window, intersects the second line at the point (2, 1), intersects the first line at the point (2, 3), and exits the top of the viewing window. The region on the graph between the 3 lines is shaded.
  6. A graph. A semicircle, 2 horizontal lines and a shaded region are graphed on an x y coordinate plane. The center of the semi circle is at the origin. The semicircle passes through the following points: (0, negative 4), (4, 0), and (0, 4). The first line enters the top left of the viewing window in the second quadrant, goes horizontally to the right, passes through the point (0, 3), intersects the circle and exits the top right of the viewing window. The second line enters the bottom left of the viewing window in the third quadrant, goes horizontally to the right, passes through the point (0, negative 1), intersects the circle and exits the bottom right of the viewing window. The region on the graph between the semicircle and the 2 horizontal lines is shaded.
  9. 96
  10. 2 ln 2 − 1
  12. (c), 16π
  13. 18
  14. 2π
  15. 4
  16. 30 ln 6
  17. 15π/4
  18. (23/2 − 1)/18
  19. sin 8
  20. π/8

  27. 4k/9
  31. ab3π/4; a3bπ/4; b/2; a/2
  32. ka4/6
  33. 16k/3
  34. a3

Exercises 9.11, Page 552

  1. 27π/2
  2. (4π − 3)/6
  3. 25π/3
  4. (2π/3)(153/2 − 73/2)
  9. πa4 k/4
  10. (ka/12)(15 − 4π)
  11. πa4 k/2
  12. 4k
  13. 9π
  14. (π/4)(e − 1)
  15. 3π/8
  16. 250
  17. approximately 1450 m3

Exercises 9.12, Page 557

  1. 3
  2. 0
  3. 75π
  4. 48π
  8. (b − a) × (area of region bounded by C)
  1. 3a2 π/8
  1. 45π/2
  2. π
  3. 27π/2
  4. 3π/2
  1. 3π

Exercises 9.13, Page 564

  2. 10π/3
  3. (π/6)(173/2 − 1)
  4. 25π/6
  5. 2a2(π − 2)
  6. 8a2
  7. 2πa(c2 − c1)
  9. 0
  10. 972π
  11. (35/2 − 27/2 + 1)/15
  12. 9(173/2 − 1)
  15. 18
  16. 28π
  17. 8π
  18. 5π/2
  19. −8πa3
  20. 4πkq
  21. (1, , 2)

Exercises 9.14, Page 570

  1. −40π
  4. −3
  5. −3π/2
  6. π
  7. −152π
  8. 112
  9. Take the surface to be z = 0; 81π/4.

Exercises 9.15, Page 579

  1. 48
  2. 36
  3. π − 2
  4. e2 − e
  5. 50

  7. (a)

    (b)

    (c)

  8. A graph. A prism is graphed on three dimensional x y z coordinate plane. The solid is bound by the x y plane, the x z plane, the y z plane, the plane y = 4, and a fifth plane. The fifth plane is formed by 2 lines segments. The first line segment begins at the point (2, 0, 0) and ends at the point (0, 0, 3). The second line segment begins at the point (2, 4, 0) and ends at the point (0, 4, 3).
  9. A graph. A cylinder is graphed on a three dimensional x y z coordinate plane. The central axis of the cylinder passes through the vertical z axis. The center of the circular base at the bottom is at the origin. The circular base at the bottom passes through the following points: (1, 0, 0), (0, negative 1, 0), (negative 1, 0, 0), (0, 1, 0). The center of the circular base at the top is at the point (0, 0, 5). The circular base at the top passes through the following points: (1, 0, 5), (0, negative 1, 5), (negative 1, 0, 5), (0, 1, 5).
  10. A graph. A solid is graphed on a three dimensional x y z coordinate plane. The solid is bound by 2 curves. The curve at the bottom is a parabola graphed on the x y plane. The parabola begins and ends at 2 points on the line y = 2, with a low point at the origin. The curve at the top is graphed on an inclined plane formed by the line y = 2 on the x y plane and the point (0, 0, 2). The curve begins at the same point as the first curve on the line y = 2, curves upward, reaches a high point at the point (0, 0, 2), curves downward and ends at the same point as the first curve on the line y = 2.
  12. 16π
  16. 2560k/3;
  17. k/30
  18. (−10/, 10/, 5)
  19. (/2, , −4)
  20. (, −π/4, −9)
  21. (2, 2π/3, 2)
  22. r2 + z2 = 25
  23. r2 − z2 = 1
  24. z = x2 + y2
  25. x = 5
  26. (2π/3)(64 − 123/2)
  27. 625π/2
  28. (0, 0, 3a/8)
  29. 8πk/3
  30. (/3, , 0); (, π/6, 0)
  31. (−4, 4, 4); (4, 3π/4, 4)
  32. (5, π/2, 5π/4)
  33. (, π/4, π/6)
  34. ρ = 8
  35. ϕ = π/6, ϕ = 5π/6
  36. x2 + y2 + z2 = 100
  37. z = 2
  38. 9π(2 − )
  39. 2π/9
  40. (0, 0, )
  41. πk

Exercises 9.16, Page 586

  2. 12a5 π/5
  3. 256π
  4. 62π/5
  6. 128

Exercises 9.17, Page 592

  1. (0, 0), (−2, 8), (16, 20), (14, 28)
  2. A graph. 3 lines and a shaded region are graphed on an x y coordinate plane. The first line begins at the origin, and ends at the point (4, 2). The second line begins at the point (4, 2), and ends at the point (6, negative 4). The third line begins at the point (6, negative 4), and ends at the origin. The region on the graph between the 3 lines is shaded and labeled R.
  3. A graph. A closed curve and a shaded region is graphed on an x y coordinate plane. The closed curve begins at the point (negative 4, 0), goes up and to the right rapidly, reaches a high point at the point (negative 3, 2), goes down to the right, passes through the points (0, 1), (1, 0), then goes horizontally to the left along the x axis, and ends at the starting point (negative 4, 0). The region on the graph inside the closed curve is shaded and labeled R.
  4. −2v
  5. −u−2
  6. A graph. 3 lines and a shaded region are graphed on an x y coordinate plane. The first line begins at the origin, and ends at the point (1, 1). The second line begins at the point (1, 1), and ends at the point (0, 1). The third line begins at the point (1, 1), and ends at the origin. The region on the graph between the 3 lines is shaded and labeled R.

    (0, 0) is the image of every point on the boundary u = 0.

  7. 16
  13. 126
  15. 15π/2

Chapter 9 in Review, Page 593

  1. true
  2. true
  3. false
  4. true
  5. false
  6. false
  7. true
  8. true
  9. true
  11. v (1) = 6i + j + 2k, v (4) = 6i + j + 8k, a (t) = 2k for all t
  12. i + 4j + (3π/4)k
  13. A graph. A u-shaped curve is graphed on a three dimensional x y z coordinate plane.
  14. (6x2 − 2y2 − 8xy)/
  15. 2; −2/; 4
  16. 4πx + 3y − 12z = 4π − 6

  18. 41k/1512
  19. 8π
  20. 6xy
  21. 0
  22. 56 π3/3
  23. 12
  24. 2 + 2/3π
  25. π2/2
  26. (ln 3)(173/2 − 53/2)/12
  27. −4πc
  28. 0
  29. 125π
  30. 3π
  32. 0
  33. π

Exercises 10.1, Page 606

  1. X′ = X, where X =
  2. X′ = X +
  3. X′ = X, where X =
  5. = 4x + 2y + et

    = −x + 3y − et

  6. = x − y + 2z + e−t − 3t

    = 3x − 4y + z + 2e−t + t

    = −2x + 5y + 6z + 2e−t − t

  1. Yes; W(X1, X2) = −2e−8t ≠ 0 implies that X1 and X2 are linearly independent on (−∞, ∞).
  2. No; W(X1, X2, X3) = 0 for every t. The solution vectors are linearly dependent on (−∞, ∞). Note that X3 = 2X1 + X2.

Exercises 10.2, Page 618

  1. X = c1e5t + c2e−t
  2. X = c1e−3t + c2et
  3. X = c1e8t + c2e−10t
  4. X = c1et + c2e2t + c3e−t
  5. X = c1e−t + c2e3t + c3e−2t
  6. X = c1e−t + c2e−t/2

    + c3e−3t/2

  7. X = 3et/2 + 2e−t/2
  8. (a)

    (b)

    (d) approximately 34.3 minutes

  1. X = c1 + c2
  2. X = c1e2t + c2
  3. X = c1et + c2e2t + c3e2t
  4. X = c1 + c2e5t

    + c3

  5. X = c1

  6. X = −7 e4t + 13 e4t
  7. Corresponding to the eigenvalue λ1 = 2 of multiplicity five, eigenvectors are

    K1 = , K2 = , K3 =.

  8. X = c1e4t + c2e4t
  9. X = c1e4t + c2e4t
  10. X = c1 + c2
  11. X = c1 + c2 + c3
  12. X = c1et + c2et + c3et
  13. X = e2t + c2e−2t

    + c3e−2t

  14. X = −et −

  15. (a)

    (b)

Exercises 10.3, Page 622

  1. X =
  2. X =
  3. X =
  4. X =
  5. X =
  6. (a)

    Since M is a diagonal matrix with m1 and m2 nonzero, it has an inverse.

    (b)

    (c)

Exercises 10.4, Page 628

  6. (a)

    (b)

    (c) 10; 30; as t → ∞ the total amount of salt in the system of mixing tanks approaches a constant 40 lb

Exercises 10.5, Page 634

  1. eAt = ; e−At =
  2. eAt =
  5. X = c3 et + c4 e2t +

Chapter 10 in Review, Page 635

  6. (b)

Exercises 11.1, Page 644

  1. x′ = y

    y′ = −9 sin x; critical points at (±nπ, 0)

  2. x′ = y

    y′ = x2 + y(x3 − 1); critical point at (0, 0)

  3. x′ = y

    y′ = ϵx3 − x;

    critical points at (0, 0),

  4. (0, 0) and (−1, −1)
  5. (0, 0) and (, )
  6. (0, 0), (10, 0), (0, 16), and (4, 12)
  7. (0, y), y arbitrary
  8. (0, 0), (0, 1), (0, −1), (1, 0), (−1, 0)
  9. (a) x = c1e5t − c2e−t

    y = 2c1e5t + c2e−t

    (b) x = −2e−t

    y = 2e−t

  10. (a) x = c1(4 cos 3t − 3 sin 3t) + c2(4 sin 3t + 3 cos 3t)

    y = c1(5 cos 3t) + c2(5 sin 3t)

    (b) x = 4 cos 3t − 3 sin 3t

    y = 5 cos 3t

  11. (a) x = c1(sin t − cos t)e4t + c2(−sin t − cos t)e4t

    y = 2c1(cos t)e4t + 2c2(sin t)e4t

    (b) x = (sin t − cos t)e4t

    y = 2(cos t)e4t

  12. r = , θ = t + c2; r = 4 , θ = t; the solution spirals toward the origin as t increases.
  13. r = , θ = t + c2; r = 1, θ = t (or x = cos t and y = sin t) is the solution that satisfies X(0) = (1, 0); r = , θ = t is the solution that satisfies X(0) = (2, 0). This solution spirals toward the circle r = 1 as t increases.
  14. There are no critical points and therefore no periodic solutions.
  15. There appears to be a periodic solution enclosing the critical point (0, 0).

Exercises 11.2, Page 651

  1. (a) If X(0) = X0 lies on the line y = 2x, then X(t) approaches (0, 0) along this line. For all other initial conditions, X(t) approaches (0, 0) from the direction determined by the line y = −x/2.
  2. (a) All solutions are unstable spirals that become unbounded as t increases.
  3. (a) All solutions approach (0, 0) from the direction specified by the line y = x.
  4. (a) If X(0) = X0 lies on the line y = 3x, then X(t) approaches (0, 0) along this line. For all other initial conditions, X(t) becomes unbounded and y = x serves as the asymptote.
  5. saddle point
  6. saddle point
  7. degenerate stable node
  8. stable spiral
  9. |μ| < 1
  10. μ < −1 for a saddle point; −1 < μ < 3 for an unstable spiral point
  1. (a) (−3, 4)

    (b) unstable node or saddle point

    (c) (0, 0) is a saddle point.

  2. (a) (, 2)

    (b) unstable spiral point

    (c) (0, 0) is an unstable spiral point.

Exercises 11.3, Page 659

  1. r =
  2. x = 0 is unstable; x = n + 1 is asymptotically stable.
  3. T = T0 is unstable.
  4. x = α is unstable; x = β is asymptotically stable.
  5. P = a/b is asymptotically stable; P = c is unstable.
  6. (, 1) is a stable spiral point.
  7. (, 0) and (, 0) are saddle points; (, −) is a stable spiral point.
  8. (1, 1) is a stable node; (1, −1) is a saddle point; (2, 2) is a saddle point; (2, −2) is an unstable spiral point.
  9. (0, −1) is a saddle point; (0, 0) is unclassified; (0, 1) is stable but we are unable to classify further.
  10. (0, 0) is an unstable node; (10, 0) is a saddle point; (0, 16) is a saddle point; (4, 12) is a stable node.
  11. θ = 0 is a saddle point. It is not possible to classify either θ = π/3 or θ = −π/3.
  12. It is not possible to classify x = 0.
  13. It is not possible to classify x = 0, but x = 1/ and x = −1/ are each saddle points.
  1. (a) (0, 0) is a stable spiral point.
  1. (a) (1, 0), (−1, 0)
  2. |v0| <
  3. If β > 0, (0, 0) is the only critical point and is stable. If β < 0, (0, 0), (, 0), and (−, 0), where 2 = −α/β, are critical points. (0, 0) is stable, while (, 0) and (−, 0) are each saddle points.
  4. (b) (5π/6, 0) is a saddle point.

    (c) (π/6, 0) is a center.

Exercises 11.4, Page 666

  1. (a) First show that y2 = + g ln .
  1. (a) The new critical point is (d/c − ϵ2/c, a/b + ϵ1/b).

    (b) yes

  2. (0, 0) is an unstable node, (0, 100) is a stable node, (50, 0) is a stable node, and (20, 40) is a saddle point.
  1. (a) (0, 0) is the only critical point.

Exercises 11.5, Page 674

  1. The system has no critical points.
  3. = −μ + 9y2 > 0 if μ < 0
  4. The single critical point (0, 0) is a saddle point.
  5. δ(x, y) = e−y/2
  6. = 4(1 − x2 − 3y2) > 0 for x2 + 3y2 < 1
  7. Use δ(x, y) = 1/(xy) and show that

  8. If n = (−2x, −2y), show that V · n = 2(x − y)2 + 2y4.
  9. Yes; the sole critical point (0, 0) lies outside the invariant region ≤ x2 + y2 ≤ 1, and so Theorem 11.5.5(ii) applies.
  10. V · n = 2y2(1 − x2) ≥ 2y2(1 − r2) and ∂P/∂x + ∂Q/∂y = x2 − 1 < 0. The sole critical point is (0, 0) and this critical point is a stable spiral point. Therefore, Theorem 11.5.6(ii) applies.
  11. (a)

    (b) limt→∞ X(t) = (, ), a stable spiral point

Chapter 11 in Review, Page 676

  1. true
  2. a center or a saddle point
  3. false
  4. false
  5. true
  6. r = , θ = t; the solution curve spirals toward the origin.
  7. center; degenerate stable node
  8. stable node for μ < −2; stable spiral point for −2 < μ < 0; unstable spiral point for 0 < μ < 2; unstable node for μ > 2
  9. Show that .
  10. = 1
  11. (a) Hint: Use the Bendixson negative criterion.

    (d) In (b), (0, 0) is a stable spiral point when β < 2ml × . In (c), (, 0) and (−, 0) are stable spiral points when β < .

Exercises 12.1, Page 685

  1. T = 1
  2. T = 2π
  3. T = 2π

Exercises 12.2, Page 690

  1. ;

    converges to

  2. converges to

  6. converges to

Exercises 12.3, Page 696

  1. odd
  2. neither even nor odd
  3. even
  4. odd
  5. neither even nor odd

  3. (a)
  4. (b)

Exercises 12.4, Page 700

Exercises 12.5, Page 706

  2. (a) . . .

    (b)

    (c)

  4. (a) λn = 16n2, yn = sin (4n tan−1 x), n = 1, 2, 3, . . .

    (b) sin (4m tan−1x) sin (4n tan−1x) dx = 0, m ≠ n

Exercises 12.6, Page 712

  1. α1 = 1.277, α2 = 2.339, α3 = 3.391, α4 = 4.441
  1. f(x) = P0(x) + P1(x) + P2(x) − P4(x) + …
  1. f(x) = P0(x) + P2(x) − P4(x) + …,

    f(x) = |x| on (−1, 1)

Chapter 12 in Review, Page 713

  1. true
  2. cosine
  3. false
  4. 5.5, 1, 0
  5. true

  4. the interval (−1, 1),

Exercises 13.1, Page 720

  1. The possible cases can be summarized in one form u = c1, where c1 and c2 are constants.
  2. u = c1
  4. not separable

  6. u = (c1 cosh αx + c2 sinh αx)(c3 cosh αat + c4 sinh αat)

    u = (c5 cos αx + c6 sin αx)(c7 cos αat + c8 sin αat)

    u = (c9x + c10)(c11t + c12)

  7. u = (c1 cosh αx + c2 sinh αx)(c3 cos αy + c4 sin αy)

    u = (c5 cos αx + c6 sin αx)(c7 cosh αy + c8 sinh αy)

    u = (c9x + c10)(c11y + c12)

  8. For λ = α2 > 0 there are three possibilities:

    (i) For 0 < α2 < 1,

    .

    (ii) For α2 > 1,

    .

    (iii) For α2 = 1,

    .

    The results for the case are similar.

    For

    .

  9. elliptic
  10. parabolic
  11. hyperbolic
  12. parabolic
  13. hyperbolic

Exercises 13.2, Page 725

  1. u(x, 0) = f(x), 0 < x < L

  2. u(x, 0) = f(x), 0 < x < L

  3. u(0, t) = sin(πt/L), u(L, t) = 0, t > 0

    u(x, 0) = f(x), 0 < x < L

  4. u(0, t) = 0, u(L, t) = 0, t > 0

Exercises 13.3, Page 727

  6. u(x, t) = ,

    where ,

Exercises 13.4, Page 732

  2. u(x, t) = sin at sin x

  8. where An = (x) sin nx dx and qn =

  1. u(x, t) = t + sin x cos at
  2. u(x, t) = sin 2x sin 2at

  3. where

Exercises 13.5, Page 738

  5. where

  7. where

  8. u = u1 + u2 where

  9. max temperature is u = 1

Exercises 13.6, Page 745

  3. where

Exercises 13.7, Page 750

  1. u(x, t) = where

    the αn are the consecutive positive roots of cot α = α/h

  2. u(x, y) = sinh αny sin αnx, where

    An = sin αnx dx

    and the αn are the consecutive positive roots of tan αa = −α/h

  3. where An =

Exercises 13.8, Page 753

  1. u(x, y, t) = sin mx sin ny,

    where Amn = [1 − (−1)m][1 − (−1)n]

  2. u(x, y, t) = sin mx sin ny cos a

    where Amn = [(−1)m − 1][(−1)n − 1]

  3. where and

  4. Use a = b = c = 1 with f(x, y) = u0 in Problem 5 and f(x, y) = −u0 in Problem 6. Add the two solutions.

Chapter 13 in Review, Page 754

  6. (a)

Exercises 14.1, Page 760

  1. u(r, θ) = rn sin nθ
  2. u(r, θ) = rn cos nθ

  4. where

  5. where

  6. where

  7. where

  9. where

  10. where

Exercises 14.2, Page 767

  5. where

  6. where

  8. (b)

    where

Exercises 14.3, Page 771

  2. u(r, θ) = cos θ

  3. where

  4. where

  6. where

Chapter 14 in Review, Page 772

  3. where

Exercises 15.1, Page 778

  1. (a) Let τ = u2 in the integral erf ().
  1. y(t) = eπt erf(t)
  2. Use the property

Exercises 15.2, Page 783

  2. u(x, t) = f
  3. u(x, t) =

  4. (a)

  6. u(x, t) = u1 + (u0 − u1) erfc
  9. u(x, t) = 60 + 40 erfc (t − 2)
  12. u(x, t) = u0 + u0
  13. u(x, t) = u0 − u0

  14. u(x, t) = u0

Exercises 15.3, Page 791

  2. where

  10. Let x = 2 in (7). Use a trigonometric identity and replace α by x. In part (b) make the change of variable 2x = kt.

Exercises 15.4, Page 797

  5. (a) u(x, t) =

Exercises 15.5, Page 801

Exercises 15.6, Page 810

  1. 1
  1. F8 =

Chapter 15 in Review, Page 811

  5. =

  1. = 1 + e−4t sin 2x

  3. or

Exercises 16.1, Page 819

  1. u11 = , u21 =
  2. u11 = u21 = /16, u22 = u12 = 3/16
  3. u21 = u12 = 12.50, u31 = u13 = 18.75, u32 = u23 = 37.50,

    u11 = 6.25, u22 = 25.00, u33 = 56.25

  4. (b) u14 = u41 = 0.5427, u24 = u42 = 0.6707,

    u34 = u43 = 0.6402, u33 = 0.9451, u44 = 0.4451

Exercises 16.2, Page 824

The tables in this section give a selection of the total number of approximations.

  1. A table consists of 11 rows and 8 columns. The column headers are as follows: Time, x = 0.25, x = 0.50, x = 0.75, x = 1.00, x = 1.25, x = 1.50, x = 1.75. The table entries are as follows: (row 1). Time, 0.000. x = 0.25, 1.0000. x = 0.50, 1.0000. x = 0.75, 1.0000. x = 1.00, 1.0000. x = 1.25, 0.0000. x = 1.50, 0.0000. x = 1.75, 0.0000. (row 2). Time, 0.100. x = 0.25, 0.3728. x = 0.50, 0.6288. x = 0.75, 0.6800. x = 1.00, 0.5904. x = 1.25, 0.3840. x = 1.50, 0.2176. x = 1.75, 0.0768. (row 3). Time, 0.200. x = 0.25, 0.2248. x = 0.50, 0.3942. x = 0.75, 0.4708. x = 1.00, 0.4562. x = 1.25, 0.3699. x = 1.50, 0.2517. x = 1.75, 0.1239. (row 4). Time, 0.300. x = 0.25, 0.1530. x = 0.50, 0.2752. x = 0.75, 0.3448. x = 1.00, 0.3545. x = 1.25, 0.3101. x = 1.50, 0.2262. x = 1.75, 0.1183. (row 5). Time, 0.400. x = 0.25, 0.1115. x = 0.50, 0.2034. x = 0.75, 0.2607. x = 1.00, 0.2757. x = 1.25, 0.2488. x = 1.50, 0.1865. x = 1.75, 0.0996. (row 6). Time, 0.500. x = 0.25, 0.0841. x = 0.50, 0.1545. x = 0.75, 0.2002. x = 1.00, 0.2144. x = 1.25, 0.1961. x = 1.50, 0.1487. x = 1.75, 0.0800. (row 7). Time, 0.600. x = 0.25, 0.0645. x = 0.50, 0.1189. x = 0.75, 0.1548. x = 1.00, 0.1668. x = 1.25, 0.1534. x = 1.50, 0.1169. x = 1.75, 0.0631. (row 8). Time, 0.700. x = 0.25, 0.0499. x = 0.50, 0.0921. x = 0.75, 0.1201. x = 1.00, 0.1297. x = 1.25, 0.1196. x = 1.50, 0.0914. x = 1.75, 0.0494. (row 9). Time, 0.800. x = 0.25, 0.0387. x = 0.50, 0.0715. x = 0.75, 0.0933. x = 1.00, 0.1009. x = 1.25, 0.0931. x = 1.50, 0.0712. x = 1.75, 0.0385. (row 10). Time, 0.900. x = 0.25, 0.0301. x = 0.50, 0.0555. x = 0.75, 0.0725. x = 1.00, 0.0785. x = 1.25, 0.0725. x = 1.50, 0.0554. x = 1.75, 0.0300. (row 11). Time, 1.000. x = 0.25, 0.0234. x = 0.50, 0.0432. x = 0.75, 0.0564. x = 1.00, 0.0610. x = 1.25, 0.0564. x = 1.50, 0.0431. x = 1.75, 0.0233.
  2. A table consists of 11 rows and 8 columns. The column headers are as follows: Time, x = 0.25, x = 0.50, x = 0.75, x = 1.00, x = 1.25, x = 1.50, x = 1.75. The table entries are as follows: (row 1). Time, 0.000. x = 0.25, 1.0000. x = 0.50, 1.0000. x = 0.75, 1.0000. x = 1.00, 1.0000. x = 1.25, 0.0000. x = 1.50, 0.0000. x = 1.75, 0.0000. (row 2). Time, 0.100. x = 0.25, 0.4015. x = 0.50, 0.6577. x = 0.75, 0.7084. x = 1.00, 0.5837. x = 1.25, 0.3753. x = 1.50, 0.1871. x = 1.75, 0.0684. (row 3). Time, 0.200. x = 0.25, 0.2430. x = 0.50, 0.4198. x = 0.75, 0.4921. x = 1.00, 0.4617. x = 1.25, 0.3622. x = 1.50, 0.2362. x = 1.75, 0.1132. (row 4). Time, 0.300. x = 0.25, 0.1643. x = 0.50, 0.2924. x = 0.75, 0.3604. x = 1.00, 0.3626. x = 1.25, 0.3097. x = 1.50, 0.2208. x = 1.75, 0.1136. (row 5). Time, 0.400. x = 0.25, 0.1187. x = 0.50, 0.2150. x = 0.75, 0.2725. x = 1.00, 0.2843. x = 1.25, 0.2528. x = 1.50, 0.1871. x = 1.75, 0.0989. (row 6). Time, 0.500. x = 0.25, 0.0891. x = 0.50, 0.1630. x = 0.75, 0.2097. x = 1.00, 0.2228. x = 1.25, 0.2020. x = 1.50, 0.1521. x = 1.75, 0.0814. (row 7). Time, 0.600. x = 0.25, 0.0683. x = 0.50, 0.1256. x = 0.75, 0.1628. x = 1.00, 0.1746. x = 1.25, 0.1598. x = 1.50, 0.1214. x = 1.75, 0.0653. (row 8). Time, 0.700. x = 0.25, 0.0530. x = 0.50, 0.0976. x = 0.75, 0.1270. x = 1.00, 0.1369. x = 1.25, 0.1259. x = 1.50, 0.0959. x = 1.75, 0.0518. (row 9). Time, 0.800. x = 0.25, 0.0413. x = 0.50, 0.0762. x = 0.75, 0.0993. x = 1.00, 0.1073. x = 1.25, 0.0989. x = 1.50, 0.0755. x = 1.75, 0.0408. (row 10). Time, 0.900. x = 0.25, 0.0323. x = 0.50, 0.0596. x = 0.75, 0.0778. x = 1.00, 0.0841. x = 1.25, 0.0776. x = 1.50, 0.0593. x = 1.75, 0.0321. (row 11). Time, 1.000. x = 0.25, 0.0253. x = 0.50, 0.0466. x = 0.75, 0.0609. x = 1.00, 0.0659. x = 1.25, 0.0608. x = 1.50, 0.0465. x = 1.75, 0.0252.

    Absolute errors are approximately 2.2 × 10–2, 3.7 × 10–2, 1.3 × 10–2.

  3. A table consists of 11 rows and 8 columns. The column headers are as follows: Time, x = 0.25, x = 0.50, x = 0.75, x = 1.00, x = 1.25, x = 1.50, x = 1.75. The table entries are as follows: (row 1). Time, 0.00. x = 0.25, 1.0000. x = 0.50, 1.0000. x = 0.75, 1.0000. x = 1.00, 1.0000. x = 1.25, 0.0000. x = 1.50, 0.0000. x = 1.75, 0.0000. (row 2). Time, 0.10. x = 0.25, 0.3972. x = 0.50, 0.6551. x = 0.75, 0.7043. x = 1.00, 0.5883. x = 1.25, 0.3723. x = 1.50, 0.1955. x = 1.75, 0.0653. (row 3). Time, 0.20. x = 0.25, 0.2409. x = 0.50, 0.4171. x = 0.75, 0.4901. x = 1.00, 0.4620. x = 1.25, 0.3636. x = 1.50, 0.2385. x = 1.75, 0.1145. (row 4). Time, 0.30. x = 0.25, 0.1631. x = 0.50, 0.2908. x = 0.75, 0.3592. x = 1.00, 0.3624. x = 1.25, 0.3105. x = 1.50, 0.2220. x = 1.75, 0.1145. (row 5). Time, 0.40. x = 0.25, 0.1181. x = 0.50, 0.2141. x = 0.75, 0.2718. x = 1.00, 0.2840. x = 1.25, 0.2530. x = 1.50, 0.1876. x = 1.75, 0.0993. (row 6). Time, 0.50. x = 0.25, 0.0888. x = 0.50, 0.1625. x = 0.75, 0.2092. x = 1.00, 0.2226. x = 1.25, 0.2020. x = 1.50, 0.1523. x = 1.75, 0.0816. (row 7). Time, 0.60. x = 0.25, 0.0681. x = 0.50, 0.1253. x = 0.75, 0.1625. x = 1.00, 0.1744. x = 1.25, 0.1597. x = 1.50, 0.1214. x = 1.75, 0.0654. (row 8). Time, 0.70. x = 0.25, 0.0528. x = 0.50, 0.0974. x = 0.75, 0.1268. x = 1.00, 0.1366. x = 1.25, 0.1257. x = 1.50, 0.0959. x = 1.75, 0.0518. (row 9). Time, 0.80. x = 0.25, 0.0412. x = 0.50, 0.0760. x = 0.75, 0.0991. x = 1.00, 0.1071. x = 1.25, 0.0987. x = 1.50, 0.0754. x = 1.75, 0.0408. (row 10). Time, 0.90. x = 0.25, 0.0322. x = 0.50, 0.0594. x = 0.75, 0.0776. x = 1.00, 0.0839. x = 1.25, 0.0774. x = 1.50, 0.0592. x = 1.75, 0.0320. (row 11). Time, 1.00. x = 0.25, 0.0252. x = 0.50, 0.0465. x = 0.75, 0.0608. x = 1.00, 0.0657. x = 1.25, 0.0607. x = 1.50, 0.0464. x = 1.75, 0.0251.

    Absolute errors are approximately 1.8 × 10–2, 3.7 × 10–2, 1.3 × 10–2.

  4. (a)

    A table consists of 6 rows and 10 columns. The column headers are as follows: Time, x = 2.00, x = 4.00, x = 6.00, x = 8.00, x = 10.00, x = 12.00, x = 14.00, x = 16.00, x = 18.00. The table entries are as follows: (row 1). Time, 0.00. x = 2.00, 30.0000. x = 4.00, 30.0000. x = 6.00, 30.0000. x = 8.00, 30.0000. x = 10.00, 30.0000. x = 12.00, 30.0000. x = 14.00, 30.0000. x = 16.00, 30.0000. x = 18.00, 30.0000. (row 2). Time, 2.00. x = 2.00, 27.6450. x = 4.00, 29.9037. x = 6.00, 29.9970. x = 8.00, 29.9999. x = 10.00, 30.0000. x = 12.00, 29.9999. x = 14.00, 29.9970. x = 16.00, 29.9037. x = 18.00, 27.6450. (row 3). Time, 4.00. x = 2.00, 25.6452. x = 4.00, 29.6517. x = 6.00, 29.9805. x = 8.00, 29.9991. x = 10.00, 29.9999. x = 12.00, 29.9991. x = 14.00, 29.9805. x = 16.00, 29.6517. x = 18.00, 25.6452. (row 4). Time, 6.00. x = 2.00, 23.9347. x = 4.00, 29.2922. x = 6.00, 29.9421. x = 8.00, 29.9963. x = 10.00, 29.9996. x = 12.00, 29.9963. x = 14.00, 29.9421. x = 16.00, 29.2922. x = 18.00, 23.9347. (row 5). Time, 8.00. x = 2.00, 22.4612. x = 4.00, 28.8606. x = 6.00, 29.8782. x = 8.00, 29.9898. x = 10.00, 29.9986. x = 12.00, 29.9898. x = 14.00, 29.8782. x = 16.00, 28.8606. x = 18.00, 22.4612. (row 6). Time, 10.00. x = 2.00, 21.1829. x = 4.00, 28.3831. x = 6.00, 29.7878. x = 8.00, 29.9782. x = 10.00, 29.9964. x = 12.00, 29.9782. x = 14.00, 29.7878. x = 16.00, 28.3831. x = 18.00, 21.1829.

    (b)

    A table consists of 6 rows and 10 columns. The column headers are as follows: Time, x = 5.00, x = 10.00, x = 15.00, x = 20.00, x = 25.00, x = 30.00, x = 35.00, x = 40.00, x = 45.00. The table entries are as follows: (row 1). Time, 0.00. x = 5.00, 30.0000. x = 10.00, 30.0000. x = 15.00, 30.0000. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 30.0000. x = 40.00, 30.0000. x = 45.00, 30.0000. (row 2). Time, 2.00. x = 5.00, 29.5964. x = 10.00, 29.9973. x = 15.00, 30.0000. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 30.0000. x = 40.00, 29.9973. x = 45.00, 29.5964. (row 3). Time, 4.00. x = 5.00, 29.2036. x = 10.00, 29.9893. x = 15.00, 29.9999. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 29.9999. x = 40.00, 29.9893. x = 45.00, 29.2036. (row 4). Time, 6.00. x = 5.00, 28.8212. x = 10.00, 29.9762. x = 15.00, 29.9997. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 29.9997. x = 40.00, 29.9762. x = 45.00, 28.8213. (row 5). Time, 8.00. x = 5.00, 28.4490. x = 10.00, 29.9585. x = 15.00, 29.9992. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 29.9993. x = 40.00, 29.9585. x = 45.00, 28.4490. (row 6). Time, 10.00. x = 5.00, 28.0864. x = 10.00, 29.9363. x = 15.00, 29.9986. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 29.9986. x = 40.00, 29.9363. x = 45.00, 28.0864.

    (c)

    A table consists of 6 rows and 10 columns. The column headers are as follows: Time, x = 2.00, x = 4.00, x = 6.00, x = 8.00, x = 10.00, x = 12.00, x = 14.00, x = 16.00, x = 18.00. The table entries are as follows: (row 1). Time, 0.00. x = 2.00, 18.0000. x = 4.00, 32.0000. x = 6.00, 42.0000. x = 8.00, 48.0000. x = 10.00, 50.0000. x = 12.00, 48.0000. x = 14.00, 42.0000. x = 16.00, 32.0000. x = 18.00, 18.0000. (row 2). Time, 2.00. x = 2.00, 15.3312. x = 4.00, 28.5348. x = 6.00, 38.3465. x = 8.00, 44.3067. x = 10.00, 46.3001. x = 12.00, 44.3067. x = 14.00, 38.3465. x = 16.00, 28.5348. x = 18.00, 15.3312. (row 3). Time, 4.00. x = 2.00, 13.6371. x = 4.00, 25.6867. x = 6.00, 34.9416. x = 8.00, 40.6988. x = 10.00, 42.6453. x = 12.00, 40.6988. x = 14.00, 34.9416. x = 16.00, 25.6867. x = 18.00, 13.6371. (row 4). Time, 6.00. x = 2.00, 12.3012. x = 4.00, 23.2863. x = 6.00, 31.8624. x = 8.00, 37.2794. x = 10.00, 39.1273. x = 12.00, 37.2794. x = 14.00, 31.8624. x = 16.00, 23.2863. x = 18.00, 12.3012. (row 5). Time, 8.00. x = 2.00, 11.1659. x = 4.00, 21.1877. x = 6.00, 29.0757. x = 8.00, 34.0984. x = 10.00, 35.8202. x = 12.00, 34.0984. x = 14.00, 29.0757. x = 16.00, 21.1877. x = 18.00, 11.1659. (row 6). Time, 10.00. x = 2.00, 10.1665. x = 4.00, 19.3143. x = 6.00, 26.5439. x = 8.00, 31.1662. x = 10.00, 32.7549. x = 12.00, 31.1662. x = 14.00, 26.5439. x = 16.00, 19.3143. x = 18.00, 10.1665.

    (d)

    A table consists of 6 rows and 10 columns. The column headers are as follows: Time, x = 10.00, x = 20.00, x = 30.00, x = 40.00, x = 50.00, x = 60.00, x = 70.00, x = 80.00, x = 90.00. The table entries are as follows: (row 1). Time, 0.00. x = 10.00, 8.0000. x = 20.00, 16.0000. x = 30.00, 24.0000. x = 40.00, 32.0000. x = 50.00, 40.0000. x = 60.00, 32.0000. x = 70.00, 24.0000. x = 80.00, 16.0000. x = 90.00, 8.0000. (row 2). Time, 2.00. x = 10.00, 8.0000. x = 20.00, 16.0000. x = 30.00, 23.9999. x = 40.00, 31.9918. x = 50.00, 39.4932. x = 60.00, 31.9918. x = 70.00, 23.9999. x = 80.00, 16.0000. x = 90.00, 8.0000. (row 3). Time, 4.00. x = 10.00, 8.0000. x = 20.00, 16.0000. x = 30.00, 23.9993. x = 40.00, 31.9686. x = 50.00, 39.0175. x = 60.00, 39.9686. x = 70.00, 23.9993. x = 80.00, 16.0000. x = 90.00, 8.0000. (row 4). Time, 6.00. x = 10.00, 8.0000. x = 20.00, 15.9999. x = 30.00, 23.9978. x = 40.00, 31.9323. x = 50.00, 38.5701. x = 60.00, 31.9323. x = 70.00, 23.9978. x = 80.00, 15.9999. x = 90.00, 8.0000. (row 5). Time, 8.00. x = 10.00, 8.0000. x = 20.00, 15.9998. x = 30.00, 23.9950. x = 40.00, 31.8844. x = 50.00, 38.1483. x = 60.00, 31.8844. x = 70.00, 23.9950. x = 80.00, 15.9998. x = 90.00, 8.0000. (row 6). Time, 10.00. x = 10.00, 8.0000. x = 20.00, 15.9996. x = 30.00, 23.9908. x = 40.00, 31.8265. x = 50.00, 37.7498. x = 60.00, 31.8265. x = 70.00, 23.9908. x = 80.00, 15.9996. x = 90.00, 8.0000.

  5. (a)

    A table consists of 6 rows and 10 columns. The column headers are as follows: Time, x = 2.00, x = 4.00, x = 6.00, x = 8.00, x = 10.00, x = 12.00, x = 14.00, x = 16.00, x = 18.00. The table entries are as follows: (row 1). Time, 0.00. x = 2.00, 30.0000. x = 4.00, 30.0000. x = 6.00, 30.0000. x = 8.00, 30.0000. x = 10.00, 30.0000. x = 12.00, 30.0000. x = 14.00, 30.0000. x = 16.00, 30.0000. x = 18.00, 30.0000. (row 2). Time, 2.00. x = 2.00, 27.6450. x = 4.00, 29.9037. x = 6.00, 29.9970. x = 8.00, 29.9999. x = 10.00, 30.0000. x = 12.00, 30.0000. x = 14.00, 29.9990. x = 16.00, 29.9679. x = 18.00, 29.2150. (row 3). Time, 4.00. x = 2.00, 25.6452. x = 4.00, 29.6517. x = 6.00, 29.9805. x = 8.00, 29.9991. x = 10.00, 30.0000. x = 12.00, 29.9997. x = 14.00, 29.9935. x = 16.00, 29.8839. x = 18.00, 28.5484. (row 4). Time, 6.00. x = 2.00, 23.9347. x = 4.00, 29.2922. x = 6.00, 29.9421. x = 8.00, 29.9963. x = 10.00, 29.9997. x = 12.00, 29.9988. x = 14.00, 29.9807. x = 16.00, 29.7641. x = 18.00, 27.9782. (row 5). Time, 8.00. x = 2.00, 22.4612. x = 4.00, 28.8606. x = 6.00, 29.8782. x = 8.00, 29.9899. x = 10.00, 29.9991. x = 12.00, 29.9966. x = 14.00, 29.9594. x = 16.00, 29.6202. x = 18.00, 27.4870. (row 6). Time, 10.00. x = 2.00, 21.1829. x = 4.00, 28.3831. x = 6.00, 29.7878. x = 8.00, 29.9783. x = 10.00, 29.9976. x = 12.00, 29.9927. x = 14.00, 29.9293. x = 16.00, 29.4610. x = 18.00, 27.0610.

    (b)

    A table consists of 6 rows and 10 columns. The column headers are as follows: Time, x = 5.00, x = 10.00, x = 15.00, x = 20.00, x = 25.00, x = 30.00, x = 35.00, x = 40.00, x = 45.00. The table entries are as follows: (row 1). Time, 0.00. x = 5.00, 30.0000. x = 10.00, 30.0000. x = 15.00, 30.0000. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 30.0000. x = 40.00, 30.0000. x = 45.00, 30.0000. (row 2). Time, 2.00. x = 5.00, 29.5964. x = 10.00, 29.9973. x = 15.00, 30.0000. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 30.0000. x = 40.00, 29.9991. x = 45.00, 29.8655. (row 3). Time, 4.00. x = 5.00, 29.2036. x = 10.00, 29.9893. x = 15.00, 29.9999. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 30.0000. x = 40.00, 29.9964. x = 45.00, 29.7345. (row 4). Time, 6.00. x = 5.00, 28.8212. x = 10.00, 29.9762. x = 15.00, 29.9997. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 29.9999. x = 40.00, 29.9921. x = 45.00, 29.6071. (row 5). Time, 8.00. x = 5.00, 28.4490. x = 10.00, 29.9585. x = 15.00, 29.9992. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 29.9997. x = 40.00, 29.9862. x = 45.00, 29.4830. (row 6). Time, 10.00. x = 5.00, 28.0864. x = 10.00, 29.9363. x = 15.00, 29.9986. x = 20.00, 30.0000. x = 25.00, 30.0000. x = 30.00, 30.0000. x = 35.00, 29.9995. x = 40.00, 29.9788. x = 45.00, 29.3621.

    (c)

    A table consists of 6 rows and 10 columns. The column headers are as follows: Time, x = 2.00, x = 4.00, x = 6.00, x = 8.00, x = 10.00, x = 12.00, x = 14.00, x = 16.00, x = 18.00. The table entries are as follows: (row 1). Time, 0.00. x = 2.00, 18.0000. x = 4.00, 32.0000. x = 6.00, 42.0000. x = 8.00, 48.0000. x = 10.00, 50.0000. x = 12.00, 48.0000. x = 14.00, 42.0000. x = 16.00, 32.0000. x = 18.00, 18.0000. (row 2). Time, 2.00. x = 2.00, 15.3312. x = 4.00, 28.5350. x = 6.00, 38.3477. x = 8.00, 44.3130. x = 10.00, 46.3327. x = 12.00, 44.4671. x = 14.00, 39.0872. x = 16.00, 31.5755. x = 18.00, 24.6930. (row 3). Time, 4.00. x = 2.00, 13.6381. x = 4.00, 25.6913. x = 6.00, 34.9606. x = 8.00, 40.7728. x = 10.00, 42.9127. x = 12.00, 41.5716. x = 14.00, 37.4340. x = 16.00, 31.7086. x = 18.00, 25.6986. (row 4). Time, 6.00. x = 2.00, 12.3088. x = 4.00, 23.3146. x = 6.00, 31.9546. x = 8.00, 37.5566. x = 10.00, 39.8880. x = 12.00, 39.1565. x = 14.00, 36.9745. x = 16.00, 31.2134. x = 18.00, 25.7128. (row 5). Time, 8.00. x = 2.00, 11.1946. x = 4.00, 21.2785. x = 6.00, 29.3217. x = 8.00, 34.7092. x = 10.00, 37.2109. x = 12.00, 36.9834. x = 14.00, 34.5032. x = 16.00, 30.4279. x = 18.00, 25.4167. (row 6). Time, 10.00. x = 2.00, 10.2377. x = 4.00, 19.5150. x = 6.00, 27.0178. x = 8.00, 32.1929. x = 10.00, 34.8117. x = 12.00, 34.9710. x = 14.00, 33.0338. x = 16.00, 29.5224. x = 18.00, 25.0019.

    (d)

    A table consists of 6 rows and 10 columns. The column headers are as follows: Time, x = 10.00, x = 20.00, x = 30.00, x = 40.00, x = 50.00, x = 60.00, x = 70.00, x = 80.00, x = 90.00. The table entries are as follows: (row 1). Time, 0.00. x = 10.00, 8.0000. x = 20.00, 16.0000. x = 30.00, 24.0000. x = 40.00, 32.0000. x = 50.00, 40.0000. x = 60.00, 32.0000. x = 70.00, 24.0000. x = 80.00, 16.0000. x = 90.00, 8.0000. (row 2). Time, 2.00. x = 10.00, 8.0000. x = 20.00, 16.0000. x = 30.00, 23.9999. x = 40.00, 31.9918. x = 50.00, 39.4932. x = 60.00, 31.9918. x = 70.00, 24.0000. x = 80.00, 16.0102. x = 90.00, 8.6333. (row 3). Time, 4.00. x = 10.00, 8.0000. x = 20.00, 16.0000. x = 30.00, 23.9993. x = 40.00, 31.9686. x = 50.00, 39.0175. x = 60.00, 39.9687. x = 70.00, 24.0002. x = 80.00, 16.0391. x = 90.00, 9.2272. (row 4). Time, 6.00. x = 10.00, 8.0000. x = 20.00, 15.9999. x = 30.00, 23.9978. x = 40.00, 31.9323. x = 50.00, 38.5701. x = 60.00, 31.9324. x = 70.00, 24.0005. x = 80.00, 16.0845. x = 90.00, 9.7846. (row 5). Time, 8.00. x = 10.00, 8.0000. x = 20.00, 15.9998. x = 30.00, 23.9950. x = 40.00, 31.8844. x = 50.00, 38.1483. x = 60.00, 31.8846. x = 70.00, 24.0012. x = 80.00, 16.1441. x = 90.00, 10.3084. (row 6). Time, 10.00. x = 10.00, 8.0000. x = 20.00, 15.9996. x = 30.00, 23.9908. x = 40.00, 31.8265. x = 50.00, 37.7499. x = 60.00, 31.8269. x = 70.00, 24.0023. x = 80.00, 16.2160. x = 90.00, 10.8012.

  6. (a) ψ(x) = x + 20

    (b)

    A table consists of 20 rows and 5 columns. The column headers are as follows: Time, x = 4.00, x = 8.00, x = 12.00, x = 16.00. The table entries are as follows: (row 1). Time, 0.00. x = 4.00, 50.0000. x = 8.00, 50.0000. x = 12.00, 50.0000. x = 16.00, 50.0000. (row 2). Time, 10.00. x = 4.00, 32.7433. x = 8.00, 44.2679. x = 12.00, 45.4228. x = 16.00, 38.8160. (row 3). Time, 20.00. x = 4.00, 29.9946. x = 8.00, 36.2354. x = 12.00, 38.3148. x = 16.00, 35.8160. (row 4). Time, 30.00. x = 4.00, 26.9487. x = 8.00, 32.1409. x = 12.00, 34.0874. x = 16.00, 32.9644. (row 5). Time, 50.00. x = 4.00, 24.1178. x = 8.00, 27.4348. x = 12.00, 29.4296. x = 16.00, 30.1207. (row 6). Time, 70.00. x = 4.00, 22.8995. x = 8.00, 25.4560. x = 12.00, 27.4554. x = 16.00, 28.8998. (row 7). Time, 90.00. x = 4.00, 22.3817. x = 8.00, 24.6176. x = 12.00, 26.6175. x = 16.00, 28.3817. (row 8). Time, 110.00. x = 4.00, 22.1619. x = 8.00, 24.2620. x = 12.00, 26.2620. x = 16.00, 28.1619. (row 9). Time, 130.00. x = 4.00, 22.0687. x = 8.00, 24.1112. x = 12.00, 26.1112. x = 16.00, 28.0687. (row 10). Time, 150.00. x = 4.00, 22.0291. x = 8.00, 24.0472. x = 12.00, 26.0472. x = 16.00, 28.0291. (row 11). Time, 170.00. x = 4.00, 22.0124. x = 8.00, 24.0200. x = 12.00, 26.0200. x = 16.00, 28.0124. (row 12). Time, 190.00. x = 4.00, 22.0052. x = 8.00, 24.0085. x = 12.00, 26.0085. x = 16.00, 28.0052. (row 13). Time, 210.00. x = 4.00, 22.0022. x = 8.00, 24.0036. x = 12.00, 26.0036. x = 16.00, 28.0022. (row 14). Time, 230.00. x = 4.00, 22.0009. x = 8.00, 24.0015. x = 12.00, 26.0015. x = 16.00, 28.0009. (row 15). Time, 250.00. x = 4.00, 22.0004. x = 8.00, 24.0007. x = 12.00, 26.0007. x = 16.00, 28.0004. (row 16). Time, 270.00. x = 4.00, 22.0002. x = 8.00, 24.0003. x = 12.00, 26.0003. x = 16.00, 28.0002. (row 17). Time, 290.00. x = 4.00, 22.0001. x = 8.00, 24.0001. x = 12.00, 26.0001. x = 16.00, 28.0001. (row 18). Time, 310.00. x = 4.00, 22.0000. x = 8.00, 24.0001. x = 12.00, 26.0001. x = 16.00, 28.0000. (row 19). Time, 330.00. x = 4.00, 22.000. x = 8.00, 24.0000. x = 12.00, 26.0000. x = 16.00, 28.0000. (row 20). Time, 350.00. x = 4.00, 22.000. x = 8.00, 24.0000. x = 12.00, 26.0000. x = 16.00, 28.0000.

Exercises 16.3, Page 827

  1. (a)

    A table consists of 9 rows and 10 columns. The column headers are as follows: Time, x = 0.1, x = 0.2, x = 0.3, x = 0.4, x = 0.5, x = 0.6, x = 0.7, x = 0.8, x = 0.9. The table entries are as follows: (row 1). Time, 0.00. x = 0.1, 0.0000. x = 0.2, 0.0000. x = 0.3, 0.0000. x = 0.4, 0.0000. x = 0.5, 0.0000. x = 0.6, 0.5000. x = 0.7, 0.5000. x = 0.8, 0.5000. x = 0.9, 0.5000. (row 2). Time, 0.12. x = 0.1, 0.0000. x = 0.2, 0.0000. x = 0.3, 0.0082. x = 0.4, 0.1126. x = 0.5, 0.3411. x = 0.6, 0.1589. x = 0.7, 0.3792. x = 0.8, 0.3710. x = 0.9, 0.0462. (row 3). Time, 0.24. x = 0.1, 0.0071. x = 0.2, 0.0657. x = 0.3, 0.2447. x = 0.4, 0.3159. x = 0.5, 0.1735. x = 0.6, 0.2463. x = 0.7, negative 0.1266. x = 0.8, negative 0.3056. x = 0.9, negative 0.0625. (row 4). Time, 0.36. x = 0.1, 0.1623. x = 0.2, 0.3197. x = 0.3, 0.2458. x = 0.4, 0.1657. x = 0.5, 0.0877. x = 0.6, negative 0.2853. x = 0.7, negative 0.2843. x = 0.8, negative 0.2104. x = 0.9, negative 0.2887. (row 5). Time, 0.48. x = 0.1, 0.1965. x = 0.2, 0.1410. x = 0.3, 0.1149. x = 0.4, negative 0.1216. x = 0.5, negative 0.3593. x = 0.6, negative 0.2381. x = 0.7, negative 0.1977. x = 0.8, negative 0.1715. x = 0.9, negative 0.0800. (row 6). Time, 0.60. x = 0.1, negative 0.2194. x = 0.2, negative 0.2069. x = 0.3, negative 0.3875. x = 0.4, negative 0.3411. x = 0.5, negative 0.1901. x = 0.6, negative 0.1662. x = 0.7, negative 0.0666. x = 0.8, negative 0.1140. x = 0.9, negative 0.0446. (row 7). Time, 0.72. x = 0.1, negative 0.3003. x = 0.2, negative 0.6865. x = 0.3, negative 0.5097. x = 0.4, negative 0.3230. x = 0.5, negative 0.1585. x = 0.6, 0.0156. x = 0.7, 0.0893. x = 0.8, negative 0.0874. x = 0.9, 0.0384. (row 8). Time, 0.84. x = 0.1, negative 0.2647. x = 0.2, negative 0.1633. x = 0.3, negative 0.3546. x = 0.4, negative 0.3214. x = 0.5, negative 0.1763. x = 0.6, negative 0.0954. x = 0.7, negative 0.1249. x = 0.8, 0.0665. x = 0.9, negative 0.0386. (row 9). Time, 0.96. x = 0.1, 0.3012. x = 0.2, 0.1081. x = 0.3, 0.1380. x = 0.4, negative 0.0487. x = 0.5, negative 0.2974. x = 0.6, negative 0.3407. x = 0.7, negative 0.1250. x = 0.8, negative 0.1548. x = 0.9, 0.0092.

    (b)

    A table consists of 6 rows and 4 columns. The column headers are as follows: Time, x = 0.25, x = 0.50, x = 0.75. The table entries are as follows: (row 1). Time, 0.00. x = 0.25, 0.1875. x = 0.50, 0.2500. x = 0.75, 0.1875. (row 2). Time, 0.20. x = 0.25, 0.1491. x = 0.50, 0.2100. x = 0.75, 0.1491. (row 3). Time, 0.40. x = 0.25, 0.0556. x = 0.50, 0.0938. x = 0.75, 0.0556. (row 4). Time, 0.60. x = 0.25, negative 0.0501. x = 0.50, negative 0.0682. x = 0.75, negative 0.0501. (row 5). Time, 0.80. x = 0.25, negative 0.1361. x = 0.50, negative 0.2072. x = 0.75, negative 0.1361. (row 6). Time, 1.00. x = 0.25, negative 0.1802. x = 0.50, negative 0.2591. x = 0.75, negative 0.1802.

    (c)

    A table consists of 6 rows and 5 columns. The column headers are as follows: Time, x = 0.4, x = 0.8, x = 1.2, x = 1.6. The table entries are as follows: (row 1). Time, 0.00. x = 0.4, 0.0032. x = 0.8, 0.5273. x = 1.2, 0.5273. x = 1.6, 0.0032. (row 2). Time, 0.20. x = 0.4, 0.0652. x = 0.8, 0.4638. x = 1.2, 0.4638. x = 1.6, 0.0652. (row 3). Time, 0.40. x = 0.4, 0.2065. x = 0.8, 0.3035. x = 1.2, 0.3035. x = 1.6, 0.2065. (row 4). Time, 0.60. x = 0.4, 0.3208. x = 0.8, 0.1190. x = 1.2, 0.1190. x = 1.6, 0.3208. (row 5). Time, 0.80. x = 0.4, 0.3094. x = 0.8, negative 0.0180. x = 1.2, negative 0.0180. x = 1.6, 0.3094. (row 6). Time, 1.00. x = 0.4, 0.1450. x = 0.8, negative 0.0768. x = 1.2, negative 0.0768. x = 1.6, 0.1450.

  2. (a)

    A table consists of 6 rows and 5 columns. The column headers are as follows: Time, x = 0.2, x = 0.4, x = 0.6, x = 0.8. The table entries are as follows: (row 1). Time, 0.00. x = 0.2, 0.5878. x = 0.4, 0.9511. x = 0.6, 0.9511. x = 0.8, 0.5878. (row 2). Time, 0.10. x = 0.2, 0.5599. x = 0.4, 0.9059. x = 0.6, 0.9059. x = 0.8, 0.5599. (row 3). Time, 0.20. x = 0.2, 0.4788. x = 0.4, 0.7748. x = 0.6, 0.7748. x = 0.8, 0.4788. (row 4). Time, 0.30. x = 0.2, 0.3524. x = 0.4, 0.5701. x = 0.6, 0.5701. x = 0.8, 0.3524. (row 5). Time, 0.40. x = 0.2, 0.1924. x = 0.4, 0.3113. x = 0.6, 0.3113. x = 0.8, 0.1924. (row 6). Time, 0.50. x = 0.2, 0.0142. x = 0.4, 0.0230. x = 0.6, 0.0230. x = 0.8, 0.0142.

    (b)

    A table consists of 11 rows and 5 columns. The column headers are as follows: Time, x = 0.2, x = 0.4, x = 0.6, x = 0.8. The table entries are as follows: (row 1). Time, 0.00. x = 0.2, 0.5878. x = 0.4, 0.9511. x = 0.6, 0.9511. x = 0.8, 0.5878. (row 2). Time, 0.05. x = 0.2, 0.5808. x = 0.4, 0.9397. x = 0.6, 0.9397. x = 0.8, 0.5808. (row 3). Time, 0.10. x = 0.2, 0.5599. x = 0.4, 0.9060. x = 0.6, 0.9060. x = 0.8, 0.5599. (row 4). Time, 0.15. x = 0.2, 0.5257. x = 0.4, 0.8507. x = 0.6, 0.8507. x = 0.8, 0.5257. (row 5). Time, 0.20. x = 0.2, 0.4790. x = 0.4, 0.7750. x = 0.6, 0.7750. x = 0.8, 0.4790. (row 6). Time, 0.25. x = 0.2, 0.4209. x = 0.4, 0.6810. x = 0.6, 0.6810. x = 0.8, 0.4209. (row 7). Time, 0.30. x = 0.2, 0.3527. x = 0.4, 0.5706. x = 0.6, 0.5706. x = 0.8, 0.3527. (row 8). Time, 0.35. x = 0.2, 0.2761. x = 0.4, 0.4467. x = 0.6, 0.4467. x = 0.8, 0.2761. (row 9). Time, 0.40. x = 0.2, 0.1929. x = 0.4, 0.3122. x = 0.6, 0.3122. x = 0.8, 0.1929. (row 10). Time, 0.45. x = 0.2, 0.1052. x = 0.4, 0.1701. x = 0.6, 0.1701. x = 0.8, 0.1052. (row 11). Time, 0.50. x = 0.2, 0.0149. x = 0.4, 0.0241. x = 0.6, 0.0241. x = 0.8, 0.0149.

  3. A table consists of 17 rows and 6 columns. The column headers are as follows: Time, x = 10, x = 20, x = 30, x = 40, x = 50. The table entries are as follows: (row 1). Time, 0.00000. x = 10, 0.1000. x = 20, 0.2000. x = 30, 0.3000. x = 40, 0.2000. x = 50, 0.1000. (row 2). Time, 0.60134. x = 10, 0.0984. x = 20, 0.1688. x = 30, 0.1406. x = 40, 0.1688. x = 50, 0.0984. (row 3). Time, 1.20268. x = 10, 0.0226. x = 20, negative 0.0121. x = 30, 0.0085. x = 40, negative 0.0121. x = 50, 0.0226. (row 4). Time, 1.80401. x = 10, negative 0.1271. x = 20, negative 0.1347. x = 30, negative 0.1566. x = 40, negative 0.1347. x = 50, negative 0.1271. (row 5). Time, 2.40535. x = 10, negative 0.0920. x = 20, negative 0.2292. x = 30, negative 0.2571. x = 40, negative 0.2292. x = 50, negative 0.0920. (row 6). Time, 3.00669. x = 10, negative 0.0932. x = 20, negative 0.1445. x = 30, negative 0.2018. x = 40, negative 0.1445. x = 50, negative 0.0932. (row 7). Time, 3.60803. x = 10, negative 0.0284. x = 20, negative 0.0205. x = 30, 0.0336. x = 40, negative 0.0205. x = 50, negative 0.0284. (row 8). Time, 4.20936. x = 10, 0.1064. x = 20, 0.1555. x = 30, 0.1265. x = 40, 0.1555. x = 50, 0.1064. (row 9). Time, 4.81070. x = 10, 0.1273. x = 20, 0.2060. x = 30, 0.2612. x = 40, 0.2060. x = 50, 0.1273. (row 10). Time, 5.41204. x = 10, 0.0625. x = 20, 0.1689. x = 30, 0.2038. x = 40, 0.1689. x = 50, 0.0625. (row 11). Time, 6.01338. x = 10, 0.0436. x = 20, 0.0086. x = 30, negative 0.0080. x = 40, 0.0086. x = 50, 0.0436. (row 12). Time, 6.61472. x = 10, negative 0.0931. x = 20, negative 0.1364. x = 30, negative 0.1578. x = 40, negative 0.1364. x = 50, negative 0.0931. (row 13). Time, 7.21605. x = 10, negative 0.1436. x = 20, negative 0.2173. x = 30, negative 0.2240. x = 40, negative 0.2173. x = 50, negative 0.1436. (row 14). Time, 7.81739. x = 10, negative 0.0625. x = 20, negative 0.1644. x = 30, negative 0.2247. x = 40, negative 0.1644. x = 50, negative 0.0625. (row 15). Time, 8.41873. x = 10, negative 0.0287. x = 20, negative 0.0192. x = 30, negative 0.0085. x = 40, negative 0.0192. x = 50, negative 0.0287. (row 16). Time, 9.02007. x = 10, 0.0654. x = 20, 0.1332. x = 30, 0.1755. x = 40, 0.1332. x = 50, 0.0654. (row 17). Time, 9.62140. x = 10, 0.1540. x = 20, 0.2189. x = 30, 0.2089. x = 40, 0.2189. x = 50, 0.1540.

    Note: Time is expressed in milliseconds.

Chapter 16 in Review, Page 828

  1. u11 = 0.8929, u21 = 3.5714, u31 = 13.3929
  2. (a)

    A table consists of 6 rows and 4 columns. The column headers are as follows: x = 0.20, x = 0.40, x = 0.60, x = 0.80. The table entries are as follows: (row 1). x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.6000. x = 0.80, 0.8000. (row 2). x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.6000. x = 0.80, 0.5500. (row 3). x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.5375. x = 0.80, 0.4250. (row 4). x = 0.20, 0.2000. x = 0.40, 0.3844. x = 0.60, 0.4750. x = 0.80, 0.3469. (row 5). x = 0.20, 0.1961. x = 0.40, 0.3609. x = 0.60, 0.4203. x = 0.80, 0.2922. (row 6). x = 0.20, 0.1883. x = 0.40, 0.3346. x = 0.60, 0.3734. x = 0.80, 0.2512.

    (b)

    A table consists of 6 rows and 4 columns. The column headers are as follows: x = 0.20, x = 0.40, x = 0.60, x = 0.80. The table entries are as follows: (row 1). x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.6000. x = 0.80, 0.8000. (row 2). x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.6000. x = 0.80, 0.8000. (row 3). x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.6000. x = 0.80, 0.5500. (row 4). x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.5375. x = 0.80, 0.4250. (row 5). x = 0.20, 0.2000. x = 0.40, 0.3844. x = 0.60, 0.4750. x = 0.80, 0.3469. (row 6). x = 0.20, 0.1961. x = 0.40, 0.3609. x = 0.60, 0.4203. x = 0.80, 0.2922.

    (c) Yes; the table in part (b) is the table in part (a) shifted downward.

Exercises 17.1, Page 834

  1. 1
  2. −1
  3. 3 + 3i
  4. −10 + i
  5. 7 − 13i
  6. −7 + 5i
  7. 11 − 10i
  8. −5 + 12i
  9. −2i
  10. − − i
  11. 8 − i
  12. − i
  13. 20i
  14. + i
  15. + i
  17. x/(x2 + y2)
  18. −2y − 4
  23. 11 − 6i

Exercises 17.2, Page 838

  1. 2(cos 0 + i sin 0) or 2(cos 2π + i sin 2π)
  6. − − i
  7. 5.5433 + 2.2961i
  9. ; 40.9808 + 10.9808i
  11. −512
  12. i
  13. −i
  14. w0 = 2, w1 = −1 + i, w2 = −1 −i
  19. cos 2θ = cos2 θ − sin2 θ, sin 2θ = 2 sin θ cos θ

Exercises 17.3, Page 841

  1. A graph. A vertical line is graphed on an x y coordinate plane. The line, labeled x = 5, enters the bottom right of the viewing window in the fourth quadrant, goes up vertically, passes through the point (5, 0), and exits the top right of the viewing window.
  2. A graph. A horizontal line is graphed on an x y coordinate plane. The line, labeled y = negative 3, enters the bottom left of the viewing window in the third quadrant, goes horizontally to the right, passes through the point (0, negative 3), and exits the bottom right of the viewing window.
  3. A graph. A circle is graphed on an x y coordinate plane. The center of the circle is at the point (0, 3). The circle passes through the following points: (negative 2, 3), (0, 5), (2, 3), (0, 1).
  4. A graph. A circle is graphed on an x y coordinate plane. The center of the circle is at the marked point (4, negative 3). The circle passes through the following points: (negative 1, negative 3), (4, 2), (9, negative 3), (4, negative 8).
  5. domain
    A graph. A dashed vertical line and a shaded region is graphed on an x y coordinate plane. The dashed line enters the bottom of the viewing window in the third quadrant, goes up vertically, passes through the point (negative 1, 0), and exits the top of the viewing window. The region on the graph to the left of the dashed vertical line is shaded.
  6. domain
    A graph. A dashed horizontal line and a shaded region is graphed on an x y coordinate plane. The dashed line enters the top left of the viewing window in the second quadrant, goes horizontally to the right, passes through the point (0, 3), and exits the top right of the viewing window. The region on the graph above the dashed vertical line is shaded.
  7. domain
    A graph. 2 dashed vertical lines and a shaded region are graphed on an x y coordinate plane. The first dashed line enters the bottom of the viewing window in the fourth quadrant, goes up vertically, passes through the point (3, 0), and exits the top of the viewing window. The second dashed line enters the bottom right of the viewing window in the fourth quadrant to the right of the first dashed line, goes up vertically, passes through the point (5, 0), and exits the top right of the viewing window. The region on the graph between the 2 dashed vertical lines is shaded.
  8. not a domain
    A graph. 2 dashed lines and 2 shaded regions are graphed on an x y coordinate plane. The first dashed line is upward sloping, enters the bottom left of the viewing window in the third quadrant, goes up to the right, passes through the origin, and exits the top right of the viewing window. The second dashed line is downward sloping, enters the top left of the viewing window in the second quadrant, goes down to the right, intersects the first line at the origin, and exits the bottom right of the viewing window. The regions on the graph on both sides of the origin and between the 2 dashed lines are shaded.
  9. not a domain
    A graph. A downward sloping line and a shaded region is graphed on an x y coordinate plane. The line enters the top left of the viewing window in the second quadrant, goes down to the right, and ends at the origin. The region on the graph to the right of the line and above the x axis is shaded.
  10. domain
    A graph. A dashed circle and a shaded region is graphed on an x y coordinate plane. The center of the dashed circle is at the point (0, 1). The circle passes through the following points: (negative 1, 1), (0, 2), (1, 1), (0, 0). The region on the graph outside the dashed circle is shaded.
  11. domain
    A graph. 2 concentric dashed circles and a shaded region are graphed on an x y coordinate plane. The centers of the dashed circles are at the point (0, 1). The first dashed circle passes through the following points: (negative 2, 1), (0, 3), (2, 1), (0, negative 1). The second dashed circle passes through the following points: (negative 3, 1), (0, 4), (3, 1), (0, negative 2). The region on the graph between the 2 dashed circles is shaded.
  12. the line y = −x
  13. the hyperbola x2 − y2 = 1

Exercises 17.4, Page 845

  1. A graph. A curve is graphed on a u v coordinate plane. The curve, labeled u = v^2 over 16 minus 4, enters the bottom of the viewing window in the fourth quadrant, passes through the point (0, negative 8), goes up to the left with increasing steepness until the point (negative 4, 0), changes direction and continues going up to the right symmetrically, passes through the point (0, 8), and exits the top of the viewing window.
  2. A graph. A horizontal line is graphed on a u v coordinate plane. The line, labeled u <= 0, v = 0, enters the bottom left of the viewing window, goes horizontally to the right along the horizontal u axis, and ends at the marked origin.
  3. A graph. A vertical line is graphed on a u v coordinate plane. The line, labeled v >= 0, u = 0, begins at the marked origin, goes vertically up along the vertical v axis, and exits the top of the viewing window.
  4. f(z) = (6x − 5) + i(6y + 9)
  5. f(z) = (x2 − y2 − 3x) + i(2xy − 3y + 4)
  6. f(z) = (x3 − 3xy2 − 4x) + i(3x2 y − y3 − 4y)
  8. −4 + i; 3 − 9i; 1 + 86i
  9. 14 − 20i; −13 + 43i; 3 − 26i
  10. 6 − 5i
  11. −4i
  1. 12z2 − (6 + 2i)z − 5
  2. 6z2 − 14z − 4 + 16i
  3. 6z(z2 − 4i)2
  5. 3i
  6. 2i, −2i
  1. x(t) = c1e2t and y(t) = c2e2t; the streamlines lie on lines through the origin.
  2. y = cx; the streamlines are lines through the origin.
  3. A graph. A curve is graphed on a u v coordinate plane. The curve is symmetric with respect to the horizontal u axis. It enters the bottom of the viewing window in the fourth quadrant, passes through the point (0, negative 8), goes up to the left with increasing steepness until the point (negative 1, negative 3), changes direction and continues going up to the right with decreasing steepness until the origin, turns sharply to the left and goes up with increasing steepness until the point (negative 1, 3), changes direction and continues going up to the right with decreasing steepness, passes through the point (0, 8), and exits the top of the viewing window.

Exercises 17.5, Page 850

  1. a = 1, b = 3
  1. f′ (z) = ex cos y + iex sin y
  2. f(z) = x + i(y + C)
  3. f(z) = x2 − y2 + i(2xy + C)
  4. f(z) = loge(x2 + y2) + i
  5. A graph. 8 hyperbolas, each consisting of 2 disconnected and symmetric curves, are graphed on an x y coordinate plane. The first set of 4 hyperbolas are labeled u = c subscript 1. The second set of 4 hyperbolas are labeled v = c subscript 2. (set 1). The first 2 hyperbolas in the first set are symmetric with respect to the x axis. The first 2 disconnected curves enter the top left of the viewing window in the second quadrant, go down to the right with decreasing steepness, reach a low point on the positive y axis, go up to the right symmetrically, and exit the top right of the viewing window. The second 2 disconnected curves enter the bottom left of the viewing window in the third quadrant, go up to the right with decreasing steepness, reach a high point on the negative y axis, go down to the right symmetrically, and exit the bottom right of the viewing window. The second 2 hyperbolas in the first set are symmetric with respect to the y axis. The first 2 disconnected curves enter the top left of the viewing window in the second quadrant, under the first 2 disconnected curves of the first 2 hyperbolas, go down to the right with increasing steepness until the negative x axis, change direction and continue going down to the left symmetrically, and exit the bottom left of the viewing window, above the second 2 disconnected curves of the first 2 hyperbolas. The second 2 disconnected curves enter the top right of the viewing window in the first quadrant, under the first 2 disconnected curves of the first 2 hyperbolas, go down to the left with increasing steepness until the positive x axis, change direction and continue going down to the right symmetrically, and exit the bottom right of the viewing window, above the second 2 disconnected curves of the first 2 hyperbolas. (set 2). The 4 hyperbolas in the second set are symmetric with respect to the origin. The first 2 disconnected curves of the first 2 hyperbolas enter the left side of the viewing window in the second quadrant just above the x axis, go up to the right with increasing steepness, and exit the top of the viewing window, to the left of the y axis. The second 2 disconnected curves of the first 2 hyperbolas enter the bottom of the viewing window in the fourth quadrant just to the right of y axis, go up to the right with decreasing steepness, and exit the right side of the viewing window, just under the x axis. The first 2 disconnected curves of the second 2 hyperbolas enter the left side of the viewing window in the third quadrant just under the x axis, go down to the right with increasing steepness, and exit the bottom of the viewing window, to the left of the y axis. The second 2 disconnected curves of the second 2 hyperbolas enter the top of the viewing window in the first quadrant just to the right of y axis, go down to the right with decreasing steepness, and exit the right side of the viewing window, just above the x axis.
  6. the x-axis and the circle |z| = 1

Exercises 17.6, Page 856

  3. −eπ
  4. −1.8650 + 4.0752i
  5. 0.2837 − 0.9589i
  6. −0.9659 + 0.2588i
  7. ey(cos x − i sin x)
  8. (cos 2xy + i sin 2xy)
  1. 1.6094 + i(π + 2nπ)
  2. 1.0397 + i(3π/4 + 2nπ)
  3. 1.0397 + i(π/3 + 2nπ)
  4. 2.1383 −(π/4)i
  5. 2.5649 + 2.7468i
  6. 3.4657 − (π/3)i
  7. 1.3863 + i(π/2 + 2nπ)
  8. 3 + i(−π/2 + 2nπ)
  9. e(2−8n)π
  10. e−2nπ(0.2740 + 0.5837i)
  11. e2
  1. no; no; no

Exercises 17.7, Page 859

  1. 10.0677
  2. 1.0911 + 0.8310i
  3. 0.7616i
  4. −0.6481
  5. −1
  6. 0.5876 + 1.3363i
  2. (−π/2 + 2nπ)i
  3. π/4 + nπ
  4. 2nπ ± 2i

Exercises 17.8, Page 862

  2. nπ
  3. 2nπ ± i loge(2 +)
  4. ±π/3 + 2nπ
  5. π/4 + nπ
  6. (−1)n loge 3 + nπi

Chapter 17 in Review, Page 862

  1. 0; 32
  2. −
  4. false
  5. 0.6931 + i(π/2 + 2nπ)
  6. −0.3097 + 0.8577i
  7. false
  9. 58 − 4i
  10. −8 + 8i
  11. A graph. A hyperbola and a shaded region is graphed on an x y coordinate plane. The 2 disconnected curves are symmetric with respect to the origin. The first curve enters the left side of the viewing window in the third quadrant, just under the x axis, goes down to the right with increasing steepness, and exits the bottom of the viewing window, to the left of the y axis. The second curve enters the top of the viewing window in the first quadrant, to the right of the y axis, goes down to the right with decreasing steepness, and exits the right side of the viewing window, just above the x axis. The region on the graph, above the first curve and under the second curve, is shaded.
  12. A graph. A circle and a shaded region is graphed on an x y coordinate plane. The center of the circle is at the origin. The circle passes through the following points: (negative 1, 0), (0, 1), (1, 0), (0, negative 1). The region on the graph outside the circle is shaded.
  13. an ellipse with foci (0, −2) and (0, 2)
  14. 1.0696 − 0.2127i, 0.2127 + 1.0696i, −1.0696 + 0.2127i, − 0.2127 − 1.0696i
  15. 5i
  16. the parabola v = u2 − 2u
  17. 1, −1
  18. pure imaginary numbers
  19. f′ (z) = (−2y − 5) + 2xi

Exercises 18.1, Page 869

  1. −28 + 84i
  2. −48 + i
  3. (2 + π)i
  4. πi
  5. − + i
  6. −e −1
  8. 0
  9. i
  10. 0
  2. circulation = 0, net flux =
  3. circulation = 0, net flux = 0

Exercises 18.2, Page 873

  3. 0

Exercises 18.3, Page 878

  1. −
  4. 0
  7. 2.3504i
  8. 0
  9. πi
  10. i
  11. 11.4928 + 0.9667i
  12. −0.9056 + 1.7699i

Exercises 18.4, Page 884

  1. 8πi
  2. −2πi
  3. −π(20 + 8i)
  4. −2π; 2π
  5. −8π
  6. −2πe−1 i
  7. πi
  8. −5πi; −5πi; 9πi; 0
  9. −π(3 + i); π(3 + i)
  10. π( + 12i)
  11. 0
  12. −πi

Chapter 18 in Review, Page 885

  1. true
  2. true
  3. 0
  4. π(6π − i)
  5. true
  6. 0 if n ≠ −1, 2πi if n = −1
  7. −
  8. + i
  9. 0
  10. −14.2144 + 22.9637i
  11. 2πi
  12. −πi
  13. πi
  14. 2π
  15. 2nπi

Exercises 19.1, Page 893

  1. 5i, −5, −5i, 5, 5i
  2. 0, 2, 0, 2, 0
  3. converges
  4. converges
  5. diverges
  6. limn→∞ Re(zn) = 2 and limn→∞ Im(zn) =
  7. The series converges to 1/(1 + 2i).
  8. divergent
  9. convergent, − + i
  10. convergent, − i
  11. |z − 2i| =, R =
  12. |z − 1 − i| = 2, R = 2
  13. |z − i| = 1/, R = 1/
  14. |z − 4 − 3i| = 25, R = 25
  15. The series converges at z = −2 + i.

Exercises 19.2, Page 897

  12. z + z3 + z5 + . . .

  15. A graph. 2 circles are graphed on an x y coordinate plane. The center of the first circle is at the point (negative 1, 0).The circle passes through the following points: (negative 2, 0), (negative 1, 1), (0, 0), (negative 1, negative 1). The center of the second circle is at the point (0, 1). The second circle intersects the first circle at the points (negative 2, 0) and (negative 1, negative 1). The region inside the first circle that falls inside the second circle is shaded.
  16. (a) The distance from z0 to the branch cut is 1 unit.

    (c) The series converges within the circle

    |z + 1 − i| =. Although the series converges in the shaded region, it does not converge to (or represent) Ln z in this region.

    A graph. 2 concentric circles are graphed on an x y coordinate plane. The centers of the circles are at the marked point labeled negative 1 + i. The smaller circle is dashed and is tangent to the x and y axes. The larger circle passes through the origin. The region inside the larger circle and under the x axis is shaded.
  17. 1.1 + 0.12i

Exercises 19.3, Page 905

  12. − 3 + 6(z − 2) − 10(z − 2)2 + . . .
  13. − 4 − 4z − 4z2 − . . .

Exercises 19.4, Page 908

  1. Define f(0) = 2.
  2. −2 + i is a zero of order 2.
  3. −i and i are zeros of order 1; 0 is a zero of order 2.
  4. 2nπi, n = 0, ±1, . . . , are zeros of order 1.
  5. order 5
  6. order 1
  7. −1 ± 2i are simple poles.
  8. −2 is a simple pole; −i is a pole of order 4.
  9. (2n + 1)π/2, n = 0, ±1, . . . , are simple poles.
  10. 0 is a pole of order 2.
  11. 2nπi, n = 0, ±1, . . . , are simple poles.
  12. 0 is a removable singularity; 1 is a simple pole.
  13. nonisolated

Exercises 19.5, Page 913

  2. −3
  3. 0
  4. Res (f(z), −4i) = , Res (f(z), 4i) =
  5. Res (f(z), 1) = , Res (f(z), −2) = −,

    Res (f(z), 0) = −

  6. Res (f(z), −1) = 6, Res (f(z), −2) = −31,

    Res (f(z), −3) = 30

  7. Res (f(z), 0) = −3/π4 , Res (f(z), π) = (π2 − 6)/2π4
  8. Res (f(z), (2n + 1)π/2) = (−1)n+1, n = 0, ±1, ±2, . . .
  9. 0; 2πi/9; 0
  10. πi; πi; 0
  12. 0
  13. 2πi cosh 1
  14. −4i
  15. 6i
  16. − + i

Exercises 19.6, Page 919

  1. 4π/
  2. 0
  3. π/
  4. π/4
  5. π/6
  6. π
  7. π/16
  8. 3π/8
  9. π/2
  10. π/
  11. πe−1
  12. πe−1
  13. πe−3

Chapter 19 in Review, Page 919

  1. true
  2. false
  3. true
  4. true

  14. −πi
  16. 7π/50

Exercises 20.1, Page 926

  1. the line v = −u
  2. the line v = 2
  3. open line segment from 0 to πi
  4. the ray θ =
  5. the line u = 1
  6. the fourth quadrant
  7. the wedge π/4 ≤ Arg w ≤ π/2
  8. the circle with center w = 4i and radius r = 1
  9. the strip −1 ≤ u ≤ 0
  10. the wedge 0 ≤ Arg w ≤ 3π/4
  11. w = −i(z − i) = −iz − 1
  12. w = 2(z − 1)
  13. w = −z4
  14. w = e3z/2
  15. w = −z + i

Exercises 20.2, Page 931

  1. conformal at all points except z = ±1
  2. conformal at all points except z = πi ± 2nπi
  3. conformal at all points outside the interval [−1, 1] on the x-axis
  4. The image is the region shown in Figure 20.2.2(b). A horizontal segment z(t) = t + ib, 0 < t < π, is mapped onto the lower or upper portion of the ellipse

    according to whether b > 0 or b < 0.

  5. The image of the region is the wedge 0 ≤ Arg w ≤ π/4. The image of the line segment [−π/2, π/2] is the union of the line segments joining eiπ/4 to 0 and 0 to 1.
  6. w = cos(πz/2) using H-4
    A graph. A horizontal line and a shaded region is graphed on a u v coordinate plane. The horizontal line enters the left side of the viewing window, goes to the right along the u axis, passes through the point B prime(1, 0), and ends at the point A prime. The region on the graph above the line is shaded and labeled R prime.
  7. w = using H-5 and w = z1/4
    A graph. 2 lines and a shaded region is graphed on a u v coordinate plane. The first line, labeled v - u, begins at the origin, goes up to the right, passes through the labeled point B prime and ends at the point A prime. The second line begins at the origin, goes horizontally to the right along the u axis. The region on the graph between the 2 lines is shaded and labeled R prime. The following information is given under the graph: B prime = e^(i pi over 4).
  8. w = using H-6 and w = z1/2
    A graph. A shaded region is graphed on a u v coordinate plane. The origin is labeled C prime. A point labeled B prime is placed on the v axis at v = I above the origin. A point labeled A prime is placed on the v axis above the point A prime. The region on the graph above the u axis, to the right of the v axis, and under the point A prime, is shaded and labeled R prime.
  9. w = sin(−iLn z − π/2); A′B′ is the real interval (−∞, −1].
  10. u = Arg (z4) or u(r, θ) = θ

Exercises 20.3, Page 938

  1. T(0) = ∞, T(1) = i, t(∞) = 0; |w| = 1 and the line v = ; |w| ≥ 1
  2. T(0) = −1, T(1) = ∞, T(∞) = 1; the line u = 0 and the circle |w − 1| = 2; the half-plane u ≤ 0

  9. . The level curves are the images of the circles |w| = r, 1 < r < 2, under the linear fractional transformation T(w) = (w + 2)/(w − 1). Since the circles do not pass through the pole at w = 1, the images are circles.
  10. Construct the linear fractional transformation that sends 1, i, −i to 0, 1, −1.
  11. Simplify T2(T1(z)) = .

Exercises 20.4, Page 942

  1. first quadrant
  2. A graph. 3 lines and a shaded region are graphed on a u v coordinate plane. The first line enters the left side of the viewing window, goes horizontally to the right along the u axis and ends at the origin. The second line begins at the origin, goes vertically up along the v axis, and ends at the point v = a i. The third line begins on the v axis at the point v = a i, goes horizontally to the right, and exits the right side of the viewing window. The region on the graph above the 3 lines is shaded.
  3. f′ (z) = A(z + 1)−1/2z−1/2(z − 1)−1/2 for some constant A
  4. f′ (z) = A(z + 1)−1/3z−1/3 for some constant A
  5. Show that f′ (z) = and conclude that f(z) = cosh−1z.
  6. Show that f′ (z) → A/z as w1 → ∞ and conclude that f(z) = Ln z.
  7. Show that f′ (z) → A(z + 1)−1/2z(z − 1)−1/2 = Az/(z2 − 1)1/2 as u1 → 0.

Exercises 20.5, Page 946

  6. u(0, 0) = , u(−0.5, 0) = 0.5693, u(0.5, 0) = 0.1516
  7. Show that u(0, 0) = .
  8. u(r, θ) = r sin θ + r cos θ or u(x, y) = x + y
    A graph. A circle is graphed on an x y coordinate plane. The center of the circle is at the origin. The circle passes through the following points: (negative 1, 0), (0, 1), (1, 0), (0, negative 1). 13 downward sloping, parallel and equally spaced out line segments inside the circle, divide the circle into several strips. The line at the center passes through the center of the circle, and is labeled 0. The second line above the central line is labeled 0.4. The fourth line above the central line is labeled 0.8. The sixth line above the central line is labeled 1.2. The second line under the central line is labeled negative 0.4. The fourth line under the central line is labeled negative 0.8. The sixth line under the central line is labeled negative 1.2.

Exercises 20.6, Page 951

  1. g(z) = is analytic everywhere and G(z) = z is a complex potential. The equipotential lines are the lines x cos θ0 + y sin θ0 = c.
    A graph. 7 downward sloping, parallel and equally spaced out lines are graphed on an x y coordinate plane. The central line passes through the origin and is labeled 0. The first line to the right of the central line is labeled 2. The second line to the right of the central line is labeled 4. The third line to the right of the central line is labeled 6. The first line to the left of the central line is labeled negative 2. The second line to the left of the central line is labeled negative 4. The third line to the left of the central line is labeled negative 6. An arrow is begins on the y axis and points up to the right in the direction perpendicular to the parallel lines. The following information is given in the graph: theta subscript 0 = pi over 6.
  2. g(z) = 1/z is analytic for z ≠ 0 and G(z) = Ln z is analytic except for z = x ≤ 0. The equipotential lines are the circles x2 + y2 = e2c.
    A graph. 4 concentric circles are graphed on an x y coordinate plane. The centers of the circles are at the origin. The circles, starting with the smallest and innermost, are labeled in the following way: 0, 0.5, 0.75, 1.
  3. ϕ = Arg z or ϕ(r, θ) = θ, and G(z) = Ln z is a complex potential. The equipotential lines are the rays θ = c and F = .
  4. The equipotential lines are the images of the rays θ = θ0 under the successive transformations ζ = w1/2 and z = (ζ + 1)/(−ζ + 1). The transformation ζ = w1/2 maps the ray θ = θ0 to the ray θ = θ0/2 in the ζ-plane, and z = (ζ + 1)/(−ζ + 1) maps this ray onto an arc of a circle that passes through z = −1 and z = 1.
  5. (a) ψ(x, y) = 4xy(x2 − y2) or, in polar coordinates, ψ(r, θ) = r4 sin 4θ. Note that ψ = 0 on the boundary of R.

    (b) V = = 4(x3 − 3xy2, y3 − 3x2y)

    (c)

    A graph. 2 lines and 7 disconnected curves are graphed on an x y coordinate plane. The first line begins at the origin, goes horizontally to the right along the x axis, and exits the bottom right of the viewing window. The second line begins at the origin, goes up to the right, and exits the top right of the viewing window. Each of the 7 curves enter the top right of the viewing window in the first quadrant one under the other and just under the second line, go down to the left, change direction at different points and go down to the right, and exit the bottom right of the viewing window one under the other, just above the first line.
  6. (a) ψ(x, y) = cos x sinh y and ψ = 0 on the boundary of R.

    (b) V = = (cos x cosh y, sin x sinh y)

    (c)

    A graph. A composite curve consisting of 3 lines, and 7 other disconnected curves are graphed on an x y coordinate plane. The first line begins on the x axis at the point (negative pi over 2, 0), goes vertically up and exits the top left of the viewing window. The second line begins on the x axis at the point (negative pi over 2, 0), goes horizontally to the right along the x axis, and ends at the point (pi over 2, 2). The third line begins on the x axis at the point (pi over 2, 0), goes vertically up and exits the top right of the viewing window. The 7 curves enter the top left of the viewing window in the second quadrant, to the right of the first line, go down to the right with decreasing steepness, reach low points one under the other above the second line, then go up to the right symmetrically, and exit the top right of the viewing window, to the left of the third line.
  7. (a) ψ(x, y) = 2xy − or, in polar coordinates, ψ(r, θ) = (r2 − 1/r2) sin 2θ. Note that ψ = 0 on the boundary of R.

    (b) V =

    (c)

    A graph. A composite curve consisting of 2 lines and a curve, and 9 other independent curves are graphed on an x y coordinate plane. The first line of the curve begins on the y axis at the point (0, 1), goes vertically up along the y axis and exits the top of the viewing window. The curve begins on the y axis at the point (0, 1), goes down to the right following a circular path, and ends on the x axis at the point (1, 0). The second line begins on the x axis at the point (1, 0), goes horizontally to the right along the x axis, and exits the bottom right of the viewing window. The 9 independent curves enter the top of the viewing window side by side and to the right of the composite curve, go down to the right with decreasing steepness, and exit the bottom right of the viewing window one under the other and above the composite curve.
  8. (a) f(t) = πi − [loge|t + 1| + loge|t − 1|

    + i Arg (t + 1) + i Arg (t − 1)] and so

    Hence, Im (G(z)) = ψ(x, y) = 0 on the boundary of R.

    (b)

    (c)

    A graph. 3 horizontal lines and 8 curves are graphed on an x y coordinate plane. The first line enters the bottom left of the viewing window, goes to the right along the x axis and exits the bottom right of the viewing window. The second line, labeled y = pi over 2, begins on the y axis at the point (0, pi over 2), goes horizontally to the right, and exits the right side of the viewing window. The third line, labeled y = pi, enters the top left of the viewing window in the second quadrant, goes horizontally to the right, passes through the point (0, pi), and exits the right side of the viewing window. The 8 curves enter the left side of the viewing window above the first line, go up to the right with increasing steepness till the midway mark between the first and third horizontal lines, then change direction at different points and go to the left with decreasing steepness, and exit the top left of the viewing window, under the third horizontal line. The changing of direction of the curves is sharper for the curves that are closer to the second horizontal line.
  9. (a) f(t) = ((t2 − 1)1/2 + cosh−1 t) = ((t2 − 1)1/2 + Ln (t + (t2 − 1)1/2)) and so Im (f(t)) =

    and Re (f(t)) = 0 for −1 < t < 1.

    Hence, Im (G(z)) = ψ(x, y) = 0 on the boundary of R.

    (b)

    (c)

    A graph. A composite curve consisting of 3 lines, and 6 curves are graphed on an x y coordinate plane. The first line enters the left side of the viewing window in the second quadrant, goes horizontally to the right and ends on the y axis. The second line begins on the y axis at the point the first line ends, goes vertically down along the y axis, and ends at the origin. The third line begins at the origin, goes horizontally to the right along the x axis, and exits the bottom right of the viewing window. The 6 curves enter the left side of the viewing window in the second quadrant one under the other and above the 3 lines, go down the right and exit the bottom right of the viewing window one under the other, and still above the 3 lines. The descent of the curves around the y axis is more rapid for the curves that are closer to the 3 lines.
  10. z = 0 in Example 5; z = 1, z = −1 in Example 6
  11. The streamlines are the branches of the family of hyperbolas x2 + Bxy − y2 − 1 = 0 that lie in the first quadrant. Each member of the family passes through (1, 0).
  12. Hint: For z in the upper half-plane,

    k [Arg (z − 1) − Arg (z + 1)] = k Arg .

Chapter 20 in Review, Page 953

  1. v = 4
  2. the wedge 0 ≤ Arg w ≤ 2π/3
  3. true
  4. 0, 1, ∞
  5. false
  6. The image of the first quadrant is the strip 0 < v < π/2. Rays θ = θ0 are mapped onto horizontal lines v = θ0 in the w-plane.
  7. w =
    A graph. A circle is graphed on a u v coordinate plane. The center of the circle is at the origin. The circle passes through the following points: (negative 1, 0), (0, 1), (1, 0), (0, negative 1). A semicircular path on the circle begins at the labeled point A prime(0, 1), goes down to the left along the left side of the circle, and ends at the labeled point B prime(0, negative 1). The region inside the circle is shaded and labeled R prime.
  8. u = 2 − 2y/(x2 + y2)
  9. (a) Note that α1 → 0, α2 → 2π, and α3 → 0 as u1 → ∞.

    (b) Hint: Write f(t) = A[loge|t + 1| + loge|t − 1| + i Arg (t + 1) + i Arg (t − 1)] + B.

  10. G(z) = f−1 (z) maps R to the strip 0 ≤ v ≤ π, and U(u, v) = v/π is the solution to the transferred boundary problem. Hence, ϕ(x, y) = (1/π)Im (G(z)) = (1/π)ψ(x, y), and so the equipotential lines ϕ(x, y) = c are the streamlines ψ(x, y) = cπ.

Exercises Appendix A, Page APP-8

  1. 120
  1. π
  1. (a)

    (b)

    (c)