answers to odd-numbered problems - purdue universitycaiz/ma527-cai/appendix2.pdf · 2012-09-19 ·...
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A4
A P P E N D I X 2
Answers to Odd-Numbered Problems
Problem Set 1.1, page 8
1. 3.
5. 7.
9. 11.
13. 15.
17. exp
19. Integrate (start from rest), then
Problem Set 1.2, page 11
11. Straight lines parallel to the x-axis 13.
15. gives the limit
[ ]
17. Errors of steps 1, 5, 10: 0.0052, 0.0382, 0.1245, approximately
19. (error 0.0093), (error 0.0189)
Problem Set 1.3, page 18
1. If you add a constant later, you may not get a solution.
Example:
3.
5. , ellipses 7.
9. 11. , hyperbola
13.
15. 17.
19.
21. 23.
25.
27.
29. No. Use Newton’s law of cooling.
31.
33.
. Eight times.� ϭ (1>0.15) ln 1000 ϭ 7.3 # 2p¢S ϭ 0.15S¢�, dS>d� ϭ 0.15S, S ϭ S0e0.15� ϭ 1000S0,
y ϭ ax, yr ϭ g(y>x) ϭ a ϭ const, independent of the point (x, y)
e�k #10 ϭ 12, k ϭ 1
10, ln 12, e�kt0 ϭ 0.01, t ϭ (ln 100)>k ϭ 66 [min]
T ϭ 22 Ϫ 17e�0.5306t ϭ 21.9 3°C4 when t ϭ 9.68 min
PV ϭ c ϭ const69.6% of y0
y0ekt ϭ 2y0, ek ϭ 2 (1 week), e2k ϭ 22 (2 weeks), e4k ϭ 24
y ϭ x arctan (x3 Ϫ 1)y2 ϩ 4x2 ϭ c ϭ 25
dy>sin2 y ϭ dx>cosh2 x, Ϫcot y ϭ tanh x ϩ c, c ϭ 0, y ϭ Ϫarccot (tanh x)
y ϭ 24>xy ϭ x>(c Ϫ x)
y ϭ x arctan (x2 ϩ c)y2 ϩ 36x2 ϭ c
cos2 y dy ϭ dx, 12 y ϩ 1
4 sin 2y ϩ c ϭ x
yr ϭ y, ln ƒ y ƒ ϭ x ϩ c, y ϭ exϩc ϭ �cex but not ex ϩ c (with c � 0)
x10 ϭ 0.2196x5 ϭ 0.0286
meter>sec
>9.8 ϭ 3.1vr ϭ 0mvr ϭ mg Ϫ bv2, vr ϭ 9.8 Ϫ v2, v(0) ϭ 10,
y ϭ x
y(t) ϭ 12 gt 2 ϩ y0, where y(0) ϭ y0 ϭ 0
ys ϭ g twice, yr(t) ϭ gt ϩ v0, yr(0) ϭ v0 ϭ 0
(Ϫ1.4 # 10�11t) ϭ 12, t ϭ 1011(ln 2)>1.4 [sec]
y ϭ 0 and y ϭ 1 because yr ϭ 0 for these yy ϭ 1>(1 ϩ 3e�x)
y ϭ (x ϩ 12)exy ϭ 1.65e�4x ϩ 0.35
y ϭ1
5.13 sinh 5.13x ϩ cy ϭ 2e�x(sin x Ϫ cos x) ϩ c
y ϭ cexy ϭ1
p cos 2px ϩ c
Problem Set 1.4, page 26
1. Exact, 3. Exact,
5. Not exact, 7.
9. Exact, . Ans.
11.
13. . Ans.
15.
Problem Set 1.5, page 34
3. 5.
7. 9.
11. 13.
15.
17.
19. Solution of
21.
. Thus, gives (4). We shall
see that this method extends to higher-order ODEs (Secs. 2.10 and 3.3).
23.
25.
27.
31.
33.
35.
. Ans. About 3 years
37.
39. . Constant solutions
Problem Set 1.6, page 38
1. 3.
5.
7. 9.
11.
13. .
Sketch or graph these curves.
15. . Trajectories . Now
. This agrees with the trajectory ODE
in u if . But these
are just the Cauchy–Riemann equations.
ux ϭ vy (equal denominators) and uy ϭ Ϫvx (equal numerators)
v ϭ c�, vx dx ϩ vy dy ϭ 0, yr ϭ Ϫvx>vy
y�r ϭ uy�>uxu ϭ c, ux dx ϩ uy dy ϭ 0, yr ϭ Ϫux>uy
yr ϭ Ϫ4x>9y. Trajectories y�r ϭ 9y�>4x, y� ϭ c�x9>4 (c� Ͼ 0)
y� ϭ c�x
yr ϭ Ϫ2xy, y�r ϭ 1>(2xy�), x ϭ c�ey�2
2y�2 Ϫ x2 ϭ c�y>x ϭ c, yr>x ϭ y>x2, yr ϭ y>x, y�r ϭ Ϫx>y�, y�2 ϩ x2 ϭ c�, circles
y Ϫ cosh (x Ϫ c) Ϫ c ϭ 0x2>(c2 ϩ 9) ϩ y2>c2 Ϫ 1 ϭ 0
y ϭ A>(ceAt ϩ B), y(0) Ͼ A>B if c Ͻ 0, y(0) Ͻ A>B if c Ͼ 0.(extinction).
yr Ͼ 0 if y Ͼ A>B (unlimited growth), yr Ͻ 0 if 0 Ͻ y Ͻ A>By ϭ A>B,
y ϭ 0,yr ϭ By2 Ϫ Ay ϭ By(y Ϫ A>B), A Ͼ 0, B Ͼ 0
yr ϭ y Ϫ y2 Ϫ 0.2y, y ϭ 1>(1.25 Ϫ 0.75e�0.8t), limit 0.8, limit 1
t ϭ (ln 3)>0.3889 ϭ 2.82
e�0.3889t ϭ (0.09 Ϫ 0.045)>0.135 ϭ 1>3,
y ϭ 0.135e�0.3889t ϩ 0.045 ϭ 0.18>2,
yr ϭ 175(0.0001 Ϫ y>450), y(0) ϭ 450 � 0.0004 ϭ 0.18,
yr ϭ A Ϫ ky, y(0) ϭ 0, y ϭ A(1 Ϫ e�kt)>kT ϭ 240ekt ϩ 60, T(10) ϭ 200, k ϭ Ϫ0.0539, t ϭ 102 min
dx>dy ϭ 6ey Ϫ 2x, x ϭ ce�2y ϩ 2ey
y ϭ 1>u, u ϭ ce�3.2x ϩ 10>3.2
y2 ϭ 1 ϩ 8e�x2
y ϭ uyhϭ r, ur ϭ r>y* ϭ re�p dx, u ϭ �e�p dx r dx ϩ c
ury* ϩ uy*r ϩ puy* ϭ ury* ϩ u(y*r ϩ py*) ϭ ury* ϩ u # 0y ϭ uy*, yr ϩ py ϭ
cy1r ϩ pcy1 ϭ c(yr1 ϩ py1) ϭ cr
(y1 ϩ y2)r ϩ p(y1 ϩ y2) ϭ (y1r ϩ py1) ϩ (y2r ϩ py2) ϭ r ϩ 0 ϭ r
(y1 ϩ y2)r ϩ p(y1 ϩ y2) ϭ (y1r ϩ py1) ϩ (y2r ϩ py2) ϭ 0 ϩ 0 ϭ 0
Separate. y Ϫ 2.5 ϭ c cosh4 1.5xy ϭ 2 ϩ c sin x
y ϭ (x Ϫ 2.5>e)ecos xy ϭ x2(c ϩ ex)
y ϭ (x ϩ c)eϪkxy ϭ cex Ϫ 5.2
b ϭ k, ax2 ϩ 2kxy ϩ ly2 ϭ c
ex Ϫ y ϩ ey ϭ cu ϭ ex ϩ k(y), uy ϭ kr ϭ Ϫ1 ϩ ey, k ϭ Ϫy ϩ ey
F ϭ sinh x, sinh2 x cos y ϭ c
e2x cos y ϭ 1u ϭ e2x cos y ϩ k(y), uy ϭ Ϫe2x sin y ϩ kr, kr ϭ 0
F ϭ ex2
, ex2
tan y ϭ cy ϭ 2x2 ϩ cx
y ϭ arccos (c>cos x)2x ϭ 2x, x2y ϭ c, y ϭ c>x2
App. 2 Answers to Odd-Numbered Problems A5
A6 App. 2 Answers to Odd-Numbered Problems
Problem Set 1.7, page 42
1. ; hence is continuous and is thus
bounded in the closed interval .
3. In ; just take b in large, namely,
5. R has sides 2a and 2b and center . In
and . Solution by , etc., .
7. .
9. No. At a common point they would both satisfy the “initial condition”
, violating uniqueness.
Chapter 1 Review Questions and Problems, page 43
11. 13.
15.
17.
19. 21.
23. 25.
27. [days]
29. . 43.7 days from
Problem Set 2.1, page 53
1. 3.
5.
7.
9. 11.
13. 15.
17. 19.
Problem Set 2.2, page 59
1. 3.
5. 7.
9. 11.
13. 15.
17. 19.
21. 23.
25. 27.
29. 31. Independent
33. , say.
Hence independent
35. Dependent since
37. y1 ϭ e�x, y2 ϭ 0.001ex ϩ e�x
sin 2x ϭ 2 sin x cos x
c1x2 ϩ c2x2 ln x ϭ 0 with x ϭ 1 gives c1 ϭ 0; then c2 ϭ 0 for x ϭ 2
y ϭ1
1p e�0.27x sin (1px)
y ϭ (4.5 Ϫ x)e�pxy ϭ 2e�x
y ϭ 6e2x ϩ 4e�3xy ϭ 4.6 cos 5x Ϫ 0.24 sin 5x
ys ϩ 4yr ϩ 5y ϭ 0ys ϩ 215yr ϩ 5y ϭ 0
y ϭ e�0.27x (A cos (1px) ϩ B sin (1px))y ϭ (c1 ϩ c2x)e5x>3y ϭ c1e�x>2 ϩ c2e3x>2y ϭ c1e�2.6x ϩ c2e0.8x
y ϭ c1 ϩ c2e�4.5xy ϭ (c1 ϩ c2x)e�px
y ϭ c1e�2.8x ϩ c2e�3.2xy ϭ c1e�2.5x ϩ c2e2.5x
y ϭ 15eϪx Ϫ sin xy ϭ Ϫ0.75x3>2 Ϫ 2.25x �1>2y ϭ 3 cos 2.5x Ϫ sin 2.5xy(t) ϭ c1e�t ϩ kt ϩ c2
y ϭ c1e2x ϩ c2y2 ϭ x3 ln x
(dz>dy)z ϭ Ϫz3 sin y, Ϫ1>z ϭ Ϫdx>dy ϭ cos y ϩ c�1, x ϭ Ϫsin y ϩ c1y ϩ c2
y ϭ (c1x ϩ c2)Ϫ1>2y ϭ c1eϪx ϩ c2F(x, z, zr) ϭ 0
ekt ϭ 0.5, ekt ϭ 0.01ek ϭ 0.9, 6.6 days
ek ϭ 1.25, (ln 2)>ln 1.25 ϭ 3.1, (ln 3)>ln 1.25 ϭ 4.9
3 sin x ϩ 13 sin y ϭ 0y ϭ sin (x ϩ 1
4p)
F ϭ x, x3ey ϩ x2y ϭ c25y2 Ϫ 4x2 ϭ c
y ϭ ceϪ2.5x ϩ 0.640x Ϫ 0.256
y ϭ ceϪx ϩ 0.01 cos 10x ϩ 0.1 sin 10x
y ϭ 1>(ceϪ4x ϩ 4)y ϭ ceϪ2x
y(x1) ϭ y1
(x1, y1)
ƒ1 ϩ y2 ƒ Ϲ K ϭ 1 ϩ b2, a ϭ b>K, da>db ϭ 0, b ϭ 1, a ϭ 12
y ϭ 1>(3 Ϫ 2x)dy>y2 ϭ 2 dxaopt ϭ b>K ϭ 18
da>db ϭ 0 gives b ϭ 1,f ϭ 2y2 Ϲ 2(b ϩ 1)2 ϭ K, a ϭ b>K ϭ b>(2(b ϩ 1)2),
R, (1, 1) since y(1) ϭ 1
b ϭ aK.a ϭ b>Kƒ x Ϫ x0 ƒ Ͻ a
ƒ x Ϫ x0 ƒ Ϲ a
0f>0y ϭ Ϫp(x)yr ϭ f (x, y) ϭ r(x) Ϫ p(x)y
Problem Set 2.3, page 61
1.
3.
5.
7.
9.
11.
15. Combine the two conditions to get .
The converse is simple.
Problem Set 2.4, page 69
1. . At integer t (if ), because of periodicity.
3. (i) Lower by a factor , (ii) higher by
5. 0.3183, 0.4775,
7. (tangential component of ),
9. where is the volume of the
water that causes the restoring force .
. Frequency .
13. ;
(ii)
15.
17. The positive solutions , that is, (max), (min). etc
19. from .
Problem Set 2.5, page 73
3. 5.
7. 9.
11.
13. 15.
17. 19.
Problem Set 2.6, page 79
3. 5. 7.
9.
11.
13.
15.
Problem Set 2.7, page 84
1. 3.
5. 7.
9.
11. y ϭ cos(13x) ϩ 6x2 Ϫ 4
y ϭ c1e4x ϩ c2eϪ4x ϩ 1.2xe4x Ϫ 2ex
y ϭ c1e�x>2 ϩ c2e�3x>2 ϩ 45 ex ϩ 6x Ϫ 16y ϭ (c1 ϩ c2x)e�2x ϩ 1
2 e�x sin x
y ϭ c1e�2x ϩ c2e�x ϩ 6x2 Ϫ 18x ϩ 21y ϭ c1e�x ϩ c2e�4x Ϫ 5e�3x
ys Ϫ 3.24y ϭ 0, W ϭ 1.8, y ϭ 14.2 cosh 1.8x ϩ 9.1 sinh 1.8x
ys ϩ 2yr ϭ 0, W ϭ Ϫ2e�2x, y ϭ 0.5(1 ϩ e�2x)
ys ϩ 5y ϩ 6.34 ϭ 0, W ϭ 0.3eϪ5x, 3eϪ2.5 cos 0.3x
ys ϩ 25y ϭ 0, W ϭ 5, y ϭ 3 cos 5x Ϫ sin 5x
W ϭ aW ϭ Ϫx4W ϭ Ϫ2.2e�3x
y ϭ Ϫ0.525x5 ϩ 0.625xϪ3y ϭ cos (ln x) ϩ sin (ln x)
y ϭ (3.6 ϩ 4.0 ln x)>xy ϭ xϪ3>2y ϭ x2(c1 cos (16 ln x) ϩ c2 sin (16 ln x))
y ϭ (c1 ϩ c2 ln x)x0.6y ϭ c1x2 ϩ c2x3
1x (c1 cos (ln x) ϩ c2 sin (ln x))y ϭ (c1 ϩ c2 ln x)x�1.8
exp (Ϫ10 � 3c>2m) ϭ 120.0231 ϭ (ln 2)>30 3kg>sec4
5p>4p>4of tan t ϭ 1
v* ϭ 3v02 Ϫ c2>(4m2)41>2 ϭ v031 Ϫ c2>(4mk)41>2 � v0(1 Ϫ c2>8mk) ϭ 2.9583
v0 ϭ Ϫ2, Ϫ32, Ϫ4
3, Ϫ54, Ϫ6
5
y ϭ [y0 ϩ (v0 ϩ ay0) t]eϪa˛t, y ϭ [1 ϩ (v0 ϩ 1)t]eϪt
v0>2p ϭ 0.4 3sec�14ys ϩ v02 y ϭ 0, v02 ϭ ag>m ϭ ag ϭ 0.000628gagy with g ϭ 9800 nt (ϭ weight>meter3)
m ϭ 1 kg, ay ϭ p # 0.012 # 2y meter3mys ϭ Ϫa�gy,
us ϩ v02u ϭ 0, v0>(2p) ϭ 1g>L >(2p)
W ϭ mgmLus ϭ Ϫmg sin u � Ϫmgu
1(k1 ϩ k2)>m>(2p) ϭ 0.5738
1212
v0 ϭ pyr ϭ y0 cos v0t ϩ (v0>v0) sin v0t
L(cy ϩ kw) ϭ L(cy) ϩ L(kw) ϭ cLy ϩ kLw
(D Ϫ 1.6I )(D Ϫ 2.4I ), y ϭ c1e1.6x ϩ c2e2.4x
(D Ϫ 2.1I )2, y ϭ (c1 ϩ c2x)e2.1x
(2D Ϫ I )(2D ϩ I ), y ϭ c1e0.5x ϩ c2e�0.5x
0, 5e2x, 0
0, 0, (D Ϫ 2I )(Ϫ4e�2x) ϭ 8e�2x ϩ 8e�2x
4e2x, Ϫe�x ϩ 8e2x, Ϫcos x Ϫ 2 sin x
App. 2 Answers to Odd-Numbered Problems A7
A8 App. 2 Answers to Odd-Numbered Problems
13. 15.
17.
Problem Set 2.8, page 91
3.
5.
7.
9.
11.
13.
15.
17.
19.
25. CAS Experiment. The choice of needs experimentation, inspection of the curves
obtained, and then changes on a trail-and-error basis. It is interesting to see how in
the case of beats the period gets increasingly longer and the maximum amplitude gets
increasingly larger as approaches the resonance frequency.
Problem Set 2.9, page 98
1.
3.
5.
7.
9. 11.
13.
15.
17.
19.
Problem Set 2.10, page 102
1.
3. 5.
7. 9.
11. 13.
Chapter 2 Review Questions and Problems, page 102
7. 9.
11. 13.
15. 17.
19. 21.
23. I ϭ Ϫ0.01093 cos 415t ϩ 0.05273 sin 415t A
y ϭ Ϫ4x ϩ 2x3 ϩ 1>xy ϭ 5 cos 4x Ϫ 34 sin 4x ϩ ex
y ϭ (c1 ϩ c2x)e1.5x ϩ 0.25x2e1.5xy ϭ c1e2x ϩ c2e�x>2 Ϫ 3x ϩ x2
y ϭ c1x�4 ϩ c2x3y ϭ (c1 ϩ c2x)e0.8x
y ϭ eϪ3x(A cos 5x ϩ B sin 5x)y ϭ c1e�4.5x ϩ c2e�3.5x
y ϭ c1xϪ3 ϩ c2x3 ϩ 3x5y ϭ c1x2 ϩ c2x3 ϩ 1>(2x4)
y ϭ (c1 ϩ c2x)ex ϩ 4x7>2exy ϭ (c1 ϩ c2x)e2x ϩ x �2e2x
y ϭ A cos x ϩ B sin x ϩ 12x (cos x ϩ sin x)y ϭ c1x ϩ c2x2 Ϫ x sin x
y ϭ A cos 3x ϩ B sin 3x ϩ 19 (cos 3x) ln ƒ cos 3x ƒ ϩ 1
3 x sin 3x
R ϭ 2 �, L ϭ 1 H, C ϭ 112 F, E ϭ 4.4 sin 10t V
E(0) ϭ 600, Ir(0) ϭ 600, I ϭ e�3t(Ϫ100 cos 4t ϩ 75 sin 4t) ϩ 100 cos t
R Ͼ Rcrit ϭ 22L>C is Case I, etc.
I ϭ e�5t(A cos 10t ϩ B sin 10t) Ϫ 400 cos 25t ϩ 200 sin 25t A
I ϭ 5.5 cos 10t ϩ 16.5 sin 10t AI ϭ 0
I0 is maximum when S ϭ 0; thus, C ϭ 1>(v2L).
I ϭ 2(cos t Ϫ cos 20t)>399
LIr ϩ RI ϭ E, I ϭ (E>R) ϩ ce�Rt>L ϭ 4.8 ϩ ce�40t
I ϭ ce�t>(RC)RIr ϩ I>C ϭ 0,
v>(2p)
v
y ϭ e�t(0.4 cos t ϩ 0.8 sin t) ϩ eϪt>2(Ϫ0.4 cos 12 t ϩ 0.8 sin 12 t)
y ϭ 13 sin t Ϫ 1
15 sin 3t Ϫ 1105 sin 5t
y ϭ e�2t(A cos 2t ϩ B sin 2t) ϩ 14 sin 2t
y ϭ A cos t ϩ B sin t Ϫ (cos vt)>(v2 Ϫ 1)
y ϭ A cos 12t ϩ B sin 12t ϩ t (sin 12t Ϫ cos 12t)>(212)
y ϭ e�1.5t(A cos t ϩ B sin t) ϩ 0.8 cos t ϩ 0.4 sin t
yp ϭ 25 ϩ 43 cos 3t ϩ sin 3t
yp ϭ Ϫ1.28 cos 4.5t ϩ 0.36 sin 4.5t
yp ϭ 1.0625 cos 2t ϩ 3.1875 sin 2t
y ϭ e�0.1x (1.5 cos 0.5x Ϫ sin 0.5x) ϩ 2e0.5x
y ϭ ln xy ϭ ex>4 Ϫ 2ex>2 ϩ 15 e�x ϩ ex
25.
27.
29.
Problem Set 3.1, page 111
9. Linearly independent 11. Linearly independent
13. Linearly independent 15. Linearly dependent
Problem Set 3.2, page 116
1. 3.
5.
7.
9. 11.
13.
Problem Set 3.3, page 122
1.
3.
5.
7.
9. 11.
13.
Chapter 3 Review Questions and Problems, page 122
7.
9.
11. 13.
15. 17.
19.
Problem Set 4.1, page 136
1. Yes
5.
7. 9.
11.
13.
15. (a) For example, gives , . (b) , 0.
(d) gives the critical case. C about 0.18506.a22 ϭ Ϫ4 ϩ 216.4 ϭ 1.05964
Ϫ2.4Ϫ0.000167Ϫ2.39993C ϭ 1000
y1r ϭ y2, y2r ϭ 24y1 Ϫ 2y2, y1 ϭ c1e4t ϩ c2e�6t ϭ y, y2 ϭ yr
y1r ϭ y2, y2r ϭ y1 ϩ ϩ 154 y2, y ϭ c1[1 4]T e4t ϩ c2 [1 Ϫ1
4 ]T e �t>4c1 ϭ 10, c2 ϭ 5c1 ϭ 1, c2 ϭ Ϫ5
y1r ϭ 0.02(Ϫy1 ϩ y2), y2r ϭ 0.02(y1 Ϫ 2y2 ϩ y3), y3r ϭ 0.02(y2 Ϫ y3)
y ϭ 4e�4x ϩ 5e�5x
y ϭ 2eϪ2x cos 4x ϩ 0.05 x Ϫ 0.06y ϭ c1x ϩ c2x1>2 ϩ c3x3>2 Ϫ 103
y ϭ (c1 ϩ c2x ϩ c3x2)e�2x ϩ x2 Ϫ 3x ϩ 3y ϭ (c1 ϩ c2x ϩ c3x2)e�1.5x
y ϭ c1 cosh 2x ϩ c2 sinh 2x ϩ c3 cos 2x ϩ c4 sin 2x ϩ cosh x
y ϭ c1 ϩ e�2x(A cos 3x ϩ B sin 3x)
y ϭ 2 Ϫ 2 sin x ϩ cos x
y ϭ eϪ3x(Ϫ1.4 cos x Ϫ sin x)y ϭ cos x ϩ 12 sin 4x
y ϭ (c1 ϩ c2x ϩ c3x2)e3x Ϫ 14 (cos 3x Ϫ sin 3x)
y ϭ c1x2 ϩ c2x ϩ c3xϪ1 Ϫ 112 xϪ2
y ϭ c1 cos x ϩ c2 sin x ϩ c3 cos 3x ϩ c4 sin 3x ϩ 0.1 sinh 2x
y ϭ (c1 ϩ c2x ϩ c3x2)e�x ϩ 18 ex Ϫ x ϩ 2
y ϭ e0.25x ϩ 4.3eϪ0.7x ϩ 12.1 cos 0.1x Ϫ 0.6 sin 0.1x
y ϭ cosh 5x Ϫ cos 4xy ϭ 4e�x ϩ 5e�x>2 cos 3x
y ϭ 2.398 ϩ e�1.6x(1.002 cos 1.5x Ϫ 1.998 sin 1.5x)
y ϭ A1 cos x ϩ B1 sin x ϩ A2 cos 3x ϩ B2 sin 3x
y ϭ c1 ϩ c2x ϩ c3 cos 2x ϩ c4 sin 2xy ϭ c1 ϩ c2 cos 5x ϩ c3 sin 5x
v ϭ 3.1 is close to v0 ϭ 2k>m ϭ 3, y ϭ 25(cos 3t Ϫ cos 3.1t).
RLC - circuit with R ϭ 20 �, L ϭ 4 H, C ϭ 0.1 F, E ϭ Ϫ25 cos 4t V
I ϭ 173 (50 sin 4t Ϫ 110 cos 4t) A
App. 2 Answers to Odd-Numbered Problems A9
Problem Set 4.3, page 147
1.
3.
5.
7.
9.
11.
13.
15.
17.
, ,
. Note that .
19. ,
Problem Set 4.4, page 151
1. Unstable improper node, ,
3. Center, always stable, ,
5. Stable spiral, ,
7. Saddle point, always unstable, ,
9. Unstable node, ,
11. . Stable and attractive spirals
15. (was 0), , spiral point, unstable.
17. For instance, (a) , (b) , (c) , (d) , (e) 4.
Problem Set 4.5, page 159
5. Center at . At set . Then . Saddle point at .
7. , , , stable and attractive spiral point;
, , , , saddle point
9. saddle point, and centers
11. saddle points; centers.
Use .
13. centers; saddle points
15. By multiplication, . By integration,
, where .
Problem Set 4.6, page 163
3.
5. y2 ϭ c1e5t Ϫ 2c2e2t ϩ 1.12t ϩ 0.53y1 ϭ c1e5t ϩ c2e2t Ϫ 0.43t Ϫ 0.24,
y2 ϭ Ϫc1e�t ϩ c2et Ϫ e3ty1 ϭ c1e�t ϩ c2et,
c* ϭ 12 c2 Ϫ 8y2
2 ϭ 4y12 Ϫ 1
2 y14 ϩ c* ϭ 1
2 (c ϩ 4 Ϫ y12)(c Ϫ 4 ϩ y1
2)
y2y2r ϭ (4y1 Ϫ y13)y1r
y1 ϭ (2n ϩ 1)p ϩ y~1r, (p Ϯ 2np, 0)(Ϯ2np, 0)
Ϫcos (Ϯ12p ϩ y~1) ϭ sin (Ϯy~1) � Ϯy~1
(Ϫ12p Ϯ 2np, 0)(1
2p Ϯ 2np, 0)
(3, 0)(Ϫ3, 0)(0, 0)
y~2r ϭ Ϫy~1 Ϫ y~2y~1r ϭ Ϫy~1 Ϫ 3y~2y2 ϭ 2 ϩ y~2y1 ϭ Ϫ2 ϩ y~1
(Ϫ2, 2),y2r ϭ Ϫy1 Ϫ y2y1r ϭ Ϫy1 ϩ y2(0, 0)
(2, 0)y~2r ϭ y~1y1 ϭ 2 ϩ y~1(2, 0)(0, 0)
ϭ1ϭ Ϫ 12Ϫ1Ϫ2
¢Ͻ0p ϭ 0.2 � 0
y ϭ e�t (A cos t ϩ B sin t)
y2 ϭ 2c1e6t Ϫ 2c2e2ty1 ϭ c1e6t ϩ c2e2t
y2 ϭ Ϫc1e�t ϩ c2e3ty1 ϭ c1e�t ϩ c2e3t
y2 ϭ e�2t(B cos 2t Ϫ A sin 2t)y1 ϭ e�2t(A cos 2t ϩ B sin 2t)
y2 ϭ 3B cos 3t Ϫ 3A sin 3ty1 ϭ A cos 3t ϩ B sin 3t
y2 ϭ c2e2ty1 ϭ c1et
I2 ϭ Ϫ3c1e�t Ϫ c2e�3tI1 ϭ c1e�t ϩ 3c2e�3t
r 2 ϭ y12 ϩ y2
2 ϭ e�2t(A2 ϩ B2)y2 ϭ y1r ϩ y1 ϭ e�t(B cos t Ϫ A sin t)
y1 ϭ e�t(A cos t ϩ B sin t)y1s ϩ 2y1r ϩ 2y1 ϭ 0
y2 ϭ y1r ϩ y1, y2r ϭ y1s ϩ y1r ϭ Ϫy1 Ϫ y2 ϭ Ϫy1 Ϫ (y1r ϩ y1),
y2 ϭ 12 et
y1 ϭ 12 et
y1 ϭ 2 sinh t, y2 ϭ 2 cosh t
y2 ϭ 4et Ϫ 4e�t>2y1 ϭ Ϫ20et ϩ 8e�t>2y3 ϭ c1e�18t Ϫ 2c2e9t Ϫ 1
2 c3e18ty2 ϭ c1e�18t ϩ c2e9t ϩ c3e18t
y1 ϭ 12 c1e�18t ϩ 2c2e9t Ϫ c3e18t
y3 ϭ c2 cos 12t Ϫ c3 sin 12t ϩ c1
y2 ϭ c212 sin 12t ϩ c312 cos 12t
y1 ϭ Ϫc2 cos 12t ϩ c3 sin 12t ϩ c1
y2 ϭ Ϫ2c1 ϩ 5c2e14.5t
y1 ϭ 5c1 ϩ 2c2e14.5t
y1 ϭ 2c1e2t ϩ 2c2, y2 ϭ c1e2t Ϫ c2
y1 ϭ c1e�2t ϩ c2e2t, y2 ϭ Ϫ3c1e�2t ϩ c2e2t
A10 App. 2 Answers to Odd-Numbered Problems
7. ,
9. The formula for v shows that these various choices differ by multiples of the eigen-
vector for , which can be absorbed into, or taken out of, in the general
solution .
11. ,
13. ,
15. ,
17. ,
19. ,
Chapter 4 Review Questions and Problems, page 164
11. , . Saddle point
13. ;
asymptotically stable spiral point
15. , . Stable node
17. , . Stable and attractive
spiral point
19. Unstable spiral point
21.
23.
25.
27. saddle point; centers
29. center when n is even and saddle point when n is odd
Problem Set 5.1, page 174
3.
5.
7.
9.
11.
13.
15.
17.
19. ; but is too large to give good
values. Exact:
Problem Set 5.2, page 179
5. ,
P7(x) ϭ 116 (429x7 Ϫ 693x5 ϩ 315x3 Ϫ 35x)
P6(x) ϭ 116 (231x6 Ϫ 315x4 ϩ 105x2 Ϫ 5)
y ϭ (x Ϫ 2)2exx ϭ 2s ϭ 4 Ϫ x2 Ϫ 1
3 x3 ϩ 130 x5, s˛(2) ϭ Ϫ8
5
s ϭ 1 ϩ x Ϫ x2 Ϫ 56 x3 ϩ 2
3 x4 ϩ 1124 x5, s˛(1
2) ϭ 923768
a�
mϭ1
(m ϩ 1)(m ϩ 2)
(m ϩ 1)2 ϩ 1xm, a
�
mϭ5
(m Ϫ 4)2
(m Ϫ 3)!xm
a0(1 Ϫ 12 x2 Ϫ 1
24 x4 ϩ 13720 x6 ϩ Á ) ϩ a1(x Ϫ 1
6 x3 Ϫ 124 x5 ϩ 5
1008 x7 ϩ Á )
a0(1 Ϫ 112 x4 Ϫ 1
60 x5 Ϫ Á ) ϩ a1(x ϩ 12 x2 ϩ 1
6 x3 ϩ 124 x4 Ϫ 1
24 x5 Ϫ Á )
y ϭ a0 ϩ a1x Ϫ 12 a0x2 Ϫ 1
6 a1x3 ϩ Á ϭ a0 cos x ϩ a1 sin x
y ϭ a0(1 Ϫ x2 ϩ x4>2! Ϫ x6>3! ϩ Ϫ Á ) ϭ a0e�x2
23>22 ƒ k ƒ
(np, 0)
(Ϫ1, 0), (1, 0)(0, 0)
I2 ϭ (Ϫ6 Ϫ 32.5t)e�5t ϩ 6 cos t ϩ 2.5 sin t
I1 ϭ (19 ϩ 32.5t)e�5t Ϫ 19 cos t ϩ 62.5 sin t,
2.5(I2r Ϫ I1r) ϩ 25I2 ϭ 0,I1r ϩ 2.5(I1 Ϫ I2) ϭ 169 sin t,
y2 ϭ Ϫc1e�t ϩ c2e3ty1 ϭ 2c1e�t ϩ 2c2e3t ϩ cos t Ϫ sin t,
y2 ϭ Ϫc1e�4t ϩ c2e4t Ϫ 4ty1 ϭ c1e�4t ϩ c2e4t Ϫ 1 Ϫ 8t 2,
y2 ϭ e�t(B cos 2t Ϫ A sin 2t)y1 ϭ e�t(A cos 2t ϩ B sin 2t)
y2 ϭ c1e�5t Ϫ c2e�ty1 ϭ c1e�5t ϩ c2e�t
y1 ϭ e�4t(A cos t ϩ B sin t), y2 ϭ 15 e�4t[(B Ϫ 2A) cos t Ϫ (A ϩ 2B) sin t]
y2 ϭ 2c1e4t Ϫ 2c2e�4ty1 ϭ c1e4t ϩ c2e�4t
c2 ϭ Ϫ67.948c1 ϭ 17.948
l1 ϭ Ϫ0.9 ϩ 10.41, l2 ϭ Ϫ0.9 Ϫ 10.41
I2 ϭ (1.1 ϩ 10.41)c1el1t ϩ (1.1 Ϫ 10.41)c2el2t,
I1 ϭ 2c1el1t ϩ 2c2el2t ϩ 100
y2 ϭ Ϫ4e�t ϩ ty1 ϭ 4e�t Ϫ 4et ϩ e2t
y2 ϭ 2 cos 2t Ϫ 2 sin 2t ϩ sin ty1 ϭ cos 2t ϩ sin 2t ϩ 4 cos t
y2 ϭ Ϫ83 sinh t Ϫ 4
3 cosh t ϩ 43 e2ty1 ϭ Ϫ8
3 cosh t Ϫ 43 sinh t ϩ 11
3 e2t
y(h)c1l ϭ Ϫ2
y2 ϭ Ϫc1et Ϫ 5c2e2t ϩ 5t ϩ 7.5 ϩ e�ty1 ϭ c1et ϩ 4c2e2t Ϫ 3t Ϫ 4 Ϫ 2e�t
App. 2 Answers to Odd-Numbered Problems A11
11. Set .
15. , , ,
Problem Set 5.3, page 186
3. ,
5.
7.
9. ,
11. ,
13. ,
15.
17.
19.
Problem Set 5.4, page 195
3.
5.
7.
9.
13. implies and
somewhere between and by Rolle’s theorem.
Now use (21b) to get there. Conversely, ,
thus implies in between by Rolle’s
theorem and (21a) with .
15. By Rolle, at least once between two zeros of . Use by (21b)
with . Together at least once between two zeros of . Also use
by (21a) with and Rolle.
19. Use (21b) with (21a) with (21d) with respectively.
21. Integrate (21a).
23. Use (21a) with partial integration, (21b) with partial integration.
25. Use (21d) to get
Problem Set 5.5, page 200
1.
3.
5. c1J0(1x) ϩ c2Y0(1x)
c1J2>3(x2) ϩ c2Y2>3(x2)
c1J4(x) ϩ c2Y4(x)
ϭ Ϫ2J4(x) Ϫ 2J2(x) Ϫ J0(x) ϩ c.
�J5(x)dx ϭ Ϫ2J4(x) ϩ�J3(x)dx ϭ Ϫ2J4(x) Ϫ 2J2(x) ϩ �J1(x)dx
� ϭ 0,� ϭ 1,
� ϭ 2,� ϭ 1,� ϭ 0,
� ϭ 1(xJ1)r ϭ xJ0
J0J1 ϭ 0� ϭ 0
Jr0 ϭ ϪJ1J0Jr0 ϭ 0
� ϭ n ϩ 1
Jn(x) ϭ 0x3nϩ1Jnϩ1(x3) ϭ x4
nϩ1Jnϩ1(x4) ϭ 0
Jnϩ1(x3) ϭ Jnϩ1(x4) ϭ 0Jnϩ1(x) ϭ 0
x2x1[x�nJn(x)]r ϭ 0
x1�nJn(x1) ϭ x2
�nJn(x2) ϭ 0Jn(x1) ϭ Jn(x2) ϭ 0
x��(c1J�(x) ϩ c2J��(x)), � � 0, Ϯ1, Ϯ2, Ác1J1>2(1
2 x) ϩ c2J�1>2(12 x) ϭ x�1>2(c�1 sin 12 x ϩ c�2 cos 12 x)
c1J�(lx) ϩ c2J��(lx), � � 0, Ϯ1, Ϯ2, Ác1J0(1x)
y ϭ c1F(2, Ϫ2, Ϫ12; t Ϫ 2) ϩ c2(t Ϫ 2)3>2F(7
2, Ϫ12, 52; t Ϫ 2)
y ϭ A(1 Ϫ 8x ϩ 325 x2) ϩ Bx3>4F(7
4, Ϫ54, 74; x)
y ϭ AF(1, 1, Ϫ12; x) ϩ Bx3>2F(5
2, 52, 52; x)
y2 ϭ ex ln xy1 ϭ ex
y2 ϭ ex>xy1 ϭ ex
y2 ϭ 1 ϩ xy11x
1120 x5 Ϫ 1
120 x6 ϩ ÁϪ 112 x4 ϩy2 ϭ x ϩ 1
6 x3
124 x4 Ϫ 1
30 x5 ϩ 1144 x6 Ϫ Á ,y1 ϭ 1 ϩ 1
2 x2 Ϫ 16 x3 ϩ
b0 ϭ 1, c0 ϭ 0, r 2 ϭ 0, y1 ϭ e�x, y2 ϭ e�x ln x
y2 ϭ1x
Ϫx
2!ϩ
x3
4!Ϫ ϩ Á ϭ
cos xx
y1 ϭ 1 Ϫx2
3!ϩ
x4
5!Ϫ ϩ Á ϭ
sin xx
P42 ϭ (1 Ϫ x2)(105x2 Ϫ15)>2
P22 ϭ 3(1 Ϫ x2)P2
1 ϭ 3x21 Ϫ x2P11 ϭ 21 Ϫ x2
y ϭ c1Pn(x>a) ϩ c2Qn(x>a)x ϭ az
A12 App. 2 Answers to Odd-Numbered Problems
7.
9.
11. Set and use (10).
13. Use (20) in Sec. 5.4.
Chapter 5 Review Questions and Problems, page 200
11.
13. ; Euler–Cauchy with instead of x
15.
17.
19.
Problem Set 6.1, page 210
1. 3.
5. 7.
9. 11.
13. 15.
19. Use .
23. Set .
25. 27.
29. 31.
33. 35.
37. 39.
41. 43.
45.
Problem Set 6.2, page 216
1.
3.
5.
7. 9.
11.
13. t ϭ t� Ϫ 1, Y�
ϭ 4>(s Ϫ 6), y� ϭ 4e6t, y ϭ 4e6(tϩ1)
y ϭ (1 ϩ t)e�1.5t ϩ 4t 3 Ϫ 16t 2 ϩ 32t
24>s4 Ϫ 32>s3 ϩ 32>s2,Y ϭ 1>(s ϩ 1.5) ϩ 1>(s ϩ 1.5) 2 ϩ
(s ϩ 1.5)2Y ϭ s ϩ 31.5 ϩ 3 ϩ 54>s4 ϩ 64>s,
y ϭ et Ϫ e3t ϩ 2ty ϭ 12 e3t ϩ 5
2 e�4t ϩ 12 e�3t
(s2 Ϫ 14)Y ϭ 12s, y ϭ 12 cosh 12 t
y ϭ 10e3t ϩ e�2t(s Ϫ 3)(s ϩ 2) ϭ 11s ϩ 28 Ϫ 11 ϭ 11s ϩ 17, Y ϭ 10>(s Ϫ 3) ϩ 1>(s ϩ 2),
y ϭ 1.25e�5.2t Ϫ 1.25 cos 2t ϩ 3.25 sin 2t
(k0 ϩ k1t)e�at
e3t(2 cos 3t ϩ 53 sin 3t)e�5pt sinh pt
72 t 3e�t22pte�pt
0.5 # 2p
(s ϩ 4.5)2 ϩ 4p2
2
(s ϩ 3)3
l�1a 4
s Ϫ 2Ϫ
3
s ϩ 1b ϭ 4e2t Ϫ 3e�t2t 3 Ϫ 1.9t 5
1
L2 cos
npt
L0.2 cos 1.8t ϩ sin 1.8t
�ϱ
0
e�(s>c)pf (p) dp>c ϭ F(s>c)>cct ϭ p. Then l( f (ct)) ϭ �ϱ
0
e�stf (ct) dt ϭ
eat ϭ cosh at ϩ sinh at
e�s Ϫ 1
2s2Ϫ
e�s
2sϩ
1
s
(1 Ϫ eϪs)2
s
1 Ϫ e�bs
s2Ϫ
be�bs
s
1
sϩ
e�s Ϫ 1
s2
(v cos u ϩ s sin u)>(s2 ϩ v2)1>((s Ϫ 2)2 Ϫ 1)
s>(s2 ϩ p2)3>s2 ϩ 12>s
1x J1(1x), 1xY1(1x)
ex, 1 ϩ x
J25 (x), J�25(x)
x Ϫ 1(x Ϫ 1)�5, (x Ϫ 1)7cos 2x, sin 2x
H (1) ϭ kH (2)
x3(c1J3(x) ϩ c2Y3(x))
1x (c1J1>4(12 kx2) ϩ c2Y1>4(1
2 kx2))
App. 2 Answers to Odd-Numbered Problems A13
15.
17. 19.
21.
23. 25.
27. 29.
Problem Set 6.3, page 223
3.
5.
7.
9.
11. 13.
15. 17.
19. 21.
23.
25.
27.
29.
31.
33.
35.
37.
39.
Problem Set 6.4, page 230
3.
5.
7.
9.
e�2tϩ30(cos (t Ϫ 10) Ϫ 7 sin (t Ϫ 10))40.1u(t Ϫ 10)3Ϫe�t ϩy ϭ 0.1[et ϩ e�2t(Ϫcos t ϩ 7 sin t)] ϩ
y ϭ e�t ϩ 4e�3t sin 12 t ϩ 12 u(t Ϫ 1
2)e�3(t�1>2) sin (12 t Ϫ 1
4)
sin t (0 Ͻ t Ͻ p); 0 (p Ͻ t Ͻ 2p); Ϫsin t (t Ͼ 2p)
y ϭ 8 cos 2t ϩ 12 u(t Ϫ p) sin 2t
i ϭ 1000e�t sin t Ϫ 1000u(t Ϫ 2)e�tϩ2 sin (t Ϫ 2)
ir ϩ 2i ϩ 2�t
0
i(t) dt ϭ 1000(1 Ϫ u(t Ϫ 2)), I ϭ 1000(1 Ϫ e�2s)>(s2 ϩ 2s ϩ 2),
i ϭ 4 cos t Ϫ 4 cos 240t Ϫ 4u(t Ϫ p)[cos t ϩ cos (140 (t Ϫ p))]
(0.5s2 ϩ 20)I ϭ 78s(1 ϩ e�ps)>(s2 ϩ 1),
i ϭ (10 sin 10t ϩ 100 sin t)(u(t Ϫ p) Ϫ u(t Ϫ 3p))
if t Ͼ 21 Ϫ e�10(t�2)
10I ϩ100
sI ϭ
100
s2e�2s, I ϭ e�2sa1
sϪ
1
s ϩ 10b, i ϭ 0 if t Ͻ 2 and
R(sQ Ϫ CV0) ϩ Q>C ϭ 0, q ϭ CV0e�t>(RC)Rqr ϩ q>C ϭ 0, Q ϭ l(q), q(0) ϭ CV0, i ϭ qr(t),
ϩ 20u(t Ϫ 1)[Ϫe�5t ϩ e�250tϩ245]i ϭ 20(e�5t Ϫ e�250t)
0.1ir ϩ 25i ϭ 490e�5t[1 Ϫ u(t Ϫ 1)],
ϩ 49 cos (2t Ϫ 10) ϩ 10 sin (2t Ϫ 10) if t Ͼ 5cos 2t
t ϭ 1 ϩ t�, y�s ϩ 4y� ϭ 8(1 ϩ t�)2(1 Ϫ u( t� Ϫ 4)), cos 2t ϩ 2t 2 Ϫ 1 if t Ͻ 5,
t Ϫ sin t (0 Ͻ t Ͻ 1), cos (t Ϫ 1) ϩ sin (t Ϫ 1) Ϫ sin t (t Ͼ 1)
et Ϫ sin t (0 Ͻ t Ͻ 2p), et Ϫ 12 sin 2t (t Ͼ 2p)
sin 3t ϩ sin t (0 Ͻ t Ͻ p); 43 sin 3t (t Ͼ p)13(et Ϫ 1)3e�5t
e�t cos t (0 Ͻ t Ͻ 2p)(t Ϫ 3)3u(t Ϫ 3)>62[1 ϩ u(t Ϫ p)] sin 3t(se�ps>2 ϩ e�ps)>(s2 ϩ 1)
e�3s>2 a 2
s3ϩ
3
s2ϩ
94
sb
1
s ϩ p(e�2(sϩp) Ϫ e�4(sϩp))
aeta1 Ϫ uat Ϫ1
2pbbb ϭ
1
s Ϫ 1 (1 Ϫ e�ps>2ϩp>2)
l((t Ϫ 2)u(t Ϫ 2)) ϭ e�2s>s2
1
a2(e�at Ϫ 1) ϩ
t
a19 (1 ϩ t Ϫ cos 3t Ϫ 1
3 sin 3t)
(1 Ϫ cos vt)>v212(1 Ϫ e�t>4)
l( f r) ϭ l(sinh 2t) ϭ sl( f ) Ϫ 1. Answer: (s2 Ϫ 2)>(s3 Ϫ 4s)
2v2
s(s2 ϩ 4v2)
1
(s ϩ a)2
t ϭ t� ϩ 1.5, (s Ϫ 1)(s ϩ 4)Y�
ϭ 4s ϩ 17 ϩ 6>(s Ϫ 2), y ϭ 3et�1.5 ϩ e2(t�1.5)
A14 App. 2 Answers to Odd-Numbered Problems
11.
15.
Problem Set 6.5, page 237
1. t 3.
5. 7.
9. 11.
13.
17. 19.
21. 23.
25.
Problem Set 6.6, page 241
3. 5.
7. 9.
11. 15.
17.
19.
Problem Set 6.7, page 246
3.
5.
7.
9.
11.
13.
15.
19.
Chapter 6 Review Questions and Problems, page 251
11. 13.
15. 17. Sec. 6.6; 2s2>(s2 ϩ 1)2e�3sϩ3>2>(s Ϫ 12)
12(1 Ϫ cos pt), p2>(2s3 ϩ 2p2s)
5s
s2 Ϫ 4Ϫ
3
s2 Ϫ 1
i2 ϭ Ϫ26e�2t ϩ 8e�8t ϩ 18 cos t ϩ 12 sin t
i1 ϭ Ϫ26e�2t Ϫ 16e�8t ϩ 42 cos t ϩ 15 sin t,
4i1 ϩ 8(i1 Ϫ i2) ϩ 2i1r ϭ 390 cos t, 8i2 ϩ 8(i2 Ϫ i1) ϩ 4i2r ϭ 0,
y1 ϭ et, y2 ϭ e�t, y3 ϭ et Ϫ e�t
y1 ϭ Ϫ4et ϩ sin 10t ϩ 4 cos t, y2 ϭ 4et Ϫ sin 10t ϩ 4 cos t
y1 ϭ et ϩ e2t, y2 ϭ e2t
y1 ϭ (3 ϩ 4t)e3t, y2 ϭ (1 Ϫ 4t)e3t
y2 ϭ Ϫe�2t ϩ et ϩ 13 u(t Ϫ 1)(Ϫe3�2t ϩ et)
y1 ϭ Ϫe�2t ϩ 4et ϩ 13 u(t Ϫ 1)(Ϫe3�2t ϩ et),
y2 ϭ cos t ϩ sin t Ϫ 1 ϩ u(t Ϫ 1)[1 Ϫ cos (t Ϫ 1) Ϫ sin (t Ϫ 1)]
y1 ϭ Ϫcos t ϩ sin t ϩ 1 ϩ u(t Ϫ 1)[Ϫ1 ϩ cos (t Ϫ 1) Ϫ sin (t Ϫ 1)]
y1 ϭ Ϫe�5t ϩ 4e2t, y2 ϭ e�5t ϩ 3e2t
3ln (s2 ϩ 1) Ϫ 2 ln (s Ϫ 1)4r ϭ 2s>(s2 ϩ 1) Ϫ 2>(s Ϫ 1); 2(Ϫcos t ϩ et)>tln s Ϫ ln (s Ϫ 1); (Ϫ1 ϩ et)>t
F(s) ϭ Ϫ1
2a 1
s2 Ϫ 9br, f (t) ϭ 1
6 t sinh 3t4s2 Ϫ p2
(s2 ϩ 14p
2)2
p(3s2 Ϫ p2)
(s2 ϩ p2)3
2s3 ϩ 24s
(s2 Ϫ 4)3
s2 Ϫ v2
(s2 ϩ v2)2
12
(s ϩ 3)2
1.5t sin 6t
4.5(cosh 3t Ϫ 1)(vt Ϫ sin vt)>v2
t sin pte4t Ϫ e�1.5t
y(t) ϩ 2�t
0
et�t y(t) dt ϭ tet, y ϭ sinh t
y ϭ cos ty Ϫ 1 * y ϭ 1, y ϭ et
et Ϫ t Ϫ 112 t sin vt
(et Ϫ e�t)>2 ϭ sinh t
ke�ps>(s Ϫ se�ps) (s Ͼ 0)
u(t Ϫ 2)(e�2(t�2) Ϫ e�3(t�2))
y ϭ Ϫe�3t ϩ e�2t ϩ 16 u(t Ϫ 1)(1 Ϫ 3e�2(t�1) ϩ 2e�3(t�1)) ϩ
App. 2 Answers to Odd-Numbered Problems A15
19. 21.
23. 25.
27. 29.
31.
33.
35.
37.
39.
41.
43.
45.
Problem Set 7.1, page 261
3.
5.
7. No, no, yes, no, no
9. , undefined
11. , same, , same
13. , same, , undefined
15. , same, undefined, undefined 17.
Problem Set 7.2, page 270
5. 10,
7. 0, I, c10
0
0d , c1
0
1
0d
n(n ϩ 1)>2
D Ϫ4.5
Ϫ27.0
9.0
TD 5.5
33.0
Ϫ11.0
TϪDD 70
Ϫ28
14
28
56
0
TD 5.4
Ϫ4.2
Ϫ0.6
0.6
2.4
0.6
TD 0
34
28
26
32
Ϫ10
TD 0
18
3
6
15
0
12
15
Ϫ9
T , D 0
2.5
Ϫ1
2.5
1.5
2
1
2
Ϫ1
T , D 0
20.5
2
8.5
16.5
2
13
17
Ϫ10
TB ϭ 1
5 A, 110 A
3 � 3, 3 � 4, 3 � 6, 2 � 2, 2 � 3, 3 � 2
i1 ϭ Ϫ8e�2t ϩ 5e�0.8t ϩ 3, i2 ϭ Ϫ4e�2t ϩ 4e�0.8t5i1r ϩ 20(i1 Ϫ i2) ϭ 60, 30i2r ϩ 20(i2r Ϫ i1r) ϩ 20i2 ϭ 0,
i(t) ϭ e�4t( 326 cos 3t Ϫ 10
39 sin 3t) Ϫ 326 cos 10t ϩ 8
65 sin 10t
1 Ϫ e�t (0 Ͻ t Ͻ 4), (e4 Ϫ 1)e�t (t Ͼ 4)
y1 ϭ (1>110) sin 110t, y2 ϭ Ϫ(1>110) sin 110t
y2 ϭ Ϫsin t Ϫ 2u(t Ϫ p) cos2 12 t ϩ u(t Ϫ 2p) sin t
y1 ϭ cos t Ϫ u(t Ϫ p) sin t ϩ 2u(t Ϫ 2p) sin2 12 t,
y1 ϭ 4et Ϫ e�2t, y2 ϭ et Ϫ e�2t
0 (0 Ϲ t Ϲ 2), 1 Ϫ 2e�(t�2) ϩ e�2(t�2) (t Ͼ 2)
e�t ϩ u(t Ϫ p)[1.2 cos t Ϫ 3.6 sin t ϩ 2e�tϩp Ϫ 0.8e2t�2p]
y ϭ e�2t(13 cos t ϩ 11 sin t) ϩ 10t Ϫ 8e�2t(3 cos t Ϫ 2 sin t)
3t 2 ϩ t 3sin (vt ϩ u)
tu(t Ϫ 1)12>(s2(s ϩ 3))
A16 App. 2 Answers to Odd-Numbered Problems
11. , same, , same
13. , undefined,
15. Undefined, , same
17. , undefined, , undefined
19. Undefined, , same
25. (d) etc.
(e) Answer. If
29.
Problem Set 7.3, page 280
1. 3.
5. 7. arb.,
9. arb.
11. arb., arb.
13. 17.
19.
21. No
23. , thus
Problem Set 7.4, page 287
1. 3.
5.
[0 0 1]}
3; {[2 Ϫ1 4], [0 1 Ϫ46], [0 0 1]}; {[2 0 1], [0 3 23],
3; {[3 5 0], [0 3 5], [0 0 1]}1; [2 Ϫ1 3]; [2 Ϫ1]T
C3H8 ϩ 5O2 : 3CO2 ϩ 4H2O
C: 3x1 Ϫ x3 ϭ 0, H: 8x1 Ϫ 2x4 ϭ 0, O: 2x2 Ϫ 2x3 Ϫ x4 ϭ 0
x2 ϭ 1600 Ϫ x1, x3 ϭ 600 ϩ x1, x4 ϭ 1000 Ϫ x1.
I1 ϭ (R1 ϩ R2)E0>(R1R2) A, I2 ϭ E0>R1 A, I3 ϭ E0>R2 A
I1 ϭ 2, I2 ϭ 6, I3 ϭ 8w ϭ 4, x ϭ 0, y ϭ 2, z ϭ 6
y ϭ 2t2 Ϫ t1, z ϭ t2w ϭ 1, x ϭ t1
x ϭ 3t Ϫ 1, y ϭ Ϫt ϩ 4, z ϭ t
z ϭ 2tx ϭ Ϫ3t, y ϭ tx ϭ 6, y ϭ Ϫ7
x ϭ 1, y ϭ 3, z ϭ Ϫ5x ϭ Ϫ2, y ϭ 0.5
p ϭ [85 62 30]T, v ϭ [44,920 30,940]T
AB ϭ ϪBA.
AB ϭ (AB)T ϭ BTAT ϭ BA;
D 10.5
0
Ϫ3
T , D 7
Ϫ3
1
TD 22
4
Ϫ12
TDϪ30
45
5
Ϫ18
9
Ϫ7
TD 8
Ϫ4
Ϫ3
T , [7 Ϫ1 3]
cϪ9
Ϫ5
3
Ϫ1
4
0dD12
0
2
13
Ϫ6
0
Ϫ6
4
T , DϪ9
3
4
Ϫ5
Ϫ1
0
TD 10
Ϫ14
Ϫ2
Ϫ5
7
Ϫ4
Ϫ15
Ϫ33
Ϫ4
TD 10
Ϫ5
Ϫ5
Ϫ14
7
Ϫ1
Ϫ6
Ϫ12
Ϫ4
TApp. 2 Answers to Odd-Numbered Problems A17
7.
9.
11. (c) 1 17. No
19. Yes 21. No
23. Yes 25. Yes
27.
29. No 31. No
33. 1, solution of the given system , basis
35.
Problem Set 7.7, page 300
7. 9. 1
11. 40 13. 289
15. 17. 2
19. 2 21.
23. 25.
Problem Set 7.8, page 308
1. 3.
5. 7.
9. 11.
15. . Multiply by A from the right.
Problem Set 7.9, page 318
1.
3. 5. No
7. Dimension 2, basis 9. 3; basis
11.
13. x1 ϭ 2y1 Ϫ 3y2, x2 ϭ Ϫ10y1 ϩ 16y2 ϩ y3, x3 ϭ Ϫ7y1 ϩ 11y2 ϩ y3
x1 ϭ 5y1 Ϫ y2, x2 ϭ 3y1 Ϫ y2
c10
0
Ϫ1d , c0
0
1
0d , c0
1
0
0dxe�x, e�x
1, [1 11 Ϫ7]T
[1 0]T, [0 1]T; [1 0]T, [0 Ϫ1]T; [1 1]T, [Ϫ1 1]T
AA�1 ϭ I, (AA�1)�1 ϭ (A�1)�1A�1 ϭ I
(A2)�1 ϭ (A�1)2 ϭ c3.760
2.400
22.272
15.280dD018
0
0
0
14
12
0
0
TA�1 ϭ AD 1
Ϫ2
3
0
1
Ϫ4
0
0
1
TD 54
2
Ϫ30
0.9
0.2
Ϫ0.5
Ϫ3.4
Ϫ0.2
2
Tc1.20
0.50
4.64
3.60d
w ϭ 3, x ϭ 0, y ϭ 2, z ϭ Ϫ2x ϭ 0, y ϭ 4, z ϭ Ϫ1
x ϭ 3.5, y ϭ Ϫ1.0
Ϫ64
cos (a ϩ b)
1, [4 2 43 1]
[1 103 3]c[1 10
3 3]
2, [Ϫ2 0 1], [0 2 1]
3; [9 0 1 0], [0 9 8 9], [0 0 1 0]
2; [8 0 4 0], [0 2 0 4]; [8 0 4], [0 2 0]
A18 App. 2 Answers to Odd-Numbered Problems
15. 17.
19. 1 21.
23.
25.
Chapter 7 Review Questions and Problems, page 318
11.
13.
15. 197, 0
17.
19. 21.
23. arb. 25.
27. 29. Ranks 2, 2,
31. Ranks 2, 2, 1 33.
35.
Problem Set 8.1, page 329
1. 3, 3.
5.
7.
9.
11.
13. defect 2
15.
17. Eigenvalues i, . Corresponding eigenvectors are complex,
indicating that no direction is preserved under a rotation.
19. A point onto the -axis goes onto itself,
a point on the -axis onto the origin.
23. Use that real entries imply real coefficients of the characteristic polynomial.
x1
x2c0 0
0 1d ; 1, c0
1d ; 0, c1
0d .
Ϫic0 Ϫ1
1 0d .
Ϫ5, [Ϫ3 Ϫ3 1 1]T, 3, [3 Ϫ3 1 Ϫ1]T
(l ϩ 1)2(l2 ϩ 2l Ϫ 15); Ϫ1, [1 0 0 0]T, [0 1 0 0]T;
Ϫ(l Ϫ 9)3; 9, [2 Ϫ2 1]T,
6, [1 2 2]T; 9, [2 1 Ϫ2]T
Ϫ(l3 Ϫ 18l2 ϩ 99l Ϫ 162)>(l Ϫ 3) ϭ Ϫ(l2 Ϫ 15l ϩ 54); 3, [2 Ϫ2 1]T;
0.8 ϩ 0.6i, [1 Ϫi]T; 0.8 Ϫ 0.6i, [1 i]T
l2 ϭ 0, [1 0]T
Ϫ3i, [1 Ϫi]; 3i, [1 i], i ϭ 1Ϫ1
Ϫ4, [2 9]T; 3, [1 1]T[1 0]T; Ϫ0.6, [0 1]T
I1 ϭ 4 A, I2 ϭ 5 A, I3 ϭ 1 A
I1 ϭ 16.5 A, I2 ϭ 11 A, I3 ϭ 5.5 A
ϱx ϭ 10, y ϭ Ϫ2
x ϭ 0.4, y ϭ Ϫ1.3, z ϭ 1.7x ϭ 6, y ϭ 2t ϩ 2, z ϭ t
x ϭ 4, y ϭ Ϫ2, z ϭ 8D Ϫ2
Ϫ12
Ϫ12
Ϫ12
16
Ϫ9
Ϫ12
Ϫ9
Ϫ14
TϪ5, det A2 ϭ (det A)2 ϭ 25, 0
[21 Ϫ8 Ϫ31]T, [21 Ϫ8 31]
D Ϫ1
Ϫ18
Ϫ13
6
8
Ϫ2
1
Ϫ7
Ϫ7
T , D 1
Ϫ6
Ϫ1
18
Ϫ8
7
13
2
7
Ta ϭ [5 3 2]T, b ϭ [3 2 Ϫ1]T, 90 ϩ 14 ϭ 2(38 ϩ 14)
a ϭ [3 1 Ϫ4]T, b ϭ [Ϫ4 8 Ϫ1]T, �a ϩ b � ϭ 2107 Ϲ 5.099 ϩ 9
k ϭ Ϫ20
25226
App. 2 Answers to Odd-Numbered Problems A19
Problem Set 8.2, page 333
1. 1.5,
3. 1,
5. 0.5, directions and
7.
9.
11. 1.8
13.
15.
17. .
19. From and Prob. 18 follows and
. Adding on both sides, we see that
has the eigenvalue . From this the statement follows.
Problem Set 8.3, page 338
1.
3. !
5.
7.
9.
15. No 17.
19. No since .
Problem Set 8.4, page 345
1.
3.
5.
9.
11. cϪ2
3
1
Ϫ1d A c1
3
1
2d ϭ c2
0
0
Ϫ5d
c 15
Ϫ25
25
15
d A c12
Ϫ2
1d ϭ c5
0
0
0d
D400
3
Ϫ5
Ϫ5
Ϫ9
15
15
T , 0, D031
T ; 4, D100
T ; 10, DϪ1
1
1
T ; x ϭ D301
T , D010
T , D 1
Ϫ1
1
Tc3.008
5.456
Ϫ0.544
6.992d , 4, cϪ17
31d ; 6, cϪ2
11d ; x ϭ c25
25d , c10
5d
cϪ25
Ϫ50
12
25d , Ϫ5, c3
5d ; 5, c2
5d ; x ϭ cϪ2
4d , c2
1d
det A ϭ det (AT) ϭ det (ϪA) ϭ (Ϫ1)3det (A) ϭ Ϫdet (A) ϭ 0
A�1 ϭ (ϪAT)�1 ϭ Ϫ(A�1)T
1, [0 1 0]T; i, [1 0 i]T; Ϫi, [1 0 Ϫi]T, orthogonal
0, Ϯ25i, skew–symmetric
1, [0 2 1]T; 6, [1 0 0]T, [0 1 Ϫ2]T; symmetric
2 Ϯ 0.8i, [1 Ϯi]. Not skew–symmetric
0.8 Ϯ 0.6i, [1 Ϯi]T; orthogonal
kpljp ϩ kqlj
qkpAp ϩ kqAqkqAqxj ϭ kqlj
qxj (p м 0, q м 0, integer)
kpApxj ϭ kpljpxjAxj ϭ ljxj (xj � 0)
Axj ϭ ljxj (xj � 0), (A Ϫ kI)xj ϭ ljxj Ϫ kxj ϭ (lj Ϫ k)xj
x ϭ (I Ϫ A)Ϫ1y ϭ [0.6747 0.7128 0.7543]T
c [10 18 25]T
[11 12 16]T
[5 8]T
45°Ϫ45°[1 Ϫ1]T; 1.5, [1 1]T;
[Ϫ1>16 1]T, 112.2°; 8, [1 1>16]T, 22.2°
[1 Ϫ1]T, Ϫ45°; 4.5, [1 1]T, 45°
A20 App. 2 Answers to Odd-Numbered Problems
13.
15.
17.
19.
21. , hyperbola
23. , hyperbolaC ϭ cϪ11
42
42
24d , 52y1
2 Ϫ 39y22 ϭ 156, x ϭ
1
213c23
3
Ϫ2dy
C ϭ c 1
Ϫ6
Ϫ6
1d , 7y1
2 Ϫ 5y22 ϭ 70, x ϭ
1
22cϪ1
1
1
1dy
C ϭ c 3
11
11
3d , 14y1
2 Ϫ 8y22 ϭ 0, x ϭ
1
22c11
1
Ϫ1dy; pair of straight lines
C ϭ c73
3
7d , 4y1
2 ϩ 10y22 ϭ 200, x ϭ
1
22c 1
Ϫ1
1
1dy, ellipse
D 13
Ϫ13
0
13
16
Ϫ12
13
16
12
T A D111
Ϫ2
1
1
0
Ϫ1
1
T ϭ D10
0
0
0
1
0
0
0
5
TD 1
Ϫ2
1
0
1
Ϫ2
0
0
1
T A D123
0
1
2
0
0
1
T ϭ D400
0
Ϫ2
0
0
0
1
TApp. 2 Answers to Odd-Numbered Problems A21
Problem Set 8.5, page 351
1. Hermitian, 5,
3. Unitary,
5. Skew-Hermitian, unitary,
7. Eigenvalues eigenvectors
9. Hermitian, 16 11. Skew-Hermitian,
13.
15. (H Hermitian, S skew-Hermitian)
19.
if and only if HS ϭ SH.
AAT
Ϫ ATA ϭ (H ϩ S)(H Ϫ S) Ϫ (H Ϫ S)(H ϩ S) ϭ 2(ϪHS ϩ SH) ϭ 0
A ϭ H ϩ S, H ϭ 12 (A ϩ A
T), S ϭ 1
2 (A Ϫ AT)
(ABC)˛
Tϭ C
TB
TA
Tϭ C�1(ϪB)A
Ϫ6i
[1 0]T, resp.[0 1]T,
[1 Ϫ1]T, [1 1]T; [1 Ϫi]T, [1 i]T;Ϫ1, 1;
Ϫi, [0 Ϫ1 1]T, i, [1 0 0]T, [0 1 1]T
(1 Ϫ i13)>2, [Ϫ1 1]T; (1 ϩ i13)>2, [1 1]T
[Ϫi 1]T, 7, [i 1]T
Chapter 8 Review Questions and Problems, page 352
11.
13.
15.
17. Ϫ1, 1; A ϭ1
16c 5
Ϫ3
Ϫ3
5d c23
39
2
1d ϭ
1
8cϪ1
63
1
1d
0, [2 Ϫ2 1]T; 9i, [Ϫ1 ϩ 3i 1 ϩ 3i 4]T; Ϫ9i, [Ϫ1 Ϫ 3i 1 Ϫ 3i 4]T
3, [1 5]T; 7, [1 1]T
3, [1 1]T; 2, [1 Ϫ1]T
19.
21.
23. , hyperbola
25. , ellipse
Problem Set 9.1, page 360
1.
3.
5. , position vector of Q
7. 9.
11. 13.
15. 17. [12, 8, 0]
21. 23. [0, 0, 5], 5
25. 27.
29. arbitrary 31.
33. Nothing
35.
37.
Problem Set 9.2, page 367
1. 44, 44, 0 3.
5.
7. ; cf. (6)
9. 300; cf. (5a) and (5b) 13. Use (1) and
15.
17.
19. is negative! Why?
21. Yes, because 23. arccos
27. is the angle between the unit vectors a and b. Use (2).
29. and
31. 33.
35. A square.
37. 0. Why?
39. If or if a and b are orthogonal.ƒa ƒ ϭ ƒb ƒ
(a ϩ b) • (a Ϫ b) ϭ ƒa ƒ 2 Ϫ ƒb ƒ 2 ϭ 0, ƒa ƒ ϭ ƒb ƒ .
Ϯ[35, Ϫ 4
5]a1 ϭ Ϫ 283
123.7°g ϭ arccos (12>(6113)) ϭ 0.9828 ϭ 56.3°
b Ϫ a
0.5976 ϭ 53.3°W ϭ (p ϩ q) • d ϭ p • d ϩ q • d.
[0, 4, 3] • [Ϫ3, Ϫ2, 1] ϭ Ϫ5
[2, 5, 0] • [2, 2, 2] ϭ 14
ϭ 2 ƒa ƒ 2 ϩ 2 ƒb ƒ 2ƒa ϩ b ƒ 2 ϩ ƒa Ϫ b ƒ 2 ϭ a • a ϩ 2a • b ϩ b • b ϩ (a • a Ϫ 2a • b ϩ b • b)
ƒ cos g ƒ Ϲ 1.
ƒϪ24 ƒ ϭ 24, ƒa ƒ ƒ c ƒ ϭ 135186 ϭ 13010 ϭ 54.86
ƒ [2, 9, 9] ƒ ϭ 1166 ϭ 12.88 Ͻ 180 ϩ 186 ϭ 18.22
135, 1320, 186
l ϭ 1000, k ϭ 10000 ϩ l Ϫ 1000 ϭ 0,
Ϫk ϩ l ϩ 0 ϭ 0,u ϩ v ϩ p ϭ [Ϫk, 0] ϩ [l, l] ϩ [0, Ϫ1000] ϭ 0,
vB Ϫ vA ϭ [Ϫ19, 0] Ϫ [22>12, 22>12] ϭ [Ϫ19 Ϫ 22>12, Ϫ22>12]
ƒp ϩ q ϩ u ƒ Ϲ 18.
k ϭ 10v ϭ [v1, v2, 3], v1, v2
p ϭ [0, 0, Ϫ5][6, 2, Ϫ14] ϭ 2u, 1236
[4, 9, Ϫ3], 1106
7[9, Ϫ7, 8] ϭ [63, Ϫ49, 56]
[1, 5, 8][6, 4, 0], [32, 1, 0], [Ϫ3, Ϫ2, 0]
Q : (0, 0, Ϫ8), ƒv ƒ ϭ 8Q: (4, 0, 12), ƒv ƒ ϭ 116.25
2, 1, Ϫ2; u ϭ [ 23, 13, Ϫ2
3 ]
8.5, Ϫ4.0, 1.7; 191.14, [0.890, Ϫ0.419, 0.178]
5, 1, 0; 126; [5>126, 1>126, 0]
C ϭ c3.7
1.6
1.6
1.3d , 4.5y1
2 ϩ 0.5y22 ϭ 4.5, x ϭ
1
15c21
1
Ϫ2dy
C ϭ c 4
12
12
Ϫ14d , 10y1
2 Ϫ 20y22 ϭ 20, x ϭ
1
15c21
1
�2dy
1
3D110
1
Ϫ1
1
Ϫ1
0
1
T A D 1
1
Ϫ1
2
Ϫ1
1
1
1
2
T ϭ D400
0
Ϫ20
0
0
0
22
T1
3c 2
Ϫ1
Ϫ1
2d A c2
1
1
2d ϭ cϪ0.9
0
0
0.6d
A22 App. 2 Answers to Odd-Numbered Problems
Problem Set 9.3, page 374
5. instead of m, tendency to rotate in the opposite sense.
7.
9. Zero volume in Fig. 191, which can happen in several ways.
11. 13.
15. 0 17.
19.
21.
23. 0, 0, 13
25. clockwise
27. 29.
31. 33.
Problem Set 9.4, page 380
1. Hyperbolas
3. Parallel straight lines (planes in space)
5. Circles, centers on the y-axis
7. Ellipses 9. Parallel planes
11. Elliptic cylinders 13. Paraboloids
Problem Set 9.5, page 390
1. Circle, center , radius 2 3. Cubic parabola
5. Ellipse 7. Helix
9. A “Lissajous curve” 11.
13. 15.
17. 19.
21. Use
25. At P,
27. 29.
31. 33. Start from .
35.
37.
39.
, and
41.
and
43.
45. ,
[mph]
49. , etc.r(t) ϭ [t, y(t), 0], rr ϭ [1, yr, 0] r • rr ϭ 1 ϩ yr226.61 # 108 ϭ 25,700 [ft>sec] ϭ 17,500
ƒv ƒ ϭ 1gR ϭg ϭ ƒa ƒ ϭ v2R ϭ ƒv ƒ 2>RR ϭ 3960 ϩ 80 mi ϭ 2.133 # 107 ft,
ƒa ƒ ϭ v2R ϭ ƒv ƒ 2>R ϭ 5.98 # 10�6 [km>sec2]
R ϭ 30 # 365 # 86,400>2p ϭ 151 # 106 [km],1 year ϭ 365 # 86,400 sec,
atan ϭ
12 sin 2t
4 ϩ sin2 tv.Ϫ4 cos 2t],a ϭ [Ϫcos t, Ϫ4 sin 2t,
ƒv ƒ 2 ϭ 4 ϩ sin2 t,v ϭ [Ϫsin t, 2 cos 2t, Ϫ2 sin 2t],
atan ϭ6 sin 3t
5 Ϫ 4 cos 3tv.Ϫsin t ϩ 4 sin 2t]a ϭ [Ϫcos t Ϫ 4 cos 2t,
ƒv ƒ 2 ϭ 5 Ϫ 4 cos 3t,cos t Ϫ 2 cos 2t],v ϭ [Ϫsin t Ϫ 2 sin 2t,
v(0) ϭ (v ϩ 1) Ri, a(0) ϭ Ϫv2Rj
v ϭ rr ϭ [1, 2t, 0], ƒv ƒ ϭ 21 ϩ 4t 2, a ϭ [0, 2, 0]
r(t) ϭ [t, f (t)]2rr • rr ϭ a, l ϭ ap>22rr • rr ϭ cosh t, l ϭ sinh l ϭ 1.175q(w) ϭ [2 ϩ w, 12 Ϫ 1
4w, 0]
rr ϭ [Ϫ8, 0, 6]. q(w) ϭ [6 Ϫ 8w, i, 8 ϩ 6w].u ϭ [Ϫsin t, 0, cos t].
sin (Ϫa) ϭ Ϫsin a.
r ϭ [cosh t, (13>2) sinh t, Ϫ2]r ϭ [12 cos t, sin t, sin t]
r ϭ [t, 4t Ϫ 1, 5t]r ϭ [2 ϩ t, 1 ϩ 2t, 3]
r ϭ [3 ϩ 113 cos t, 2 ϩ 113 sin t, 1]
x ϭ 0, z ϭ y3(3, 0)
y ϭ 34 x ϩ c
474>6 ϭ 793x ϩ 2y Ϫ z ϭ 5
12 ƒ [Ϫ12, 2, 6] ƒ ϭ 146[6, 2, 0] � [1, 2, 0] ϭ [0, 0, 10]
m ϭ [Ϫ2, Ϫ2, 0] � [2, 3, 0] ϭ [0, 0, Ϫ10], m ϭ 10
[Ϫ48, Ϫ72, Ϫ168], 121248 ϭ 189.0, 189.0
1, Ϫ1
[Ϫ32, Ϫ58, 34], [Ϫ42, Ϫ63, 19]
[6, 2, 7], [Ϫ6, Ϫ2, Ϫ7][0, 0, 7], [0, 0, Ϫ7], Ϫ4
ƒv ƒ ϭ ƒ [0, 20, 0] � [8, 6, 0] ƒ ϭ ƒ [0, 0, Ϫ160] ƒ ϭ 160
Ϫm
App. 2 Answers to Odd-Numbered Problems A23
51.
53.
Problem Set 9.7, page 402
1. 3.
5. 7. Use the chain rule.
9. Apply the quotient rule to each component and collect terms.
11.
13.
15. 17. For P on the x- and y-axes.
19. 21.
23. Points with 25.
31. 33.
35. 37.
39.
41. 43.
45.
Problem Set 9.8, page 405
1. 3. 0, after simplification; solenoidal
5. 7.
9. (b) , etc.
11. , and
. Hence as t increases from 0 to 1, this “shear flow”
transforms the cube into a parallelepiped of volume l.
13. because do not depend on x, y, z, respectively.
15. 17. 0
19.
Problem Set 9.9, page 408
3. Use the definitions and direct calculation.
5. 7.
9. incompressible,
11. incompressible,
13. irrotational, compressible, Sketch it.
15. same (why?)
17. (why?),
19. same (why?)[Ϫ2z Ϫ y, Ϫ2x Ϫ z, Ϫ2y Ϫ x],
Ϫy Ϫ z Ϫ xϪyz Ϫ zx Ϫ xy, 0
[Ϫ1, Ϫ1, Ϫ1],
r ϭ [c1et, c2et, c3e�t].div v ϭ 1,curl v ϭ 0,
x2 ϩ 12y2 ϭ c, z ϭ c3
xr ϭ y, yr ϭ Ϫ2x, 2xxr ϩ yyr ϭ 0,curl v ϭ [0, 0, Ϫ3],
y ϭ 3c32t ϩ c2yr ϭ 3z2 ϭ 3c3
2,z ϭ c3,
x ϭ c1,v ϭ rr ϭ [xr, yr, zr] ϭ [0, 3z2, 0],curl v ϭ [Ϫ6z, 0, 0]
e�x[cos y, sin y, 0][x (z2 Ϫ y2), y (x2 Ϫ z2), z (y2 Ϫ x2)]
2>(x2 ϩ y2 ϩ z2)2
Ϫ2 cos 2x ϩ 2 cos 2y
v1, v2, v3div (w � r) ϭ 0
xr ϭ y ϭ c2, x ϭ c2t ϩ c1
[v1, v2, v3] ϭ rr ϭ [xr, yr, zr] ϭ [y, 0, 0], zr ϭ 0, z ϭ c3, yr ϭ 0, y ϭ c2
( fv1)x ϩ ( fv2)y ϩ ( fv3)z ϭ f [(v1)x ϩ (v2)y ϩ (v3)z] ϩ fxv1 ϩ fyv2 ϩ fzv3
Ϫ2ex (cos y)z9x2y2z2; 1296
2x ϩ 8y ϩ 18z; 7
f ϭ �v1 dx ϩ �v2 dy ϩ �v3 dz
f ϭ xyz28>3[1, 1, 1] • [Ϫ3>125, 0, Ϫ4>125]>13 ϭ Ϫ7>(12513)
[2, 1] • [1, Ϫ1]>15 ϭ 1>15[Ϫ2x, Ϫ2y, 1], [Ϫ6, Ϫ8, 1]
[12x, 4y, 2z], [60, 20, 10]f ϭ [32x, Ϫ2y], f (P) ϭ [160, Ϫ2]
Ϫ�T(P) ϭ [0, 4, Ϫ1]y ϭ 0, Ϯp, Ϯ2p, Á .
[0, Ϫe][Ϫ1.25, 0]
[8x, 18y, 2z], [40, Ϫ18, Ϫ22]
[2x>(x2 ϩ y2), 2y>(x2 ϩ y2)], [0.16, 0.12]
[y, x], [5, Ϫ4]
[4x3, 4y3]
[Ϫy>x2, 1>x][2y Ϫ 1, 2x ϩ 2]
3>(1 ϩ 9t 2 ϩ 9t 4)
dr
dsϭ
dr
dt >>ds
dt,
d2r
ds2 ϭd2r
dt 2 >>ads
dtb2 ϩ Á ,
d3r
ds3 ϭd3r
dt 3 >>ads
dtb3 ϩ Á
A24 App. 2 Answers to Odd-Numbered Problems
Chapter 9 Review Questions and Problems, page 409
11. 1080, 1080, 65
13.
15. undefined
17. 125,
19.
21.
23.
25. 27.
29. tendency of clockwise rotation 31. 4
33. 1,
35. 0, same (why?),
37. 39.
Problem Set 10.1, page 418
3. 4
5.
7. “Exponential helix,” 9. 23.5, 0
11. 15.
17. 19.
Problem Set 10.2, page 425
3. 5.
7.
9.
13. 15. Dependent, etc.
17. Dependent, etc. 19.
Problem Set 10.3, page 432
3. 5.
7. 9.
11.
13. 15.
17.
19.
Problem Set 10.4, page 438
1. 3.
5.
7. 0. Why? 9.
13. 2w ϭ cosh x, y ϭ x>2 Á 2, 12 cosh 4 Ϫ 1
2
165
2x Ϫ 2y, 2x(1 Ϫ x2) Ϫ (2 Ϫ x2)2 ϩ 1, x ϭ Ϫ1 Á 1, Ϫ 5615
9(e2 Ϫ 1) Ϫ 83 (e3 Ϫ 1)(Ϫ1 Ϫ 1) � p>4 ϭ Ϫp>2
Ix ϭ (a ϩ b)h3>24, Iy ϭ h(a4 Ϫ b4)>(48(a Ϫ b))
Ix ϭ bh3>12, Iy ϭ b3h>4x ϭ 0, y ϭ 4r>3px ϭ 2b>3, y ϭ h>3
z ϭ 1 Ϫ r 2, dx dy ϭ r dr du, Answer: p>236 ϩ 27y2, 144cosh 2x Ϫ cosh x, 1
2 sinh 4 Ϫ sinh 2
�1
0
[x Ϫ x3 Ϫ (x2 Ϫ x5)] dx ϭ1
128y3>3, 54
sin (a2 ϩ 2b2 ϩ c2)4 � 0,
x2 � Ϫ4y2,ea2
cos 2b
ex cosh y ϩ ez sinh y, e Ϫ (cosh 1 ϩ sinh 1) ϭ 0
cosh 1 Ϫ 2 ϭ Ϫ0.457
exy sin z, e Ϫ 0sin 12 x cos 2y, 1 Ϫ 1>12 ϭ 0.293
144t 4, 1843.2[4 cos t, ϩ sin t, sin t, 4 cos t], [2, 2, 0]
18p, 43 (4p)3, 18p2e�t ϩ 2te�t2
, Ϫ2e�2 Ϫ e�4 ϩ 3
(e6p Ϫ 1)>3r ϭ [2 cos t, 2 sin t], 0 Ϲ t Ϲ p>2; 8
5
9>1225 ϭ 35[0, Ϫ2, 0]
2(y2 ϩ x2 Ϫ xz)
Ϫ2y
[0, 0, Ϫ14],
v • w> ƒw ƒ ϭ 22>18 ϭ 7.78[5, 2, 0] • [4 Ϫ 1, 3 Ϫ 1, 0] ϭ 19
g1 ϭ arccos (Ϫ10>165 # 40) ϭ 1.7682 ϭ Ϫ101.3°, g2 ϭ 23.7°
[Ϫ2, Ϫ6, Ϫ13]
[70, Ϫ40, Ϫ50], 0, 2352 ϩ 202 ϩ 252 ϭ 12250
Ϫ125Ϫ125,
[Ϫ1260, Ϫ1830, Ϫ300], [Ϫ210, 120, Ϫ540],
[Ϫ10, Ϫ30, 0], [10, 30, 0], 0, 40
Ϫ10,
App. 2 Answers to Odd-Numbered Problems A25
15. 17.
19.
Problem Set 10.5, page 442
1. Straight lines, k
3. , circles, straight lines,
5. , circles, parabolas,
7. ,
ellipses
11.
13. Set and .
15.
17.
19.
Problem Set 10.6, page 450
1.
3. from (3), Sec. 10.5. Answer:
5.
7.
9. Integrate to
get
13.
15. . Answer: 54.4
21.
23.
25. B the z-axis,
.
Problem Set 10.7, page 457
1. 224
3.
5.
7.
9. 11.
13. 15.
17. 19.
21. 23.
25. Do Prob. 20 as the last one.
h5p>10(a4>4) � 2p � h ϭ ha4p>28abc(b2 ϩ c2)>3h4p>21>p ϩ 5
24 ϭ 0.5266div F ϭ Ϫsin z, 0
12(e Ϫ 1>e) ϭ 24 sinh 1div F ϭ 2x ϩ 2z, 48
[r cos u cos v, cos u sin v, r sin u], dV ϭ r 2 cos u dr du dv, s ϭ v, 2p2a3>312 (sin 2x) (1 Ϫ cos 2x), 1
8, 34
Ϫe�1�z ϩ e�y�z, Ϫ2e�1�z ϩ e�z, 2e�3 Ϫ e�2 Ϫ 2e�1 ϩ 1
IK ϭ IB ϩ 12 # 4p ϭ 20.9
IB ϭ 8p>3,[cos u cos v, cos u sin v, sin u], dA ϭ (cos u) du dv,
[u cos v, u sin v, u], �2p
0
�h
0
u2⅐ u12 du dv ϭ
p
12h4
Ixϭy ϭ ��S
[12 (x Ϫ y)2 ϩ z2] s dA
G(r) ϭ (1 ϩ 9u4)3>2, ƒN ƒ ϭ (1 ϩ 9u4)1>27p3>16 ϭ 88.6
2(1 Ϫ 1>12)(cosh 5 Ϫ 1) ϭ 42.885.
2 sinh v sin ur ϭ [2 cos u, 2 sin u, v], 0 Ϲ u Ϲ p>4, 0 Ϲ v Ϲ 5.
F • N ϭ [0, sin u, cos v] • [1, Ϫ2u, 0], 4 ϩ (Ϫ2 ϩ p2>16 Ϫ p>2)12 ϭ Ϫ0.1775
F(r) • N ϭ Ϫu3, Ϫ128p
13F(r) • N ϭ cos3 v cos u sin u
F(r) • N ϭ [Ϫu2, v2, 0] • [Ϫ3, 2, 1] ϭ 3u2 ϩ 2v2, 29.5
[cosh u, sinh u, v], [cosh u, Ϫsinh u, 0]
a2 cos2 v sin u, a2 cos v sin v][a2 cos2 v cos u,
[a cos v cos u, Ϫ2.8 ϩ a cos v sin u, 3.2 ϩ a sin v], a ϭ 1.5;
[2 ϩ 5 cos u, Ϫ1 ϩ 5 sin u, v], [5 cos u, 5 sin u, 0]
y ϭ vx ϭ u
[u~, v~, u~2, ϩ v~2], N~
ϭ [Ϫ2u~, Ϫ2v~, 1]
x2>a2 ϩ y2>b2 ϩ z2>c2 ϭ 1, [bc cos2 v cos u, ac cos2 v sin u, ab sin v cos v]
[Ϫ2u2 cos v, Ϫ2u2 sin v, u]z ϭ x2 ϩ y2
[Ϫcu cos v, Ϫcu sin v, u]z ϭ c2x2 ϩ y2
ƒgrad w ƒ 2 ϭ e2x, 52 (e4 Ϫ 1)
2w ϭ 6x Ϫ 6y, Ϫ 38.42w ϭ 6xy, 3x(10 Ϫ x2)2 Ϫ 3x, 486
A26 App. 2 Answers to Odd-Numbered Problems
Problem Set 10.8, page 462
1. , no contributions. etc.
Integrals . Sum 0
3. The volume integral of is . The surface
integral of over is Others 0.
5. The volume integral of is 0; others 0.
7. use Sec. 10.7, etc.
9. and
11.
Problem Set 10.9, page 468
1.
3.
5.
7.
9. The sides contribute a, 0.
11. 13. 5k,
15.
17.
19.
Answer:
Chapter 10 Review Questions and Problems, page 469
11.
Or using exactness.
13. Not exact, 15. 0 since
17. By Stokes, 19.
21.
23.
25. 27.
29. Answer: 21 31.
33. Direct integration, 35.
Problem Set 11.1, page 482
1. 5. There is no smallest
13.
15.
17. p
2ϩ
4
pacos x ϩ
1
9 cos 3x ϩ
1
25 cos 5x ϩ Á b
13 sin 3x ϩ Á )
43p
2 ϩ 4 (cos x ϩ 14 cos 2x ϩ 1
9 cos 3x ϩ Á ) Ϫ 4p (sin x ϩ 12 sin 2x ϩ
4
p (cos x ϩ
1
9 cos 3x ϩ
1
25 cos 5x ϩ Á ) ϩ 2 (sin x ϩ
1
3 sin 3x ϩ
1
5 sin 5x ϩ Á )
p Ͼ 0.2p, 2p, p, p, 1, 1, 12, 12
72p2243
24 sinh 1 ϭ 28.205div F ϭ 20 ϩ 6z2.
288(a ϩ b ϩ c)pM ϭ 4k>15, x ϭ 516, y ϭ 4
7
M ϭ 6320, x ϭ 8
7 ϭ 1.14, y ϭ 11849 ϭ 2.41
M ϭ 8, x ϭ 85, y ϭ 16
5
F ϭ grad (y2 ϩ xz), 2pϮ18p
curl F ϭ 0curl F ϭ (5 cos x)k, Ϯ10
Ϫ4528>3.
r ϭ [4 Ϫ 10t, 2 ϩ 8t], F(r) • dr ϭ [2(4 Ϫ 10t)2, Ϫ4(2t ϩ 8t)2] • [Ϫ10, 8] dt;
1>2[Ϫez, 1, 0] • [Ϫu cos v, Ϫu sin v, u].
r ϭ [u cos v, u sin v, u], 0 Ϲ u Ϲ 1, 0 Ϲ v Ϲ p>2,
r ϭ [cos u, sin u, v], [Ϫ3v2, 0, 0] • [cos u, sin u, 0], Ϫ1
[0, Ϫ1, 2x Ϫ 2y] • [0, 0, 1], 13
80pϪ2p; curl F ϭ 0
3a2>2, Ϫa,
[Ϫez, Ϫex, Ϫey] • [Ϫ2x, 0, 1], Ϯ(e4 Ϫ 2e ϩ 1)
[0, 2z, 32 ] • [0, 0, 1] ϭ 32 , Ϯ3
2 a2
[2e�z cos y, Ϫe�z, 0] • [0, Ϫy, 1] ϭ ye�z, Ϯ(2 – 2>1e)
S: z ϭ y (0 Ϲ x Ϲ 1, 0 Ϲ y Ϲ 4), [0, 2z, Ϫ2z] • [0, Ϫ1, 1], Ϯ20
r ϭ a, � ϭ 0, cos � ϭ 1, v ϭ 13 a � (4pa2)
Ϫ2p �12 (a2 Ϫ r 2)3>2
�23 ƒ
0
aϭ 2
3pa3
dx dy ϭ r dr du,z ϭ 2a2 Ϫ x2 Ϫ y2 ϭ 2a2 Ϫ r 2,z ϭ 0
(2*),F ϭ [x, 0, 0], div F ϭ 1,
8(x ϭ 1), Ϫ8(y ϭ 1),6y2� 4 Ϫ 2x2
� 12
8y3>3 ϭ 83.x ϭ 1f 0g>0n ϭ f � 2x ϭ 2f ϭ 8y2
8y3>3 ϭ 838y2 ϩ [0, 8y] � [2x, 0] ϭ 8y2
x ϭ a: (Ϫ2a)bc, y ϭ b: (Ϫ2b)ac, z ϭ c: (4c) ab
x ϭ a: 0f>0n ϭ 0f>0x ϭ Ϫ2x ϭ Ϫ2a,x ϭ 0, y ϭ 0, z ϭ 0
App. 2 Answers to Odd-Numbered Problems A27
19.
21.
Problem Set 11.2, page 490
1. Neither, even, odd, odd, neither 3. Even 5. Even
9. Odd,
11. Even,
13. Rectifier,
15. Odd,
17. Even,
19.
23. (a) 1, (b)
25. (a)
(b)
27. (a)
(b)
29. Rectifier,
(a) (b)
Problem Set 11.3, page 494
3. The output becomes a pure cosine series.
5. For this is similar to Fig. 54 in Sec. 2.8, whereas for the phase shift
the sense is the same for all n.
BnAn
sin x2
pϪ
4
pa 1
1 # 3 cos x ϩ
1
3 # 5 cos 3x ϩ
1
5 # 7 cos 5x ϩ Á b ,
L ϭ p,
a1
5ϩ
2
25pb sin 5x ϩ
1
6 sin 6x ϩ Á
a1 ϩ2
pb sin x ϩ
1
2sin 2x ϩ a1
3Ϫ
2
9pb sin 3x ϩ
1
4sin 4x ϩ
1
18 cos 6x ϩ
1
49 cos 7x ϩ
1
81 cos 9x Ϫ
1
50 cos 10x ϩ
1
121 cos 11x ϩ Á b
3p
8ϩ
2
pacos x Ϫ
1
2cos 2x ϩ
1
9 cos 3x ϩ
1
25 cos 5x ϪL ϭ p,
2 (sin x ϩ 12 sin 2x ϩ 1
3 sin 3x ϩ 14 sin 4x ϩ Á )
p
2ϩ
4
pacos x ϩ
1
9 cos 3x ϩ
1
25 cos 5x ϩ Á b ,L ϭ p,
4
pasin px
4ϩ
1
3 sin
3px
4ϩ
1
5 sin
5px
4ϩ Á bL ϭ 4,
38 ϩ 1
2 cos 2x ϩ 18 cos 4x
L ϭ 1,1
2ϩ
4
p2 acos px ϩ1
9 cos 3px ϩ
1
25 cos 5px ϩ Á b
L ϭ p,4
pasin x Ϫ
1
9 sin 3x ϩ
1
25 sin 5x Ϫ ϩ Á b
1
pa1
2sin 2px Ϫ
1
4 sin 4px ϩ
1
6 sin 6px Ϫ
1
8 sin 8px ϩ Ϫ Á b
L ϭ1
2,
1
8Ϫ
1
p2 acos 2px ϩ1
9 cos 6px ϩ
1
25 cos 10px ϩ Á b ϩ
L ϭ 1,1
3Ϫ
4
p2 acos px Ϫ1
4cos 2px ϩ
1
9 cos 3px Ϫ ϩ Á b
L ϭ 2, 4
pasin px
2ϩ
1
3 sin
3px
2ϩ
1
5 sin
5px
2ϩ Á b
2(sin x ϩ 12 sin 2x ϩ 1
3 sin 3x ϩ 14 sin 4x ϩ 1
5 sin 5x ϩ Á )
1
3 sin 3x Ϫ ϩ Á
p
4Ϫ
2
pacos x ϩ
1
9 cos 3x ϩ
1
25 cos 5x ϩ Á b ϩ sin x Ϫ
1
2sin 2x ϩ
A28 App. 2 Answers to Odd-Numbered Problems
7.
Note the change of sign.
11.
13.
15. (n odd),
with as in
Prob. 13.
17.
19.
Section 11.4, page 498
3.
5. 0.6243,
7.
Why is so large?
Section 11.5, page 503
3. Set 5. etc.
7.
9.
11.
13.
Section 11.6, page 509
1.
3.
9.
Rounding seems to have considerable influence in Probs. 8–13.m0 ϭ 9.
Ϫ0.4775P1(x) Ϫ 0.6908P3(x) ϩ 1.844P5(x) Ϫ 0.8236P7(x) ϩ 0.1658P9(x) ϩ Á ,
45 P0(x) Ϫ 4
7 P2(x) Ϫ 835 P4(x)
8 (P1(x) Ϫ P3(x) ϩ P5(x))
p ϭ e8x, q ϭ 0, r ϭ e8x, lm ϭ m2, ym ϭ e�4x sin mx, m ϭ 1, 2, Álm ϭ m2, m ϭ 1, 2, Á , ym ϭ x sin (m ln ƒ x ƒ )
l ϭ [(2m ϩ 1)p>(2L)]2, m ϭ 0, 1, Á , ym ϭ sin ((2m ϩ 1)px>(2L))
lm ϭ (mp>10)2, m ϭ 1, 2, Á ; ym ϭ sin (mpx>10)
x ϭ cos u, dx ϭ Ϫsin u du,x ϭ ct ϩ k.
E*E* ϭ 674.8, 454.7, 336.4, 265.6, 219.0.
F ϭ 2[(p2 Ϫ 6) sin x Ϫ 18 (4p2 Ϫ 6) sin 2x ϩ 1
27 (9p2 Ϫ 6) sin 3x Ϫ ϩ Á ];
0.4206 (0.1272 when N ϭ 20)
F ϭ4
pasin x ϩ
1
3 sin 3x ϩ
1
5 sin 5x ϩ Á b , E* ϭ 1.1902, 1.1902, 0.6243,
0.0748, 0.0119, 0.0119, 0.0037
F ϭp
2Ϫ
4
pacos x ϩ
1
9 cos 3x ϩ
1
25 cos 5x ϩ Á b , E* ϭ 0.0748,
Bn ϭ (Ϫ1)nϩ124,000
nDn
, Dn ϭ (10 Ϫ n2)2 ϩ 100n2
I (t) ϭ a
�
nϭ1
(An cos nt ϩ Bn sin nt), An ϭ (Ϫ1)nϩ12400 (10 Ϫ n2)
n2Dn
,
Bn ϭ 10nan>Dn, an ϭ Ϫ400>(n2p), Dn ϭ (n2 Ϫ 10)2 ϩ 100n2An ϭ (10 Ϫ n2)an>Dn,I ϭ 50 ϩ A1 cos t ϩ B1 sin t ϩ A3 cos 3t ϩ B3 sin 3t ϩ Á ,
DnBn ϭ (Ϫ1)nϩ1 # 12(1 Ϫ n2)>(n3Dn)An ϭ (Ϫ1)n # 12nc>n3Dn,
y ϭ a
�
nϭ1
(An cos nt ϩ Bn sin nt),bn ϭ (Ϫ1)nϩ1 # 12 >n3
Bn ϭ [(1 Ϫ n2)bn ϩ ncan]>Dn, Dn ϭ (1 Ϫ n2)2 ϩ n2c2
y ϭ a
N
nϭ1
(An cos nt ϩ Bn sin nt), An ϭ [(1 Ϫ n2)an Ϫ nbnc]>Dn,
1
v2 Ϫ 121
sin 5t ϩ Á by ϭ C1 cos vt ϩ C2 sin vt ϩ
4
pa 1
v2 Ϫ 9
sin t ϩ1
v2 Ϫ 49
sin 3t ϩ
0.8, 0.01.Ϫ5.26, 4.76,
y ϭ C1 cos vt ϩ C2 sin vt ϩ a (v) sin t, a (v) ϭ 1>(v2 Ϫ 1) ϭ Ϫ1.33,
App. 2 Answers to Odd-Numbered Problems A29
11.
13.
15.
Section 11.7, page 517
1. gives
(see Example 3), etc.
3. Use (11);
5.
7.
9.
11.
15. For the value of equals 0.28,
0.029, 0.0103, 0.0065, (rounded).
17.
19.
Section 11.8, page 522
1.
3.
5.
7. Yes. No 9.
11.
13.
Problem Set 11.9, page 533
3. if otherwise
5.
7. 9.
11. 13. by formula 9e�w2>2i12>p (cos w Ϫ 1)>w12>p(cos w ϩ w sin w Ϫ 1)>w2(e�iaw(1 ϩ iaw) Ϫ 1)>(12pw2)
[e(1�iw)a Ϫ e�(1�iw)a]>(12p(1 Ϫ iw))
a Ͻ b; 0i (e�ibw Ϫ e�iaw)>(w12p)
fs(e�x) ϭ
1
waϪfc(e�x) ϩB
2
p# 1b ϭ
1
waB
2
p# 1
w2 ϩ 1ϩ B
2
pb ϭ B
2
p
w
w2 ϩ 1
12>p ((2 Ϫ w2) cos w ϩ 2w sin w Ϫ 2)>w3
12>p w>(a2 ϩ w2)
fcˆ (w) ϭ B
2
p
(w2 Ϫ 2) sin w ϩ 2w cos w
w3
fcˆ (w) ϭ 1(2>p) (cos 2w ϩ 2w sin 2w Ϫ 1)>w2
(2 sin w Ϫ sin 2w)>wfcˆ (w) ϭ 1(2>p)
2
p ��
0
w Ϫ e (w cos w Ϫ sin w)
1 ϩ w2 sin xw dw
2p �
�
0
1 Ϫ cos ww
sin xwdw
Ϫ0.0064Ϫ0.0099,Ϫ0.026,
Ϫ0.15,Si (np) Ϫ p>2n ϭ 1, 2, 11, 12, 31, 32, 49, 50
2
p ��
0
cos pw ϩ 1
1 Ϫ w2 cos xw dw
A (w) ϭ2
p ��
0
cos wv
1 ϩ v2 dv ϭ e�w (w Ͼ 0)
2p �
�
0
sin w cos xw
wdw
B (w) ϭ2
p �1
0
1
2pv sin wvdv ϭ
sin w Ϫ w cos w
w2
B ϭ2
p ��
0
p
2 sin wvdv ϭ
1 Ϫ cos pw
w
A ϭ ��
0
e�v cos wv dv ϭ1
1 ϩ w2, B ϭ
w
1 ϩ w2f (x) ϭ pe�x(x Ͼ 0)
(c) am ϭ (2>J 12(a0,m)) (J1(a0,m)>a0,m) ϭ 2>(a0,mJ1(a0,m))
0.1212P0(x) Ϫ 0.7955P2(x) ϩ 0.9600P4(x) Ϫ 0.3360P6(x) ϩ Á , m0 ϭ 8
0.7854P0(x) Ϫ 0.3540P2(x) ϩ 0.0830P4(x) Ϫ Á , m0 ϭ 4
A30 App. 2 Answers to Odd-Numbered Problems
17. No, the assumptions in Theorem 3 are not satisfied.
19.
21.
Chapter 11 Review Questions and Problems, page 537
11.
13.
15. respectively 17. Cf. Sec. 11.1.
19.
21.
23. 0.82, 0.50, 0.36, 0.28, 0.23
25. 0.0076, 0.0076, 0.0012, 0.0012, 0.0004
27.
29.
Problem Set 12.1, page 542
1.
3. 5.
7. Any c and 9.
15. 17.
19.
21.
23. (Euler–Cauchy)
25. a, b, c, k arbitrary constants
Problem Set 12.3, page 551
5.
7.
9. 0.8
p2 acos pt sin px Ϫ1
9 cos 3pt sin 3px ϩ
1
25 cos 5pt sin 5px Ϫ ϩ Á b
8k
p3 acos pt sin px ϩ1
27 cos 3pt sin 3px ϩ
1
125 cos 5pt sin 5px ϩ Á b
k cos 3pt sin 3px
u(x, y) ϭ axy ϩ bx ϩ cy ϩ k;
u ϭ c1(y)x ϩ c2(y)>x2
u ϭ e�3y(a(x) cos 2y ϩ b(x) sin 2y) ϩ 0.1e3y
u ϭ c(x) e�y3>3u ϭ a(y) cos 4px ϩ b(y) sin 4pxu ϭ 110 Ϫ (110>ln 100) ln (x2 ϩ y2)
c ϭ p>25v
c ϭ a>bc ϭ 2
L(c1u1 ϩ c2u2) ϭ c1L(u1) ϩ c2L(u2) ϭ c1# 0 ϩ c2
# 0 ϭ 0
12>p (cos aw Ϫ cos w ϩ aw sin aw Ϫ w sin w)>w2
1
p ��
0
(cos w ϩ w sin w Ϫ 1)cos wx ϩ (sin w Ϫ w cos w) sin wx
w2 dw
Ϫ1
16# cos 4t
v2 Ϫ 16ϩ Ϫ Á b
y ϭ C1 cos vt ϩ C2 sin vt ϩp2
v2Ϫ 12a cos t
v2 Ϫ 1Ϫ
1
4# cos 2t
v2 Ϫ 4ϩ
1
9# cos 3t
v2 Ϫ 9
1
2Ϫ
4
p2 acos px ϩ1
9 cos 3px ϩ Á b , 2
pasin px Ϫ
1
2sin 2px ϩ Ϫ Á b
cosh x, sinh x (Ϫ5 Ͻ x Ͻ 5),
1
pasin px Ϫ
1
2sin 2px ϩ
1
3 sin 3px Ϫ ϩ Á b
1
4Ϫ
2
p2 acos px ϩ1
9 cos 3px ϩ
1
25 cos 5px ϩ Á b ϩ
1 ϩ4
pasin px
2ϩ
1
3 sin
3px
2ϩ
1
5 sin
5px
2ϩ Á b
c1 1
1 Ϫ1d c f1
f2d ϭ c f1 ϩ f2
f1 Ϫ f2d
[ f1 ϩ f2 ϩ f3 ϩ f4, f1 Ϫ if2 Ϫ f3 ϩ if4, f1 Ϫ f2 ϩ f3 Ϫ f4, f1 ϩ if2 Ϫ f3 Ϫ if4]
App. 2 Answers to Odd-Numbered Problems A31
11.
13.
No terms with
17.
19. (a) (b) (c) (d)
from (a), (c). Insert this. The coefficient determinant resulting from (b), (d) must be
zero to have a nontrivial solution. This gives (22).
Problem Set 12.4, page 556
3.
9. Elliptic,
11. Parabolic,
13. Hyperbolic,
15. Hyperbolic,
17. Elliptic, Real or imaginary parts of any
function u of this form are solutions. Why?
Problem Set 12.6, page 566
3. differ in rapidity of decay.
5.
7.
9. where satisfies the boundary conditions of the text,
so that
11.
etc.
13.
15.
17.
19.
21. u ϭ100
p a�
nϭ1
1
(2n Ϫ 1) sinh (2n Ϫ 1)p sin
(2n Ϫ 1)px
24 sinh
(2n Ϫ 1)py
24
u ϭ 1000 (sin 12px sinh 12py)>sinh p
ϪKp
L a�
nϭ1
nBn e�ln2t
1
2ϩ
4
p2 acos x e�t ϩ1
9 cos 3x e�9t ϩ
1
25 cos 5x e�25t ϩ Á b
u ϭ 1
p ϭ np>L,
F ϭ A cos px ϩ B sin px, Fr(0) ϭ Bp ϭ 0, B ϭ 0, Fr(L) ϭ ϪAp sin pL ϭ 0,
uII
ϭ a�
nϭ1
Bn sin npx
Le�(cnp>L)2t, Bn ϭ
2
L �L
0
[ f (x) Ϫ uI(x)] sin
npx
Ldx.
uII ϭ u Ϫ uIu ϭ uI ϩ uII,
u ϭ800
p3asin 0.1px e�0.01752p2t ϩ
1
33 sin 0.3px e�0.01752(3p)2t ϩ Á b
u ϭ sin 0.1px e�1.752p2t>100
u1 ϭ sin x e�t, u2 ϭ sin 2x e�4t, u3 ϭ sin 3x e�9t
u ϭ f1(y Ϫ (2 Ϫ i)x) ϩ f2(y Ϫ (2 ϩ i)x).
xyr2 ϩ yyr ϭ 0, y ϭ v, xy ϭ w, uw ϭ z, u ϭ1y
f1(xy) ϩ f2(y)
u ϭ f1(y Ϫ 4x) ϩ f2(y Ϫ x)
u ϭ x f1(x Ϫ y) ϩ f2(x Ϫ y)
u ϭ f1(y ϩ 2ix) ϩ f2(y Ϫ 2ix)
c2 ϭ 300>[0.9>(2 # 9.80)] ϭ 80.832 [m2>sec2]
ux(L, t) ϭ 0. C ϭ ϪA, D ϭ ϪBux(0, t) ϭ 0,u(L, t) ϭ 0,u(0, t) ϭ 0,
u ϭ8L2
p3 acos c c apLb2t d sin
px
Lϩ
1
33 cos c c a3p
Lb2t d sin
3px
Lϩ Á b
n ϭ 4, 8, 12, Á .ϩ4 Ϫ 5p
125 cos 5pt sin 5px ϩ Á b .
4
p3 a(4 Ϫ p) cos pt sin px ϩ cos 2pt sin 2px ϩ4 ϩ 3p
27 cos 3pt sin 3px
ϩ1
25 (2 ϩ 12) cos 5pt sin 5px Ϫ ϩ Á b
2
p2 a(2 Ϫ 12) cos pt sin px Ϫ1
9 (2 ϩ 12) cos 3pt sin 3px
A32 App. 2 Answers to Odd-Numbered Problems
23.
25.
Problem Set 12.7, page 574
3.
5. etc.
7. if and 0 if
9. Set in (21) to get erf
13. In (12) the argument is 0 (the point where f jumps) when
This gives the lower limit of integration.
15. Set in (21).
Problem Set 12.9, page 584
1. (a), (b) It is multiplied by (c) Half
5. if m odd, 0 if m even
7.
11.
13.
17. (corresponding eigenfunctions and ), etc.
19.
Problem Set 12.10, page 591
5.
7. 55p Ϫ440
p (r cos u ϩ
1
9r 3 cos 3u ϩ
1
25r 5 cos 5u ϩ Á )
110 ϩ440
p (r cos u Ϫ
1
3r 3 cos 3u ϩ
1
5r 5 cos 5u Ϫ ϩ Á )
cos aptB36
a2ϩ
4
b2b sin
6px
a sin
4py
b
F16,14F4,16cp1260
6.4
p2 a
�
mϭ1
a
�
nϭ1m,n odd
1
m3n3 cos (t2m2 ϩ n2) sin mx sin ny
u ϭ 0.1 cos 120t sin 2x sin 4y
Bmn ϭ (Ϫ1)mϩn4ab>(mnp2)
Bmn ϭ (Ϫ1)nϩ18>(mnp2)
12.
w ϭ s>12
z ϭ Ϫx>(2c1t).x ϩ 2cz1t
(Ϫx) ϭ Ϫerf x.w ϭ Ϫv
u ϭ �1
0
cos px e�c2p2t dp
p Ͼ 1,0 Ͻ p Ͻ 1A ϭ2
p ��
0
sin v
v cos pv dv ϭ
2
p# p
2ϭ 1
A ϭ2
p �1
0
v cos pv dv ϭ2
p# cos p ϩ p sin p Ϫ 1
p2 ,
A ϭ2
p�
�
0
cos pv
1 ϩ v2dv ϭ
2
p# p
2e�p, u ϭ �
�
0
e�p�c2p2t cos px dp
a�
nϭ1
An sin npx
a sinh
np(b Ϫ y)
a, An ϭ
2
a sinh (npb>a) �a
0
f (x) sin npx
adx
A0 ϭ1
242 �24
0
f (y) dy, An ϭ1
12�
24
0
f (y) cos npy
24dy
u ϭ A0x ϩ a�
nϭ1
An
sinh (npx>24)
sinh np cos
npy
24,
App. 2 Answers to Odd-Numbered Problems A33
11. Solve the problem in the disk subject to (given) on the upper semicircle
and on the lower semicircle.
13. Increase by a factor 15.
17. No 25. See Table A1 in App. 5.
Problem Set 12.11, page 598
5.
9.
13. is smaller than the potential in Prob. 12 for
17.
19.
25. Set Then
Substitute this and etc. into (7) [written in terms of ] and divide by
Problem Set 12.12, page 602
5.
7. take to get and from
11. Set Use z as a new variable of integration. Use
Chapter 12 Review Questions and Problems, page 603
17. 19. Hyperbolic,
21. Hyperbolic, 23.
25.
27.
29.
39.
Problem Set 13.1, page 612
1. 3.
5. 9.
11. 13.
15. 17.
19.
Problem Set 13.2, page 618
1.
3. 2(cos 12p ϩ i sin 12p), 2(cos 12p Ϫ i sin 12p)
12 (cos 14p ϩ i sin 14p)
(x2 Ϫ y2)>(x2 ϩ y2), 2xy>(x2 ϩ y2)
Ϫ4x2y23 Ϫ i
Ϫ120 Ϫ 40iϪ8 Ϫ 6i
Ϫ117, 4x Ϫ iy ϭ Ϫ(x ϩ iy), x ϭ 0
4.8 Ϫ 1.4i1>i ϭ i>i2 ϭ Ϫi, 1>i3 ϭ i>i4 ϭ i
u ϭ (u1 Ϫ u0)(ln r)>ln (r1>r0) ϩ (u0 ln r1 Ϫ u1 ln r0)>ln (r1>r0)
100 cos 2x e�4t
34 sin 0.01px e�0.001143t Ϫ 1
4 sin 0.03px e�0.01029t
sin 0.01px e�0.001143t
34 cos 2t sin x Ϫ 1
4 cos 6t sin 3xf1(y ϩ 2x) ϩ f2(y Ϫ 2x)
f1(x) ϩ f2(y ϩ x)u ϭ c1(x)e�3y ϩ c2(x)e2y Ϫ 3
erf (ϱ) ϭ 1.x2>(4c2t) ϭ z2.
w(x, 0) ϭ x(c Ϫ 1) ϭ 0.
c ϭ 1g ϭ ce�t ϩ t Ϫ 1f (x) ϭ xw ϭ f (x)g(t), xf rg ϩ fg#ϭ xt,
W ϭc(s)
x sϩ
x
s2(s ϩ 1), W(0, s) ϭ 0, c(s) ϭ 0, w(x, t) ϭ x(t Ϫ 1 ϩ e�t)
r 5.ru�� ϭ rv��
urr ϭ (2vr ϩ rvrr)(1>r4) ϩ (v ϩ rvr)(2>r3), urr ϩ (2>r)ur ϭ r 5(vrr ϩ (2>r)vr).
u(r, u, �) ϭ rv(r, u, �), ur ϭ (v ϩ rvr)(Ϫ1>r2), 1>r ϭ r.
cos 2� ϭ 2 cos2 � Ϫ 1, 2w2 Ϫ 1 ϭ 43 P2(w) Ϫ 1
3, u ϭ 43r 2P2(cos �) Ϫ 1
3
u ϭ 1
2 Ͻ r Ͻ 4.u ϭ 320>r ϩ 60
ur ϭ c~>r 2, u ϭ c>r ϩ k
2u ϭ us ϩ 2ur>r ϭ 0, us>ur ϭ Ϫ2>r, ln ƒur ƒ ϭ Ϫ2 ln ƒ r ƒ ϩ c1,
A4 ϭ A6 ϭ A8 ϭ A10 ϭ 0, A5 ϭ 605>16, A7 ϭ Ϫ4125>128, A9 ϭ 7315>256
a11>(2p) ϭ 0.6098;
T ϭ 6.826rR2f 1212
u ϭ4u0
pa r
a sin u ϩ
1
3a3 r 3 sin 3u ϩ1
5a5r 5 sin 5u ϩ Á b
Ϫu0
u0r Ͻ a
A34 App. 2 Answers to Odd-Numbered Problems
5. 7.
9. 11.
13. 15.
17. 21.
23.
25.
27.
29. 31.
33. Multiply out and use
(Prob. 34).
Hence Taking
square roots gives (6).
35.
Problem Set 13.3, page 624
1. Closed disk, center radius
3. Annulus (circular ring), center radii and
5. Domain between the bisecting straight lines of the first quadrant and the fourth
quadrant.
7. Half-plane extending from the vertical straight line to the right.
11.
15. Yes, since
17. Yes, because and as
19. Now hence
21. 23.
Problem Set 13.4, page 629
1.
(a)
(b)
Multiply (a) by (b) by , and add. Etc.
3. Yes 5. No,
7. Yes, when Use (7). 9. Yes, when
11. Yes 13. c real
15. (c real) 17. (c real)
19. No 21.
23. 27. implies
29. Use (4), (5), and (1).
Problem Set 13.5, page 632
3. 5.
7. 9.
11. 13. 12epi>46.3epi
5ei arctan (3>4) ϭ 5e0.644ie12i ϭ 4.113i
e2(Ϫ1) ϭ Ϫ7.389e2pie�2p ϭ e�2p ϭ 0.001867
if ϭ Ϫv ϩ iu.f ϭ u ϩ iva ϭ 0, v ϭ 12 b(y2 Ϫ x2) ϩ c
a ϭ p, v ϭ epx sin py
f (z) ϭ z2 ϩ z ϩ cf (z) ϭ 1>z ϩ c
f (z) ϭ Ϫ12 i(z2 ϩ c),
z � 0, Ϫ2pi, 2piz � 0.
f (z) ϭ (z2)
sin ucos u,
0 ϭ uy ϩ vx ϭ ur sin u ϩ uu(cos u)>r ϩ vr cos u ϩ vu(Ϫsin u)>r0 ϭ ux Ϫ vy ϭ ur cos u ϩ uu(Ϫsin u)>r Ϫ vr sin u Ϫ vu (cos u)>r
rx ϭ x>r ϭ cos u, ry ϭ sin u, ux ϭ Ϫ(sin u)>r, uy ϭ (cos u)>r
3iz2>(z ϩ i)4, Ϫ3i>16n(1 Ϫ z) �n�1i, ni
f r(3 ϩ 4i) ϭ 8 # 37 ϭ 17,496.z Ϫ 4i ϭ 3,f r(z) ϭ 8(z Ϫ 4i)7.
r: 0.1 Ϫ ƒ z ƒ : 1Re z ϭ r cos u: 0
Im( ƒ z ƒ 2>z) ϭ Im( ƒ z ƒ 2 z>(zz)) ϭ Im z ϭ Ϫr sin u: 0.
v(x, y) ϭ y((1 Ϫ x)2 ϩ y2), v(1, Ϫ1) ϭ Ϫ1
u(x, y) ϭ (1 Ϫ x)>((1 Ϫ x)2 ϩ y2), u(1, Ϫ1) ϭ 0,
x ϭ Ϫ1
3pp4 Ϫ 2i,
32Ϫ1 ϩ 5i,
[(x1 ϩ x2)2 ϩ (y1 ϩ y2)2] ϩ [(x1 Ϫ x2)2 ϩ (y1 Ϫ y2)2] ϭ 2(x12 ϩ y1
2 ϩ x22 ϩ y2
2)
ƒ z1 ϩ z1 ƒ 2 Ϲ ( ƒ z1 ƒ ϩ ƒ z2 ƒ )2.ϭ ( ƒ z1 ƒ ϩ ƒ z2 ƒ )2.ϩ 2 ƒ z1 ƒ ƒ z2 ƒ ϩ ƒ z2 ƒ 2z1z1 ϩ z1z2 ϩ z2z1 ϩ z2z2 ϭ ƒ z1 ƒ 2 ϩ 2 Re z1z2 ϩ ƒ z2 ƒ 2 Ϲ ƒ z1 ƒ 2Re z1z2 Ϲ ƒ z1z2 ƒƒ z1 ϩ z2 ƒ 2 ϭ (z1 ϩ z2)( z1 ϩ z2 ) ϭ (z1 ϩ z2)( z 1 ϩ z 2).
Ϯ(1 Ϫ i), Ϯ(2 ϩ 2i)i, Ϫ1 Ϫ i
cos 15p Ϯ i sin 15p, cos 35p Ϯ i sin 35p, Ϫ1
cos (18p ϩ 1
2 kp) ϩ i sin (18p ϩ 1
2 kp), k ϭ 0, 1, 2, 3
6, Ϫ3 Ϯ 313i
26 2 (cos 112 kp ϩ i sin 1
12p), k ϭ 1, 9, 172 ϩ 2i
Ϫ3iϪ1024. Answer: p
Ϯarctan ( 43 ) ϭ Ϯ0.92733p>4
21 ϩ 14p
2 (cos arctan 12p ϩ i sin arctan 12p)12 (cos p ϩ i sin p)
App. 2 Answers to Odd-Numbered Problems A35
15.
17.
19.
Problem Set 13.6, page 636
1. Use (11), then (5) for and simplify. 7.
9. Both Why? 11. both
15. Insert the definitions on the left, multiply out, and simplify.
17. 19.
Problem Set 13.7, page 640
5. 7.
9. 11.
13.
15.
17.
19.
21.
23.
25.
27.
Chapter 13 Review Questions and Problems, page 641
1. 3.
11. 13.
15. i 17.
19. 21.
23. 25.
27. 29.
31.
33.
35.
Problem Set 14.1, page 651
1. Straight segment from (2, 1) to (5, 2.5).
3. Parabola from (1, 2) to (2, 8).
5. Circle through (0, 0), center radius oriented clockwise.
7. Semicircle, center 2, radius 4.
9. Cubic parabola
11.
13. z(t) ϭ 2 Ϫ i ϩ 2eit (0 Ϲ t Ϲ p)
z(t) ϭ t ϩ (2 ϩ t)i (Ϫ1 Ϲ t Ϲ 1)
y ϭ x3 (Ϫ2 Ϲ x Ϲ 2)
110,(3, Ϫ1),
y ϭ x2
cosh p cos p ϩ i sinh p sin p ϭ Ϫ11.592
i tanh 1 ϭ 0.7616i
cos 3 cosh 1 ϩ i sin 3 sinh 1 ϭ Ϫ1.528 ϩ 0.166i
f (z) ϭ e�z2>2f (z) ϭ e�2z
f (z) ϭ Ϫiz2>2(Ϯ1 Ϯ i)>12
Ϯ3, Ϯ3i15e�pi>2412e�3pi>40.16 Ϫ 0.12iϪ5 ϩ 12i
27.46e0.9929i, 7.616e1.976i2 Ϫ 3i
e(2�i) Ln(�1) ϭ e(2�i)pi ϭ ep ϭ 23.14
e(3�i)(ln 3ϩpi) ϭ 27ep(cos (3p Ϫ ln 3) ϩ i sin (3p Ϫ ln 3)) ϭ Ϫ284.2 ϩ 556.4i
e(1�i) Ln (1ϩi) ϭ eln12ϩpi>4�i ln12ϩp>4 ϭ 2.8079 ϩ 1.3179i
e0.6e0.4i ϭ e0.6 (cos 0.4 ϩ i sin 0.4) ϭ 1.678 ϩ 0.710i
e4�3i ϭ e4 (cos 3 Ϫ i sin 3) ϭ Ϫ54.05 Ϫ 7.70i
ln (i2) ϭ ln (Ϫ1) ϭ (1 Ϯ 2n)pi, 2 ln i ϭ (1 Ϯ 4n)pi, n ϭ 0, 1, Á
ln ƒ ei ƒ ϩ i arctan sin 1
cos 1Ϯ 2npi ϭ 0 ϩ i ϩ 2npi, n ϭ 0, 1, Á
Ϯ2npi, n ϭ 0, 1, Áln e ϩ pi>2 ϭ 1 ϩ pi>2i arctan (0.8>0.6) ϭ 0.927i
12 ln 32 Ϫ pi>4 ϭ 1.733 Ϫ 0.785iln 11 ϩ pi
z ϭ Ϯnpiz ϭ Ϯ(2n ϩ 1)i>2
i sinh p ϭ 11.55i,Ϫ0.642 Ϫ 1.069i.
cosh 1 ϭ 1.543, i sinh 1 ϭ 1.175ieiy,
z ϭ 2npi, n ϭ 0, 1, ÁRe (exp (z3)) ϭ exp (x3 Ϫ 3xy2) cos (3x2y Ϫ y3)
exp (x2 Ϫ y2) cos 2xy, exp (x2 Ϫ y2) sin 2xy
A36 App. 2 Answers to Odd-Numbered Problems
15.
17. Circle
19.
21.
23.
25.
27.
29. on the axes.
integrated:
35.
Problem Set 14.2, page 659
1. Use (12), Sec. 14.1, with 3. Yes 5. 5
7. (a) Yes. (b) No, we would have to move the contour across
9. 0, yes 11. no
13. 0, yes 15. no
17. 0, no 19. 0, yes
21. 23. hence
25. 0 (Why?) 27. 0 (Why?)
29. 0
Problem Set 14.3, page 663
1. 3. 0
5. 7.
11. 13.
15. since
and
17.
19.
Problem Set 14.4, page 667
1. 3.
5.
7.
9.
11.
ϭ 2.821i
ϭ 12pi(sin 12 ϩ 3
2 cos 12 )
2pi
4((1 ϩ z)sin z)r `
zϭ1>2ϭ 1
2pi(sin z ϩ (1 ϩ z) cos z) ƒ zϭ1>2
Ϫ2pi(tan pz)r `zϭ0
ϭϪ2pi � p
cos2 pz`zϭ0
ϭ Ϫ2p2i
(2pi>(2n)!) (cos z)(2n) ƒ zϭ0 ϭ (2pi>(2n)!)(Ϫ1)n cos 0 ϭ (Ϫ1)n2pi>(2n)!
2pi
3!(cosh 2z)t ϭ
pi
3� 8 sinh 1 ϭ 9.845i
(2pi>(n Ϫ 1)!)e0(2pi>3!)(Ϫcos 0) ϭ Ϫpi>3
2pie2i>(2i) ϭ pe2i
2piLn (z ϩ 1)
z ϩ i`zϭi
ϭ 2piLn (1 ϩ i)
2iϭ p(ln12 ϩ ip>4) ϭ 1.089 ϩ 2.467i
sinh pi ϭ i sin p ϭ 0.
cosh pi ϭ cos p ϭ Ϫ12pi cosh (Ϫp2 Ϫ pi) ϭ Ϫ2pi cosh p2 ϭ Ϫ60,739i
2pi(z ϩ 2) ƒ zϭ2 ϭ 8pi2pi �1
z ϩ 2i`zϭ2i
ϭp
2
2pi(i>2)3>2 ϭ p>82pi(cos 3z)>6 ƒ zϭ0 ϭ pi>32piz2>(z Ϫ 1) ƒ zϭ�1 ϭ Ϫpi
2pi ϩ 2pi ϭ 4pi.1>z ϩ 1>(z Ϫ 1),2pi
Ϫp,
pi,
Ϯ2i.
m ϭ 2.
ƒRe z ƒ ϭ ƒ x ƒ Ϲ 3 ϭ M on C, L ϭ 18
(Ϫ1 ϩ i)>3.y(Ϫ1 ϩ i)(Im z2)#z ϭ 2(1 Ϫ t)
z ϭ 1 ϩ (Ϫ1 ϩ i)t (0 Ϲ t Ϲ 1),Im z2 ϭ 2xy ϭ 0
tan 14pi Ϫ tan 14 ϭ i tanh 14 Ϫ 1
12 exp z2 ƒ
i
1 ϭ 12 (e�1 Ϫ e1) ϭ Ϫsinh 1
e2pi Ϫ epi ϭ 1 Ϫ (Ϫ1) ϭ 2
z(t) ϭ (1 ϩ i)t (1 Ϲ t Ϲ 3), Re z ϭ t, zr (t) ϭ 1 ϩ i. Answer: 4 ϩ 4i
z(t) ϭ t ϩ (1 Ϫ 14 t 2)i (Ϫ2 Ϲ t Ϲ 2)
z(t) ϭ Ϫa Ϫ ib ϩ re�it (0 Ϲ t Ϲ 2p)
z(t) ϭ 2 cosh t ϩ i sinh t (Ϫϱ Ͻ t Ͻ ϱ)
App. 2 Answers to Odd-Numbered Problems A37
13. 15. 0. Why?
17. 0 by Cauchy’s integral theorem for a doubly connected domain; see (6) in Sec. 14.2.
19.
Chapter 14 Review Questions and Problems, page 668
21.
23. by Cauchy’s integral formula.
25.
27. 0 since and
29.
Problem Set 15.1, page 679
1. bounded, divergent,
3. by algebra; convergent to
5. Bounded, divergent,
7. Unbounded, hence divergent
9. Convergent to 0, hence bounded
17. Divergent; use 19. Convergent; use
21. Convergent 23. Convergent
25. Divergent
29. By absolute convergence and Cauchy’s convergence principle, for given we
have for every and
hence by Sec. 13.2, hence convergence by Cauchy’s
principle.
Problem Set 15.2, page 684
1. No! Nonnegative integer powers of z (or only!
3. At the center, in a disk, in the whole plane
5. hence
7. 9. 11.
13. 15. 2i, 1 17.
Problem Set 15.3, page 689
3. Apply l’Hôpital’s rule to
5. 2 7. 9.
11. 13. 1 15.
Problem Set 15.4, page 697
3. 2z2 Ϫ(2z2)3
3!ϩ Á ϭ 2z2 Ϫ
4
3z6 ϩ
4
15z10 Ϫ ϩ Á , R ϭ ϱ
3427
3
1>1213
ln f ϭ (ln n)>n.f ϭ 2n n.
1>12Ϫi, 12
0,2265i,13p>2, ϱ
ƒ z ƒ Ͻ 1R.Sanz2n ϭ San(z2)n, ƒ z2 ƒ Ͻ R ϭ lim ƒan>anϩ1 ƒ ;
z Ϫ z0)
(6*),ƒ znϩ1 ϩ Á ϩ znϩp ƒ Ͻ P
ƒ znϩ1 ƒ ϩ Á ϩ ƒ znϩp ƒ Ͻ P,
p ϭ 1, 2, Án Ͼ N(P)
P Ͼ 0
S1>n2.1>ln n Ͼ 1>n.
Ϯ1 ϩ 10i
Ϫpi>2zn ϭ Ϫ12pi>(1 ϩ 2>(ni))
Ϯ1, Ϯizn ϭ (2i>2)n;
Ϫ4pi
y ϭ xz2 ϩ z Ϫ 2 ϭ 2(x2 Ϫ y2)
Ϫ2pi(tan pz)r|zϭ1 ϭ Ϫ2p2i>cos2 pz|zϭ1 ϭ Ϫ2p2i
2pi(ez)(4) ƒ zϭ0 ϭ iez>12 ƒ zϭ0 ϭ pi>12
12 cosh (Ϫ1
4p2) Ϫ 1
2 ϭ 2.469
(2pi>2!)4�3(e3z)s ƒ zϭpi>4 ϭ Ϫ9p(1 ϩ i)>(6412)
2pi # 1z`zϭ2
ϭ pi
A38 App. 2 Answers to Odd-Numbered Problems
5.
7.
9.
11.
13.
17. Team Project. (a)
(c) Use that the terms of are all positive, so that the sum cannot be zero.
19.
21.
23.
25.
Problem Set 15.5, page 704
3.
5.
7. Nowhere
9.
11. and converges. Use Theorem 5.
13. for all z, and converges. Use Theorem 5.
15. by Theorem 2 in Sec. 15.2; use Theorem 1.
17. use Theorem 1.
Chapter 15 Review Questions and Problems, page 706
11. 1 13. 3
15. 17.
19.
21.
23.
25.
27.
29. ln 3 ϩ1
3(z Ϫ 3) Ϫ
1
2 � 9(z Ϫ 3)2 ϩ
1
3 � 27(z Ϫ 3)3 Ϫ ϩ Á , R ϭ 3
cos [(z Ϫ 12p) ϩ 1
2p] ϭ Ϫ(z Ϫ 12p) ϩ 1
6(z Ϫ 12p)3 Ϫ ϩ Á ϭ Ϫsin (z Ϫ 1
2p)
a�
nϭ1
(Ϫ1)nϩ1
n!z2n�2, R ϭ ϱ
1
2ϩ
1
2cos 2z ϭ 1 ϩ
1
2 a�
nϭ1
(Ϫ1)n
(2n)! (2z)2n, R ϭ ϱ
a�
nϭ0
z4n
(2n ϩ 1)!, R ϭ ϱ
ϱ, cosh 1z
ϱ, e2z12
R ϭ 1>1p Ͼ 0.56;
R ϭ 4
S1>n2ƒ sinn ƒ z ƒ ƒ Ϲ 1
S1>n2ƒ zn ƒ Ϲ 1
ƒ z Ϫ 2i ƒ Ϲ 2 Ϫ d, d Ͼ 0
ƒ z ϩ 12 i ƒ Ϲ 1
4 Ϫ d, d Ͼ 0
ƒ z ϩ i ƒ Ϲ 13 Ϫ d, d Ͼ 0
2az Ϫ1
2ib ϩ
23
3!az Ϫ
1
2ib
3
ϩ25
5!az Ϫ
1
2ib
5
ϩ Á , R ϭ ϱ
Ϫ14 Ϫ 2
8 i(z Ϫ i) ϩ 316 (z Ϫ i)2 ϩ 4
32 i(z Ϫ i)3 Ϫ 564 (z Ϫ i)4 ϩ Á , R ϭ 2
1 Ϫ1
2! az Ϫ
1
2pb
2
ϩ1
4! az Ϫ
1
2pb
4
Ϫ1
6! az Ϫ
1
2pb
6
ϩ Ϫ Á , R ϭ ϱ
12 ϩ 1
2 i ϩ 12 i(z Ϫ i) ϩ (Ϫ1
4 ϩ 14 i)(z Ϫ i)2 Ϫ 1
4 (z Ϫ i)3 ϩ Á , R ϭ 12
(sin iy)>(iy)
(Ln (1 ϩ z))r ϭ 1 Ϫ z ϩ z2 Ϫ ϩ Á ϭ 1>(1 ϩ z).
(2>1p)(z Ϫ z3>3 ϩ z5>(2!5) Ϫ z7>(3!7) ϩ Á ), R ϭ ϱ
z3>(1!3) Ϫ z7>(3!7) ϩ z11>(5!11) Ϫ ϩ Á , R ϭ ϱ
�z
0
a1 Ϫ1
2t 2 ϩ
1
8t 4 Ϫ ϩ Áb dt ϭ z Ϫ
1
6z3 ϩ
1
40z5 Ϫ ϩ Á , R ϭ ϱ
1
2ϩ
1
2cos z ϭ 1 Ϫ
1
2 � 2!z2 ϩ
1
2 � 4!z4 Ϫ
1
2 � 6!z6 ϩ Ϫ Á , R ϭ ϱ
12 Ϫ 1
4 z4 ϩ 18 z8 Ϫ 1
16 z12 ϩ 132 z16 Ϫ ϩ Á , R ϭ 24 2
App. 2 Answers to Odd-Numbered Problems A39
A40 App. 2 Answers to Odd-Numbered Problems
Problem Set 16.1, page 714
1.
3.
5.
7.
9.
11.
13.
15.
19.
21.
23.
25.
Section 16.2, page 719
1. fourth order 3. fourth order
5. second order 7. simple
9. simple
11. hence
13. Second-order poles at i and
15. Simple pole at essential singularity at
17. Fourth-order poles at essential singularity at
19. simple zeros. Answer: simple poles at
essential singularity at
21. essential singularities, simple poles
Section 16.3, page 725
3. at 0 5.
7. at 9. at
11. at
15. Simple pole at inside C, residue Answer:
17. Simple poles at residue and at residue
Answer:
19.
21. Answer: 2p5i>24z�5 cos pz ϭ Á ϩ p4>(4!z) Ϫ ϩ Á .
2pi (sinh 12 i)>2 ϭ Ϫp sin 12
Ϫ4pi sinh p>2e�p>2>sin p>2 ϭ e�p>2.
Ϫp>2,ep>2>(Ϫsin p>2),p>2,
ϪiϪ1>(2p).14
z ϭ pi(ez)s>2! ƒ zϭpi ϭ Ϫ12
Ϯ2npiϪ10, Ϯ1, Á1>pϮ4i at ϯ i
415
Ϯ2npi, n ϭ 0, 1, Á ,1, ϱ
ϱ
Ϯ2npi,ez(1 Ϫ ez) ϭ 0, ez ϭ 1, z ϭ Ϯ2npi
ϱϮnpi, n ϭ 0, 1, Á ,
1 ϩ iϱ,
Ϫ2i
f 2(z) ϭ (z Ϫ z0)2ng2(z).f (z) ϭ (z Ϫ z0)ng(z), g(z0) � 0,
12 sin 4z, z ϭ 0, Ϯp>4, Ϯp>2, Á ,
Ϯ(2 ϩ 2i), Ϯi,Ϯ1, Ϯ2, Á ,
Ϫ81i,0 Ϯ 2p, Ϯ4p, Á ,
i
(z Ϫ i)2 ϩ1
z Ϫ iϩ i ϩ (z Ϫ i)
ƒ z ƒ Ͼ 1z8 ϩ z12 ϩ z16 ϩ Á , ƒ z ƒ Ͻ 1, Ϫz4 Ϫ 1 Ϫ z�4 Ϫ z�8 Ϫ Á ,
ƒ z ϩ 12p ƒ Ͼ 0
Ϫ(z ϩ 12p)�1 cos (z ϩ 1
2p) ϭ Ϫ(z ϩ 12p)�1 ϩ 1
2 (z ϩ 12p) Ϫ 1
24 (z ϩ 12p)3 ϩ Ϫ Á ,
ƒ z ƒ Ͼ 1a�
nϭ0
z2n, ƒ z ƒ Ͻ 1, Ϫ a�
nϭ0
1
z2nϩ2,
0 Ͻ ƒ z Ϫ p ƒ Ͻ ϱ
(Ϫcos (z Ϫ p))(z Ϫ p)�2 ϭ Ϫ(z Ϫ p)�2 ϩ 12 Ϫ 1
24 (z Ϫ p)2 ϩ Ϫ Á ,
0 Ͻ ƒ z Ϫ i ƒ Ͻ 1Ϫ3(z Ϫ i)�1 Ϫ 6i ϩ 10(z Ϫ i) ϩ Á ,
ϭ i(z Ϫ i)�2i�3 a1 ϩz Ϫ i
ib
�3
(z Ϫ i)�2 ϭ a�
nϭ0
aϪ3
nb i�3�n(z Ϫ i)n�2
[pi ϩ (z Ϫ pi)]2
(z Ϫ pi)4ϭ
(pi)2
(z Ϫ pi)4ϩ
2pi
(z Ϫ pi)3ϩ
1
(z Ϫ pi)2
0 Ͻ ƒ z Ϫ 1 ƒ Ͻ ϱ
exp [1 ϩ (z Ϫ 1)] (z Ϫ 1)�2 ϭ e � [(z Ϫ 1)�2 ϩ (z Ϫ 1)�1 ϩ 12 ϩ 1
6 (z Ϫ 1) ϩ Á ],
z3 ϩ 12 z ϩ 1
24 z�1 ϩ 1720 z3 ϩ Á , 0 Ͻ ƒ z ƒ Ͻ ϱ
z�2 ϩ z�1 ϩ 1 ϩ z ϩ z2 ϩ Á , 0 Ͻ ƒ z ƒ Ͻ 1
z�3 ϩ z�1 ϩ 12 z ϩ 1
6 z3 ϩ 124 z5 ϩ Á , 0 Ͻ ƒ z ƒ Ͻ ϱ
z�4 Ϫ 12 z�2 ϩ 1
24 Ϫ 1720 z2 ϩ Ϫ Á , 0 Ͻ ƒ z ƒ Ͻ ϱ
23. Residues at at Answer:
25. Simple poles inside C at residues
respectively. Answer:
Problem Set 16.4, page 733
1. 3.
5. 7.
9. 0. Why? (Make a sketch.) 11.
13. 0. Why? 15.
17. 0. Why?
19. Simple poles at (and
21. Simple poles at 1 and residues i and Answer:
23. 25. 0
27. Let Use (4) in Sec. 16.3 to form the sum of
the residues and show that this sum is 0; here
Chapter 16 Review Questions and Problems, page 733
11. 13.
15. 17. 0 (n even), (n odd)
19. 21.
23. 0. Why? 25. Answer:
Problem Set 17.1, page 741
5. Only in size
7.
9. Parallel displacement; each point is moved 2 to the right and 1 up.
11. 13.
15. 17. Annulus
19.
21.
23.
25. at
29. on the unit circle,
31. on the unit circle,
33. for the y-axis,
35. on the unit circle,
Problem Set 17.2, page 745
7. 9.
11. 13. z ϭ 0, Ϯ12 , Ϯ ϭ Ϯi>2z ϭ 0, 1>(a ϩ ib)
z ϭ4w ϩ i
Ϫ3iw ϩ 1z ϭ
w ϩ i
2w
J ϭ 1> ƒ z ƒ 2M ϭ 1> ƒ z ƒ ϭ 1
J ϭ e2xx ϭ 0,M ϭ ex ϭ 1
J ϭ 1> ƒ z ƒ 4ƒwr ƒ ϭ 1> ƒ z ƒ 2 ϭ 1
J ϭ ƒ z ƒ 2M ϭ ƒ z ƒ ϭ 1
z ϭ 0, Ϯpi, Ϯ2pi, Ásinh z ϭ 0
z ϭ (Ϫ1 Ϯ 13)>2z3 ϩ az2 ϩ bz ϩ c, z ϭ Ϫ 1
3 (a Ϯ 2a2 Ϫ 3b)
0 Ͻ u Ͻ ln 4, p>4 Ͻ v Ϲ 3p>412 Ϲ ƒw ƒ Ϲ 4u м 1
Ϫ5 Ϲ Re z Ϲ Ϫ2ƒw ƒ Ϲ 14 , Ϫp>4 Ͻ Arg w Ͻ p>4
x ϭ c, w ϭ Ϫy ϩ ic; y ϭ k, w ϭ Ϫk ϩ ix
p>e.Reszϭi
eiz>(z2 ϩ 1) ϭ 1>(2ie).
p>60p>6(Ϫ1)(n�1)>22pi>(n Ϫ 1)!2pi(25z2)r ƒ zϭ5 ϭ 500pi
2pi(Ϫ10 Ϫ 10)6pi
k Ͼ 1.1>qr(a1) ϩ Á ϩ 1>qr(ak)
q(z) ϭ (z Ϫ a1)(z Ϫ a2) Á (z Ϫ ak).
Ϫp>2
p
5 (cos 1 Ϫ e�2)Ϫi.Ϯ2pi,
Ϫi); 2pi �14 i ϩ pi(Ϫ1
4 ϩ 14 ) ϭ Ϫ1
2pϮ1, i
p>3p>22ap>2a2 Ϫ 15p>12
p>122p>2k2 Ϫ 1
2pi �410
110 , 1
10 , 110 , 1
10 ,
(2i cosh 2i)>(4z3 ϩ 26z) ƒ zϭ2i ϭ2i, Ϫ2i, 3i, Ϫ3i,
5piz ϭ 13 .z ϭ 1
2 , 212
App. 2 Answers to Odd-Numbered Problems A41
15. 17. 19.
Problem Set 17.3, page 750
3. Apply the inverse g of f on both sides of to get
9. a rotation. Sketch to see. 11.
13. almost by inspection 15.
17. 19.
Problem Set 17.4, page 754
1. Circle 3. Annulus
5. w-plane without 7.
9.
11.
13. Elliptic annulus bounded by and
15.
17. is the image of R under Answer:
19. Hyperbolas when and
(thus when
21. Interior of in the fourth quadrant, or map
by (why?).
23.
25. The images of the five points in the figure can be obtained directly from the
function w.
Problem Set 17.5, page 756
1. w moves once around the circle
3. Four sheets, branch point at
5. three sheets
7. n sheets
9. two sheets
Chapter 17 Review Questions and Problems, page 756
11. 13. Horizontal strip
15. same (why?) 17.
19. 21.
23. 25. Rotation
27. 29.
31. 33.
35. 37.
39. w ϭ z2>(2c)
w ϭ iz2 ϩ 1w ϭ e4z
z ϭ 0, Ϯi, Ϯ3iz ϭ 2 Ϯ16
z ϭ 0w ϭ 1>zw ϭ izw ϭ
10z ϩ 5i
z ϩ 2i
w ϭ 1 ϩ iv, v Ͻ 013 Ͻ ƒw ƒ Ͻ 1
2 , v Ͻ 0
ƒw ƒ Ͼ 1u ϭ 1 Ϫ 14 v2,
Ϫ8 Ͻ v Ͻ 81 Ͻ ƒw ƒ Ͻ 4, ƒ arg w ƒ Ͻ p>4
1z (z Ϫ i)(z ϩ i), 0, Ϯi,
z0,
Ϫi>4,
z ϭ Ϫ1
ƒw ƒ ϭ 12 .
v Ͻ 0
w ϭ sin zp>2 Ͻ x Ͻ p, 0 Ͻ y Ͻ 2
u2>cosh2 2 ϩ v2>sinh2 2 ϭ 1
c ϭ 0, p.ƒu ƒ м 1), v ϭ 0u ϭ Ϯcosh y
c � 0, p,u2>cos2 c Ϫ v2>sin2 c ϭ cosh2 c Ϫ sinh2 c ϭ 1
et ϭ ez2>2.t ϭ z2>2.0 Ͻ Im t Ͻ p
cosh z ϭ cos iz ϭ sin (iz ϩ 12p)
u2>cosh2 3 ϩ v2>sinh2 3 ϭ 1
u2>cosh2 1 ϩ v2>sinh2 1 ϭ 1
u2>cosh2 2 ϩ v2>sinh2 2 Ͻ 1, u Ͼ 0, v Ͼ 0
Ϯ(2n ϩ 1)p>2, n ϭ 0, 1, Á1 Ͻ ƒw ƒ Ͻ e, v Ͼ 0w ϭ 0
1>1e Ϲ ƒw ƒ Ϲ 1eƒw ƒ ϭ ec
w ϭ (z4 Ϫ i)(Ϫiz4 ϩ 1)w ϭ (2z Ϫ i)>(Ϫiz Ϫ 2)
w ϭ 1>z Ϫ 1w ϭ 1>z,
w ϭ (z ϩ i)>(z Ϫ i)w ϭ iz,
g(z1) ϭ g( f (z1)) ϭ z1.z1 ϭ f (z1)
w ϭaz ϩ b
Ϫbz ϩ aw ϭ
az
cz ϩ az ϭ i, 2i
A42 App. 2 Answers to Odd-Numbered Problems
Problem Set 18.1, page 762
1.
3.
5.
7.
13. Use Fig. 391 in Sec. 17.4 with the z- and w-planes interchanged and
15.
Problem Set 18.2, page 766
3. maps R onto the strip and
5. (a) (b)
7. See Fig. 392 in Sec. 17.4.
9.
13. Corresponding rays in the w-plane make equal angles, and the mapping is conformal.
15. Apply
17. by (3) in Sec. 17.3.
19.
Problem Set 18.3, page 769
1. Rotate through
5.
7.
9.
11.
13. from and Prob. 11.
15.
17. Re (arcsin z)
Problem Set 18.4, page 776
1. V(z) continuously differentiable.
3. is maximum at namely, 2.y ϭ Ϯ1,ƒFr(iy) ƒ ϭ 1 ϩ 1>y2, ƒ y ƒ м 1,
Re F(z) ϭ 100 ϩ (200>p)
Ϫ20 ϩ (320>p) Arg z ϭ Re aϪ20 Ϫ320i
p Ln zbw ϭ z2100
p [Arg (z2 Ϫ 1) Ϫ Arg (z2 ϩ 1)]
100p (Arg (z Ϫ 1) Ϫ Arg (z ϩ 1)) ϭ Re a100i
p Ln z ϩ 1
z Ϫ 1b
T1
p aarctan y
x Ϫ bϪ arctan
y
x Ϫ ab ϭ Re a iT1
p Ln z Ϫ a
z Ϫ bb
T1 ϩ2p (T2 Ϫ T1) arctan
y
xϭ Re aT1 Ϫ
2i
p (T2 Ϫ T1) Ln zb80p arctan
y
xϭ Re aϪ 80i
p Ln zbp>2.(80>d )y ϩ 20.
£ ϭ5p Arg (z Ϫ 2), F ϭ Ϫ
5i
p Ln (z Ϫ 2)
z ϭ (2Z Ϫ i)>(ϪiZ Ϫ 2)
w ϭ z2.
cos2 x (y ϭ 0), cos2 x cosh2 1 Ϫ sin2 x sinh2 1 (y ϭ 1)
Ϫsinh y (x ϭ p2 ),£ (x, y) ϭ cos2 x cosh2 y Ϫ sin2 x sinh2 y; cosh2 y (x ϭ 0),
sinh2 1 (y ϭ 1), Ϫsinh2 y (x ϭ 0, p).
sin2 x cosh2 1 Ϫ cos2 xsin2 x (y ϭ 0),£ ϭ Re (sin2 z),
x2 Ϫ y2 ϭ c, xy ϭ c, ex cos y ϭ c(x Ϫ 2) (2x Ϫ 1) ϩ 2y2
(x Ϫ 2)2 ϩ y2ϭ c,
U2 ϩ (U1 Ϫ U2) (1 Ϫ xy).
£* ϭ U2 ϩ (U1 Ϫ U2) (1 ϩ 12 u) ϭϪ2 Ϲ u Ϲ 0;w ϭ iz2
£ ϭ 220 (x3 Ϫ 3xy2) ϭ Re (220z3)
cos z ϭ sin (z ϩ 12p).
£ (r) ϭ Re (32 Ϫ z)
£ (x) ϭ Re (375 ϩ 25z)
£ ϭ Re a30 Ϫ20
ln 10 Ln zb
2.5 mm ϭ 0.25 cm; £ ϭ Re 110 (1 ϩ (Ln z)>ln 4)
App. 2 Answers to Odd-Numbered Problems A43
5. Calculate or note that div grad and curl grad is the zero vector; see Sec. 9.8 and
Problem Set 9.7.
7. Horizontal parallel flow to the right.
9.
11. Uniform parallel flow upward,
13.
15.
17. Use that gives and interchanging the roles of the z- and
w-planes.
19. or
Problem Set 18.5, page 781
5.
7.
9.
11.
13.
15.
17.
Problem Set 18.6, page 784
1. Use (2). etc.
3. Use (2). etc.
5. No, because is not analytic.
7.
9.
13.
15.
17.
19. No. Make up a counterexample.
ƒF(z) ƒ 2 ϭ 4(2 Ϫ 2 cos 2u), z ϭ p>2, 3p>2, Max ϭ 4
Max ϭ sinh 2 ϭ 3.627
1 � sin2 2y, z ϭ 1,ƒF(z) ƒ 2 ϭ sinh2 2x cos2 2y ϩ cosh2 2x sin2 2y ϭ sinh2 2x ϩ
ƒF(z) ƒ ϭ [cos2 x ϩ sinh2 y]1>2, z ϭ Ϯi, Max ϭ [1 ϩ sinh2 1]1>2 ϭ 1.543
ϭ1p �
32
� 2p
£ (1, 1) ϭ 3 ϭ1p �
1
0
�2p
0
(3 ϩ r cos a ϩ r sin a ϩ r 2 cos a sin a)r dr da
ϭ1
p �1
0
�2p
0
(Ϫ3r ϩ Á ) dr da ϭ
1
paϪ 3
2b � 2p
£ (2, Ϫ2) ϭ Ϫ3 ϭ1p �
1
0
�2p
0
(1 ϩ r cos a) (Ϫ3 ϩ r sin a)r dr da
ƒ z ƒ
F(4) ϭ 100F(z0 ϩ eia) ϭ (2 ϩ 3eia)2,
F(52) ϭ 343
8F(z0 ϩ eia) ϭ (72 ϩ eia)3,
£ ϭ1
3Ϫ
4
p2 ar cos u Ϫ1
4r 2 cos 2u ϩ
1
9r 3 cos 3u Ϫ ϩ Á b
£ ϭ1
2ϩ
2
par cos u Ϫ
1
3r 3 cos 3u ϩ
1
5r 5 cos 5u Ϫ ϩ Á b
£ ϭ2
pr sin u ϩ
1
2r 2 sin 2u Ϫ
2
9pr 3 sin 3u Ϫ
1
4r 4 sin 4u ϩ ϩ Ϫ Á
£ ϭ2
par sin u Ϫ
1
2r 2 sin 2u ϩ
1
3r 3 sin 3u Ϫ ϩ Á b
£ ϭ 3 Ϫ 4r 2 cos 2u ϩ r 4 cos 4u
£ ϭ 12 a ϩ 1
2 ar 8 cos 8u
£ ϭ 32 r 3 sin 3u
x2 ϩ (y Ϫ k)2 ϭ k2y>(x2 ϩ y2) ϭ c
z ϭ cos ww ϭ arccos z
F(z) ϭ z>r0 ϩ r0>zF(z) ϭ z3
V ϭ F r ϭ iK, V1 ϭ 0, V2 ϭ K
F(z) ϭ z4
2 ϭ
A44 App. 2 Answers to Odd-Numbered Problems
Chapter 18 Review Questions and Problems, page 785
11.
13.
17.
19.
21. parallel flow
23.
25.
Problem Set 19.1, page 796
1.
3. 6.3698, 6.794, 8.15, impossible
5. Add first, then round.
7. (6S-exact)
9.
11.
13.
hence
15. is 6S-exact.
19. In the present case, (b) is slightly more accurate than (a) (which may produce
nonsensical results; cf. Prob. 20).
21.
23. The algorithm in Prob. 22 repeats 0011 infinitely often.
25. The beginning is 0.09375
27.
etc.
29.
Problem Set 19.2, page 807
3.
5. Convergence to 4.7 for all these starting values.
7. converges to 0.58853 (5S-exact) in 14 steps.
9. (6S-exact)
11. 4S-exact
13. This follows from the intermediate value theorem of calculus.
15.
17. Convergence to respectively. Reason seen easily from the
graph of f.
x ϭ 4.7, 4.7, 0.8, Ϫ0.5,
x3 ϭ 0.450184
g ϭ 4>x ϩ x3>16 Ϫ x5>576; x0 ϭ 2, xn ϭ 2.39165 (n м 6), 2.405
x ϭ x4 Ϫ 0.12; x0 ϭ 0, x3 ϭ Ϫ0.119794
x ϭ x>(ex sin x); 0.5, 0.63256, Á
g ϭ 0.5 cos x, x ϭ 0.450184 (ϭ x10, exact to 6S)
Ϫ0.126 # 10�2, Ϫ0.402 # 10�3; Ϫ0.266 # 10�6, Ϫ0.847 # 10�7
I11 ϭ 0.2102 (0.2103),
I12 ϭ 0.1951 (0.1951),I14 ϭ 0.1812 (0.1705 4S-exact), I13 ϭ 0.1812 (0.1820),
(n ϭ 1).n ϭ 26.
c4# 24 ϩ Á ϩ c0
# 20 ϭ (1 0 1 1 1.)2, NOT (1 1 1 0 1.)2
(a) 1.38629 Ϫ 1.38604 ϭ 0.00025, (b) ln 1.00025 ϭ 0.000249969
` aa1
a2Ϫ
a�1
a�2b^ ` a1
a2` � ` P1
a1Ϫ
P2
a2` Ϲ ƒ Pr1 ƒ ϩ ƒ Pr2 ƒ Ϲ br1 ϩ br2
a1
a2
ϭa�1 ϩ P1
a�2 ϩ P2
ϭa�1 ϩ P1
a�2
a1 ϪP2
a�2
ϩP2
2
a�22 Ϫ ϩ Á b �
a�1
a�2
ϩP1
a�2
ϪP2
a�2
#a�1
a�2
,
Ϲ ƒ Px ƒ ϩ ƒ Py ƒ ϭ bx ϩ by
ƒ P ƒ ϭ ƒ x ϩ y Ϫ (x� ϩ y�) ƒ ϭ ƒ (x Ϫ x�) ϩ (y Ϫ y�) ƒ ϭ ƒ Px ϩ Py ƒ
29.97, 0.035; 29.97, 0.03337; 30, 0.0; 30, 0.033
29.9667, 0.0335; 29.9667, 0.0333704
0.84175 # 102, Ϫ0.52868 # 103, 0.92414 # 10�3, Ϫ0.36201 # 106
Fr(z) ϭ z ϩ 1 ϭ x ϩ 1 Ϫ iy
T(x, y) ϭ x (2y ϩ 1) ϭ const
£ ϭ x ϩ y ϭ const, V ϭ Fr(z) ϭ 1 Ϫ i,
30(1 Ϫ (2>p) Arg (z Ϫ 1))
2(1 Ϫ (2>p) Arg z)
£ ϭ Re (220 Ϫ 95.54 Ln z) ϭ 220 Ϫ220
ln 10 ln r ϭ 220 Ϫ 95.54 ln r.
£ ϭ 10(1 Ϫ x ϩ y), F ϭ 10 Ϫ 10(1 ϩ i)z
App. 2 Answers to Odd-Numbered Problems A45
19.
21.
23. (6S-exact)
25. (a) ALGORITHM BISECT Bisection Method
This algorithm computes the solution c of ( f continuous) within the
tolerance given an initial interval such that
INPUT: Continuous function f, initial interval tolerance maximum
number of iterations N.
OUTPUT: A solution c (within the tolerance or a message of failure.
For do:
If then OUTPUT c Stop. [Procedure completed]
Else if then set and
Else set and
If then OUTPUT c. Stop. [Procedure completed]
End
OUTPUT and a message “Failure”. Stop.
[Unsuccessful completion; N iterations did not give an interval of length not
exceeding the tolerance.]
End BISECT
Note that gives as an approximation of the zero and
as a corresponding error bound.
(b) 0.739085; (c) 1.30980, 0.429494
27. (6S-exact)
29. 0.904557 (6S-exact)
Problem Set 19.3, page 819
1.
3.
5.
7.
9.
11.
13.
15.
17.
ϭ 0.3293
r ϭ Ϫ1.5, p2(0.3) ϭ 0.6039 ϩ (Ϫ1.5) # 0.1755 ϩ 12 (Ϫ1.5)(Ϫ0.5) # (Ϫ0.0302)
p3(x) ϭ 2.1972 ϩ (x Ϫ 9) # 0.1082 ϩ (x Ϫ 9)(x Ϫ 9.5) # 0.005235
2x2 Ϫ 4x ϩ 2
p2(0.5) ϭ 0.943654, p3(1.5) ϭ 0.510116, p3(2.5) ϭ Ϫ0.047991
p3(x) ϭ 1 ϩ 0.039740x Ϫ 0.335187x2 ϩ 0.060645x3;L3 ϭ 16 x(x Ϫ 1)(x Ϫ 2);
L0 ϭ Ϫ16 (x Ϫ 1)(x Ϫ 2)(x Ϫ 3), L1 ϭ 1
2 x(x Ϫ 2)(x Ϫ 3), L2 ϭ Ϫ12 x(x Ϫ 1)(x Ϫ 3),
(5S-exact 0.71116)
p2(x) ϭ Ϫ0.44304x2 ϩ 1.30896x Ϫ 0.023220, p2(0.75) ϭ 0.70929
0.9053 (0.0186), 0.9911 (Ϫ0.0672)
p2(x) ϭ 1.1640x Ϫ 0.3357x2; Ϫ0.5089 (error 0.1262), 0.4053 (Ϫ0.0226),
0.4678 (0.0046)
0.8033 (error Ϫ0.0245), 0.4872 (error Ϫ0.0148); quadratic: 0.7839 (Ϫ0.0051),
ϩ(x Ϫ 1)(x Ϫ 1.02)
0.04 # 0.02# 0.9784 ϭ x2 Ϫ 2.580x ϩ 2.580; 0.9943, 0.9835
p2(x) ϭ(x Ϫ 1.02)(x Ϫ 1.04)
(Ϫ0.02)(Ϫ0.04)# 1.0000 ϩ
(x Ϫ 1)(x Ϫ 1.04)
0.02 (Ϫ0.02)# 0.9888
ϭ 0.1086 # 9.3 ϩ 1.230 ϭ 2.2297
L0(x) ϭ Ϫ2x ϩ 19, L1(x) ϭ 2x Ϫ 18, p1(9.3) ϭ L0(9.3) # f0 ϩ L1(9.3) # f1
x2 ϭ 1.5, x3 ϭ 1.76471, Á , x7 ϭ 1.83424
(bN Ϫ aN)>2(aN ϩ bN)>2[aN, bN]
[aN, bN]
ƒanϩ1 Ϫ bnϩ1 ƒ Ͻ P ƒ c ƒ
bnϩ1 ϭ bn.anϩ1 ϭ c,
bnϩ1 ϭ c.anϩ1 ϭ anf (an) f (bn) Ͻ 0
f (c) ϭ 0
c ϭ 12 (an ϩ bn)
n ϭ 0, 1, Á , N Ϫ 1
P),
P,[a0, b0],
f (a0) f (b0) Ͻ 0.[a0, b0]P,
f (x) ϭ 0
( f, a0, b0, P, N )
x0 ϭ 4.5, x4 ϭ 4.73004
1.834243 (ϭ x4), 0.656620 (ϭ x4), Ϫ2.49086 (ϭ x4)
0.5, 0.375, 0.377968, 0.377964; (b) 1>17
A46 App. 2 Answers to Odd-Numbered Problems
Problem Set 19.4, page 826
9. at (due to roundoff;
should be 0).
11.
13.
15.
17. Use the fact that the third derivative of a cubic polynomial is constant, so that
is piecewise constant, hence constant throughout under the present assumption.
Now integrate three times.
19. Curvature if is small.
Problem Set 19.5, page 839
1. 0.747131, which is larger than 0.746824. Why?
3. (exact)
5.
7. 0.693254 (6S-exact 0.693147)
9. 0.073930 (6S-exact 0.073928)
11. 0.785392 (6S-exact 0.785398)
13.
15. (a) (b)
hence 14.
17. 0.94614588, (8S-exact 0.94608307)
19. 0.9460831 (7S-exact)
21. 0.9774586 (7S-exact 0.9774377)
23. Set
25.
27.
29.
Chapter 19 Review Questions and Problems, page 841
17.
19.
21. The same as that of
23.
(error less than 1 unit of the last digit)
25.
27. 0.824
29.
31.
33.
35. (a) (b) (exact)(0.33 Ϫ 2 � 0.23 ϩ 0.13)>0.01 ϭ 1.2(0.43 Ϫ 2 � 0.23 ϩ 0)>0.04 ϭ 1.2,
0.90443, 0.90452 (5S-exact 0.90452)
0.26, M2 ϭ 6, M*2 ϭ 0, Ϫ0.02 Ϲ P Ϲ 0, 0.01
Ϫx ϩ x3, 2(x Ϫ 1) ϩ 3(x Ϫ 1)2 Ϫ (x Ϫ 1)3
x ϭ x4 Ϫ 0.1, Ϫ0.1, Ϫ0.999, Ϫ0.99900399
ϭ 0.05006
x ϭ 20 Ϯ 1398 ϭ 20.00 Ϯ 19.95, x1 ϭ 39.95, x2 ϭ 0.05, x2 ϭ 2>39.95
�a.
44.885 Ϲ s Ϲ 44.995
4.375, 4.50, 6.0, impossible
5(0.1040 Ϫ 12 � 0.1760 ϩ 1
3 � 0.1344 Ϫ 14 � 0.0384) ϭ 0.256
0.08, 0.32, 0.176, 0.256 (exact)
x ϭ 12 (t ϩ 1), dx ϭ 1
2 dt, 0.746824127 (9S-exact 0.746824133)
x ϭ 12 (t ϩ 1), 0.2642411177 (10S-exact), 1 Ϫ 2>e
0.94608693
ƒCM4 ƒ ϭ 24>(180 � (2m)4) ϭ 10�5>2, 2m ϭ 12.8,
f iv ϭ 24>x5, M4 ϭ 24,M2 ϭ 2, ƒKM2 ƒ ϭ 2>(12n2) ϭ 10�5>2, n ϭ 183.
(0.785398126 Ϫ 0.785392156)>15 ϭ 0.39792 # 10�6
P0.5 � 0.03452 (P0.5 ϭ 0.03307), P0.25 � 0.00829 (P0.25 ϭ 0.00820)
0.5, 0.375, 0.34375, 0.335
ƒ f r ƒf s>(1 ϩ f r2)3>2 � f s
gt
4 ϩ 32(x Ϫ 4) ϩ 25(x Ϫ 4)2 Ϫ 11(x Ϫ 4)34 ϩ x2 Ϫ x3, Ϫ8(x Ϫ 2) Ϫ 5(x Ϫ 2)2 ϩ 5(x Ϫ 2)3,
Ϫ 6(x Ϫ 2)3Ϫ1 ϩ 2(x Ϫ 2) ϩ 5(x Ϫ 2)21 Ϫ x2, Ϫ2(x Ϫ 1) Ϫ (x Ϫ 1)2 ϩ 2(x Ϫ 1)3,
1 Ϫ 54 x2 ϩ 1
4 x4
x ϭ 5.8[Ϫ1.39 (x Ϫ 5)2 ϩ 0.58 (x Ϫ 5)3]s ϭ 0.004
App. 2 Answers to Odd-Numbered Problems A47
Problem Set 20.1, page 851
1. 3. No solution 5.
7.
9.
11.
13.
15.
Problem Set 20.2, page 857
1.
3.
5. D1 0 0
6 1 0
3 9 1
T D3 9 6
0 Ϫ6 3
0 0 Ϫ3
T , x1 ϭ Ϫ 115
x2 ϭ 415
x3 ϭ 25
D1 0 0
2 1 0
2 5 1
T D5 4 1
0 1 2
0 0 3
T , x1 ϭ 0.4
x2 ϭ 0.8
x3 ϭ 1.6
c1 0
3 1d c4 5
0 Ϫ1d , x1 ϭ Ϫ4
x2 ϭ 6
x1 ϭ 4.2, x2 ϭ 0, x3 ϭ Ϫ1.8, x4 ϭ 2.0
EϪ1 Ϫ3.1 2.5 0 Ϫ8.7
0 2.2 1.5 Ϫ3.3 Ϫ9.3
0 0 Ϫ1.493182 Ϫ0.825 1.03773
0 0 0 6.13826 12.2765
Ux1 ϭ 0.142856, x2 ϭ 0.692307, x3 ϭ Ϫ0.173912
D5 0 6 Ϫ0.329193
0 Ϫ4 Ϫ3.6 Ϫ2.143144
0 0 2.3 Ϫ0.4
Tx1 ϭ t1 arbitrary, x2 ϭ (3.4>6.12)t1, x3 ϭ 0
D3.4 Ϫ6.12 Ϫ2.72 0
0 0 4.32 0
0 0 0 0
Tx1 ϭ 6.78, x2 ϭ Ϫ11.3, x3 ϭ 15.82
D13 Ϫ8 0 178.54
0 6 13 137.86
0 0 Ϫ16 Ϫ253.12
Tx1 ϭ 3.908, x2 ϭ Ϫ1.998, x3 ϭ 2.557
DϪ3 6 Ϫ9 Ϫ46.725
0 9 Ϫ13 Ϫ51.223
0 0 Ϫ2.88889 Ϫ7.38689
Tx1 ϭ 2, x2 ϭ 1x1 ϭ 7.3, x2 ϭ Ϫ3.2
A48 App. 2 Answers to Odd-Numbered Problems
7.
9.
11.
13. No, since
15.
17.
19.
Problem Set 20.3, page 863
5. Exact 7.
9. Exact
11. (a)
(b)
13. steps; spectral radius approximately
15. (Jacobi, Step 5);
(Gauss–Seidel)
19.
Problem Set 20.4, page 871
1.
3.
5. 7. ab ϩ bc ϩ ca ϭ 05, 15, 1, [1 1 1 1 1]
5.9, 113.81 ϭ 3.716, 3, 13 [0.2 0.6 Ϫ2.1 3.0]
18, 1110 ϭ 10.49, 8, [0.125 Ϫ0.375 1 0 Ϫ0.75 0]
1306 ϭ 17.49, 12, 12
[2.00004 0.998059 4.00072]T[1.99934 1.00043 3.99684]T
0.09, 0.35, 0.72, 0.85,8, Ϫ16, 43, 86
x(3)T ϭ [0.50333 0.49985 0.49968]
x(3)T ϭ [0.49983 0.50001 0.500017],
2, 1, 4
x1 ϭ 2, x2 ϭ Ϫ4, x3 ϭ 80.5, 0.5, 0.5
1
16E 21 Ϫ6 Ϫ14 6
Ϫ6 36 Ϫ12 Ϫ4
Ϫ14 Ϫ12 20 Ϫ4
6 Ϫ4 Ϫ4 4
U1
36D 584 104 Ϫ66
104 20 Ϫ12
Ϫ66 Ϫ12 9
TcϪ3.5 1.25
3.0 Ϫ1.0d
xT (ϪA)x ϭ ϪxTAx Ͻ 0; yes; yes; no
E 1 0 0 0
Ϫ1 2 0 0
3 Ϫ1 3 0
2 0 Ϫ1 4
U E1 Ϫ1 3 2
0 2 Ϫ1 0
0 0 3 Ϫ1
0 0 0 4
U , x1 ϭ 2
x2 ϭ Ϫ3
x3 ϭ 4
x4 ϭ Ϫ1
D0.1 0 0
0 0.4 0
0.3 0.2 0.1
T D0.1 0 0.3
0 0.4 0.2
0 0 0.1
T , x1 ϭ 2
x2 ϭ Ϫ1
x3 ϭ 4
1
D 3 0 0
2 3 0
4 1 3
T D3 2 4
0 3 1
0 0 3
T , x1 ϭ 0.6
x2 ϭ 1.2
x3 ϭ 0.4
App. 2 Answers to Odd-Numbered Problems A49
9. 11.
13.
15.
17.
19. extremely ill-conditioned
21. Small residual but large deviation of .
23.
Problem Set 20.5, page 875
1. 3.
5. 9.
11.
13.
Problem Set 20.7, page 884
1. Spectrum
3. Centers Skew-symmetric, hence
5.
7.
9. They lie in the intervals with endpoints Why?
11.
13.
15.
17. Show that
19. 0 lies in no Gerschgorin disk, by (3) with hence
Problem Set 20.8, page 887
1.
3.
5. Same answer as in Prob. 3, possibly except for small roundoff errors.
7. eigenvalues (4S) 1.697,
3.382, 5.303, 5.618
9.
11.
(Step 8)
Problem Set 20.9, page 896
1. D 0.98 Ϫ0.4418 0
Ϫ0.4418 0.8702 0.3718
0 0.3718 0.4898
Tƒ P ƒ Ϲ 1.633, Á , 0.7024
q ϭ 1, Á , Ϫ2.8993 approximates Ϫ3 (0 of the given matrix),
P2 Ϲ yTy>xTx Ϫ (yTx>xTx)2 ϭ l2 Ϫ l2 ϭ 0
x ϭ lxTx, yTy ϭ l2xTx,y ϭ Ax ϭ lx, yT
q ϭ 5.5, 5.5738, 5.6018; ƒ P ƒ Ϲ 0.5, 0.3115, 0.1899;
q Ϯ d ϭ 4 Ϯ 1.633, 4.786 Ϯ 0.619, 4.917 Ϯ 0.398
q ϭ 10, 10.9908, 10.9999; ƒ P ƒ Ϲ 3, 0.3028, 0.0275
det A ϭ l1Á ln � 0.Ͼ;
AAT
ϭ ATA.
10.52 ϭ 0.7211
1122 ϭ 11.05
r (A) Ϲ Row sum norm �A �ϱ ϭ maxjak
ƒajk ƒ ϭ maxj
( ƒajj ƒ ϩ Gerschgorin radius)
ajj Ϯ (n Ϫ 1) # 10�5.
t11 ϭ 100, t22 ϭ t33 ϭ 1
2, 3, 8; radii 1 ϩ 12, 1, 12; actually (4S) 1.163, 3.511, 8.326
l ϭ i�, Ϫ0.7 Ϲ � Ϲ 0.7.0; radii 0.5, 0.7, 0.4.
{Ϫ1, 4, 9}5, 0, 7; radii 6, 4, 6.
Ϫ 1.778x2 ϩ 2.852x32.552 ϩ 16.23x, Ϫ4.114 ϩ 13.73x ϩ 2.500x2, 2.730 ϩ 1.466x
1.89 Ϫ 0.739x ϩ 0.207x2
Ϫ11.36 ϩ 5.45x Ϫ 0.589x2s ϭ 90t Ϫ 675, vav ϭ 90 km>hr
1.48 ϩ 0.09x1.846 Ϫ 1.038x
27, 748, 28,375, 943,656, 29,070,279
x~[0.145 0.120],
[Ϫ2 4]T, [Ϫ144.0 184.0]T, � ϭ 25,921,
167 Ϲ 21 # 15 ϭ 315
� ϭ 20 # 20 ϭ 400; ill-conditioned
� ϭ 19 # 13 ϭ 247; ill-conditioned
� ϭ (5 ϩ 15)(1 ϩ 1>15) ϭ 6 ϩ 215� ϭ 5 # 12 ϭ 2.5
A50 App. 2 Answers to Odd-Numbered Problems
3.
5.
7. Eigenvalues 16, 6, 2
9. Eigenvalues (4S)
Chapter 20 Review Questions and Problems, page 896
15.
17.
19.
21.
Exact:
23.
Exact:
25. 27. 30
29. 5 31.
33. 35.
37. Centers respectively. Eigenvalues (3S)
39. Centers respectively; eigenvalues 0, 4.446, Ϫ9.4460, Ϫ1, Ϫ4; radii 9, 6, 7,
2.63, 40.8, 96.615, 35, 90; radii 30, 35, 25,
1.514 ϩ 1.129x Ϫ 0.214x25 # 2163 ϭ 5
3
115 # 0.4458 ϭ 51.27
42, 1674 ϭ 25.96, 21
[2 1 4]T
D1.700
1.180
4.043
T , D1.986
0.999
4.002
T , D2.000
1.000
4.000
T[6.4 3.6 1.0]T
D5.750
3.600
0.838
T , D6.400
3.559
1.000
T , D6.390
3.600
0.997
TD 0.28193 Ϫ0.15904 Ϫ0.00482
Ϫ0.15904 0.12048 Ϫ0.00241
Ϫ0.00482 Ϫ0.00241 0.01205
T[Ϫ2 0 5]T
[3.9 4.3 1.8]T
D141.4 1.166 0
1.166 68.66 0.1661
0 0.1661 Ϫ30.04
TD141.1 4.926 0
4.926 68.97 0.8691
0 0.8691 Ϫ30.03
T , D141.3 2.400 0
2.400 68.72 0.3797
0 0.3797 Ϫ30.04
T ,141.4, 68.64, Ϫ30.04
D 15.8299 Ϫ1.2932 0
Ϫ1.2932 6.1692 0.0625
0 0.0625 2.0010
TD 11.2903 Ϫ5.0173 0
Ϫ5.0173 10.6144 0.7499
0 0.7499 2.0952
T , D 14.9028 Ϫ3.1265 0
Ϫ3.1265 7.0883 0.1966
0 0.1966 2.0089
T ,
E 3 Ϫ67.59 0 0
Ϫ67.59 143.5 45.35 0
0 45.35 23.34 3.126
0 0 3.126 Ϫ33.87
UD 7 Ϫ3.6056 0
Ϫ3.6056 13.462 3.6923
0 3.6923 3.5385
TApp. 2 Answers to Odd-Numbered Problems A51
Problem Set 21.1, page 910
1. (errors of
3. (set (errors of
5. 0.0013, 0.0042 (errors of
7. 0.00029, 0.01187 (errors of
9. Errors 0.03547 and 0.28715 of and much larger
11. error
(use RK with
13. error
15. 0.18, 0.74, 1.73, 3.28, 5.59, 9.04, 14.3, 22.8,
36.8, 61.4
17. 0.2, 3.1, 10.7, 23.2, 28.5,
19. Errors for Euler–Cauchy 0.02002, 0.06286, 0.05074; for improved Euler–Cauchy
0.012086, 0.009601; for Runge–Kutta. 0.0000011, 0.000016, 0.000536
Problem Set 21.2, page 915
1.
3.
5. RK error smaller in absolute value,
(for
7. 0.232490 (0.34), 0.236787 (0.44),
0.240075 (0.42), 0.242570 (0.35), 0.244453 (0.25), 0.245867 (0.16), 0.246926 (0.09)
9.
13. from to
15. (a) 0, 0.02, 0.0884, 0.215848, (poor)
(b) By 30–
Problem Set 21.3, page 922
1. errors of (of from 0.002 to 0.5
(from to 0.1), monotone
3. error
exact
5.
(error 0.005), 0.61 (0.01), 0.429 (0.012), 0.2561 (0.0142), 0.0905 (0.0160)
7. By about a factor
9. Errors of (of from to (from to
11.
continuation will give an “ellipse.”(Ϫ0.81);(Ϫ0.59),
Ϫ0.91), (Ϫ0.42,Ϫ0.97), (Ϫ0.23,Á , 0.92), (0.39, 0.98), (0.20, 1),y2) ϭ (0,(y1,
0.6 # 10�5)0.3 # 10�51.3 # 10�50.3 # 10�5y2)y1
Pn(y2) # 106 ϭ 0.08, Á , 0.27
105. Pn (y1) # 106 ϭ Ϫ0.082, Á , Ϫ0.27,
y1r ϭ y2, y2r ϭ y1 ϩ x, y1 (0) ϭ 1, y2 (0) ϭ Ϫ2, y ϭ y1 ϭ e�x Ϫ x, y ϭ 0.8
y ϭ cos 12 xϪ0.005, Ϫ0.01, Ϫ0.015, Ϫ0.02, Ϫ0.0229;
y1r ϭ y2, y2r ϭ Ϫ14 y1, y ϭ y1 ϭ 1, 0.99, 0.97, 0.94, 0.9005,
Ϫ0.01
y2)y1y1 ϭ Ϫe�2x ϩ 4ex, y2 ϭ Ϫe�2x ϩ ex;
50%
y4 ϭ 0.417818, y5 ϭ 0.708887
0.7: Ϫ5, Ϫ11, Ϫ19, Ϫ31, Ϫ41x ϭ 0.3y ϭ exp (x2). Errors # 105
0.133156 (Ϫ74)
0.095411 (Ϫ54),0.066096 (Ϫ39),0.043810 (Ϫ26),0.027370 (Ϫ17),
0.015749 (Ϫ10),y10 (error # 107) 0.008032 (Ϫ4),Á ,y4,y ϭ exp (x3) Ϫ 1,
y ϭ 1>(4 ϩ e�3x), y4, Á , y10 (error # 105)
x ϭ 0.4, 0.6, 0.8, 1.0)
error # 105 ϭ 0.4, 0.3, 0.2, 5.6
1.557626 (Ϫ22)
1.260288 (Ϫ13),1.029714 (Ϫ7.5), 0.842332 (Ϫ4.4),0.684161 (Ϫ2.4),
0.546315 (Ϫ1.2),y10 (error # 105) 0.422798 (Ϫ0.49),Á ,y4,y ϭ tan x,
P10 ϭ Ϫ1.8 # 10�6y10 ϭ 2.718284,y10* ϭ 2.718276,
P5 ϭ Ϫ3.8 # 10�8,y5 ϭ 1.648722,y5* ϭ 1.648717,y ϭ ex,
Ϫ0.000455,
Ϫ3489, ϩ80444
Ϫ1656,Ϫ376,Ϫ32.3,yr ϭ 1>(2 Ϫ x4); error # 109:
y ϭ 3 cos x Ϫ 2 cos2 x; error # 107:
0.83 # 10�7, 0.16 # 10�6, Á , Ϫ0.56 # 10�6, ϩ0.13 # 10�5y ϭ tan x;
h ϭ 0.2)P ϭ 0.0002>15 ϭ 1.3 # 10�5ϩ9 # 10�6;Ϫ4 # 10�8, Á , Ϫ6 # 10�7,Ϫ10�8,y ϭ 1>(1 Ϫ x2>2);
y10y5
y5, y10)y ϭ 1>(1 Ϫ x2>2),
y5, y10)y ϭ ex,
y5, y10)y Ϫ x ϭ u), 0.00929, 0.01885y ϭ x Ϫ tanh x
y5, y10)y ϭ 5e�0.2x, 0.00458, 0.00830
A52 App. 2 Answers to Odd-Numbered Problems
Problem Set 21.4, page 930
3.
5. 105, 155, 105, 115; Step 5: 104.94, 154.97, 104.97, 114.98
7. 0, 0, 0, 0. All equipotential lines meet at the corners (why?).
Step 5: 0.29298, 0.14649, 0.14649, 0.073245
9. 0.108253, 0.108253, 0.324760, 0.324760; Step 10: 0.108538, 0.108396,
0.324902, 0.324831
11. (a) . (b) Reduce to 4 equations by symmetry.
13. at the others
15. otherwise
17. (0.1083, 0.3248 are 4S-values
of the solution of the linear system of the problem.)
Problem Set 21.5, page 935
5.
7. A, as in Example 1, right sides
Solution
13.
Here with from the stencil.
15. (4S)
Problem Set 21.6, page 941
5. 0, 0.6625, 1.25, 1.7125, 2, 2.1, 2, 1.7125, 1.25, 0.6625, 0
7. Substantially less accurate, 0.15, 0.25 0.100, 0.163
9. Step 5 gives 0, 0.06279, 0.09336, 0.08364, 0.04707, 0.
11. Step 2: 0 (exact 0), 0.0453 (0.0422), 0.0672 (0.0658), 0.0671 (0.0628), 0.0394
(0.0373), 0 (0)
13. 0.3301, 0.5706, 0.4522, 0.2380 0.06538, 0.10603, 0.10565, 0.6543
15.
Problem Set 21.7, page 944
1.
3. For we obtain
etc.
5.
7. 0.190, 0.308, 0.308, 0.190, (3S-exact: 0.178, 0.288, 0.288, 0.178)
Á (t ϭ 0.2)1.357,
1.296,1.135,0.935,Á (t ϭ 0.1); 0, 0.575,1.834,1.679,1.271,0.766,0.354,0,
Ϫ0.08, Ϫ0.16 (t ϭ 0.6),
0.24, 0.40 (t ϭ 0.2), 0.08, 0.16 (t ϭ 0.4),x ϭ 0.2, 0.4
u (x, 1) ϭ 0, Ϫ0.05, Ϫ0.10, Ϫ0.15, Ϫ0.20, 0
0.1018, 0.1673, 0.1673, 0.1018 (t ϭ 0.04), 0.0219, 0.0355, Á (t ϭ 0.20)
(t ϭ 0.20)
(t ϭ 0.04),
(t ϭ 0.08)(t ϭ 0.04),
b ϭ [Ϫ200, Ϫ100, Ϫ100, 0]T; u11 ϭ 73.68, u21 ϭ u12 ϭ 47.37, u22 ϭ 15.79
43Ϫ14
3 ϭ Ϫ43 (1 ϩ 2.5)
u12 ϭ 1.u21 ϭ 4,u11 ϭ u22 ϭ 2,2u21 ϩ 2u12 Ϫ 12u22 ϭ Ϫ14,
u11 Ϫ 4u12 ϩ u22 ϭ 0,u11 Ϫ 4u21 ϩ u22 ϭ Ϫ12,Ϫ4u11 ϩ u21 ϩ u12 ϭ Ϫ3,
u11 ϭ u21 ϭ 125.7, u21 ϭ u22 ϭ 157.1
Ϫ220, Ϫ220, Ϫ220, Ϫ220.
u11 ϭ 0.766, u21 ϭ 1.109, u12 ϭ 1.957, u22 ϭ 3.293
13, u11 ϭ u21 ϭ 0.0849, u12 ϭ u22 ϭ 0.3170.
u21 ϭ u23 ϭ 0.25, u12 ϭ u32 ϭ Ϫ0.25, ujk ϭ 0
u12 ϭ u32 ϭ 31.25, u21 ϭ u23 ϭ 18.75, ujk ϭ 25
u13 ϭ u23 ϭ u33 ϭ 0
u12 ϭ u32 ϭ Ϫu14 ϭ Ϫu34 ϭ Ϫ64.22, u22 ϭ Ϫu24 ϭ Ϫ53.98,
u11 ϭ u31 ϭ Ϫu15 ϭ Ϫu35 ϭ Ϫ92.92, u21 ϭ Ϫu25 ϭ Ϫ87.45,
u11 ϭ Ϫu12 ϭ Ϫ66
Ϫ3u11 ϩ u12 ϭ Ϫ200, u11 Ϫ 3u12 ϭ Ϫ100
App. 2 Answers to Odd-Numbered Problems A53
Chapter 21 Review Questions and Problems, page 945
17. 0.038, 0.125 (errors of and
19.
21.
23.
25.
27.
errors between and
29. 3.93, 15.71, 58.93
31. 0, 0.04, 0.08, 0.12, 0.15, 0.16, 0.15, 0.12, 0.08, 0.04, 0 3 time steps)
33.
35. 0.043330, 0.077321, 0.089952, 0.058488 0.010956, 0.017720, 0.017747,
0.010964
Problem Set 22.1, page 953
3.
9. Step 5: value 0.000016
Problem Set 22.2, page 957
7. No
9. is the unused time on respectively.
11.
13.
15.
17. (copper),
19.
21.
Problem Set 22.3, page 961
3.
5. Eliminate in Column 3, so that 20 goes.
7.
9. on the segment from (3, 0, 0) to (0, 0, 2)
11. We minimize! The augmented matrix is
T0 ϭ D1 1.8 2.1 0 0 0
0 15 30 1 0 150
0 600 500 0 1 3900
T .fmax ϭ 6
fmax ϭ f (6021 , 0, 1500
105 , 0) ϭ 22007
fmin ϭ f (0, 12 ) ϭ Ϫ10.
f (120>11, 60>11) ϭ 480>11
37,500
fmax ϭ f (210, 60) ϭx1>3 ϩ x2>2 Ϲ 100, x1>3 ϩ x2>6 Ϲ 80, f ϭ 150x1 ϩ 100x2,
f ϭ x1 ϩ x2, 2x1 ϩ 3x2 Ϲ 1200, 4x1 ϩ 2x2 Ϲ 1600, fmax ϭ f (300, 200) ϭ 500
fmax ϭ f (45, 30) ϭ 8400
0.5x1 ϩ 0.25x2 Ϲ 30, f ϭ 120x1 ϩ 100x2,0.5x1 ϩ 0.75x2 Ϲ 45
f (9, 6) ϭ 360
f (Ϫ113 , 26
3 ) ϭ 198 13
f (2.5, 2.5) ϭ 100
M1, M2,x3, x4
(0.11247, Ϫ0.00012),
f (x) ϭ 2(x1 Ϫ 1)2 ϩ (x2 ϩ 2)2 Ϫ 6; Step 3: (1.037, Ϫ1.926), value Ϫ5.992
(t ϭ 0.20)
(t ϭ 0.04),
u (P22) ϭ 60u (P12) ϭ u (P32) ϭ 90,
u (P21) ϭ u (P13) ϭ u (P23) ϭ u (P33) ϭ 30,u (P11) ϭ u (P31) ϭ 270,
(t ϭ 0.3.
10�510�6
y1r ϭ y2, y2r ϭ 2ex Ϫ y1, y ϭ ex Ϫ cos x, y ϭ y1 ϭ 0, 0.241, 0.571, Á ;
y1r ϭ y2, y2r ϭ x2y1, y ϭ y1 ϭ 1, 1, 1, 1.0001, 1.0006, 1.002
Ϫ2.5 # 10�5)
y ϭ sin x, y0.8 ϭ 0.717366, y1.0 ϭ 0.841496 (errors Ϫ1.0 # 10�5,
0.5463023 (1.8 # 10�7)0.4227930 (2.3 # 10�7),
0.3093360 (2.1 # 10�7),0.1003346 (0.8 # 10�7) 0.2027099 (1.6 # 10�7),
1.5538 (0.0036)1.2593 (0.0009),1.0299 (Ϫ0.0002),0.84295 (Ϫ0.00066),
0.68490 (Ϫ0.00076),0.54702 (Ϫ0.00072),0.42341 (Ϫ0.00062),
0.30981 (Ϫ0.00048),0.20304 (Ϫ0.00033),0.10050 (Ϫ0.00017),y ϭ tan x; 0 (0),
y10)y5y ϭ ex,
A54 App. 2 Answers to Odd-Numbered Problems
The pivot is 600. The calculation gives
The next pivot is The calculation gives
Hence has the maximum value so that f has the minimum value 13.5, at
the point
13.
Problem Set 22.4, page 968
1.
3.
5.
7.
9.
Chapter 22 Review Questions and Problems, page 968
9. Step 5: Slower. Why?
11. Of course! Step 5:
17.
19.
Problem Set 23.1, page 974
9. 11.
13. D0 1 1
0 0 1
1 1 0
TE0 1 1 1
0 0 0 0
1 0 0 0
0 0 0 0
UD0 1 0
0 0 1
1 0 0
Tf (3, 6) ϭ Ϫ54
f (2, 4) ϭ 100
[Ϫ1.003 1.897]T
[0.353 Ϫ0.028]T.
f (4, 0, 12) ϭ 9
f (1, 1, 0) ϭ 13
f (10, 5) ϭ 5500
f (20, 20) ϭ 40
f (6, 3) ϭ 84
fmax ϭ f (5, 4, 6) ϭ 478
(x1, x2) ϭ a2400
600,
105>235>2 b ϭ (4, 3).
Ϫ13.5,Ϫf
T2 ϭ D1 0 0 Ϫ 6175 Ϫ 3
1400 Ϫ272
0 0 352 1 Ϫ 1
40105
2
0 600 0 Ϫ2007
127 2400
T Row 1 Ϫ 1.235 Row 2
Row 2
Row 3 Ϫ 100035 Row 2
352 .
T1 ϭ D1 0 610 0 Ϫ 3
1000 Ϫ11710
0 0 352 1 Ϫ 1
401052
0 600 500 0 1 3900
T Row 1 Ϫ 1.8600 Row 3
Row 2 Ϫ 15600 Row 3
Row 3
App. 2 Answers to Odd-Numbered Problems A55
15. 1 2
3 4
17. If G is complete.
Edge
19.
Problem Set 23.2, page 979
1. 5 3. 4
5. The idea is to go backward. There is a adjacent to and labeled etc.
Now the only vertex labeled 0 is s. Hence implies so that
is a path that has length k.
15. Delete the edge
17. No
Problem Set 23.3, page 983
1.
5.
7.
9.
Problem Set 23.4, page 987
21. 4 Ϫ 3 Ϫ 5 L ϭ 10
1
13. 5 Ϫ 3 Ϫ 6
ì
êL ϭ 17
2 Ϫ 4
2
5. 1 ì
ê 3 L ϭ 124 ì
ê
5
9. Yes
2
11. 1 Ϫ 3 Ϫ 4 ì
êL ϭ 38
5 Ϫ 6
13. New York–Washington–Chicago–Dalles–Denver–Los Angeles
15. G is connected. If G were not a tree, it would have a cycle, but this cycle would
provide two paths between any pair of its vertices, contradicting the uniqueness.
(1, 5), (2, 3), (2, 6), (3, 4), (3, 5); L2 ϭ 9, L3 ϭ 7, L4 ϭ 8, L5 ϭ 4, L6 ϭ 14
(1, 2), (2, 4), (3, 4); L2 ϭ 10, L3 ϭ 15, L4 ϭ 13
(1, 2), (2, 4), (3, 4), (3, 5); L2 ϭ 2, L3 ϭ 4, L4 ϭ 3, L5 ϭ 6
(1, 2), (2, 4), (4, 3); L2 ϭ 12, L3 ϭ 36, L4 ϭ 28
(2, 4).
s: vkv0 Ϫ v1 Ϫ Á Ϫ vk�1 Ϫ vk
v0 ϭ s,l(v0) ϭ 0
k Ϫ 1,vkvk�1
EϪ1 Ϫ1 1 Ϫ1
1 0 0 0
0 1 Ϫ1 0
0 0 0 1
U1
2
3
4
e1
e2
e3
e4
A56 App. 2 Answers to Odd-Numbered Problems
Ver
tex
ì
ê
19. If we add an edge to T, then since T is connected, there is a path in T
which, together with forms a cycle.
Problem Set 23.5, page 990
1. If G is a tree.
3. A shortest spanning tree of the largest connected graph that contains vertex 1.
7.
9.
11.
Problem Set 23.6, page 997
1.
3.
5.
7.
9. One is interested in flows from s to t, not in the opposite direction.
13.
15.
17.
is unique.
19. For instance,
is unique.
Problem Set 23.7, page 1000
3.
5. By considering only edges with one labeled end and one unlabeled end
7. where 6 is
the given flow
9. where 4
is the given flow
15.
Problem Set 23.8, page 1005
1. No 3. No
5. Yes,
7. Yes, 11.
13. is augmenting and gives and
is of maximum cardinality.
15. is augmenting and gives and
is of maximum cardinality.
19. 3 21. 2
23. 3 25. K4
(1, 4), (3, 6), (7, 8)
1 � 4 Ϫ 3 � 6 Ϫ 7 � 81 Ϫ 4 � 3 Ϫ 6 � 7 Ϫ 8
(3, 7), (5, 4)
(1, 2),1 � 2 Ϫ 3 � 7 Ϫ 5 � 41 Ϫ 2 � 3 Ϫ 7 � 5 Ϫ 4
1 Ϫ 2 � 3 Ϫ 7 � 5 Ϫ 4S ϭ {1, 3, 5}
S ϭ {1, 4, 5, 8}
S ϭ {1, 2, 4, 5}, T ϭ {3, 6}, cap (S, T) ϭ 14
1 Ϫ 2 Ϫ 4 Ϫ 6, ¢t ϭ 2; 1 Ϫ 3 Ϫ 5 Ϫ 6, ¢t ϭ 1; f ϭ 4 ϩ 2 ϩ 1 ϭ 7,
1 Ϫ 2 Ϫ 5, ¢t ϭ 2; 1 Ϫ 4 Ϫ 2 Ϫ 5, ¢t ϭ 1; f ϭ 6 ϩ 2 ϩ 1 ϭ 9,
(2, 3) and (5, 6)
f ϭ 3 ϩ 5 ϩ 7 ϭ 15, f ϭ 15
f12 ϭ 10, f24 ϭ f45 ϭ 7, f13 ϭ f25 ϭ 5, f35 ϭ 3, f32 ϭ 2,
f ϭ 17
f13 ϭ f35 ϭ 8, f14 ϭ f45 ϭ 5, f12 ϭ f24 ϭ f46 ϭ 4, f56 ϭ 13, f ϭ 4 ϩ 13 ϭ 17,
1 Ϫ 2 Ϫ 5, ¢f ϭ 2; 1 Ϫ 4 Ϫ 2 Ϫ 5, ¢f ϭ 2, etc.
P1: 1 Ϫ 2 Ϫ 4 Ϫ 5, ¢f ϭ 2; P2: 1 Ϫ 2 Ϫ 5, ¢f ϭ 3; P3: 1 Ϫ 3 Ϫ 5, ¢f ϭ 4
¢12 ϭ 5, ¢24 ϭ 8, ¢45 ϭ 2; ¢12 ϭ 5, ¢25 ϭ 3; ¢13 ϭ 4, ¢35 ϭ 9
S ϭ {1, 4}, 8 ϩ 6 ϭ 14
{3, 6, 7}, 8 ϩ 4 ϩ 4 ϭ 16
{4, 5, 6}, 10 ϩ 5 ϩ 13 ϭ 28
{3, 6}, 11 ϩ 3 ϭ 14
(1, 4), (4, 3), (4, 5), (1, 2); L ϭ 12
(1, 4), (4, 3), (4, 2), (3, 5); L ϭ 20
(1, 4), (1, 3), (1, 2), (2, 6), (3, 5); L ϭ 32
(u, v),
u: v(u, v)
App. 2 Answers to Odd-Numbered Problems A57
Chapter 23 Review Questions and Problems, page 1006
11.
13. To vertex
From vertex
15.
17.
Vertex Incident Edges
1 (1, 2), (1, 4)
2 (2, 1), (2, 4)
3 (3, 4)
4 (4, 1), (4, 2), (4, 3)
19.
23.
Problem Set 24.1, page 1015
1. 3.
5. 7.
9. 11.
13. 15.
17. 3.54, 1.29
Problem Set 24.2, page 1017
1. outcomes: RRR, RRL, RLR, LRR, RLL, LRL, LLR, LLL
3. outcomes first number (second number) referring
to the first die (second die)
5. Infinitely many outcomes
7. The space of ordered pairs of numbers
9. 10 outcomes:
11. Yes
17. implies by the definition of union. Conversely. implies
that because always and if we must have equality
in the previous relation.
A � B,B � A �B,A �B ϭ B
A � BA � BA �B ϭ B
D ND NND Á NNNNNNNNND
H TH TTH TTTH Á (H ϭ Head, T ϭ Tail)
(1, 1), (1, 2), Á , (6, 6),62 ϭ 36
23
x ϭ 1.355, s ϭ 0.136, IQR ϭ 0.15x ϭ 144.67, s ϭ 8.9735, IQR ϭ 16
x ϭ 19.875, s ϭ 0.835, IQR ϭ 1.5qL ϭ 89.9, qM ϭ 91.0, qU ϭ 91.8
qL ϭ 1.3, qM ϭ 1.4, qU ϭ 1.45qL ϭ 199, qM ϭ 201, qU ϭ 201
qL ϭ 138, qM ϭ 144, qU ϭ 154qL ϭ 19, qM ϭ 20, qU ϭ 20.5
(1, 6), (4, 5), (2, 3), (7, 8)
(1, 2), (1, 4), (2, 3); L2 ϭ 2, L3 ϭ 5, L4 ϭ 5
1 2
4 3
E0 1 0 1
1 0 1 0
0 1 0 1
1 0 1 0
U1
2
3
4
1 2 3 4
E0 0 1 1
0 0 1 1
1 1 0 0
1 1 0 0
UA58 App. 2 Answers to Odd-Numbered Problems
Problem Set 24.3, page 1024
1. by Theorem 1
3.
5.
7. Small sample from a large population containing many items in each class we are
interested in (defectives and nondefectives, etc.)
9.
11. (a) (c) same as (a).
Why?
13.
15.
17. hence by disjointedness of B
and
Problem Set 24.4, page 1028
1. In ways
3.
5. 7.
9. In ways 11.
13. (b)
15. P (No two people have a birthday in common)
Answer: which is surprisingly large.
Problem Set 24.5, page 1034
1. by (6)
3. by (10),
5. No, because of (6)
7. because of (6) and
9.
11.
13. if if
if if
Answer: 500 cans,
15. etc.
Problem Set 24.6, page 1038
1. 3. cf. Example 2
5. 7.
9. 11.
13. 15.
17. Product of the 2 numbers. cents
19. (0 ϩ 1 � 3 ϩ 3 � 8 ϩ 1 � 27)>8 ϭ 54>8 ϭ 6 � 75
E (X ) ϭ 12.25, 12X ϭ
12 , 1
20 , (X Ϫ 12 )120$643.50
c ϭ 0.073750, 1, 0.002
C ϭ 12 , � ϭ 2, s2 ϭ 4� ϭ 1
4 , s2 ϭ 116
� ϭ p, s2 ϭ p2>3;k ϭ 12 , � ϭ 4
3 , s2 ϭ 29
X Ͼ b, X м b, X Ͻ c, X Ϲ c,
P ϭ 0.125, 0
x Ϲ 10 Ϲ x Ͻ 1, F(x) ϭ 1F(x) ϭ 1 Ϫ 12 (x Ϫ 1)2
Ϫ1 Ϲ x Ͻ 0x Ͻ Ϫ1, F(x) ϭ 12(x ϩ 1)2F(x) ϭ 0
0.53 ϭ 12.5%
k ϭ 5; 50%
1 ϩ 8 ϩ 27 ϩ 64 ϭ 100k ϭ 1100
P(0 Ϲ X Ϲ 2) ϭ 12k ϭ 1
4
k ϭ 155
41%,
ϭ 365 � 364 Á 346>36520 ϭ 0.59.
1>(12n)
9 � 8 ϭ 726!>6 ϭ 120
210, 70, 112, 28A103 B A52 B A62 B ϭ 18,000
26 �
15 �
44 �
33 �
22 �
11 ϭ 4
6 �35 �
24 �
13 �
22 �
11 ϭ 4!2!
6! ϭ 26 �
15 ϭ 1
15
10! ϭ 3,628,800
A�BcP(A) ϭ P(B) ϩ P(A�Bc) м P (B)A ϭ B � (A�Bc),
1 Ϫ 0.8754 ϭ 0.4138 Ͻ 1 Ϫ 0.752 ϭ 0.4375 Ͻ 0.5 (c Ͻ b Ͻ a)
1 Ϫ 0.963 ϭ 11.5%
(a) ϩ (b) ϩ (c) ϭ 1.
100200 �
99199 ϭ 24.874%, (b)
100200 �
100199 ϩ 100
200 �100199 ϭ 50.25%,
498500 �
497499 �
496498 �
495497 �
494496 � 0.98008
89
(a) 0.93 ϭ 72.9%, (b)90
100 ⅐8999 ⅐
8898 ϭ 72.65%
1 Ϫ 4>216 ϭ 98.15%,
App. 2 Answers to Odd-Numbered Problems A59
Problem Set 24.7, page 1044
3.
5.
7.
9. Answer:
11.
13.
15.
Problem Set 24.8, page 1050
1. 3.
5. 7.
9. About 11. hours
13. About 683 (Fig. 521a)
Problem Set 24.9, page 1059
1. 3.
5. if
7. 27.45 mm, 0.38 mm
11. 25.26 cm, 0.0078 cm 13.
15. The distributions in Prob. 17 and Example 1
17. No
Chapter 24 Review Questions and Problems, page 1060
11.
13.
21. Sum over j from 1.
17.
19.
21. 23.
25.
Problem Set 25.2, page 1067
1. In Example 1, so and is as before.
3.
5.
7.
9.
11.
13.
15. Variability larger than perhaps expected
u ϭ 1
u ϭ n>S x j ϭ 1>xl ϭ f ϭ p(1 Ϫ p)x�1, etc., p ϭ 1>x7>12
l ϭ pk(1 Ϫ p)n�k, p ϭ k>n, k ϭ number of successes in n trails
� ϭ x ϭ 15.3n� ϭ nx,
/ ϭ e�n��(x1ϩÁϩxn)>(x1! Á xn!), 0 ln />0� ϭ Ϫn ϩ (x1 ϩ Á ϩ xn)>� ϭ 0,
s� 2an
jϭ1
x j ϭ 0. 0 ln />0/ ϭ 0� ϭ 0
0.1587, 0.6306, 0.5, 0.4950
1, 12f (x) ϭ 2�x, x ϭ 1, 2, Á
f (x) ϭ A50x B0.03x0.9750�x � 1.5xe�1.5>x!
x ϭ 6, s ϭ 3.65
xmin Ϲ x j Ϲ xmax.
x ϭ 111.9, s ϭ 4.0125, s2 ϭ 16.1
QL ϭ 110, QM ϭ 112, QU ϭ 115
50%
a2 Ͻ y Ͻ b2f2(y) ϭ 1>(b2 Ϫ a2)
29 , 19 , 12
18 , 3
16 , 38
t ϭ 108458%
31.1%, 95.4%15.9%
45.065, 56.978, 2.0220.1587, 0.5, 0.6915, 0.6247
1 Ϫ e�0.2 ϭ 18%
42%, 47.2%, 10.5%, 0.3%
1314 %
9%f (x) ϭ 0.5xe�0.5>x!, f (0) ϩ f (1) ϭ e�0.5(1.0 ϩ 0.5) ϭ 0.91.
0.265
A5x B 0.55, 0.03125, 0.15625, 1 Ϫ f (0) ϭ 0.96875, 0.96875
38%
A60 App. 2 Answers to Odd-Numbered Problems
Problem Set 25.3, page 1077
3. Shorter by a factor 5.
7.
9.
11.
13.
15. degrees of freedom.
Hence
17.
19. is normal with mean 105 and variance 1.25.
Answer:
Problem Set 25.4, page 1086
3. accept the hypothesis.
5. do not reject the hypothesis.
7. accept the hypothesis.
9.
11. Alternative
(Table A9, Appendix 5). Reject the hypothesis
13. Two-sided. (Table A9, Appendix 5),
no difference
15. (Table A10. Appendix 5), accept the
hypothesis
17. By (12), Assert that B is better.
Problem Set 25.5, page 1091
1.
3. 27
5. Choose 4 times the original sample size
9.
11.
13. In about of the cases
15. is negative in (b) and we set
Problem Set 25.6, page 1095
1. 3.
5. 7.
9. 11.
13. (by the normal approximation)
15. (1 Ϫ u)5, 3u(1 Ϫ u)5�14r ϭ 0, u ϭ 16, AOQL ϭ 6.7%
a9
xϭ0
a100
xb 0.12x 0.88100�x ϭ 22%
(1 Ϫ 12 )3 ϩ 3 # 1
2 (1 Ϫ 12 )2 ϭ 1
2(1 Ϫ u)n ϩ nu(1 Ϫ u)n�1
19.5%, 14.7%e�25u(1 ϩ 25u), P(A; 1.5) ϭ 94.5, a ϭ 5.5%
0.8187, 0.6703, 0.13530.9825, 0.9384, 0.4060
UCL ϭ � ϩ 31� ϭ 9.3.
LCL ϭ 0, CL ϭ � ϭ 3.6,LCL ϭ � Ϫ 31�
30% (5%)
LCL ϭ np Ϫ 31np(1 Ϫ p), CL ϭ np, UCL ϭ np ϩ 31np(1 Ϫ p)
2.5810.0004>12 ϭ 0.036, LCL ϭ 3.464, UCL ϭ 3.536
LCL ϭ 1 Ϫ 2.58 # 0.02>2 ϭ 0.974, UCL ϭ 1.026
t0 ϭ 116(20.2 Ϫ 19.6)>10.16 ϩ 0.36 Ͼ c ϭ 1.70.
19 � 1.02>0.82 ϭ 29.69 Ͻ c ϭ 30.14
t ϭ (0.55 Ϫ 0)>10.546>8 ϭ 2.11 Ͻ c ϭ 2.37
� ϭ 5000 g.
� � 5000, t ϭ (4990 Ϫ 5000)>(20>150) ϭ Ϫ3.54 Ͻ c ϭ Ϫ2.01
� Ͻ 58.69 or � Ͼ 61.31
s2>n ϭ 1.8, c ϭ 57.8,
c ϭ 6090 Ͼ 6019:
t ϭ (0.286 Ϫ 0)>(4.31>17) ϭ 0.18 Ͻ c ϭ 1.94;
P (104 Ϲ Z Ϲ 106) ϭ 63%
Z ϭ X ϩ Y
CONF0.95{0.74 Ϲ s2 Ϲ 5.19}
k1 ϭ 12.41, k2 ϭ 7.10. CONF0.95 {7.10 Ϲ s2 Ϲ 12.41}.c2 ϭ 129.6.
F(c2) ϭ 0.975,c1 ϭ 74.2,F (c1) ϭ 0.025,n Ϫ 1 ϭ 99
CONF0.95{0.023 Ϲ s2 Ϲ 0.085}
k ϭ 366.66 (Table 25.2), CONF0.99{9166.7 Ϲ � Ϲ 9900}
n Ϫ 1 ϭ 5, F(c) ϭ 0.995, c ϭ 4.03, x ϭ 9533.33, s2 ϭ 49,666.67,
CONF0.99{63.72 Ϲ � Ϲ 66.28}
CONF0.95{125.3 Ϲ � Ϲ 126.7}, CONF0.95{0.1566 Ϲ p Ϲ 0.1583}
c ϭ 1.96, x ϭ 126, s2 ϭ 126 # 674>800 ϭ 106.155, k ϭ cs>1n ϭ 0.714,
4, 1612
App. 2 Answers to Odd-Numbered Problems A61
Problem Set 25.7, page 1099
3.
5.
7.
9. 42 even digits, accept.
13. (1 degree of
freedom, 95%)
15. Combining the last three nonzero values, we have since we
estimated the mean, Accept the hypothesis.
Problem Set 25.8, page 1102
3. is the probability that 7 cases in 8 trials favor A under the
hypothesis that A and B are equally good. Reject.
5.
7.
Hypothesis rejected.
9. Hypothesis Alternative
Hypothesis rejected.
11. Consider
13. transpositions, Assert that fertilizing increases yield.
15. Assert that there is an increase.
Problem Set 25.9, page 1111
1. 3.
5. 7.
9.
13.
15.
Chapter 25 Review Questions and Problems, page 1111
15. 17.
19.
21.
23. when For we obtain
If n increases, so does whereas decreases.
25. y ϭ 3.4 Ϫ 1.85x
ba,b ϭ (1 Ϫ u)6 ϭ 37.7%.
u ϭ 15%u ϭ 0.01.a ϭ 1 Ϫ (1 Ϫ u)6 ϭ 5.85%,
2.58 � 10.00024>12 ϭ 0.028, LCL ϭ 2.722, UCL ϭ 2.778
c ϭ 14.74 Ͼ 14.5, reject �0; £((14.74 Ϫ 14.50)>10.025) ϭ 0.9353
CONF0.99{27.94 Ϲ � Ϲ 34.81}� ϭ 20.325, s2 ϭ (78)s2 ϭ 3.982
CONF0.95{0.046 Ϲ �1 Ϲ 0.088}
y Ϫ 1.875 ϭ 0.067(x Ϫ 25), 3sx2 ϭ 500, q0 ϭ 0.023, K ϭ 0.021,
CONF0.95{41.7 Ϲ �1 Ϲ 44.7}.
c ϭ 3.18 (Table A9), k1 ϭ 43.2, q0 ϭ 54,878, K ϭ 1.502,
y ϭ 0.32923 ϩ 0.00032x, y(66) ϭ 0.35035
y ϭ 0.5932 ϩ 0.1138x, R ϭ 1>0.1138y ϭ Ϫ10 ϩ 0.55x
y ϭ Ϫ11,457.9 ϩ 43.2xy ϭ 0.98 ϩ 0.495x
P(T Ϲ 2) ϭ 2.8%.
P(T Ϲ 4) ϭ 0.007.n ϭ 8; 4
yj ϭ x j Ϫ �0� .
t ϭ 110 � 1.58>1.23 ϭ 4.06 Ͼ c ϭ 1.83 (a ϭ 5%).
�� Ͼ 0, x ϭ 1.58,�� ϭ 0.
x ϭ 9.67, s ϭ 11.87. t0 ϭ 9.67>(11.87>115) ϭ 3.16 Ͼ c ϭ 1.76 (a ϭ 5%).
(12)18(1 ϩ 18 ϩ 153 ϩ 816) ϭ 0.0038
(12)8 ϩ 8 # (1
2)8 ϭ 3.5%
10,0942608 � 3.87). 0
2 ϭ 12.8 Ͻ c ϭ 16.92.
K Ϫ r Ϫ 1 ϭ 9 (r ϭ 1
02 ϭ
(355 Ϫ 358.5)2
358.5ϩ
(123 Ϫ 119.5)2
119.5ϭ 0.137 Ͻ c ϭ 3.84
02 ϭ 10.264 Ͻ 11.07; yes
02 ϭ 16
10 Ͼ 11.07; yes
02 ϭ (40 Ϫ 50)2>50 ϩ (60 Ϫ 50)2>50 ϭ 4 Ͼ c ϭ 3.84; no
A62 App. 2 Answers to Odd-Numbered Problems