Download mathematical methods in the physical sciences solutions and more Exercises Educational Mathematics in PDF only on Docsity!
Answers to
Selected Problems
Chapter 1
1.1 0.0173 yd; 0.104 yd (compared to a total of 5 yd)
2.1 1 2.4 ∞ 2.7 e^2 2.9 1
4.2 an =
5 n−^1
→ 0; Sn =
5 n
; Rn =
4 · 5 n−^1
4.4 an =
3 n^
→ 0; Sn =
3 n
; Rn =
2 · 3 n^
4.6 an =
n(n + 1) → 0; Sn = 1 −
n + 1 → 1; Rn =
n + 1
5.2 Test further 5.4 D 5.5 D 5.6 Test further 5.8 Test further 5.9 D
6.5 b D 6.7 D 6.9 C 6.10 C 6.14 D 6.18 D 6.20 C 6.22 C 6.23 D 6.24 D 6.26 C 6.29 D 6.31 D 6.32 D 6.35 C 6.36 D
7.1 C 7.2 D 7.4 C 7.6 D 7.8 C
9.2 D 9.3 C 9.7 D 9.8 C 9.9 D 9.10 D 9.12 C 9.13 C 9.15 D 9.16 C 9.20 C 9.21 C 9.22 (b) D
10.1 |x| < 1 10.3 |x| ≤ 1 10.4 |x| ≤
10.5 All x 10.9 |x| < 1 10.10 |x| ≤ 1 10.11 − 5 ≤ x < 5 10.13 − 1 < x ≤ 1 10.15 − 1 < x < 5
10.17 − 2 < x ≤ 0 10.18 −
≤ x ≤ −
10.20 All x
10.21 0 ≤ x ≤ 1 10.22 No x 10.24 |x| <
10.25 nπ −
π 6
< x < nπ +
π 6
781
782 Answers to Selected Problems Chapter 2
0
n
(−1)n(2n − 1)!! (2n)!!
0
n
xn+1^ (see Example 2)
0
n
(−x^2 )n^ (see Problem 13.4)
∑^ ∞
0
(−1)nxn (2n + 1)!
∑^ ∞
0
x^2 n+ 2 n + 1
∑^ ∞
0
n
(−1)n^ x^2 n+ 2 n + 1
odd n
xn n
13.21 x^2 + 2x^4 /3 + 17x^6 / 45 · · · 13.22 1 + 2x + 5x^2 /2 + 8x^3 /3 + 65x^4 / 24 · · · 13.25 1 − x + x^2 / 3 − x^4 / 45 · · · 13.27 1 + x + x^2 / 2 − x^4 / 8 − x^5 / 15 · · · 13.28 x − x^2 /2 + x^3 / 6 − x^5 / 12 · · · 13.29 1 + x/ 2 − 3 x^2 /8 + 17x^3 / 48 · · · 13.34 x − x^2 + x^3 − 13 x^4 /12 + 5x^5 / 4 · · · 13.35 1 + x^2 /3! + 7x^4 /(3 · 5!) + 31x^6 /(3 · 7!) · · · 13.41 e^3 [1 + (x − 3) + (x − 3)^2 /2! + (x − 3)^3 /3! · · · ] 13.44 5 + (x − 25)/ 10 − (x − 25)^2 / 103 + (x − 25)^3 /(5 × 104 ) · · ·
14.8 For x < 0, error < 0 .001; for x > 0, error < 0 .002.
15.1 −x^4 / 24 − x^5 / 30 · · · ∼= − 3. 376 × 10 −^16 15.3 x^5 / 15 − 2 x^7 / 45 · · · ∼= 6. 667 × 10 −^17 15.6 12 15.8 1 / 2 15.10 − 1 15.12 1/ 3
15.14 t − t
3 3 , error^ <^10
− (^6) 15.17 cos(π/2) = 0
2 15.20 (b) 5e 15.21 (b) 0. 937548 15.22 (b) 1. 202057 15.23 (a) 1/ 2 (c) 1/ 3 15.24 (a) −π (d) 0 (f) 0 15.27 (a) 1 − v c = 1. 3 × 10 −^5 , or v = 0. 999987 c
(d) 1 − v c = 1. 3 × 10 −^11
15.28 mc^2 + 12 mv^2
15.29 (b)
F
W
x l
x^3 2 l^3
3 x^5 8 l^5
15.30 (b) T =
F
θ
θ^2 6
θ^4 360
15.31 (a) finite (b) infinite
16.6 C 16.7 D 16.9 − 1 ≤ x < 1 16.10 − 4 < x < 4 16.13 − 5 < x ≤ 1 16.15 −x^2 / 6 − x^4 / 180 − x^6 / 2835 · · · 16.16 1 − x/2 + 3x^2 / 8 − 11 x^3 /48 + 19x^4 / 128 · · · 16.19 −(x − π) + (x − π)^3 /3! − (x − π)^5 /5! · · ·
16.20 2 + x − 8 12
(x − 8)^2 25 · 32
5(x − 8)^3 28 · 34
16.31 (b) 2. 66 × 1086 terms. For N = 15, 1. 6905 < S < 1. 6952
784 Answers to Selected Problems Chapter 2
7.1 All z 7.3 All z 7.6 |z| < 1 / 3 7.7 All z 7.10 |z| < 1 7.12 |z| < 4 7.14 |z − 2 i| < 1 7.16 |z + (i − 3)| < 1 /
8.3 See Problem 17.
9.3 − 9 i 9.4 −e(1 + i
9.7 3 e^2 9.8 −
3 + i 9.10 − 2 9.11 − 1 − i 9.13 −4 + 4i 9.14 64 9.17 −(1 + i)/ 4 9.19 16 9.20 i 9.21 1 9.24 4 i 9.26 (1 + i
9.29 1 9.32 3 e^2 9.34 4 /e 9.35 21 9.38 1 /
10.3 ± 1 , ± i 10.4 ± 2 , ± 2 i 10.7 ±
2 , ± i
2 , ± 1 ± i 10.9 1 , 0. 309 ± 0. 951 i, − 0. 809 ± 0. 588 i 10.16 ±i, (±
3 ± i)/ 2 10.17 − 1 , 0. 809 ± 0. 588 i, − 0. 309 ± 0. 951 i 10.18 ±(1 + i)/
3 + i) 10.22 r =
2, θ = 45◦^ + 120◦n: 1 + i, − 1 .366 + 0. 366 i, 0. 366 − 1. 366 i 10.24 ±(
3 + i)/ 2 , ± (1 − i
3)/ 2 , ± (0.259 + 0966i), ± (0. 966 − 0. 259 i) 10.25 0.758(1 + i), − 0 .487 + 0. 955 i, − 1. 059 − 0. 168 i, − 0. 168 − 1. 059 i,
- 955 − 0. 487 i
11.3 3(1 − i)/
2 11.5 1 + i 11.8 − 41 / 9 11.9 4 i/ 3
12.25 sin x cosh y − i cos x sinh y,
sin^2 x + sinh^2 y 12.26 cosh 2 cos 3 − i sinh 2 sin 3 = − 3. 72 − 0. 51 i, 3. 76 12.28 tanh 1 = 0. 762 12.30 −i 12.32 − 4 i/ 3 12.33 i tanh 1 = 0. 762 i 12.35 − cosh 2 = − 3. 76
14.2 −iπ/2 or 3iπ/ 2 14.3 Ln 2 + iπ/ 6 14.5 Ln 2 + 5iπ/ 4 14.6 −iπ/4 or 7iπ/ 4 14.8 − 1 , (1 ± i
3)/ 2 14.10 e−π (^2) / 4
14.11 cos(Ln 2) + i sin(Ln 2) = 0.769 + 0. 639 i 14.14 0. 3198 i − 0. 2657 14.15 e−π^ sinh 1^ = 0. 0249 14.18 − 1 14.20 1 14.23 eπ/^2 = 4. 81
15.2 π/2 + nπ + (i Ln 3)/ 2 15.3 i(±π/3 + 2nπ) 15.4 i(2nπ + π/6), i(2nπ + 5π/6) 15.5 ±[π/2 + 2nπ − i Ln(3 +
8)]
15.8 π/2 + 2nπ ± i Ln 3 15.9 i(π/3 + nπ) 15.12 i(2nπ ± π/6) 15.14 2nπ + i Ln 2, (2n + 1)π − i Ln 2 15.15 nπ + 3π/8 + (i/4) Ln 2
16.3 |z| =
2; motion around a circle of radius
2, at constant speed v =
constant acceleration a =
Chapter 3 Answers to Selected Problems 785
16.5 v = |z 1 − z 2 |; a = 0 16.6 (a) Series: 3 − 2 i; parallel: 5 + i (b) Series: 2(1 + i
3 ); parallel: i
16.8 [R − i(ωCR^2 + ω^3 L^2 C − ωL)]/[(ωCR)^2 + (ω^2 LC − 1)^2 ]; this simplifies to L/(RC) at resonance. 16.9 (b) ω = 1/
LC 16.12 (1 + r^4 − 2 r^2 cos θ)−^1
17.2 (
3 + i)/ 2 17.4 i cosh 1 = 1. 54 i 17.6 −e−π 2 = − 5. 17 × 10 −^5 17.7 eπ/^2 = 4. 81 17.9 π/ 2 ± 2 nπ 17.11 i 17.13 x = 0, y = 4 17.15 |z| < 1 /e
17.26 1 17.27 (c) e−2(x−t)
2
17.28 1 + (a^2 + b^2 )^2 (2ab)−^2 sinh^2 b 17.30 ex^ cos x =
n=
2 n/^2 xn/n!
cos nπ/ 4 ex^ sin x =
n=
2 n/^2 xn/n!
sin nπ/ 4
Chapter 3
, x = 12 (z + 1), y = 1
, x = y − 11, z = 7
, inconsistent, no solution
, x = −2, y = 1, z = 1
2.17 R = 2
3.16 A = −(K + ik)/(K − ik), |A| = 1
4.12 arc cos(− 1 /
- = 3π/ 4 4.14 (a) arc cos(1/3) = 70. 5 ◦ 4.14 (c) arc cos
2 /3 = 35. 3 ◦^ 4.15 (b) 8i − 4 j + 8k 4.18 2 i − 8 j − 3 k 4.19 i + j + k 4.22 Law of cosines 4.24 A^2 B^2
5.1 r = (2i − 3 j) + (4i + 3j)t [Note that 2i − 3 j may be replaced by any point on the line; 4i + 3j may be replaced by any vector along the line. Thus, for example, r = 6i − (8i + 6j)t is just as good an answer, and similarly for all such problems.] 5.4 r = i + (2i + j)t 5.6 (x − 1)/1 = (y + 1)/(−2) = (z + 5)/2, or r = i − j − 5 k + (i − 2 j + 2k)t 5.8 x/3 = (z − 4)/(−5), y = −2; or r = − 2 j + 4k + (3i − 5 k)t 5.9 x = −1, z = 7; or r = −i + 7 k + j t
Chapter 3 Answers to Selected Problems 787
8.1 In terms of basis u = 19 (9, 0 , 7), v = 19 (0, − 9 , 13), the vectors are: u − 4 v, 5 u − 2 v, 2u + v, 3u + 6v. 8.3 Basis i, j, k 8.6 V = 3A − B 8.19 x = y = z = w = 0 8.20 x = −z, y = z 8.23 For λ = 3, x = 2y; for λ = 8, x = − 2 y 8.25 For λ = 2: x = 0, y = − 3 z; for λ = −3: x = − 5 y, z = 3y; for λ = 4: z = 3y, x = 2y 8.26 r = (3, 1 , 0) + (− 1 , 1 , 1)z
9.4 A†^ =
0 i 3 − 2 i 2 0 − 1 0 0
, A−^1 =^1
0 3 i − 6 6 i − 2
9.14 CTBAT, C−^1 M−^1 C, H
10.1 (b) d = 8 10.2 The number of basis vectors given is the dimension of the space. We list one possible basis; other bases consist of the same number of independent linear combinations of the vectors given. (b) (1, 0 , 0 , 5 , 0 , 1), (0, 1 , 0 , 0 , 6 , 4), (0, 0 , 1 , 0 , − 3 , 0) 10.3 (a) Label the vectors A, B, C, D. Then cos(A, B) = 1/
cos(A, C) =
2 /3, cos(B, D) =
10.4 (b) e 1 = (0, 0 , 0 , 1), e 2 = (1, 0 , 0 , 0), e 3 = (0, 1 , 1 , 0)/
10.5 (b) ‖A‖ = 7, ‖B‖ =
60, |Inner product of A and B| =
11.5 θ = 1.1 = 63. 4 ◦
x y
x′ y′
, not orthogonal
In the following answers, for each eigenvalue, the components of a corresponding eigenvector are listed in parentheses. 11.12 4 (1, 1) − 1 (3, −2)
The two eigenvectors corresponding to the eigenvalue 9 may be any two vectors orthogonal to (2, 2 , −1) and or- thogonal to each other.
11.26 4 (1, 1 , 1) 1 (1, − 1 , 0) 1 (1, 1 , −2)
11.27 D =
, C =
11.29 D =
, C =
11.31 D =
, C =
11.41 λ = 1, 3; U =
1 i i 1
788 Answers to Selected Problems Chapter 3
11.44 λ = 3, −7; U =
5 − 3 − 4 i 3 − 4 i 5
11.52 60◦^ rotation about −i
2 + k and reflection through the plane z = x
11.53 180◦^ rotation about i + j + k 11.56 45◦^ rotation about j − k
11.58 M^10 =
2 − 2 · 610 4 + 6^10
11.59 eM^ = e^3
cosh 1 − sinh 1 − sinh 1 cosh 1
12.2 3 x′^2 − 2 y′^2 = 24 12.3 10 x′^2 = 35 12.6 3 x′^2 +
3 y′^2 −
3 z′^2 = 12 12.15 y = 2x with ω =
3 k/m; x = − 2 y with ω =
8 k/m 12.17 x = − 2 y with ω =
2 k/m; 3x = 2y with ω =
2 k/(3m) 12.19 y = −x with ω =
3 k/m; y = 2x with ω =
3 k/(2m) 12.22 y = −x with ω =
2 k/m; y = 3x with ω =
2 k/(3m)
13.6 The cyclic group 13.11 The four matrices of the symmetry group of the rectangle are:
I =
, P =
= −P,
= −I
This group is isomorphic to the 4’s group. 13.21 SO(2) is Abelian; SO(3) is not Abelian.
14.3 x, cos x, x cos x, ex^ cos x 14.5 1, x + x^3 , x^2 , x^4 , x^5 14.6 Not a vector space 14.8 1, x^2 , x^4 , x^6
15.3 (a) (x − 4)/1 = (y + 1)/(−2) = (z − 2)/(−2); or r = (4, − 1 , 2) + (1, − 2 , −2)t (b) x − 5 y + 3z = 0 (c) 5/ 7 (d) 5
2 /3 = 2. 36 (e) arc sin 19/21 = 64. 8 ◦ 15.5 (a) y = 7, (x − 2)/3 = (z + 1)/4; or r = (2, 7 , −1) + (3, 0 , 4)t (b) x − 4 y − 9 z = 0 (c) arc sin( (^3370)
(d) 12/
98 = 1. 21 (e)
15.7 You should have found all except ATBT, BAT, ABC, ABTC, B−^1 C, and CBT, which are meaningless.
BTAC =
1 − 3 i 1 − 1 − 5 i − 1
, C−^1 A =
0 −i 1 − 1
f
= (n − 1)
[
R 1
R 2
(n − 1)d nR 1 R 2
]
15.13 Area = (^12)
P Q ×
P R
15.14 x′′^ = −x, y′′^ = −y, 180 ◦^ rotation 15.15 x′′^ = −y, y′′^ = x; 90◦^ rotation of vectors or − 90 ◦^ rotation of axes 15.18 1 (1, 1) − 2 (0, 1)
15.27 3x′^2 − y′^2 − 5 z′^2 = 15, d =
5 15.29 3x′^2 + 6y′^2 − 4 z′^2 = 54, d = 3
790 Answers to Selected Problems Chapter 5
10.2 4, 2 10.4 d = 1 10.6 d = 2 10.7 (^12)
10.10 (a) max T = 12 , min T = − (^12) (b) max T = 1, min T = − 21 (c) max T = 1, min T = − (^12)
10.12 Largest sum = 180◦ Smallest sum = 3 arc cos(1/
10.13 Largest sum = 3 arc sin(1/
- = 105. 8 ◦, smallest sum = 90◦
11.1 z = f (y + 2x) + g(y + 3x) 11.6 d^2 y/dz^2 + dy/dz − 5 y = 0 11.11 H = p q˙ − L
12.1 12 x−^1 /^2 sin x 12.3 dz/dx = − sin(cos x) tan x − sin(sin x) cot x 12.4 12 sin 2 12.7 (∂u/∂x)y = −e^4 , (∂u/∂y)x = e^4 / ln 2, (∂y/∂x)u = ln 2 12.10 dy/dx = (ex^ − 1)/x 12.12 (2x + 1)/ ln(x + x^2 ) − 2 / ln(2x) 12.14 π/(4y^3 )
13.2 (a) and (b) d = 4/
13.4 − csc θ cot θ 13.5 − 6 x, 2x^2 tan θ sec^2 θ, 4x tan θ sec^2 θ 13.9 dz/dt = 1 + (t/z)(2 − x − y), z = 0 13.10 [x ln x − (y^2 /x)]xy^ where x = r cos θ, y = r sin θ 13.13 − 1 13.14 (∂w/∂x)y = (∂f /∂x)s, t + 2(∂f /∂s)x, t + 2(∂f /∂t)x, s = f 1 + 2f 2 + 2f 3
26 / 3 13.21 T (2) = 4, T (5) = − 5
13.23 t cot t 13.25 −ex/x 13.29 dt = 3. 9
Chapter 5
2.1 3 2.3 4 2.5 14 e^2 − 125 2.7 5 / 3 2.9 6 2.11 36 2.13 7 / 4 2.15 3 / 2 2.17 12 ln 2 2.19 32 2.21 131 / 6 2.23 9 / 8 2.25 3 / 2 2.27 32 / 5 2.29 2 2.31 6 2.33 16 / 3 2.36 1 / 6 2.37 7 / 6 2.39 70 2.41 5 2.43 9 / 2 2.45 46 k/ 15 2.47 16 / 3 2.49 1 / 3
3.2 (b) M l^2 / 12 (c) M l^2 / 3 3.3 (a) M = 140 (b) ¯x = 130/ 21 (c) Im = 6. 92 M (d) I = 150M/ 7 3.5 (a) M a^2 / 3 (b) M a^2 / 12 (c) 2M a^2 / 3 3.7 (a) M = 9 (b) (¯x, y¯) = (2, 4 /3) (c) Ix = 2M , Iy = 9M/ 2 (d) Im = 13M/ 18 3.9 (a) 1/ 6 (b) (1/ 4 , 1 / 4 , 1 /4) (c) M = 1/24, ¯z = 2/ 5 3.11 (a) M = (
(b) ¯x = 0, ¯y = (313 + 15
3.14 V = 2π^2 a^2 b, A = 4π^2 ab, where a = radius of revolving circle, and b = distance to axis from center of this circle. 3.15 For area, (¯x, ¯y) = (0, 43 r/π), for arc, (¯x, y¯) = (0, 2 r/π)
Chapter 6 Answers to Selected Problems 791
3.18 s = [
2 + ln(1 +
2 )]/ 2
3.20 13 π/ 3 3.21 sx¯ = [
2 − ln(1 +
2 )]/32, sy¯ = 13/6, s as in Problem 3. 3.23 (149/ 130 , 0 , 0) 3.25 I/M has the same numerical value as ¯x in Problem 3. 3.26 2 M/ 3 3.27 149 M/ 130 3.29 2 3.30 32 / 5
4.1 (b) ¯x = ¯y = 43 a/π (c) I = M a^2 / 4 (e) ¯x = ¯y = 2a/π 4.2 (c) ¯y = 43 a/π (d) Ix = M a^2 /4, Iy = 5M a^2 /4, Iz = 3M a^2 / 2 (e) ¯y = 2a/π (f) ¯x = 6a/5, Ix = 48M a^2 /175, Iy = 288M a^2 /175, Iz = 48M a^2 / 25 (g) A = ( 23 π − (^12)
3 )a^2 4.4 (b) (0, 0 , a/2) (c) 2M a^2 / 3 (e) (0, 0 , 3 a/8) 4.5 7 π/ 3 4.11 12 π 4.12 (c) M = (16ρ/9)(3π − 4) = 9. 64 ρ
I = (128ρ/ 152 )(15π − 26) = 12. 02 ρ = 1. 25 M 4.14 π(1 − e−^1 )/ 4 4.16 u^2 + v^2 4.19 π/ 4 4.22 12(1 + 36π^2 )^1 /^2 4.24 ρGπa/ 2 4.26 (a) 75 M a^2 4.27 2 πah (where h = distance between parallel planes)
5.1 95 π
30 5.3 π(37^3 /^2 − 1)/ 6 5.5 8 π for each nappe 5.6 4 5.8 163
6 + 169 ln(
3 ) 5.9 π
5.12 M = 16
3, (¯x, y,¯ ¯z) = ( 12 , 14 , 14 ) 5.14 M = 12 π − (^43)
5.16 x¯ = 0, ¯y = 1, ¯z = [32/(9π)]
2 )/ 112 6.3 15 π/ 8 6.4 (a) 12 M R^2 (b) 32 M R^2 6.6 (a) (4π − 3
6.7 (8π − 3
3 )(4π − 3
3 )−^1 M 6.8 (b) 27/ 20 6.10 (a) (¯x, y¯) = (π/ 2 , π/8) 6.10 (c) 3M/ 8 6.12 (abc)^2 / 6 6.14 16 a^3 / 3 6.15 Ix = 158 M a^2 , Iy = 157 M a^2 6.16 x¯ = ¯y = 2a/ 5 6.18 (0, 0 , 5 h/6) 6.19 Ix = Iy = 20M h^2 /21, Iz = 10M h^2 /21, Im = 65M h^2 / 252 6.21 πGρh(2 −
2 ) 6.24 (0, 0 , 2 c/3) 6.26 12 sinh 1 6.27 e^2 − e − 1
Chapter 6
3.1 (A · B)C = 6C, (A × B) · C = A · (B × C) = −8, A × (B × C) = −4(i + 2k) 3.3 − 5 3.6 v = (2/
6 )(A × B) = (2/
6 )(i − 7 j − 3 k), r × F = (A − C) × B = 3i + 3j − k, n · (r × F) = [(A − C) × B] · C/|C| = 8/
3.7 (a) 11i + 3j − 13 k, (b) 3, (c) 17
Chapter 7 Answers to Selected Problems 793
12.1 (sin θ cos θ)C 12.7 (a) 9i + 5j − 3 k (b) 29/ 3 12.9 24 12.11 (a) grad φ = − 3 yi − 3 xj + 2zk (b) −
(c) 2x + y − 2 z + 2 = 0, r = (1, 2 , 3) + (2, 1 , −2)t 12.13 (a) 6i − j − 4 k (b) 53−^1 /^2 (6i − j − 4 k) (c) same as (a) (d) 53^1 /^2 (e) 53^1 /^2 12.18 Not conservative (a) 1/ 2 (b) 4/ 3 12.21 4 12.23 192π 12.25 − 18 π 12.27 4 12.29 10 12.31 29/ 3
Chapter 7
Amplitude Period Frequency Velocity Amplitude
2.2 2 π/ 2 2 /π 8 2.3 1 / 2 2 1 / 2 π/ 2
2.6 s = 6 cos(π/8) sin(2t) 6 cos(π/8) = 5. 54
π 1 /π 12 cos(π/8) = 11. 1 2.8 2 4 π 1 /(4π) 1
2.10 4 π 1 /π 8 2.11 q 3 1 / 60 60 I 360 π 1 / 60 60
2.13 A = maximum value of θ, ω =
g/l 2.16 t ∼= 4. 91 ∼= 281◦ 2.19 A = 1, T = 4, f = 1/4, v = 1/4, λ = 1 2.21 y = 20 sin 12 π(x − 6 t), ∂y/∂t = − 60 π cos 12 π(x − 6 t) 2.23 y = sin 880π((x/350) − t) 2.25 y = 10 sin[π(x − 3 · 108 t)/250]
3.6 sin(2x + 13 π)
4.5 π−^1 + 12 4.6 2 /π 4.8 0 4.11 1 / 2 4.14 (a) 2π/ 3 (b) π 4.15 (a) 3/ 2
x → − 2 π −π −π/ 2 0 π/ 2 π 2 π
6.2 1 / 2 0 0 1 / 2 1 / 2 0 1 / 2 6.4 − 1 0 − 1 − 1 0 0 − 1 6.6 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2
6.8 1 1 1 − 12 π 1 1 + 12 π 1 1 6.10 π 0 π/ 2 π π/ 2 0 π
794 Answers to Selected Problems Chapter 7
7.1 f (x) =
i π
∑^ ∞
−∞ odd n
n
einx
7.2 f (x) =
2 π
[
(1 − i)eix^ + (1 + i)e−ix^ − i(e^2 ix^ − e−^2 ix)
−
1 + i 3
e^3 ix^ −
1 − i 3
e−^3 ix^ +
1 − i 5
e^5 ix^ +
1 + i 5
e−^5 ix^ · · ·
]
7.7 f (x) =
π 4
∑^ ∞
−∞ odd n
n^2 π
i 2 n
einx^ +
∑^ ∞
−∞ even n=
i 2 n
einx
7.11 f (x) =
π
eix^ − e−ix 4 i
π
∑^ ∞
−∞ even n=
einx n^2 − 1
8.2 f (x) =
π
cos
πx l
cos
3 πx l
cos
5 πx l
π
sin
πx l
sin
2 πx l
sin
3 πx l
sin
5 πx l
sin
6 πx l
8.6 f (x) =
π
n
sin
nπx l
(n = 2, 6, 10, · · · )
8.11 (a) f (x) = π^2 3
∑^ ∞
1
(−1)n n^2
cos nx
(b) f (x) =
4 π^2 3
∑^ ∞
−∞
n^2
iπ n
einx, n = 0
8.14 (a) f (x) =
π
∑^ ∞
1
n(−1)n+ 4 n^2 − 1
sin 2nπx
(b) f (x) =
π
π
∑^ ∞
1
cos 2nπx 4 n^2 − 1
π
∑^ ∞
−∞
4 n^2 − 1
e^2 inπx
8.19 f (x) =
π^2
∑^ ∞
odd n=
n^2
cos 2nπx +
2 π
∑^ ∞
1
(−1)n+ n
sin 2nπx
8.20 f (x) =
8 π^2
[
cos 2 πx 3
cos 4 πx 3
cos 8 πx 3
]
8 π^2
π
sin 2 πx 3
32 π^2
2 π
sin 4 πx 3
3 π
sin 6 πx 3
128 π^2
4 π
sin 8 πx 3
9.2 (a) 12 ln | 1 − x^2 | + 12 ln |(1 − x)/(1 + x)|
9.5 f (x) =
π
∑^ ∞
odd n=
n
sin nx
796 Answers to Selected Problems Chapter 8
12.10 f (x) = 2
−∞
αa − sin αa iπα^2
eiαx^ dα
12.11 f (x) =
π
−∞
cos(απ/2) 1 − α^2
eiαx^ dα
12.13 fc(x) =
π
0
sin απ − sin(απ/2) α
cos αx dα
12.16 fc(x) =
π
0
cos(απ/2) 1 − α^2 cos αx dα
12.18 fs(x) =
π
0
sin α − α cos α α^2
sin αx dα
12.19 fs(x) =
π
0
αa − sin αa α^2
sin αx dα
12.21 g(α) = σ(2π)−^1 /^2 e−α
(^2) σ (^2) / 2
12.25 (a) f (x) =
2 π
−∞
1 + e−iαπ 1 − α^2
eiαx^ dα
12.28 (a) fc(x) =
π
0
cos 3α sin α α
cos αx dα
(b) fs(x) =
π
0
sin 3α sin α α sin αx dα
12.30 (a) fc(x) =
π
0
1 − cos 2α α^2
cos αx dα
(b) fs(x) =
π
0
2 α − sin 2α α^2
sin αx dα
13.7 f (x) = π 2
π
∑^ ∞
1 odd n
n^2
cos nx 13.8 (b) 1
13.10 (d) −1, − 1 /2, −2, − 1 13.14 (a) f (x) =
π^2
∑^ ∞
1
cos nπx n^2 (b) π^4 / 90 13.15 −π/ 4 13.23 π^2 / 8
Chapter 8
1.5 x = −Aω−^2 sin ωt + v 0 t + x 0 1.7 x = (c/F )[(m^2 c^2 + F 2 t^2 )^1 /^2 − mc]
2.2 (1 − x^2 )^1 /^2 + (1^ −^ y^2 )^1 /^2 =^ C,^ C^ =^
2.3 ln y = A(csc x − cot x), A =
2.6 2 y^2 + 1 = A(x^2 − 1)^2 , A = 1 2.7 y^2 = 8 + eK−x
2 , K = 1 2.9 yey^ = aex, a = 1 2.13 y ≡ 1, y ≡ −1, x ≡ 1, x ≡ − 1 2.19 (a) I/I 0 = e−^0.^5 = 0.6 for s = 50 ft
Half value thickness = (ln 2)/μ = 69.3 ft (b) Half life T = (ln 2)/λ 2.20 (c) τ = RC, τ = L/R. Corresponding quantities are a, λ = (ln 2)/T , μ, 1/τ. 2.22 N = N 0 eKt^ − (R/K)(eKt^ − 1) where N 0 = number of bacteria at t = 0, KN = rate of increase, R = removal rate. 2.23 T = 100[1 − (ln r)/(ln 2)]
Chapter 8 Answers to Selected Problems 797
2.26 (a) k = weight divided by terminal speed (b) t = g−^1 · (terminal speed) · (ln 100); typical terminal speeds are 0.02 to 0 .1 cm/sec, so t is of the order of 10−^4 sec. 2.27 t = 10(ln 135 )/(ln 133 ) = 6.6 min 2.29 t = 100 ln 94 = 81.1 min
2.31 ay = bx 2.33 x^2 + ny^2 = C 2.35 x(y − 1) = C
3.1 y = 12 ex^ + Ce−x^ 3.3 y = ( 12 x^2 + C)e−x
2
3.6 y = (x + C)/(x +
x^2 + 1 ) 3.8 y = 12 ln x + C/ ln x 3.9 y(1 − x^2 )^1 /^2 = x^2 + C 3.11 y = 2(sin x − 1) + Ce−^ sin^ x 3.13 x = 12 ey^ + Ce−y^ 3.14 x = y^2 /^3 + Cy−^1 /^3
3.15 S = (10^7 /2)[(1 + 3t/ 104 ) + (1 + 3t/ 104 )−^1 /^3 ], where S = number of pounds of salt, and t is in hours. 3.17 I = Ae−t/(RC)^ − V 0 ωC(sin ωt − ωRC cos ωt)/(1 + ω^2 R^2 C^2 ) 3.21 Nn = c 1 e−λ^1 t^ + c 2 e−λ^2 t^ + · · · where
c 1 =
λ 1 λ 2 · · · λn− 1 N 0 (λ 2 − λ 1 )(λ 3 − λ 1 )... (λn − λ 1 )
, c 2 =
λ 1 λ 2 · · · λn− 1 N 0 (λ 1 − λ 2 )(λ 3 − λ 2 ) · · · (λn − λ 2 )
etc. (all λ’s different) 3.22 y = x + 1 + Kex
4.1 y^1 /^3 = x − 3 + Ce−x/^3 4.4 x^2 e^3 y^ + ex^ − 13 y^3 = C
4.5 x^2 − y^2 + 2x(y + 1) = C 4.7 x = y(ln x + C) 4.9 y^2 = Ce−x (^2) /y 2 4.11 tan 12 (x + y) = x + C
4.13 y^2 = − sin^2 x + C sin^4 x 4.16 y^2 = C(C ± 2 x) 4.18 x^2 + (y − k)^2 = k^2 4.19 r = Ae−θ, r = Beθ
5.1 y = Aex^ + Be−^2 x^ 5.3 y = Ae^3 ix^ + Be−^3 ix^ or other forms as in (5.24) 5.5 y = (Ax + B)ex^ 5.7 y = Ae^3 x^ + Be^2 x 5.9 y = Ae^2 x^ sin(3x + γ) 5.11 y = (A + Bx)e−^3 x/^2 5.20 y = Ae−x^ + Beix^ 5.22 y = Aex^ + Be−^3 x^ + Ce−^5 x 5.24 y = Ae−x^ + Bex/^2 sin( 12 x
3 + γ) 5.26 y = Ae^5 x^ + (Bx + C)e−x
5.28 y = ex(A sin x + B cos x) + e−x(C sin x + D cos x) 5.29 y = (A + Bx)e−x^ + Ce^2 x^ + De−^2 x^ + E sin(2x + γ) 5.35 T = 2π
R/g ∼= 85 min.
6.1 y = Ae^2 x^ + Be−^2 x^ − 52 6.3 y = Aex^ + Be−^2 x^ + 14 e^2 x
6.5 y = Aeix^ + Be−ix^ + ex^ 6.7 y = Ae−x^ + Be^2 x^ + xe^2 x 6.9 y = (Ax + B + x^2 )e−x 6.11 y = e−x(A sin 3x + B cos 3x) + 8 sin 4x − 6 cos 4x 6.13 y = (Ax + B)ex^ − sin x 6.15 y = e−^6 x/^5 [A sin(8x/5) + B cos(8x/5)] − 5 cos 2x 6.17 y = A sin 4x + B cos 4x + 2x sin 4x 6.18 y = e−x(A sin 4x + B cos 4x) + 2e−^4 x^ cos 5x 6.20 y = Ae−^2 x^ sin(2x + γ) + 4e−x/^2 sin(5x/2) 6.22 y = A + Be−x/^2 + x^2 − 4 x 6.24 y = (A + Bx + 2x^3 )e^3 x 6.26 y = A sin x + B cos x − 2 x^2 cos x + 2x sin x 6.33 y = A sin(x + γ) + x^3 − 6 x − 1 + x sin x + (3 − 2 x)ex
Chapter 9 Answers to Selected Problems 799
11.9 y =
1 3 e
−(t−t 0 ) (^) sin 3(t − t 0 ), t > t 0 0 , t < t 0
11.11 y =
1 2 [sinh(t^ −^ t^0 )^ −^ sin(t^ −^ t^0 )],^ t > t^0 0 , t < t 0
11.13 (b) 3δ(x + 5) − 4 δ(x − 10) 11.15 (b) 0 (d) cosh 1 11.21 (b) φ(|a|)/(2|a|) (c) 1/ 2 11.23 (a) δ(x + 5)δ(y − 5)δ(z), δ(r − 5
2 )δ(θ − 34 π )δ(z)/r, δ(r − 5
2 )δ(θ − π 2 )δ(φ − 34 π )/(r sin θ) (c) δ(x + 2)δ(y)δ(z − 2
3 ), δ(r − 2)δ(θ − π)δ(z − 2
3 )/r, δ(r − 4)δ(θ − π 6 )δ(φ − π)/(r sin θ) 11.25 (c) G′′(x) = δ(x) + 5δ′(x)
12.2 y = (sin ωt − ωt cos ωt)/(2ω^2 ) 12.7 y = [a(cosh at − e−t) − sinh at]/[a(a^2 − 1)] 12.11 y = − 13 sin 2x
12.13 y =
x −
2 sin x, x < π/ 4 1 2 π^ −^ x^ −
2 cos x, x > π/ 4
12.16 y = −x ln x − x − x(ln x)^2 / 2 12.18 y = x^2 /2 + x^4 / 6
13.1 y = − 13 x−^2 + Cx 13.3 y = A + Be−x^ sin(x + γ)
13.5 x^2 + y^2 − y sin^2 x = C 13.7 3 x^2 y^3 + 1 = Ax^3 13.8 y = x(A + B ln x) + 12 x(ln x)^2 13.10 u − ln u + ln v + v−^1 = C 13.13 y = Ae−^2 x^ sin(x + γ) + e^3 x^ 13.15 y = (A + Bx)e^2 x^ + 3x^2 e^2 x 13.18 x = (y + C)e−^ sin^ y 13.20 y = Aex^ sin(2x + γ) + x + 25 + ex(1 − x cos 2x) 13.22 y = (A + Bx)e^2 x^ + C sin(3x + γ) 13.24 y^2 = ax^2 + b 13.26 y = x^2 + x 13.28 y^2 + 4(x − 1)^2 = 9 13.30 y = g/3, v = 7g/12, a = 5g/ 12 13.32 1:23 p.m. 13.33 In both (a) and (b), the temperature of the mixture at time t is given by the formula Ta(1 − e−kt) + (n + n′)−^1 (nT 0 + n′T 0 ′)e−kt. 13.38 12 ln[(a^2 + p^2 )/p^2 ] 13.41 14 (tanh 1 − sech^2 1) = 0. 0854
13.43 (sin at + at cos at)/(2a) 13.46 For e−x: gs(α) = (2/π)^1 /^2 α/(1 + α^2 ), gc(α) = (2/π)^1 /^2 /(1 + α^2 ) 13.47 y = A sin t + B cos t + sin t ln(sec t + tan t) − 1
Chapter 9
2.1 Parabola 2.2 Circle 2.3 ax = sinh(ay + b) 2.6 x + a = 43 (y^1 /^2 − 2 b)(b + y^1 /^2 )^1 /^2
3.1 dx/dy = C/
y^3 − C^2 3.3 x^4 y′^2 = C^2 (1 + x^2 y′^2 )^3 3.6 x = ay^3 /^2 − 12 y^2 + b 3.7 y = K sinh(x + C) 3.9 cot θ = A cos(φ − α) 3.12 (x − a)^2 + y^2 = C^2
800 Answers to Selected Problems Chapter 10
3.15 r cos(θ + α) = C or, in rectangular coordinates, the straight line x cos α − y sin α = C 3.18 See Problem 3.
4.6 Cycloid
m(¨r − r θ˙^2 ) = −∂V /∂r m(r ¨θ + 2 ˙r θ˙) = −(1/r)(∂V /∂θ) mz¨ = −∂V /∂z
Comment: These equations are in the form ma = F; recall from Chapter 6, equation (6.7), the polar coordinate form for F = −∇V. 5.4 l¨θ + g sin θ = 0 5.
aθ¨ − a sin θ cos θ φ˙^2 − g sin θ = 0 (d/dt)(sin^2 θ φ˙) = 0
5.8 L = 12 m(2 ˙r^2 + r^2 θ˙^2 ) − mgr
2¨r − r θ˙^2 + g = 0, (d/dt)(r^2 θ˙) = 0
5.11 L = 12 (m + Ia−^2 ) ˙z^2 − mgz (ma^2 + I)¨z + mga^2 = 0
5.12 L = 12 m( ˙r^2 + r^2 θ˙^2 ) −
[ 1
2 k(r^ −^ r^0 )
(^2) − mgr cos θ]
r ¨ − r θ˙^2 + (^) mk (r − r 0 ) − g cos θ = 0, (d/dt)(r^2 θ˙) + gr sin θ = 0
5.14 L = M x˙^2 + M gx sin α, 2 M x¨ − M g sin α = 0
5.16 L = 12 m(l + aθ)^2 θ˙^2 − mg[a sin θ − (l + aθ) cos θ]
(l + aθ)θ¨ + a θ˙^2 + g sin θ = 0 5.19 x = y with ω =
g/l ; x = −y with ω =
3 g/l 5.21 2 θ¨ + φ¨ cos(θ − φ) + φ˙^2 sin(θ − φ) + (^2) lg sin θ = 0
φ¨ + θ¨ cos(θ − φ) − θ˙^2 sin(θ − φ) + g l sin^ φ^ = 0 5.23 φ = 2θ with ω =
2 g/(3l) ; φ = − 2 θ with ω =
2 g/l
6.1 Catenary 6.3 Circular cylinder 6.5 Circle
8.4 dr/dθ = Kr
r^4 − K^2 8.6 (x − a)^2 + (y + 1)^2 = C^2 8.8 Intersection of r = 1 + cos θ with z = a + b sin(θ/2) 8.10 Intersection of y = x^2 with az = b[2x
4 x^2 + 1 + sinh−^1 2 x] + c 8.12 ey^ cos(x − a) = K 8.16 Hyperbola: r^2 cos(2θ + α) = K or (x^2 − y^2 ) cos α − 2 xy sin α = K 8.17 K ln r = cosh(Kθ + C) 8.18 Parabola: (x − y − C)^2 = 4K^2 (x + y − K^2 ) 8.20 m(¨r − r θ˙^2 ) + Kr−^2 = 0, r^2 θ˙ = const. 8.22 r−^1 m(r^2 θ¨ + 2r r˙ θ˙ − r^2 sin θ cos θ φ˙^2 ) = −r−^1 (∂V /∂θ) = Fθ = maθ , aθ = r θ¨ + 2 ˙r θ˙ − r sin θ cos θ φ˙^2 8.27 dr/dθ = r
K^2 (1 + λr)^2 − 1
Chapter 10
4.6 I =
0 6 0 − 3 0 9
; principal moments: (6, 6 , 12); principal axes along the vec- tors (1, 0 , −1) and any two orthogonal vectors in the plane z = x, say (0, 1 , 0) and (1, 0 , 1).
5.6 (a) 3 (c) 2 (e) − 1
