text stringlengths 1 1k ⌀ | source stringclasses 12
values |
|---|---|
Answers to Selected Exercises 763
10. 5. 27 (p. 555) y = c1
[ 0
1
1
]
e2t + c2
([ 1
1
0
]
e2t
2 +
[ 0
1
1
]
te2t
)
+c3
([ −1
1
0
]
e2t
8 +
[ 1
1
0
]
te2t
2 +
[ 0
1
1
]
t2e2t
2
)
10. 5. 28 (p. 555) y = c1
[ −2
1
2
]
e−6t + c2
(
−
[ 6
1
0
]
e−6t
6 +
[ −2
1
2
]
te−6t
)
+c3
(
−
[ 12
1
0
]
e−6t
36 −
[ 6
1
0
]
te−6t
6 +
[ −2... | Elementary Differential Equations with Boundary Value Problems.pdf |
]
+ c2e2t
[
3 sin t − cos t
5 sin t
]
.
10. 6. 2 (p. 565) y = c1e−t
[
5 cos 2 t + sin 2 t
13 cos 2 t
]
+ c2e−t
[
5 sin 2 t − cos 2 t
13 sin 2 t
]
.
10. 6. 3 (p. 565) y = c1e3t
[
cos 2t + sin 2 t
2 cos 2 t
]
+ c2e3t
[
sin 2t − cos 2 t
2 sin 2 t
]
.
10. 6. 4 (p. 565) y = c1e2t
[
cos 3t − sin 3 t
cos 3 t
]
+ c2e2t
[
sin 3... | Elementary Differential Equations with Boundary Value Problems.pdf |
764 Answers to Selected Exercises
10. 6. 8 (p. 566) y = c1
[ −1
1
1
]
et + c2e−t
[ − sin 2 t − cos 2 t
2 cos 2 t
2 cos 2 t
]
+ c3e−t
[ cos 2 t − sin 2t
2 sin 2 t
2 sin 2 t
]
10. 6. 9 (p.
566) y = c1e3t
[
cos 6t − 3 sin 6 t
5 cos 6 t
]
+ c2e3t
[
sin 6 t + 3 cos 6 t
5 sin 6 t
]
10. 6. 10 (p. 566) y = c1e2t
[
cos t − 3 si... | Elementary Differential Equations with Boundary Value Problems.pdf |
566) y = c1
[ 1
2
1
]
e3t + c2e6t
[ − sin 3 t
sin 3 t
cos 3 t
]
+ c3e6t
[ cos 3 t
− cos 3 t
sin 3t
]
10. 6. 16 (p.
566) y = c1
[ 1
1
1
]
et + c2et
[ 2 cos t − 2 sin t
cos t − sin t
2 cos t
]
+ c3et
[ 2 sin t + 2 cos t
cos t + sin t
2 sin t
]
10. 6. 17 (p.
566) y = et
[
5 cos 3 t + sin 3 t
2 cos 3 t + 3 sin 3 t
]
10. 6.... | Elementary Differential Equations with Boundary Value Problems.pdf |
7 cos 6 t − 4 sin 6 t
]
10. 6. 24 (p.
566) y =
[ 6
−3
3
]
e8t +
[ 10 cos 4 t − 4 sin 4 t
17 cos 4 t − sin 4 t
3 cos 4 t − 7 sin 4 t
]
10. 6. 29 (p.
567) U = 1√
2
[
−1
1
]
, V = 1√
2
[
1
1
]
10. 6. 30 (p.
567) U ≈
[
. 5257
. 8507
]
, V ≈
[
−. 8507
. 5257
] | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 765
10. 6. 31 (p. 567) U ≈
[
. 8507
. 5257
]
,
V ≈
[
−. 5257
. 8507
]
10. 6. 32 (p. 567) U ≈
[
−. 9732
. 2298
]
, V ≈
[
. 2298
. 9732
]
10. 6. 33 (p. 567) U ≈
[
. 5257
. 8507
]
, V ≈
[
−. 8507
. 5257
]
10. 6. 34 (p. 567) U ≈
[
−. 5257
. 8507
]
, V ≈
[
. 8507
. 5257
]
10. 6. 35 (p. 568) U ≈... | Elementary Differential Equations with Boundary Value Problems.pdf |
[
5 − 3et
−6 + 5 et
]
10. 7. 5 (p. 575)
[
e−5t (3 + 6 t) + e−3t (3 − 2t)
−e−5t (3 + 2 t) − e−3t (1 − 2t)
]
10. 7. 6 (p. 575)
[
t
0
]
10. 7. 7 (p. 575) − 1
6
[ 2 − 6t
7 + 6 t
1 − 12t
]
10. 7. 8 (p. 575) − 1
6
[ 3et + 4
6et − 4
10
]
10. 7. 9 (p. 575) 1
18
[ et(1 + 12 t) − e−5t (1 + 6 t)
−2et (1 − 6t) − e−5t (1 − 12t)
et(... | Elementary Differential Equations with Boundary Value Problems.pdf |
]
10. 7. 22 (p. 576) (a) y′=
0 1 · · · 0
0 0 · · · 0
.
.
.
.
.
. . . . .
.
.
0 0 · · · 1
−Pn(t)/P 0(t) −Pn−1/P 0(t) · · · −P1(t)/P 0(t)
y +
0
0
.
.
.
F (t)/P 0(t)
. | Elementary Differential Equations with Boundary Value Problems.pdf |
766 Answers to Selected Exercises
(b)
y1 y2 · · · yn
y′
1 y′
2 · · · y′
n
.
.
.
.
.
. . . . .
.
.
y(n−1)
1 y(n−1)
2 · · ·y(n−1)
n
Section 11.1 Answers, pp.
585– 586
11. 1. 2 (p. 585) λ n = n2 , yn = sin nx, n = 1 , 2, 3, . . .
11. 1. 3 (p. 585) λ 0 = 0 , y0 = 1 ; λ n = n2 , yn = cos nx, n = 1 , 2, 3, . ... | Elementary Differential Equations with Boundary Value Problems.pdf |
11. 1. 11 (p. 585) λ n = (2n − 1)2π 2
4 , yn = sin (2n − 1)πx
2 , n = 1 , 2, 3, . . .
11. 1. 12 (p. 585) λ 0 = 0 , y0 = 1 , λ n = n2 π 2
4 , y1n = cos nπx
2 , y2n = sin nπx
2 , n = 1 , 2, 3, . . .
11. 1. 13 (p. 585) λ n = n2 π 2
4 , yn = sin nπx
2 , n = 1 , 2, 3, . . .
11. 1. 14 (p. 585) λ n = (2n − 1)2π 2
36 , yn = co... | Elementary Differential Equations with Boundary Value Problems.pdf |
11. 2. 2 (p. 598) F (x) = 2 + 2
π
∞∑
n=1
(−1)n
n sin nπx ; F (x) =
{ 2, x = −1,
2 − x, −1 < x < 1,
2, x = 1
11. 2. 3 (p. 598) F (x) = −π 2 − 12
∞∑
n=1
(−1)n
n2 cos nx − 4
∞∑
n=1
(−1)n
n sin nx;
F (x) =
{ −3π 2, x = −π,
2x − 3x2 , −π < x < π,
−3π 2, x = π
11. 2. 4 (p. 598) F (x) = − 12
π 2
∞∑
n=1
(−1)n cos nπx
n2 ; F (x... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 767
11. 2. 5 (p. 598) F (x) = 2
π − 4
π
∞∑
n=1
1
4n2 − 1 cos 2nx; F (x) = |sin x|, −π ≤ x ≤ π
11. 2. 6 (p. 599) F (x) = − 1
2 sin x + 2
∞∑
n=2
(−1)n n
n2 − 1 sin nx;; F (x) = x cos x, −π ≤ x ≤ π
11. 2. 7 (p. 599) F (x) = − 2
π + π
2 cos x − 4
π
∞∑
n=1
4n2 + 1
(4n2 − 1)2 cos 2 nx;
F (x) = |... | Elementary Differential Equations with Boundary Value Problems.pdf |
π
∞∑
n=1
(−1)n n
4n2 − 1 sin 2 nπx ; F (x) =
0, −1 ≤ x < 1
2 ,
− 1
2 , x = − 1
2 ,
sin πx, − 1
2 < x < 1
2 ,
1
2 , x = 1
2 ,
0, 1
2 < x ≤ 1
11. 2. 13 (p. 599) F (x) = 1
π + 1
π cos πx − 2
π
∞∑
n=2
1
n2 − 1
(
1 − n sin nπ
2
)
cos nπx ;
F (x) =
0, −1 ≤ x < 1
2 ,
1
2 , x = −1,
|sin πx |... | Elementary Differential Equations with Boundary Value Problems.pdf |
768 Answers to Selected Exercises
F (x) =
2, x = −4,
0, −4 < x < 0,
x, 0 ≤ x < 4,
2, x = 4
11. 2. 16 (p.
599) F (x) = 1
2 + 1
π
∞∑
n=1
1
n sin 2nπx + 8
π 3
∞∑
n=0
1
(2n + 1) 3 sin(2n + 1) πx ;
F (x) =
1
2 , x = −1,
x2 , −1 < x < 0,
1
2 , x = 0 ,
1 − x2, 0 < x < 1,
1
2 , x = 1
11. 2. 17 (p. 5... | Elementary Differential Equations with Boundary Value Problems.pdf |
π
∞∑
n=1
(−1)n n
n2 − k2 sin nx
11. 2. 24 (p. 600) F (x) = sin kπ
π
[
1
k − 2k
∞∑
n=1
(−1)n
n2 − k2 cos nx
]
Section 11.3 Answers, pp. 613– 616
11. 3. 1 (p. 613) C(x) = L2
3 + 4L2
π 2
∞∑
n=1
(−1)n
n2 cos nπx
L
11. 3. 2 (p. 613) C(x) = 1
2 + 4
π 2
∞∑
n=1
1
(2n − 1)2 cos(2n − 1)πx
11. 3. 3 (p. 613) C(x) = − 2L2
3 + 4L2
π... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 769
11. 3. 7 (p. 613) C(x) = 1
3 + 4
π 2
∞∑
n=1
1
n2 cos nπx
11. 3. 8 (p. 613) C(x) = eπ − 1
π + 2
π
∞∑
n=1
[(−1)neπ − 1]
(n2 + 1) cos nx
11. 3. 9 (p. 613) C(x) = L2
6 − L2
π 2
∞∑
n=1
1
n2 cos 2nπx
L
11. 3. 10 (p. 613) C(x) = − 2L2
3 + 4L2
π 2
∞∑
n=1
1
n2 cos nπx
L
11. 3. 11 (p. 614) S(x) ... | Elementary Differential Equations with Boundary Value Problems.pdf |
π
∞∑
n=1
(−1)n
2n − 1
[
1 − 8
(2n − 1)2π 2
]
cos (2n − 1)πx
2L
11. 3. 20 (p. 614) CM (x) = − 4
π
∞∑
n=1
[
(−1)n + 2
(2n − 1)π
]
cos (2n − 1)πx
2 .
11. 3. 21 (p. 614) CM (x) = − 4
π
∞∑
n=1
1
2n − 1 cos (2n + 1) π
4 cos (2n − 1)πx
2L
11. 3. 22 (p. 614) CM (x) = 4
π
∞∑
n=1
(−1)n 2n − 1
(2n − 3)(2n + 1) cos (2n − 1)x
2
11.... | Elementary Differential Equations with Boundary Value Problems.pdf |
770 Answers to Selected Exercises
11. 3. 25 (p. 614) SM (x) = 4
π
∞∑
n=1
1
(2n − 1) sin (2n − 1)πx
2L
11. 3. 26 (p. 614) SM (x) = − 16L2
π 2
∞∑
n=1
1
(2n − 1)2
[
(−1)n + 2
(2n − 1)π
]
sin (2n − 1)πx
2L
11. 3. 27 (p. 614) SM (x) = 4
π
∞∑
n=1
1
2n − 1
[
1 − cos (2n − 1)π )
4
]
sin (2n − 1)πx
2L
11. 3. 28 (p. 614) SM (x) ... | Elementary Differential Equations with Boundary Value Problems.pdf |
L
11. 3. 36 (p. 615) S(x) = 8L2
π 3
∞∑
n=1
1
(2n − 1)3 sin (2n − 1)πx
L
11. 3. 37 (p. 615) S(x) = − 4L3
π 3
∞∑
n=1
(1 + ( −1)n2)
n3 sin nπx
L
11. 3. 38 (p. 615) S(x) = − 12L3
π 3
∞∑
n=1
(−1)n
n3 sin nπx
L
11. 3. 39 (p. 615) S(x) = 96L4
π 5
∞∑
n=1
1
(2n − 1)5 sin (2n − 1)πx
L
11. 3. 40 (p. 615) S(x) = − 720L5
π 5
∞∑
n=1... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 771
11. 3. 45 (p. 615) CM (x) = − 96L3
π 3
∞∑
n=1
1
(2n − 1)3
[
(−1)n + 2
(2n − 1)π
]
cos (2n − 1)πx
2L
11. 3. 46 (p. 615) CM (x) = 96L3
π 3
∞∑
n=1
1
(2n − 1)3
[
(−1)n3 + 4
(2n − 1)π
]
cos (2n − 1)πx
2L
11. 3. 47 (p. 615) CM (x) = 96L3
π 3
∞∑
n=1
1
(2n − 1)3
[
(−1)n5 + 8
(2n − 1)π
]
cos (2... | Elementary Differential Equations with Boundary Value Problems.pdf |
π 4
∞∑
n=1
(−1)n
(2n − 1)4 sin (2n − 1)πx
2L
11. 3. 55 (p. 616) SM (x) = 1536L4
π 4
∞∑
n=1
1
(2n − 1)4
[
(−1)n + 3
(2n − 1)π
]
sin (2n − 1)πx
2L
11. 3. 56 (p. 616) SM (x) = 384L4
π 4
∞∑
n=1
1
(2n − 1)4
[
(−1)n + 4
(2n − 1)π
]
sin (2n − 1)πx
2L
Section 12.1 Answers, pp. 626– 629
12. 1. 8 (p. 626) u(x, t ) = 8
π 3
∞∑
n=1... | Elementary Differential Equations with Boundary Value Problems.pdf |
772 Answers to Selected Exercises
12. 1. 15 (p. 626) u(x, t ) = 96
π 5
∞∑
n=1
1
(2n − 1)5 e−5(2n−1)2 π 2t sin(2n − 1)πx
12. 1. 16 (p. 626) u(x, t ) = − 240
π 5
∞∑
n=1
1 + ( −1)n2
n5 e−2n2 π 2t sin nπx .
12. 1. 17 (p. 627) u(x, t ) = 16
3 + 64
π 2
∞∑
n=1
(−1)n
n2 e−9π 2n2 t/ 16 cos nπx
4
12. 1. 18 (p. 627) u(x, t ) = − ... | Elementary Differential Equations with Boundary Value Problems.pdf |
12. 1. 25 (p. 627) u(x, t ) = 8
π
∞∑
n=1
(−1)n
(2n + 1)(2 n − 3) e−(2n−1)2 π 2t/ 4 sin (2n − 1)πx
2
12. 1. 26 (p. 627) u(x, t ) = 8
∞∑
n=1
1
(2n − 1)2
[
(−1)n + 4
(2n − 1)π
]
e−3(2n−1)2 t/ 4 sin (2n − 1)x
2
12. 1. 27 (p. 627) u(x, t ) = 128
π 3
∞∑
n=1
1
(2n − 1)3 e−5(2n−1)2 t/ 16 sin (2n − 1)πx
4
12. 1. 28 (p. 627) u(x... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 773
12. 1. 33 (p. 628) u(x, t ) = −64
∞∑
n=1
e−3(2n−1)2 t/ 4
(2n − 1)3
[
(−1)n + 3
(2n − 1)π
]
cos (2n − 1)x
2
12. 1. 34 (p. 628) u(x, t ) = − 16
π
∞∑
n=1
(−1)n
2n − 1 e−(2n−1)2 t cos (2n − 1)x
4
12. 1. 35 (p. 628) u(x, t ) = − 64
π
∞∑
n=1
(−1)n
2n − 1
[
1 − 8
(2n − 1)2π 2
]
e−9(2n−1)2 π 2... | Elementary Differential Equations with Boundary Value Problems.pdf |
2
12. 1. 41 (p. 628) u(x, t ) = − 768
π 4
∞∑
n=1
1
(2n − 1)4
[
1 + (−1)n2
(2n − 1)π
]
e−(2n−1)2π 2t/ 4 cos (2n − 1)πx
2
12. 1. 42 (p. 628) u(x, t ) = − 384
π 4
∞∑
n=1
1
(2n − 1)4
[
1 + (−1)n4
(2n − 1)π
]
e−(2n−1)2π 2t/ 4 cos (2n − 1)πx
2
12. 1. 43 (p. 628) u(x, t ) = 1
2 − 2
π
∞∑
n=1
(−1)n
2n − 1 e−(2n−1)2 π 2a2t/L 2
c... | Elementary Differential Equations with Boundary Value Problems.pdf |
π 2
∞∑
n=1
1
(2n − 1)2 e−3(2n−1)2 π 2t/ 4 cos (2n − 1)πx
2
12. 1. 51 (p. 629) u(x, t ) = x2 − x − 2 − 64
π
∞∑
n=1
(−1)n
2n − 1
[
1 − 8
(2n − 1)2 π 2
]
e−9(2n−1)2π 2 t/ 64 cos (2n − 1)πx
8 | Elementary Differential Equations with Boundary Value Problems.pdf |
774 Answers to Selected Exercises
12. 1. 52 (p. 629) u(x, t ) = sin πx + 8
π
∞∑
n=1
(−1)n
(2n + 1)(2 n − 3) e−(2n−1)2π 2 t/ 4 sin (2n − 1)πx
2
12. 1. 53 (p. 629) u(x, t ) = x3 − x + 3 + 32
π 3
∞∑
n=1
e−(2n−1)2π 2t/ 4
(2n − 1)3 sin (2n − 1)πx
2
Section 12.2 Answers, pp. 642– 649
12. 2. 1 (p. 642) u(x, t ) = 4
3π 3
∞∑
n=... | Elementary Differential Equations with Boundary Value Problems.pdf |
12. 2. 8 (p. 642) u(x, t ) = 243
2π 4
∞∑
n=1
(−1)n
n4 sin 8nπt
3 sin nπx
3
12. 2. 9 (p. 642) u(x, t ) = 48
π 6
∞∑
n=1
1
(2n − 1)6 sin 2(2n − 1)πt sin(2n − 1)πx .
12. 2. 10 (p. 642) u(x, t ) = π
2 cos
√
5 t sin x − 16
π
∞∑
n=1
n
(4n2 − 1)2 cos 2 n
√
5 t sin 2nx
12. 2. 11 (p. 642) u(x, t ) = − 240
π 5
∞∑
n=1
1 + ( −1)n2
... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 775
12. 2. 19 (p. 644) u(x, t ) = − 64
π 3
∞∑
n=1
1
(2n − 1)3
[
(−1)n + 3
(2n − 1)π
]
cos(2n − 1)πt cos (2n − 1)πx
2
12. 2. 20 (p. 644) u(x, t ) = − 512
3π 4
∞∑
n=1
(−1)n
(2n − 1)4 sin 3(2n − 1)πt
4 cos (2n − 1)πx
4
12. 2. 21 (p. 644) u(x, t ) = − 64
π 4
∞∑
n=1
1
(2n − 1)4
[
(−1)n + 3
(2n ... | Elementary Differential Equations with Boundary Value Problems.pdf |
π 4
∞∑
n=1
1
(2n − 1)4
[
1 + (−1)n4
(2n − 1)π
]
cos 3(2n − 1)πt
2 cos (2n − 1)πx
2
12. 2. 27 (p. 644) u(x, t ) = 96
π 3
∞∑
n=1
1
(2n − 1)3
[
(−1)n5 + 8
(2n − 1)π
]
cos (2n − 1)
√
7 πt
2 cos (2n − 1)πx
2
12. 2. 28 (p. 644) u(x, t ) = − 768
3π 5
∞∑
n=1
1
(2n − 1)5
[
1 + (−1)n4
(2n − 1)π
]
sin 3(2n − 1)πt
2 cos (2n − 1)πx... | Elementary Differential Equations with Boundary Value Problems.pdf |
2
12. 2. 36 (p. 645) u(x, t ) = − 96
π 3
∞∑
n=1
1
(2n − 1)3
[
1 + ( −1)n 4
(2n − 1)π
]
cos 3(2n − 1)πt
2 sin (2n − 1)πx
2
12. 2. 37 (p. 645) u(x, t ) = 8
π
∞∑
n=1
1
(2n − 1)4 sin 4(2n − 1)t sin (2n − 1)x
2
12. 2. 38 (p. 646) u(x, t ) = − 64
π 4
∞∑
n=1
1
(2n − 1)4
[
1 + ( −1)n 4
(2n − 1)π
]
sin 3(2n − 1)πt
2 sin (2n − 1... | Elementary Differential Equations with Boundary Value Problems.pdf |
776 Answers to Selected Exercises
12. 2. 39 (p. 646) u(x, t ) = 96
π 3
∞∑
n=1
1
(2n − 1)3
[
1 + ( −1)n 2
(2n − 1)π
]
cos 3(2n − 1)πt
2 sin (2n − 1)πx
2
12. 2. 40 (p. 646) u(x, t ) = 192
π
∞∑
n=1
(−1)n
(2n − 1)4 cos (2n − 1)
√
3 t
2 sin (2n − 1)x
2
12. 2. 41 (p. 646) u(x, t ) = 64
π 4
∞∑
n=1
1
(2n − 1)4
[
1 + ( −1)n 2
(... | Elementary Differential Equations with Boundary Value Problems.pdf |
2
12. 2. 46 (p. 646) u(x, t ) = 384
π 5
∞∑
n=1
1
(2n − 1)5
[
(−1)n + 4
(2n − 1)π
]
sin(2n − 1)πt sin (2n − 1)πx
2
12. 2. 47 (p. 646) u(x, t ) = 1
2 [SM f(x + at) + SM f(x − at)] + 1
2a
∫ x+at
x−at
SM g(τ ) dτ
12. 2. 50 (p. 647) u(x, t ) = 4 − 768
π 4
∞∑
n=1
1
(2n − 1)4 cos
√
5(2n − 1)πt
2 cos (2n − 1)πx
2
12. 2. 51 (p.... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 777
12. 2. 59 (p. 647) u(x, t ) = 3t
5 − 48
π 5
∞∑
n=1
2 + ( −1)n
n5 sin nπt cos nπx
12. 2. 60 (p. 647) u(x, t ) = 1
2 [Cf (x + at) + Cf (x − at)] + 1
2a
∫x+at
x−at Cg (τ ) dτ
12. 2. 63 (p. 648) (c) u(x, t ) = f (x + at) + f (x − at)
2 + 1
2a
∫ x+at
x−at
g(u) du
12. 2. 64 (p. 649) u(x, t )... | Elementary Differential Equations with Boundary Value Problems.pdf |
n=1
(−1)n+1 sinh(2n − 1)π (1 − y/ 2)
(2n − 1)2 sinh(2n − 1)π sin (2n − 1)πx
2
12. 3. 4 (p. 662) u(x, y ) = π
2
sinh(1 − y)
sinh 1 sin x − 16
π
∞∑
n=1
n sinh 2n(1 − y)
(4n2 − 1)2 sinh 2n sin 2 nx
12. 3. 5 (p. 662) u(x, y ) = 3 y + 108
π 3
∞∑
n=1
(−1)n sinh nπy/ 3
n3 cosh 2nπ/ 3 cos nπx
3
12. 3. 6 (p. 662) u(x, y ) = y
2... | Elementary Differential Equations with Boundary Value Problems.pdf |
∞∑
n=1
[
1 + ( −1)n 2
(2n − 1)π
]
cosh(2n − 1)π (x − 2)/ 4
(2n − 1)3 cosh(2n − 1)π/ 2 sin (2n − 1)πy
4
12. 3. 12 (p. 662) u(x, y ) = 96
π 3
∞∑
n=1
[
3 + ( −1)n 4
(2n − 1)π
]
cosh(2n − 1)π (x − 3)/ 2
(2n − 1)3 cosh 3(2 n − 1)π/ 2 sin (2n − 1)πy
2
12. 3. 13 (p. 663) u(x, y ) = − 16
π
∞∑
n=1
cosh(2n − 1)x/ 2
(2n − 3)(2n +... | Elementary Differential Equations with Boundary Value Problems.pdf |
778 Answers to Selected Exercises
12. 3. 15 (p. 663) u(x, y ) = − 64
π
∞∑
n=1
(−1)n cosh(2n − 1)x/ 2
(2n − 1)4 sinh(2n − 1)/ 2 cos (2n − 1)y
2 .
12. 3. 16 (p. 663) u(x, y ) = − 192
π 4
∞∑
n=1
cosh(2n − 1)πx/ 2
(2n − 1)4 sinh(2n − 1)π/ 2
[
(−1)n + 2
(2n − 1)π
]
cos (2n − 1)πy
2
12. 3. 17 (p. 663) u(x, y ) =
∞∑
n=1
α n
s... | Elementary Differential Equations with Boundary Value Problems.pdf |
2a dx
u(x, y ) = 288
π 3
∞∑
n=1
sinh(2n − 1)π (2 − y)/ 6
(2n − 1)3 sinh(2n − 1)π/ 3 sin (2n − 1)πx
6
12. 3. 20 (p. 663) u(x, y ) =
∞∑
n=1
α n
sinh(2n − 1)π (b − y)/ 2a
sinh(2n − 1)πb/ 2a sin (2n − 1)πx
2a ,
α n = 2
a
∫ a
0
f (x) sin (2n − 1)πx
2a dx
u(x, y ) = 32
π 3
∞∑
n=1
[
(−1)n5 + 18
(2n − 1)π
]
sinh(2n − 1)π (2 − ... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 779
12. 3. 24 (p. 663) u(x, y ) =
∞∑
n=1
α n
cosh nπx/b
cosh nπa/b sin nπy
b , α n = 2
b
∫ b
0
g(y) sin nπy
b dy
u(x, y ) = 96
π 5
∞∑
n=1
cosh(2n − 1)πx
(2n − 1)5 cosh(2n − 1)π sin(2n − 1)πy .
12. 3. 25 (p. 664) u(x, y ) =
∞∑
n=1
α n
cosh(2n − 1)πx/ 2b
cosh(2n − 1)πa/ 2b cos (2n − 1)πy
2b ... | Elementary Differential Equations with Boundary Value Problems.pdf |
n=1
[
1 + ( −1)n 4
(2n − 1)π
]
cosh(2n − 1)(x − 1)/ 2
(2n − 1)4 sinh(2n − 1)/ 2 sin (2n − 1)y
2 .
12. 3. 28 (p. 664) u(x, y ) = α 0(x − a) + b
π
∞∑
n=1
α n
sinh nπ (x − a)/b
n cosh nπa/b cos nπy
b , α 0 = 1
b
∫ b
0
g(y) cos nπy
b dy,
α n = 2
b
∫ b
0
g(y) cos nπy
b dy
u(x, y ) = π (x − 2)
2 − 4
π
∞∑
n=1
sinh(2n − 1)(x −... | Elementary Differential Equations with Boundary Value Problems.pdf |
780 Answers to Selected Exercises
12. 3. 32 (p. 664) u(x, y ) = − a
π
∞∑
n=1
α n
n e−nπy/a sin nπx
a , α n = 2
a
∫ a
0
f (x) sin nπx
a dx
u(x) = 4
∞∑
n=1
(1 + ( −1)n2)
n4 e−ny sin nx
12. 3. 33 (p. 664) u(x, y ) = − 2a
π
∞∑
n=1
α n
2n − 1 e−(2n−1)πy/ 2a cos (2n − 1)πx
2a , α n = 2
a
∫ a
0
f (x) cos (2n − 1)πx
2a dx
u(x,... | Elementary Differential Equations with Boundary Value Problems.pdf |
π
∞∑
n=1
Bn cosh nπy/a − An cosh nπ (y − b)/a
n sinh nπb/a cos nπx
a
+ b
π
∞∑
n=1
Dn cosh nπx/b − Cn cosh nπ (x − a)/b
n sinh nπa/b cos nπy
b
Section 12.4 Answers, pp. 672– 673
12. 4. 1 (p. 672) u(r, θ ) = α 0
ln r/ρ
ln ρ 0/ρ +
∞∑
n=1
rn ρ −n − ρ nr−n
ρ n
0 ρ −n − ρ nρ −n
0
(α n cos nθ + β n sin nθ ) α 0 = 1
2π
∫ π
−π
... | Elementary Differential Equations with Boundary Value Problems.pdf |
12. 4. 4 (p. 672) u(r, θ ) =
∞∑
n=1
α n
r(2n−1)π/ 2γ
ρ (2n−1)π/ 2γ cos (2n − 1)πθ
2γ
α n = 2
γ
∫ γ
0
f (θ ) cos (2n − 1)πθ
2γ dθ , n = 1 , 2, 3,. . .
12. 4. 5 (p. 673) u(r, θ ) = 2γρ 0
π
∞∑
n=1
α n
2n − 1
ρ −(2n−1)π/ 2γ r(2n−1)π/ 2γ + ρ (2n−1)π/ 2γ r−(2n−1)π/ 2γ
ρ −(2n−1)π/ 2γ ρ (2n−1)π/ 2γ
0 − ρ (2n−1)π/ 2γ ρ −(2n−1)π... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 781
α n = 2
γ
∫ γ
0
g(θ ) sin (2n − 1)πθ
2γ dθ , n = 1 , 2, 3,. . .
12. 4. 6 (p. 673) u(r, θ ) = α 0 +
∞∑
n=1
α n
rnπ/γ
ρ nπ/γ cos nπθ
γ α 0 = 1
γ
∫ γ
0
f (θ ) dθ ,
α n = 2
γ
∫ γ
0
f (θ ) cos nπθ
γ dθ , n = 1 , 2, 3,. . .
12. 4. 7 (p. 673) vn(r, θ ) = rn
nρ n−1 (α n cos nθ + sin nθ )
u(r, ... | Elementary Differential Equations with Boundary Value Problems.pdf |
∫ b
a
tF (t) dt = 0 y = −x
∫ 1
x
F (t) dt −
∫ x
0
tF (t) dt + c1x with c1 arbitrary
13. 1. 9 (p. 685) (a) b − a ̸= kπ (k = integer)
y = sin(x − a)
sin(b − a)
∫ b
x
F (t) sin(t − b) dt + sin(x − b)
sin(b − a)
∫ x
a
F (t) sin(t − a) dt
(b)
∫ b
a
F (t) sin(t − a) dt = 0
y = −sin(x − a)
∫ b
x
F (t) cos( t − a) dt − cos(x −... | Elementary Differential Equations with Boundary Value Problems.pdf |
782 Answers to Selected Exercises
13. 1. 12 (p. 685) y = sinh(x − a)
sinh(b − a)
∫ b
x
F (t) sinh( t − b) dt + sinh(x − b)
sinh(b − a)
∫ x
a
F (t) sinh( t − a) dt
13. 1. 13 (p. 685) y = − sinh(x − a)
cosh(b − a)
∫ b
x
F (t) cosh(t − b) dt − cosh(x − b)
cosh(b − a)
∫ x
a
F (t) sinh(t − a) dt
13. 1. 14 (p. 685) y = − cos... | Elementary Differential Equations with Boundary Value Problems.pdf |
with c1 arbitrary.
13. 1. 17 (p. 685) If ω ̸= n + 1 / 2 (n = integer), then
y = − sin ωx
ω cos ωπ
∫ π
x
F (t) cos ω (t − π ) dt − cos ω (x − π )
ω cos ωπ
∫ x
0
F (t) sin ωt dt.
If ω = n + 1 / 2 (n = integer), then
∫ π
0
F (t) sin(n + 1 / 2)t dt = 0 is necessary
for existence of a solution. In this case,
y = − sin(n + 1... | Elementary Differential Equations with Boundary Value Problems.pdf |
with c1 arbitrary.
13. 1. 19 (p. 685) If ω isn’t a positive integer, then
y = 1
ω sin ωπ
(
cos ωx
∫ π
x
F (t) cos ω (t − π ) dt + cos ω (x − π )
∫ x
0
F (t) cos ωt dt
)
. | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 783
If ω = n (positive integer), then
∫ π
0
F (t) cos nt dt = 0 is necessary for existence
of a solution. In this case,
y = − 1
n
(
cos nx
∫ π
x
F (t) sin nt dt + sin nx
∫ x
0
F (t) cos nt dt
)
+ c1 cos nx
with c1 arbitrary.
13. 1. 20 (p. 685) y1 = B1(z2)z1 − B1 (z1)z2
13. 1. 21 (p. 685) (... | Elementary Differential Equations with Boundary Value Problems.pdf |
13. 1. 22 (p. 686) G(x, t ) =
− (2 + t)(3 − x)
5 , 0 ≤ t ≤ x,
− (2 + x)(3 − t)
5 , x ≤ t ≤ 1
(a) y = x2 − x − 2
2 (b) y = 5x2 − 7x − 14
30
(c) y = 5x4 − 9x − 18
60
13. 1. 23 (p. 686) G(x, t ) =
cos t sin x
t3/ 2√x , π
2 ≤ t ≤ x,
cos x sin t
t3/ 2√x , x ≤ t ≤ π
(a) y = 1 + cos x − sin x√x (b) y = x +... | Elementary Differential Equations with Boundary Value Problems.pdf |
784 Answers to Selected Exercises
13. 1. 26 (p. 686) α (ρ + δ ) − βρ ̸= 0 G(x, t ) =
(β − αt )(ρ + δ − ρx )
α (ρ + δ ) − βρ , 0 ≤ t ≤ x,
(β − αx )(ρ + δ − ρt )
α (ρ + δ ) − βρ , x ≤ t ≤ 1
13. 1. 27 (p. 686) αδ − βρ ̸= 0 G(x, t ) =
(β cos t − α sin t)(δ cos x − ρ sin x)
αδ − βρ , 0 ≤ t ≤ x,
(β cos x ... | Elementary Differential Equations with Boundary Value Problems.pdf |
ex−t (β cos t − (α + β ) sin t)((ρ + δ ) cos x + δ sin x)
βδ + ( α + β )(ρ + δ , 0 ≤ t ≤ x,
ex−t (β cos x − (α + β ) sin x)((ρ + δ ) cos t + δ sin t)
βδ + ( α + β )(ρ + δ , x ≤ t ≤ π/ 2
13. 1. 31 (p. 686) (ρ + δ )(α − β )e(b−a) − (ρ − δ )(α + β )e(a−b) ̸= 0
G(x, t ) =
((α − β )e(t−a) − (α + β )e−(t−a))((ρ... | Elementary Differential Equations with Boundary Value Problems.pdf |
αe −xy = 0
13. 2. 7 (p. 696) ((1 − x2)y′)′+ α (α + 1) y = 0
13. 2. 9 (p. 696) λ n = n2 π 2, yn = e−x sin nπx (n = positive integer)
13. 2. 10 (p. 697) λ 0 = −1, y0 = 1 λ n = n2π 2 , yn = e−x(nπ cos nπx + sin nπx ) (n = positive
integer)
13. 2. 11 (p. 697) (a) λ = 0 is an eigenvalue y0 = 2 − x (b) none (c) 5. 0476821, 1... | Elementary Differential Equations with Boundary Value Problems.pdf |
Answers to Selected Exercises 785
13. 2. 15 (p. 697) (a) λ = 0 isn’t an eigenvalue (b) −1. 0664054 y = cosh
√
−λ x (c) 1. 5113188,
8. 8785880, 21. 2104662, 38. 4805610 y = cos
√
λ x
13. 2. 16 (p. 697) (a) λ = 0 isn’t an eigenvalue (b) −1. 0239346
y =
√
−λ cosh
√
−λ x − sinh
√
−λ x (c) 2. 0565705, 9. 3927144, 21. 716913... | Elementary Differential Equations with Boundary Value Problems.pdf |
y = 2
√
−λ cosh
√
−λ x − sinh
√
−λ x (c) 8. 8694608, 16. 5459202, 26. 4155505,
38. 4784094 y = 2
√
λ cos
√
λ x − sin
√
λ x
13. 2. 20 (p. 697) (a) λ = 0 isn’t an eigenvalue (b) −7. 9394171, −3. 1542806
y = 2
√
−λ cosh
√
−λ x − 5 sinh
√
−λ x (c) 29. 3617465, 78. 777456, 147. 8866417,
236. 7229622 y = 2
√
λ cos
√
λ x − 5 ... | Elementary Differential Equations with Boundary Value Problems.pdf |
13. 2. 27 (p. 698) (a) y = x − α (b) y = αk cosh kx − sin kx (c) y = αk cos kx − sin kx
13. 2. 29 (p. 698) (b) λ = −α 2/β 2 y = e−αx/β | Elementary Differential Equations with Boundary Value Problems.pdf |
Index
A
Abel’ s formula,
199– 202, 468
Accelerated payment, 139
Acceleration due to gravity, 151
Airy’ s equation, 319
Amplitude,
of oscillation, 271
time-varying, 279
Amplitude–phase form, 272
Aphelion distance, 300
Apogee, 300
Applications,
of first order equations, 130– 192
autonomous second order equations, 162– 179... | Elementary Differential Equations with Boundary Value Problems.pdf |
stability and instability conditions for, 170– 178
B
Beat,
275
Bernoulli’ s equation, 63– 64
Bessel functions of order ν , 360
Bessel’ s equation, 205 287, 348
of order ν , 360
of order zero, 377
ordinary point of, 319
singular point of, 319, 342
Bifurcation value, 54, 176
Birth rate, 2
Boundary conditions, 580
in heat... | Elementary Differential Equations with Boundary Value Problems.pdf |
with disinct real roots, 211– 217
with repeated real root, 212, 217
Characteristic polynomial, 210, 340, 475
Charge, 290
steady state, 293
Chebyshev polynomials, 322
Chebshev’ s equation, 322
787 | Elementary Differential Equations with Boundary Value Problems.pdf |
788 Index
Circuit, RLC. See RLC circuit
Closed Circuit, 289
Coefficient(s) See also Constant coefficient equations
computing recursively, 322
Fourier, 587
in Frobenius solutions, 352– 358
undetermined, method of, 229– 248, 475– 496
principle of superposition and, 235
Coefficient matrix, 516, 516
Competition, species, 6, 5... | Elementary Differential Equations with Boundary Value Problems.pdf |
439
Constant coefficient homogeneous linear systems of
differential equations, 529– 568
geometric properties of solutions,
when n = 2 , 536– 539, 551– 554, 562– 565
with complex eigenvalue of constant matrix, 556–
565
with defective constant matrix, 558– 556
with linearly independent eigenvetors, 529– 541
Constant solut... | Elementary Differential Equations with Boundary Value Problems.pdf |
Critical point, 163
Current, 289
steady state, 293
transient, 293
Curves, 179– 192
equipotential, 185
geometric problems, 183
isothermal, 185
one-parameter famlies of, 179– 183 subsubitem
defined, 179
differential equation for, 180
orthogonal trajectories, 190– 190, 192
finding, 186– 190
D
D’Alembert’ s solution,
637
Dam... | Elementary Differential Equations with Boundary Value Problems.pdf |
Decay constant, 130
Derivatives, Laplace transform of, 413– 415
Differential equations,
defined, 8
order of, 8
ordinary, 8
partial, 8
solutions of, 9– 11
Differentiation of power series, 308
Dirac, Paul A. M., 452 | Elementary Differential Equations with Boundary Value Problems.pdf |
Index 789
Dirac delta function, 452
Direction fields for first order equations, 16– 27
Dirichlet, Peter G. L., 662
Dirichlet condition, 662
Dirichlet problem, 662
Discontinuity,
jump, 398
removable, 408
Distributions, theory of, 453
Divergence of improper integral, 393
Divergent power series, 306
E
Eccentricity of orbit,... | Elementary Differential Equations with Boundary Value Problems.pdf |
116
Escape velocity, 159
Euler’ s equation, 343– 346, 229
Euler’ s identity, 96– 108
Euler’ s method, 96– 108
error in, 97– 100
truncation, 100– 102
improved, 109– 114
semilinear, 102– 106
step size and accuracy of, 97
Even functions, 592
Exact first order equations, 73– 82
implicit solutions of, 73– 74
procedurs for so... | Elementary Differential Equations with Boundary Value Problems.pdf |
procedurs for solving, 76
linear, 31– 44
homogeneous, 106– 35
nonhomogeneous, 35– 41
solutions of, 30
nonlinear, 41, 52, 56– 72
existence and uniqueness of solutions of, 56–
62
transformation into separables, 62– 72
numerical methods for solving. See Numerical
method
separable, 45– 55, 68– 72
constant solutions of, 48–... | Elementary Differential Equations with Boundary Value Problems.pdf |
790 Index
piecewise continuous constant equations with,
430– 439
Fourier coefficients, 587
Fourier series, 586– 616
convergence of, 589
cosine series, 603– 604
convergence of, 607
mixed, 606
defined 588
even and odd functions, 592– 589
sine series, 605
convergence of, 605
mixed, 609
Fourier solutions of partial different... | Elementary Differential Equations with Boundary Value Problems.pdf |
vibrations, 279– 283
Frequency, 279
of simple harmonic motion, 292
Frobenius solutions, 347– 390
indicial equation with distinct real roots differ-
ing by an integer, 378– 390
indicial equation with distinct real roots not dif-
fering by an integer, 351– 364
indicial equation with repeated root, 364– 378
power series i... | Elementary Differential Equations with Boundary Value Problems.pdf |
equations, 475– 480
of homogeneous linear second order equations,
198
of homogeneous linear systems of differential
equations, 521, 524
of linear higher order equations, 465, 469
of nonhomogeneous linear first order equations,
30, 40
of nonhomogeneous linear second order equa-
tions, 221, 248– 255
Geometric problems, 23... | Elementary Differential Equations with Boundary Value Problems.pdf |
Harmonic function, 82
Harmonic motion, simple, 166, 271 292 , 292
amplitude of oscillation, 272
natural frequancy of, 272
phase angle of, 272
Heat equation, 618– 629
boundary conditions in, 618
defined, 618
formal and actual solutions of, 620
initial-boundary value problems, 618, 625
initial condition, 618
nonhomogeneou... | Elementary Differential Equations with Boundary Value Problems.pdf |
Index 791
separation of variables to solve, 618
Heat flow lines, 185
Heaviside’ s method, 407, 412
Hermite’ s equation, 322
Heun’ s method, 116
Higher order constant coefficient homogeneous equa-
tions, 475– 487
characteristic polynomial of, 482– 478
fundamental sets of solutions of, 480
general solution of, 476– 482
Hom... | Elementary Differential Equations with Boundary Value Problems.pdf |
with complex eigenvalues of coefficient ma-
trix, 556– 568
with defective coefficient matrix, 542– 551
geometric properties of solutions when n =
2, 529– 539, 551– 554, 562– 565
with linearly independent eigenvectors, 529–
539 subitem fundamental set of solutions
of, 521, 524
general solution of, 521, 524
trivial and non... | Elementary Differential Equations with Boundary Value Problems.pdf |
Impulses, constant coefficient equations with, 452–
461
Independence, linear
of n function, 466
of two functions, 198
of vector functions, 525
Indicial equation, 343, 351
with distinct real roots differing by an integer,
378– 390
with distinct real roots not differing by an inte-
ger, 351– 364
with repeated root, 364– 3... | Elementary Differential Equations with Boundary Value Problems.pdf |
Interval of validity, 12
Inverse Laplace transforms, 404– 413
defined, 404
linearity property of, 405
of rational functions, 406– 413
Inverse square law force, motion under, 299– 301
Irregular singular point, 342
Isothermal curves, 185
J
Jump discontinuity,
398
K
Kepler’ s second law,
296
Kepler’ s third law, 301
Kircho... | Elementary Differential Equations with Boundary Value Problems.pdf |
792 Index
boundary conditions, 649– 651
defined, 649
formal solutions of, 651– 662
in polar coordinates, 666– 673
for semi-infinte strip, 660
Laplace transforms, 393– 461
computation of simple, 393– 396
of constant coefficient equations
with impulses 452– 461
with piecewise continuous forcing functions,
430– 439
convoluti... | Elementary Differential Equations with Boundary Value Problems.pdf |
ordinary points of, 319
singular points of, 319, 347
Limit, 398
Limit cycle, 176
Linear combination(s), 198, 465, 521
of power series, 313– 316
Linear difference equations, second order homoge-
neous, 340
Linear first order equations, 30– 44
homogeneous, 30– 35
general solution of, 33
separation of variables, 35
nonhomo... | Elementary Differential Equations with Boundary Value Problems.pdf |
undetermined coefficients for, 487– 496
variation of parameters for, 497– 505
derivation of method, 497– 499
fourth order equations, 501– 497
third order equations, 499
Wronskian of solutions of 467– 469
Linear independence 198
of n functions, 466
of two functions, 198
of vector functions, 521– 523
Linearity,
of inverse... | Elementary Differential Equations with Boundary Value Problems.pdf |
superposition principle and, 225– 227
undetermined coefficients method for, 229–
248
variation of parameters to find particular so-
lution of, 255– 264
series solutions of, 306– 390
Euler’ s equation, 343– 347
Frobenius solutions, 347– 390
near an ordinary point, 319– 339
with regular singular points, 342– 347
Linear sys... | Elementary Differential Equations with Boundary Value Problems.pdf |
Index 793
linear indeopendence of, 521, 524
trivial and nontrivial solution of, 521
Wronskian of solution set of, 523
nonhomogeneous, 516
variation of parameters for, 568– 576
solutions to initial value problem, 515– 517
Lines of force, 185
Liouville, Joseph, 689
local truncation error, 100– 102
numerical methods with ... | Elementary Differential Equations with Boundary Value Problems.pdf |
damped, 174– 176
undamped, 173– 169
spring-mass system
damped, 173– 174, 269, 279– 289
undamped, 164– 173,268
units used in, 151
Midpoint method, 109
Mixed boundary value problems, 649
Mixed Fourier cosine series, 606– 608
Mixed Fourier sine series, 609
Mixed growth and decay, 134
Mixing problems, 143– 148
Models, math... | Elementary Differential Equations with Boundary Value Problems.pdf |
undamped, 268
Multiplicity, 479
N
Natural frequency,
272
Natural length of spring, 268
Negative half plane, 552
Neumann condition, 649
Neumann problem, 649
Newton’ s law of cooling, 3, 140– 141, 148– 150
Newton’ s law of gravitation, 151, 176, 295, 510, 519
Newton’ s second law of motion, 151– 151, 163, 166,
173, 176, ... | Elementary Differential Equations with Boundary Value Problems.pdf |
248– 255
superposition principle and, 225– 223
undetermined coefficients method for, 229– 255
forcing functions with exponential factors, 242–
244
forcing functions without exponential factors,
238– 241
superposition principle and, 235
variation of parameters to find particular so-
lution of, 255– 264
Nonhomogeneous line... | Elementary Differential Equations with Boundary Value Problems.pdf |
794 Index
existence and uniqueness of solutions of, 56– 72
transformation into separable equations, 62– 72
Nonoscillatory solution, 358
Nontrivial solutions
of homogeneous linear first order equations, 30
of homogeneous linear higher order equations,
465
of homogeneous linear second order equations,
194
of homogeneous l... | Elementary Differential Equations with Boundary Value Problems.pdf |
Numerical quadrature, 119, 127
O
Odd functions,
592
One-parameter families of curves, 179– 183
defined, 179
differential equation for, 180
One-parameter families of functions, 30
Open interval of convergence, 306
Open rectangle, 56
Orbit, 301
eccentricity of, 300
elliptic, 300
period of, 301
Order of differential equati... | Elementary Differential Equations with Boundary Value Problems.pdf |
P
Partial differential equations
defined,
8
Fourier solutions of, 618– 673
heat equation, 618– 630
Laplace’ s equation, 649– 673
wave equation 630– 649
Partial fraction expansions, software packages to find,
411
Particular solutions of nonhomogeneous higher equa-
tions, 469, 487– 505
Particular solutions of nonhomogeneou... | Elementary Differential Equations with Boundary Value Problems.pdf |
Plucked string, wave equation applied to, 638– 642
Poinccaré, Henri, 162
Polar coordinates
central force in terms of, 296– 298
in amplitude-phase form, 271
Laplace’ s equation in, 666– 673
Polynomial(s)
characteristic, 210, 340, 482 | Elementary Differential Equations with Boundary Value Problems.pdf |
Index 795
of higher order constant coefficient homoge-
neous equations, 475– 478
Chebyshev, 322
indicial, 343, 351
T aylor, 309
trigonometric, 602
Polynomial operator, 475
Population growth and decay, 2
Positive half-plane, 552
Potential equation, 649
Power series, 306– 319
convergent, 306– 307
defined, 306
differentiati... | Elementary Differential Equations with Boundary Value Problems.pdf |
Recurrence relations, 322
in Frobenius solutions, 351
two term, 353– 355
Reduction of order, 213, 248– 255
Regular singular points, 342– 347
at x0 = 0 , 347– 364
Removable discontinuity, 398
Resistance, 290
Resistor, 290
Resonance, 277
Ricatti, Jacopo Francesco, 72
Ricatti equation, 72
RLC circuit, 289– 294
closed, 289... | Elementary Differential Equations with Boundary Value Problems.pdf |
linear, See linear second equations
two-point boundary value problems for, 676–
686
assumptions, 676
boundary conditions for, 676
defined, 676
Green’ s function for, 682
homogeneous and nonhomogeneous, 676
procedure for solving, 677
Second order homogeneous linear difference equa-
tion, 340
Second shifting Theorem, 425–... | Elementary Differential Equations with Boundary Value Problems.pdf |
to solve wave equation, 632
Separatrix, 171, 170
Series, power . See Power series
Series solution of linear second order equations, 306-
390
Frobenius solutions, 347– 390
near an ordinary point, 319
Shadow trajectory, 564– 565
Shifting theorem
first, 397 | Elementary Differential Equations with Boundary Value Problems.pdf |
796 Index
second, 425– 427
Simple harmonic motion, 269– 273
amplitude of oscillation, 271
natural frequency of, 272
phase angle of, 272
Simpson’ s rule, 127
Singular point, 319
irregular, 342
regular, 342– 347
Solution(s), 9– 10 See also Frobenius solutions Non-
trivial solutions Series solutions of linear
second order... | Elementary Differential Equations with Boundary Value Problems.pdf |
forced oscillation, 273– 287
Stability of equilibrium and critical point, 163– 164
Steady state, 135
Steady state charge, 293
Steady state component, 285, 447
Steady state current, 293
String motion, wave equation applied to, 630– 638
plucked, 638– 642
vibrating, 630– 638
Sturm-Liouville equation, 689
Sturm-Liouville e... | Elementary Differential Equations with Boundary Value Problems.pdf |
T aylor Series, 308
T emperature, Newton’ s law of cooling, 3 140 – 141,
148– 149
T emperature decay constant of the medium, 140
T erminal velocity, 152
Time-varying amplitude, 279
T otal impulse, 452
Trajectory(ies),
of autonomous second order equations, 162
orthogonal, 186– 190
finding, 186– 244
shadow, 564
of 2 × 2 s... | Elementary Differential Equations with Boundary Value Problems.pdf |
equations, 521
of linear higher order differential equations, 465
Truncation error(s), 96
in Euler’ s method, 100
global, 102, 109
local, 100
numerical methods with O(h3 ), 114– 116
T wo-point boundary value problems, 676– 686
assumptions, 676
boundary conditions for, 676
defined, 676
Green’ s function for, 681
homogene... | Elementary Differential Equations with Boundary Value Problems.pdf |
Index 797
Undamped autonomous second order equations, 164–
171
pendulum, 173– 169
spring-mass system, 164– 166
stability and instabilty conditions for, 170– 171
Undamped motion, 268
Underdamped motion, 279
Underdamped oscillation, 291
Undetermined coefficients
for linear higher order equations, 487– 496
forcing function... | Elementary Differential Equations with Boundary Value Problems.pdf |
third order equations, 499
for linear higher second order equations, 255
for nonhomogeneous linear systems of differen-
tial equations, 568– 577
V elocity
escape, 158– 151
terminal, 152– 156
V erhulst, Pierre, 3
V erhulst model, 3, 27, 69
Vibrating strings, wave equation applied to, 630
Vibrations
forced, 283– 287
free... | Elementary Differential Equations with Boundary Value Problems.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.