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Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
Download (direct link): elementarydifferentialequat2001.pdf
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? = ?
n=0
x2 n
2nn!x2 n+1 (2n + 1)!
3. (n + 2)an+2 - an+1 - an = 0
71 (x) = 1 + 2 (x - 1)2 + i (x - 1)3 + 1 (x - 1)4 + ••• y2(x) = (x - 1) + 2 (x - 1)2 + 2(x - 1)3 + 1 (x - 1)4 +
4. an+4 = -k2an/(n + 4)(n + 3);
k2x4 k4x8
71 (x) = 1 - — +
a2 = a3 = 0 k6 x12
3 • 4 3 • 4 • 7 • 8
to ( i \m-
= 1 + V-----(—^-
3478
3 4 7 8 11 12
\
(-1)m+1(k2 x4)m+1
m=0
3 • 4 • 7 • 8 • • • (4m + 3)(4m + 4)
k2 x5
y2(x) = x - +
k4 x9
k6 x13
4.5 4-5-8-9 4-5-8-9-12-13
\
1\
(-1)m+1(k2 x4)m+1
m=0
4 • 5 • 8 • 9 •••(4m + 4)(4m + 5)
Hint: Let n = 4m in the recurrence relation, m = 1, 2, 3,... .
5. (n + 2)(n + 1)an+2 - n(n + 1)an+1 + an = 0, n > 1;
n+1
a2 — 0 a,
y1(x) = 1 - 2x2 - 1 x - 24x4 + ••• > y2(x) = x - 6x3 - 12x4 - 24x5 + '
2 = 2 °0
?1 x4 12
6. an+2 = -(n2 - 2n + 4)an/[2(n + 1)(n + 2)], n > 2;
2
0
y1(x) = 1 - x2 + 1 x4 - 30 x6 + ••• ,
7. an+2 = —an/(n + 1), n = 0, 1, 2,...
72(x) = x - 4x3 + 160x5 - 1T2??x7 + ?
y, (x) — 1 — — +------------—---------
^ 1 1 • 3 1 • 3 • 5
x3 x5 x7
y2 (x) — x — — +-----------—---------
J2 2 2 • 4 2-4-6
\
V (-1)nx2 n
t-t 1 • 3 • 5 ••• (2n - 1)
n= 1
(-1)nx2 n+1
+------= x +> —---------------
1 2 • 4 • 6 • • • (2n)
8. an+2 = -[(n + 1)2an+1 + an + an-1]/(n + 1)(n + 2), n = 1, 2,...
a2 = ~(a0 + a1)/2
y1(x) = 1 - 2 (x - 1)2 + 1 (x - 1)3 - 112 (x - 1)4 + • • •
72(x) = (x - 1) - 2 (x - 1)2 + 6(x - 1)3 - 6(x - 1)4 + • • •
9. (n + 2)(n + 1)an+2 + (n - 2)(n - 3)an = 0; n = 0, 1, 2,...
y1 (x) = 1 — 3x2, y2(x) = x — x3/3
10. 4(n + 2)an+2 - (n - 2)an = 0; n = 0, 1, 2,...
x2
x3 x5
y1(x 1 4 , y2(x) x 12 240 2240
x2 n+1
4n (2n - 1)(2n + 1)
7
a3 = 4 a1
Answers to Problems
697
See SSM for detailed solutions to 14, 16a, 19
22b, 23, 26
1, 6, 9afh
11. 3(n + 2)an+2 - (n + 1)an = 0; n = 0, 1, 2,...
ylU ) = 1 + ^ + i! + -1- x« + ? ? ? + 3n 5' • ? (2n - 1) x2 n + ? . .
^ 6 24 432 3n- 2-4- ? ? (2n)
2 3 8 5 16 7 2-4- ? ? (2n) 2n, 1
y, (x) = x + - x + x5 + x7 +-1- —----x n+1 + •
9 135 945 3n- 3-5- ? ? (2n + 1)
12. (n + 2)(n + 1)an+2 - (n + 1)nan-1 + (n - 1)an = 0; n = 0, 1, 2,...
x2 x3 x4 xn
y1(x) = 1 + y + y + 24 + ? ? ? + n! + ? ? ? , y2 = x
13. 2(n + 2)(n + 1)an+2 + (n + 3)an = 0; n = 0, 1, 2,...
y{x) = 1 - - x2 + — x4 - — x6 + ? ? ? + (-1)n 3'5' ' (2n + 1) x2n + ? ?
J1 4 32 384 2n (2n)!
y,(x) = x - y + ?. - jL + ... + (-1)n 4-6n (2n+2) ^ n+1 + , ..
3 20 210 2n (2n + 1)!
14. 2(n + 2)(n + 1)an+2 + 3(n + 1)an+1 + (n + 3)an = 0; n = 0, 1, 2,...
y1(x) = 1 - I(x - 2)2 + §(x - 2)3 + 64(x - 2)4 + ? ? ?
y2(x) = (x - 2) - 3 (x - 2)2 + 214 (x - 2)3 + «4(x - 2)4 + ? ? ?
15. (a) y = 2 + x + x2 + 1 x3 + 1 x4 + ? ? ? (c) about |x| < 0.7
16. (a) y =-1 + 3x + x2 - 3x3 - «x4 + ? ? ? (c) about |x| < 0.7
17. (a) y = 4 - x - 4x2 + 2x3 + f x4 + ? ? ? (c) about |x| < 0.5
18. (a) y = —3 + 2x - 3x2 - 2x3 - 8x4 + ? ? ? (c) about |x| < 0.9
19. y1(x) = 1 - 5(x - 1)3 - t2(x - 1)4 + T8(x - 1)6 + ' ' '
y2(x) = (x - 1) - 1 (x - 1)4 - 20 (x - 1)5 + 28(x - 1)7 + ? ? ?
71 zx , X 1 X 2 , X(X - 4) 4 X(X - 4)(X - 8) 6 ,
21. (a) y (x) = 1 x2 4---------------------------------------------x4-------------------------------------------x6 + • • •
1 2! 4! 6!
X - 2 3 (X - 2)(X - 6) 5 (X - 2)(k - 6)(k - 10) 7 y2(x) = x - -fp +--------------------------5-x5----------------7--------------x + ?
(b) 1, x, 1 - 2x2, x - 3x3, 1 - 4x2 + 3x4, x - 4x3 + 15x5
(c) 1, 2x, 4x2 - 2, 8x3 - 12x, 16x4 - 48x2 + 12, 32x5 - 160x3 + 120x
22. (b) y = x - x3/6 +? ? ?
Section 5.3, page 253
1. 4>"(0) = -1,
2. 0"(0) = 0,
3. 0"(1) = 0,
4. 0"(0) = 0,
0 "'(0) = 0,
0 '"(0) = -2, 0 '"(1) = -6, 0 "'(0) = -a0,
0lv (0) = 3 0lv (0) = 0 0lv (1) = 42 0lv (0) = -4a1
5. p = to, p = to
6. p = 1, p = 3, p = 1
7. p = 1, p = \/3
8. p = 1
9. (a) p = to (b) p = to (c) p = to
(f) p = V2 (g) p = to (h) p = 1
(k) p = \/3 (l) p = 1 (m) p = to (n) p = to
(d) p = to (e) p = 1
(i) p = 1 (j) p = 2
698
Answers to Problems
See SSM for detailed solutions to 10a
10b, 11
18
20, 22
26, 28
a2 2 (22 - a2)a2 4 (42 - a2)(22 - a2)a
,0. (a)/lW = 1 - -V - * „ ' x4 - !----------?6---------L-
)
-x2 m------
[(2m - 2)2 - a2] ••• (22 - a2)a2
(2m)!
1 — a2 3 (32 — a2)(1 — a2) 5
Yi(x) = x + -3- x3 + ^---------------5!------2 x5 + •••
, [(2m — 1)2 — a2] — 0 — a2) x2m+1 ,
(2m + 1)!
(b) y1 (x) or y2(x) terminates with xn as a — n is even or odd.
(c) n — 0, y — 1; n — 1, y — x; n — 2, y — 1 — 2x2; n — 3, y — x — 3x3
n. y1(x) = 1 — 6x3 + Jj0x5 + 4x6 + ••• . y2(x) = x — 10x4 + iiio x6 + 504 x? + ?
p — ^
12. y1(x) = 1 — 6x3 + 1"2x4 — 45x5 + ••• > y2(x) = x — 112x4 + 20x5 — 60x6 + ••• -
13. y1(x) — 1 + x2 + 112x4 + ^x6 + •••, y2(x) — x + 6x3 + 65x5 + 56;0x7 + •••,
p = n/2
14. y1(x) = 1 + 1 v3 + 12x4 — ^x6 + •••- y2(x) = x — 1 x3 + 24x4 + 120 x5 + •••-
p=1
15. Cannot specify arbitrary initial conditions at x — 0; hence x — 0 is a singular point.
x2 xn
16. y = 1 + x + — + ••• + — + ••• — ex
2! n!
x2 x4 x6 x2n
17. y = 1 + T + + 2^ + ••• + im. +•••
18. y = 1 + x + 2x2 + 2x3 +----
19. y = 1 + x + x2 + •••+ xn + •••—----------
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