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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
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(c) A = 2.16, T = 6.65
(d) u = 0.2, A = 1.99, T = 6.31; u = 0.5, A = 2.03, T = 6.39;
U = 2, A = 2.60, T = 7.65; u = 5, A = 4.36, T = 11.60
16. (a) k = 0, (1.1994, —0.62426); k = 0.5, (0.80485, —0.13106)
(b) k0 = 0.3465 (c) T = 12.54 (d) k1 = 0.3369
Section 9.8, page 538
1ab
1. (b) k - y, ?(1) - (0, 0, 1)T; k - k2, g(2) - (20, 9 - V81 + 40r, 0)T; k - k3, ?(3) - (20, 9 + ysT+i?r, 0)T
724
See SSM for detailed solutions to 1c, 2abc, 3b
3c, 4, 5ab
CHAPTER 10
2, 3, 7, 11, 15
3, 5, 7, 10
13ab, 15ab
15a, 21 abed, 25
(c) A.J = -2.6667, ?(1) = (0, 0, 1)T; X2 = -22.8277, g(2) = (20, -25.6554, 0)T; X3 = 11.8277, ?(3) = (20, 43.6554, 0)T 2. (c) X1 = -13.8546; X2,X3 = 0.0939556 ± 10.19457 5. (a) dV/dt = -2a[rx2 + y2 + b(z - r)2 - br2]
Section 10.1, page 547
2. y = (cot \fln cosV2x + sinv/2x)/v/2
1. y =— sin X
3. y = 0 for all L; y = c2 sin x if sin L = 0.
4. y =— tan L cos x + sin x if cos L = 0; no solution if cos L = 0.
5. No solution.
6. y = (—n sinV^x + xsin\/ln)/2sin\/ln
7. No solution. 8. y = c2sin2x + 1 sin x
9. y = c1cos2x + I cos x 10. y = 2 cos x
11. Xn = [(2n - 1 )/2]2, yn (x) = sin[(2n - 1 )x/2]; n = 1, 2, 3,...
12. Xn = [(2n - 1)/2]2, yn(x) = cos[(2n - 1)x/2]; n = 1, 2, 3,...
13.
^0 = 0,
y0(x) = 1; Xn = n , yn(x) = cosnx; n = 1, 2, 3,...
14. Xn = [(2n - 1)n/2 L]2, yn (x) = cos[(2n - 1)n x/2L]; n = 1, 2, 3,...
15. X0 = 0, y0(x) = 1; 7n = (nn/L)2, yn(x) = cos(nnx/L); n = 1, 2, 3,
16. Xn = - [(2n- 1)n/2L]2, yn (x) = sin[(2n - 1)n x/2L ]; n = 1, 2, 3,...
Section 10.2, page 555
1. T = 2n/5 3. Not periodic T = 1 T=2
f (x) = 2L - x in L < x < 2L;
5.
7.
9.
10.
11.
2. T = 1 4. T = 2L 6. Not periodic 8. T = 4 f (x) = —2L - x in -3L < x < -2L
f (x) = x - 1 in 1 < x < 2; f (x) = x - 8 in 8 < x < 9
f (x) = - L - x in - L < x < 0
2L
(-1)n
13. (b) f(x) = —T)
TT • J
1 2 ^ 14. (b) f(x) = 2 -
L
n x—v
15. (b) f(x) = -- + J2
sin[(2n - 1)n x/L]
2n - 1
2cos(2n - 1)x (- 1)n+1 sin nx
n(2n - 1)2 n
1 4 cos(2n - 1)nx
16. (b) f(x) = - + V - L—
2 n2 ^ (2n - 1)2
3L
17. (b) f(x) = — + J2
n=1
18. (b) f(x) = jT
n=1
2L cos[(2n - 1)nx/L] (-1)n+1 L sin(nnx/L)
(2n - 1)2n2 nn
2
2 nn I 2 \ nn
cos — + — sin —
nn 2 \nn ) 2
4 sin[(2n - 1)nx/2]
19. (b) f(x) =
n=1
2n 1
2 ^ (-1)n+1 20. (b) f (x) = — y ------------------sin nnx
rr ‘ J tl
n
2
725
See SSM for detailed solutions to 27a
2ab, 4a
4b, 7abc
2 8 ^ (-1)" nnx
21. (b) f (x) = —\----------t 7 5— cos------
3 n2 n=1 n2 2
1 12 ^ cos[(2n - 1)nx/2] 2 ^ (-1) . nnx
22. (b) f (x) = - +---------2 > 2-------\-------> sin -
- „2 ^ (2n - 1)2 n ^ ”
2
n=1
n= 1
2
11 1 ?
23. (b) fx = n + -,?
(-1)n - 5 nnx
r cos ---------------
n2 2
\
4[1 - (-1)n] (-1)n
9 vS
24. (b) f (x) = 8 +J2
162[(-1)n - 1] 27(-1)n
nn x 108(-1) + 54 nn x
;--------->----------------=—=-sin---
3 „3„3 3
25. m = 81
26. m = 27
28. f (t) dt may not be periodic; for example, let f (t) = 1 + cos t.
Section 10.3, page 562
4 sin(2n - 1)n x
1. (a) f(x) = -J2'
n=1
2. (a) f (x) = - - ?
n=1
L 4 L "2
2n - 1 2
(-1)n
2 cos(2n - 1)x +--------------sinnx
3. (a) f(x) = - + 2
n
_ (2n - 1)2n
cos[(2n - 1)nx/L]
n=1
(2n - 1)
2
2 4 ^ (-1)n+1
4. (a) f(x) = —\-----------2 > -----2— cos nnx
3 n2 ^ n2
\n-1
5. (a)
6. (a)
1 2 ^ (— 1)n
f (x) = - + - V ------------------ cos(2n - 1)x
2 ^ — 2n — 1
n=1
^o ^—v
f (x) =--------+ > (an cos nn x + bn sin nn x)
2 n=1
1 2(-1)n
= 3 ’ 3n =
22 n n
bn =
-1 / nn,
1 /nn - 4/n3n3, n odd
7. (a) (b)
n = 10 n = 20 n = 40
n= 1
1 - cos nn (-1)
;----------- cos nx------------sin nx
max|e| = 1.6025 at x = ±n max|e| = 1.5867 at x = ±n max|e| = 1.5788 at x = ±n
(c) Not possible
8. (a)
(b)
1 CO 1
1 2 1 - cos nn
f (x) = - + T y --------------------------=-------------cos nn x
n 2 n2
2
(c) 9. (a)
(b)
(c)
n = 10 n = 20 n = 40 n 21
max|e| = 0.02020 at x = 0, ±1 max|e| = 0.01012 at x = 0, ±1 max|e| = 0.005065 at x = 0, ±1
2 ^ (-1)n+1 f (x) = — > ----------sin nn x
TT --/ 11
n=1
n = 10, 20, 40; max|e| = 1 at x ± 1
Not possible
3 3
n n
2
«4^4
n n
22 n n
n even
a
0
n
726
See SSM for detailed solutions to 12a
12b, 14
15, 16, 18a 18b
3, 6
7, 10, 13, 14
15, 18
1 TO
10. (a) f(x) = - + J2
2 n=1
6(1 - cos nn) nnx 2cosnn . nnx
22 n n
? cos 12
? sin -nn 2
(b) n = 10 n = 20 n 40
lub|e| = 1.0606 as x ^ 2 lub|e| = 1.0304 as x ^ 2 lub|e| = 1.0152 as x ^ 2
(c) Not possible
1 TO
11. (a) f(x) = - + J2
2 cos nn
2 - 2 cos nn + n2n2 cos nn
? cos nnx —
(b) n = 10 n = 20 n 40
lub|e| = 0.5193 as x ^ 1 lub|e| = 0.5099 as x ^ 1 lub|e| = 0.5050 as x ^ 1 (c) Not possible
n () er \ 12 'TO (-1)n ?
12. (a) i (x) =--------3 > —3— sin nn x
n n=1 n
(b) n = 10 n = 20 n = 40
(c) n = 4
max|e| = 0.001345 at x = ±0.9735 max|e| = 0.0003534 at x = ±0.9864 max|e| = 0.00009058 at x = ±0.9931
13. y = (a sin nt - n sin at)/a(a2 - n2), a2 = n2 y = (sin nt - nt cos nt)/2n2, a>2 = n2
TO
14. y = bn (a sin nt - n sin at)/a(a>2 - n2), a = 1, 2, 3,...
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