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# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
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25c, 27, 28, 30, 34
39, 40, 43
2, 4, 8
12, 14, 15, 18, 21
25, 27, 30, 32, 34, 36
37
2, 6, 7, 12, 15
23.
24.
25.
26.
27.
28.
30.
y ^ 0 for a < 0; y becomes unbounded for a > 1
y ^ 0 for a < 1; there is no a for which all nonzero solutions become unbounded.
(a) y = 1 (1 + 2p)e-2t + 5 (4 - 2p)et/2
(b) y = 0.71548 when t = 5 ln6 = 0.71670 (c) â = 2
(a) y = (6 + e)e-2t - (4 + e)e-3t
(b) tm = ln[(12 + 3â)/(12 + 2â)], Óø = 27(6 + â)3/(4 + â)2
(c) â = 6(1 + V3) = 16.3923 (d) tm ^ ln(3/2),
Ó + 3y + 2y = 0 is one such equation.
Óø
= ñ, t 1 + a, + ln t
29. y = ñ1 ln t + c2 + t
31. y
y
y = (1/k) ln l(k - t)/(k + t)| + c2 if c1 = k2 > 0; y = (2/k) arctan(t/k) + c2 if c1 = —k2 < 0; y = —2t-1 + c2 if c1 = 0; also y = ñ y =±3 (t - 2q)y t + c1 + c2; also y = ñ
factor.
Hint: ji(v) = v 3 is an integrating
32.
33.
34. 36. 38. 40.
42.
43.
y = c1 e- t + c2 - te- t y = c11 - ln 11 + c111 + c2 if c1 = 0; y2 = c1 t + c2
3y3 - 2c1 y + c2 = 2t; also y = c
yln |y| - y + c1 y + t = c2;also y = c y = 4 (t+1)3/2- 3 y = 3ln t - 2 ln(t2 + 1) - 5 arctan t + 2 + |ln2 + 5n
y = 112 + 3
y = 212 + c2 if c1 = 0; also y = c 35. y = c1 sin(t + c2) = k1 sin t + k2 cos t 37. t + c2 = ±§ (y — 2c1)(y + c1)1/2 39. ey = ( t + c2 ) 2 + c1 41. y = 2(1 - t)-2
Section 3.2, page 145
1. 2/ 2. 1
te
MO
1
3. e-4t 4. x2 ex
5. -e21 6. 0
7. 0 < t < æ 8. -æ < t < 1
9. 0 < t < 4 10. 0 < t < æ
11. 0 < x < 3 12. 2 < x < 2/
3
14. The equation is nonlinear. 15. The equation is nonhomogeneous.
16. No 17. 3te2t + ce2t
18. tet + ct 19. 5W (f, g)
20. -4(f cos t - sin t)
21. y() = 3 e-2t + 2 et, y2(t) = - 1 e-2t + 3 e‘
22. t- 1) + 2 e-(t-1), yt(2 = -2e -3(t-1) + 1 e-(t-1)
3( 2å
-
e
'---'l(N
-
II
y1
23. Yes 24. Yes
25. Yes x2/2 26. Yes
28. Yes, y = c1 e-x2/2 fX et2/2 dt + c2e-
29. No 0
30. 1 Ã l(f) ’ j(x) = exp f ( 1 cos x\
Yes, y = - c1 dt + c2 -j Ü + X Jdx.
l(x) _ Jx0 1
31. Yes, y = c1 x-1 + c2x 33. x2/ë" + 3xj' + (1 + x2 - v2)i =
34. (1 - x2)l - 2xj + a(a + 1)i =0 35. 1 --- xë = 0
37. The Legendre and Airy equations are self-adjoint.
Section 3.3, page 152
1. Independent 2. Dependent
3. Independent 4. Dependent
5. Dependent 6. Independent
7. Independent if origin is interior to interval; otherwise dependent
689
See SSM for detailed solutions to 20, 24, 26, 27
28
1, 5, 7
11, 14, 18, 22, 23a
23b, 25abcd, 31, 33
35, 38, 39
1,9, 12
8. Independent if origin is interior to interval; otherwise dependent
9. Independent; W is not always zero 10. Independent; W is not always zero
11. W(c171, c272) = c1c2 W(71, 72) = 0 12. W(ó 74) = ---2 W(71, 72)
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