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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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27. sin x cos 2y — 2 sin2 x = c 28. 2xy + xy3 - x3 = c
29. arcsin(y/x) — ln |x| = c; also y = x and y = = —x
30. xy2 - ln |y| = 0
31. x + ln |x| + x-1 + y — 2 ln |y| = = c; also y = 0
32. x3 y2 + xy3 = -4
Section 3.1, page 136
y = c1et + c2e
9
10
11
12
13
y = c^2
2 . y = c1e t + c2e 2t
+ c2e
7/3 4. y = c1et/2
5. y = c1 + c2e~5t 6. y = c1e3t/2 +
7. y = c1 exp[(9 + 3V5)f/2] + c2 exp[(9 - 3^5)t/2]
8. y = c1 exp[(1 + V3)f] + c2 exp[(1 - V3)f] t-
+ c2e
-3t/2
y
to as t -
-3t.
y = e
y = 5 e-t - 5 e
y = 12et/3 - 8et/2;
y =-1 - e-3t ; y
to
y — 0 as t —— to y — -to as t —to — 1 as t —to
y = 26 (13 + 5VT3) exp[(-5 + v/?3)t/2] + 26(13 - 5^/13) exp[(-5 - V?3) t/2]; y — 0 as t —to
14. y = (2/V33) exp[( 1 + V33)t/4] - (2/^33) exp[(-1 - v/33)t/4];
y —to as t —to
15. y = 1oe-9(t-1) + 190et-1; y —to as f —to
16. y = -2 e(t+2)/2 + |e-(t+2)/2 ; y —-to as t —to
17. f + /- 6y = 0 18. 2 f + 5/ + 2y = 0
19.
y = 4e7 + e 7; minimum is y = 1 at t = ln2
20. y = -et + 3et/2; maximum is y = 4 at t = ln(9/4), y = 0 at t = ln9
9
21. a = -2
22. ^ = -1
ce
2
688
Answers to Problems
See SSM for detailed solutions to 24, 25a.
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) p = 2
(a) y = (6 + p)e-2t - (4 + p)e-3t
(b) tm = ln[(12 + 3p)/(12 + 2p)], ym = 27(6 + P)3/(4 + p)2
(c) p = 6(1 + V3) = 16.3923 (d) tm ^ ln(3/2),
y" + 3y + 2y = 0 is one such equation.
Ym
= ql 1 + a, + ln f
29. y = c1 ln f + c2 + f
31. y =
y
y = (1/k) ln l(k - f)/(k + f)| + c2 if c1 = k2 > 0; y = (2/k) arctan(f/k) + c2 if c1 = —k2 < 0; y = —2f-1 + c2 if c1 = 0; also y = c y =±3 (t - 2q)y t + c1 + c2; also y = c
factor.
Hinf: ?(v) = v 3 is an integrating
32.
33.
34. 36. 38. 40.
42.
43.
y = Cj e f + c2 — te f Cj y = Cj f — ln 11 + Cj f | + c2 if Cj = 0; y2 = Cjf + Cj
3y3 — 2Cjy + C2 = 2t; also y = C
yln |y| — y + Cjy + f = C2;also y = C y = 4 (t+1)3/2 — 3
y = 3ln t — 2 ln(f2 + 1) — 5 arctan t + 2 + |ln2 + 5n
y = 1 f2 + 3
y = 2 f2 + c2 if c1 = 0; also y = c 35. y = c1 sin(f + c2) = k1 sin f + k2 cos f 37. f + c2 = ±§ (y -2c1)(y + c1)1/2 39. ey = (f + c2)2 + c1 41. y = 2(1 - f)-2
Section 3.2, page 145
1. _7 ft/2 2 2. 1
3. e-4t 4. x2 ex
5. -e2t 6. 0
7. 0 < f < <x 8. -TO < f <1
9. 0 < t < 4 10. 0 < f < <x
11. 0 < x < 3 12. 2 < x < 3n/2
14. 16. The equation is nonlinear. No 15. 17. The equation is nonhomogeneous. 3 te2t + ce2t
18. tet + ct 19. 5 W (f, g)
20. - 4(t cos t - sin t )
21. y(f) = 3 e-2f + 2 ef, y2(f) = - 1 e + 3 e
22. Y1 (f) = -5 e-3(f - 1) + § e-^-1'), y2 (f ) = e -3(t-1) + 1 e-(f-1) 2e
23. Yes 24. Yes
25. 28. Yes Yes, y = c1 e-x2/2 f X ef2/2 df + c2e- x2/2 26. Yes
29. No 0
30. 1 Yes, y = ?(x) 1 + ? *= tT , ?(x) = exp f ( 1 cos x\ rJ U + x )dx.
31. Yes, y = c1 x-1 + c2x 33. x2?" + 3x? + (1 + x2 - v2)g =
34. (1 - x2)?n - 2x ? + a(a + 1)? =0 35. ? — x? = 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
Answers to Problems
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(c1 Yv c272) = C1C2 W(7v 7)) = 0 12. W(7 74) = —2 W(71, 72) 15. ct2et
13. a1 b2 — a2b1 = 0
16. c cos t 17. c/x
18. C/(1 - X2) 20. 2/25
21. 3ve = 4.946 22. p(t) = 0 for all t
26. If t0 is an inflection point, and 7 = 0(t) is P(t0W(t0) + q (t0)$(t0) = 0. a solution, then from the differential equation
Section 3.4, page 158
1. e cos2 + ie sin2 = -1.1312 + 2.47177
2. e2 cos 3 - ie2 sin3 = -7.3151 - 1.0427i
3. -1
4. e2 cos(n/2) - ie2 sin(n/2) = —e2i = -7.38917 2 cos(ln2) - 2i sin(ln2) = 1.5385 - 1.27797 n-1 cos(2 lnn) + i n-1 sin(2 lnn) = -0.20957 + 0.23959i
8. 7 = c1 ' 10. 7 = c 12. 7 = c 14. 7 = c 16 7 = c
5.
6.
7.
9.
11.
13
15
17
18
19
20 21 22.
23.
24.
26.
35.
36.
37. 39. 41.
7 = c1et cos t + c1et sin t
7 = c
7 = c1 7 = c1
e2t + c2 e-4t
e-3t cos2t + c2e-3t sin2t
cos(f/2) + c2e—r sin(t/2) y = Cj e-t/2 cos t + c2e—1/2 sin t y = 2 sin 21; steady oscillation y = e-2t cos t + 2e—2t sin t; decaying oscillation y = — et—n/2 sin2t; growing oscillation y = (1 + 2^3) cos t — (2 — V3) sint
e‘ cos V5 f + c2et sin \/5 t e-t cos t + c2e-t sin t cos(3t/2) + c2 sin(3t/2)
et/3 + c,e-4t/3
e 2t cos(3t/2) + c2e 2t sin(3t/2)
7 = 3e 7 = V2e
t/2 cos t + 2 e
t/2
sin t ;
steady oscillation decaying oscillation
-(t-n/4) ,
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