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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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11 + \J~2 e
-(t-n/4)
sin t; decaying oscillation
(a) u = 2et/6 cos(v/23 t/6) - (2/v/23)et/6 sin(V23 t/6)
(b) t = 10.7598
(a) u = 2e-/ cos(V34 t/5) + (7/V34) e-t/5 sm(V34 t/5)
(b) T = 14.5115
25. (a) 7 = 2e-t cosV^ t + [(a + 2)/V5] e-t sinv^ t (b) a = 1.50878
(c) t ={n - arctan [2\T5/(2 + a)]}/\/5 (d) n/\T5
(b) T = 1.8763
(a) 7 = e at cos t + ae at sin t
2
x =
T = 4.3003;
e
-t2/2 dt
(c) a = 1, T = 7.4284; a Yes, 7 = c1 cos x + c2 sin x.
No
Yes, 7 = c1 e-t2/4 cos(V3 t2/4) + c2e-2/4 sm(V3 t2/4) 7 = c1 cos (ln t) + c2 sin(ln t)
7 = cj t-1 cos(2 ln t) + c2t-1 sin(2 ln t)
= 2, T= 1.5116
40.
42.
7 = c1t 1 + c2t / 7 = c/ + c2t 1
Section 3.5, page 166
7 = c1et + c2 te(
,e-"2 + c2e3t/2
7 = c1 7 = c1 7 = c1 7 = c1
e< cos 3t + c2e( sin3t
t/4
+ c2e
e2t/5 + c2te2t/5 11. 7 = 2e2t/3 - 3te2t/3,
12. 7 = 2te
13.
7 as t
2.
4.
6.
8.
10.
00
t/3
-3t/2
+ c2 te + c2te
t/3
-3t/2
7 = c1e 7 = c1e
7 = c1e3t + c2te3t 7 = c1e-3t/4 + c2te-3t/4 7 = e-t/2 cos(t/2) + cyf
t/2
sin(t/2)
7 = —e-t/3 cos 3t + 9 e-t/3 sin3f.
- to as t ? to
7 ^ 0 as t -
TO
e
690
Answers to Problems
See SSM for detailed solutions to 14, 17ab
17cd, 19, 21, 25
27, 30, 31b 33, 35, 38
42
I, 4, 6, 8
II, 13, 16
19a, 22a
?2(t+1)
14. / = 7e-2(t+1) + 5te
15. (a) / = e-3t/2 - 2 te-3t/2
y ^ 0 as t -(b) t = 5
(c) to = 16/15, yo =
--5 e-8/5
= - 0.33649
-3t/2.
b= -3
w 2
yo
= 5e-4/5 = 2.24664
(d) y = e-3t/2 + (b + §)te-
16. y = 2et/2 + (b - 1)tet/2; b = 1
17. (a) y = e-t/2 + 5 te-t/2 (b) to =
(c) y = e-t/2 + (b + 2 )te-t/2
(d) tM = 4b/(1 + 2b) ^ 2 as b ^?; yM = (1 + 2b) exp[-2b/(1 + 2b)] as b
00
18. (a) y = ae-2t/3 + ( § a - 1)te-2t/3 23. y2(t) = t3
25. y2(t) = t-1 lnt
27. y2(x) = cosx2
29. ^^(x) = x 1/4e-2^
32. y = q e-^/2fo X eSs2/2 ds + c2e-*x2/2
33. y2(t) = y() [ y-2(s) exp -f p(r) dr
Jt0 L Js0
34. y2(t) = t-1lnt
36. y2 ( x) = x
39. (b) yo + (a/b)y>o
42. y = c1t-1/2 + c2t-1/2ln t
(b) a = 2
24. y2(t) = t-2 26. y2(t) = te‘
28. y2(x) = x
3o. y2(x) = x-1/2cosx
ds
35. y2(t) = cos t2 37. y2(x) = x-1/2cosx 41. y= c112 + c2t2ln t
Section 3.6, page 178
1. y = c1e3t + c2e ‘ - e21
e 1 cos2t + c2e 1 sin2t + 17sin2t - 12cos2t
e3t + c2e-t + 16 te-t + 3 t2e-t
+ c2e-2t + 21 - 1 sin2t - 1 cos2t
cos 3t + c2 sin3t + ^(9t2 - 6t + 1)e3t + 3
e-t + c2te-t + t2e-t
e-t + c2e-t/2 + t2 - 6t + 14 - 1o sin t - H cos t cos t + c2 sin f - 31 cos2t - 9 sin 21 cos aot + c2 sin«ot + («3 - a2)-1 cos at cos aot + c2 sinaot + (1 /2«o)tsinaot
e-t/2 cos^vT5 t/2) + c2e-t/2 sin^vT5 t/2) + 1 et - 4e-t
2. y = c1
3. y = c1
4. y = c1
5. y = c1
6. y = c1
7. y = c1
8. y = c1
9. u = c 1o. u = c
H. y = c
12. y = c1
14. y = yo sin2f - 19 cos2t + 412 - 1 + fet 15. y = 4te' - 3e' + 6t3et + 4
41
e-t + c, e2t + 1 te2t + 1 e
2t
13. y = et - 1 e-2t - t - 2
2 1 3 t2
17. y = 2cos2t - 1sin2t - 31cos2t
16. y = e3t + 3e-t - 3e2t - te2t
18. y = e-t cos2t + 2e-t sin2t + te-t sin2t
19. (a) Y (t) = t ( V4 + A t
+ D sin3t + E cos 3t
(b) Ao = 2/15, A1 = -2/9, A2 = 8/27, A3 = -8/27, A4 = 16/81, Bo = -1/9,
B1 = -1/9, B2 = -2/27, D = -1/18, E = -1/18
20. (a) Y(t) = Aot + A1 + t(Bot + B1) sin t + t(Dot + D1) cos t
(b) Ao = 1, A1 = o, Bo = o, B1 = 1/4, Do = -1/4, D1 = o
21. (a) Y(t) = et(Acos2t + Bsin 21) + (Dot + D1)e2t sin t + (Eot + E1)e2t cos t
(b) A = — 1 /2o, B = — 3/2o, Do = -3/2, D1 = -5, Eo = 3/2, E1 = 1/2
22. (a) Y (t) = Ae-( + t ( Bot2 + B1t + B2)e~‘ cost + t ( Dot2 + D1t + D2)e-( sin t
(b) A = 3, Bo = -2/3, B1 = o, B2 = 1, Do = o, D1 = 1, D2 = 1
Answers to Problems
691
See SSM for detailed solutions to 28
30, 33 2
5, 11, 14
18, 22 25, 29
23. (a) Y(t) = A0t2 + A11 + A2 + t2(B0t + B1)e2t + (D0t + D^ sin2t + (E0t + E1) cos2t
(b) A0 = 1/2, A1 = 1, A2 = 3/4, B0 = 2/3, B1 = 0, D0 = 0, D1 = -1/16,
E0 = 1 /8, E1 = 1/16
24. (a) Y(t) = t(A0t2 + A1t + A2) sin2t + t(B0t2 + B1t + B2) cos2t
(b) A0 = 0, A1 = 13/16, A2 = 1/4, B0 = -1/12, B1 =0, B2 = 13/32
25. (a) Y(t) = (A0t2 + A1t + A2)e‘ sin2t +(B0t2 + B1t + B2)et cos2t
+ e-t (D cos t + E sin t) + Fet
(b) A0 = 1/52, A1 = 10/169, A2 = -1233/35152, B0 = -5/52, B1 = 13/616,
B2 = -4105/35152, D =-3/2, E = 3/2, F = 2/3
26. (a) Y(t) = t(A0t + A1)e-t cos2t + t(B0t + B1)e-t sin2t + (D0t + D1)e-2t cos t + (E0t + E1)e-2t sin t
(b) A0 = 0, A1 = 3/16, B0 = 3/8, B1 = 0, D0 = -2/5, D1 = -1/25, E0 = 1/5,
E1 = 10/25 1 0 1 0 1 0
1 N
21. y = c1 cos kt + c2 sinkt + ^ [am/(k2 - m2n2)] sinmnt
m=1
28 — I t, 0 < t < n
. y j —(1 + n/2) sin t — (n/2) cos f + (n/2)en-t, t >n
i1 - e-t sin2t - 1 e-t cos2t, 0 < t < n/2
5 10 5
-1 (1 + en/2)e-t cos2t - 1!(1 + en/2)e^t sin21, t > n/2
30 . No 33. y = c1e4t + c2e-t — 2e2^
Section 3.7, page 183
1. Y (t) = e' 2. Y (t) = -§ te-t
3. Y (t) = § t2e-t 4. Y(t) = 2tV/2
5. y = c1 cos t + c2 sin t — (cos t) ln(tan t + sec t)
cos 31 + c2 sin 31 + (sin 31) ln(tan 31 + sec 31) — 1
6. y = c 1. y = c
8. y = c
9. y = c1
10. y = c1
11. y = c1
12. y = c1
e 2t + c2te 2t - e 2t ln t cos2t + c2 sin2t + 3 (sin2t) lnsin2t - 11 cos2f cos(t/2) + c2 sin(t/2) + t sin(t/2) + 2[lncos(t/2)] cos(t/2) e‘ + c2te( - je1 ln(1 + t2) + te‘ arctan t
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