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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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CL
> 0
Section 7.6, page 390
1. x c et
. eJ cos2t sin2t A
1 \vcos2f + sin2fy 2 y- cos2f + sin2fy
, -t /2cos2f\ -t
2. x = c, e I . „ , I + ^e
1e Ë sin2^ + c2^l IZt)
3 _ I 5 cos t \ / 5 sin t \
. x cM2cosf + sin t) ^ \ - cos t + 2 sin t)
2
4
711
See SSM for detailed solutions to 7, 9
16abc, 18abc
21, 23abc, 29a
29bce
4 x = cet/2( 5C0S 3 * \+ñå×2( 5sin2f ,
1 ^(cosj t + sin| t)J 2 \3(- cos 21 + sin| t)
t cost t sint
x c e c e-
1 ' 2 cos t + sin fy 2 y- cos t + 2 sin t
6 x— c ^ -2cos3f \ c / -2sin3t
. X c^cos3t + 3sin3y c^sin3t - 3cos3t
2 t t 0 t 0
7. x — c1 1-Ì et + c2et I cos2f I + c3e4 sin2t I
1 2 2 sin 2t 3 - cos 2t
/ 2\ / -V^sin^21 \
8. x — c1 I -2 I e-2t + c2e-t I cosV2 t I
\ 1/ \- cos^2 t - V^sin^2 v
(v/2cosv/21 \
smV21 I
\/2 cos V2 t - smV2 t/
-t t cos t — 3 sin A , „ -2t / cos t — 5 sin t
9. x — e t . 10. x — e
\ cos t - sin t / V -2 cos t - 3 sin t
11.
13.
14.
15.
16.
17.
18.
19.
20.
a) r — - 1 ± i 12. (a) r — 1 ± i
a) r — a ± i (b) a — 0
a) r — (a ± Va2 - 20)/2 (b) a — -V20, 0,
a) r — ±V4 - 5a (b) a — 4/5
a) r — 5 ± 2V^ (b) a — 0,25/12
a) r — — 1 ± v- a (b) a — -1, 0
a) r — -2 ± 1V49 - 24a (b) a — 2,49/24
a) r — 1 a - 2 ± ya2 + 8a - 24 (b) a — -4 - 2^10, -4 + 2^10, 5/2
2a - 2 ± V a + 8a - 24 (b) a
a) r —-1 ± V25 + 8a (b) a — -25/8, -3
1 ( cos(V2ln t) \ 1 ( sin(V2ln t)
2 1 \V2 sin(V2ln t)) + 2 Ó -v/2cos(v/2Tn t)
22 c t 5cos(ln t) V c t 5sin(ln t)
1 ^2 cos(ln t) + sin(ln t) J 2 ó- cos(ln t) + 2 sin(ln t)
23. (a) r — -1 ± i, -4
24. (a) r — -1 ± i, ^
25. (b) f A=c e-t/2( cos(t/2)\ +c e-t/2( sin(t/2)
2 \ VJ 1 \v4sin(t/2)J 2 y-4cos(t/2)
(c) Use c1 — 2, c2 — - 3 in answer to part (b).
(d) lim I(t) — lim V(t) — 0; no
t^TO t^TO
26. (b)( Ë — c1e-^ cos f . )+ c2e-^ . sin t )
V 1 - cost - sint 2 - sint + cos t
(c) Use c1 — 2 and c2 — 3 in answer to part (b).
(d) lim I(t) — lim V(t) — 0; no
t^TO t^TO
28. (b) r — ±i*Jk/m (d) |r | is the natural frequency.
29 (a) y1 — Ó2, y2 — “2/1 + Óç, y3 — Ó4, y4 — Ó1 - 2/3
(b) r — ±i, ±V3 i
(c) y1 — y3 — sin t + 2 cos t, y2 — y4 — —2 sin t + cos t
(d) y1 — -y3 — smV3 t + 2 cos V31, y2 — —y4 — —2\fb sin \fb t + +J3cos V31
712
See SSM for detailed solutions to 2, 4
6, 10
11
1
3, 5
6. Ô(0 =
7. Ô(´) =
8. Ô(´) =
9. Ô(0 = 10. Ô(0 =
11. x = -
2
Section 7.7, page 400
1. Ô(0 =
2. Ô(0 =
3. Ô(0 =
4. Ô(0 =
5. Ô(´) =
- f å-1 ³ 4 2t f å-1 - 2 å2|\

- 2 å-‘ ³ 2 å2² f å-1 - 1 å27

1 å~ t/2 ³ 1 å-t å
2å t
-
å
1
å t/2 - 1 å-t 2 å-²/2 + 1å-

f å‘ - 1 å -t - 2 å1 + 1 å-1 \
f å‘ - 3 å -t - 2å +f å-7
1 å-3 + 4 å21 - ³
5 ^ ³ 5
- 4 å-3 + 4 å21
cos t + 2 sin t
1 å-31 + ³ å21
å-31 + 1 å21
Sin t
-5 sin f cos t — 2 sin t
/ å t cos2t -2å t sin2t\ Ó å-t sin2f å-t cos2ty
- 2 º21 + f º41
- 2 º21 + f º41
1 º2³ ____ 1 º4³
2 2
f º2³ _____ 1 º4³
å t cos t + 2å t sin f 5å-1 sin t (-2å-21 + 3å-1 5 º-21 - 4å-1 + 3 º21 \7 º-21 - 2º-1 - 2 º21
2
º-1 sin t å~- cos t - 2å-1 sin t. -2f + º-1

5 å-2f — 4 å-f 4- 13 º21
,( C7 C7 1 12
3
¦ + º-1
5 å-2f 4 å- , 1 º2³
1 ñ ^ 1 10
7 å-2t 2 å- _ 13 º2³
4å 3å 12å
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