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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.
Download (direct link): elementarydifferentialequations2001.pdf
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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 sin2^ + c2^l IZt)
3 _ I 5 cos t \ / 5 sin t \
. x cM2cosf + sin t) ^ \ - cos t + 2 sin t)
2
4
Answers to Problems
711
See SSM for detailed solutions to 7, 9
11 ad,15ab
16abc, 18abc
21, 23abc, 29a
29bce
4 x = cet/2( 5C0S 3 * \+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- 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 -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 - 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)( 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 2, y2 2/1 + , y3 4, y4 1 - 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
Answers to Problems
See SSM for detailed solutions to 2, 4
6, 10
11
1
3, 5
6. (0 =
7. () =
8. () =
9. (0 = 10. (0 =
11. x = -
2
Section 7.7, page 400
1. (0 =
2. (0 =
3. (0 =
4. (0 =
5. () =
- f -1 4 2t f -1 - 2 2|\
3
- 2 - 2 2 f -1 - 1 27
3
1 ~ t/2 1 -t
2 t
-

1
t/2 - 1 -t 2 -/2 + 1-
4
f - 1 -t - 2 1 + 1 -1 \
f - 3 -t - 2 +f -7
1 -3 + 4 21 -
5 ^ 5
- 4 -3 + 4 21
cos t + 2 sin t
1 -31 + 21
-31 + 1 21
Sin t
-5 sin f cos t 2 sin t
/ t cos2t -2 t sin2t\ -t sin2f -t cos2ty
- 2 21 + f 41
- 2 21 + f 41
1 2 ____ 1 4
2 2
f 2 _____ 1 4
t cos t + 2 t sin f 5-1 sin t (-2-21 + 3-1 5 -21 - 4-1 + 3 21 \7 -21 - 2-1 - 2 21
2
-1 sin t ~- cos t - 2-1 sin t. -2f + -1
-
5 -2f 4 -f 4- 13 21
,( C7 C7 1 12
3
+ -1
5 -2f 4 - , 1 2
1 ^ 1 10
7 -2t 2 - _ 13 2
4 3 12
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