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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
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24. X1 = -1, x(1) = ^-4j ; X2 = -1, x(2) =
Section 7.4, page 371
2. (c) W(t) = cexp J [p11 (t) + p22(t)] dt
6. (a) W(t) = t2
(b) x(1) and x(2) are linearly independent at each point except t = 0; they are linearly
independent on every interval.
(c) At least one coefficient must be discontinuous at t = 0.
0 1
v-2t-2 2t-
7. (a) W(t) = t(t - 2)e'
(b) x(1) and x(2)
are linearly independent at each point except t = 0 and t = 2; they are linearly independent on every interval.
(c) There must be at least one discontinuous coefficient at t = 0 and t = 2.
/0 1 \
(d) x' = I 2 - 2t t2 - 2 I x
V t2 - 2t t2 - 2t)
Section 7.5, page 381
(d) *=1 0,-2 o,-1 *x
L x = C1 Q) e-t + „(1) e2t
3. x = c1 (^ + c^^e-t
5. x = c1 (-?^) e-3t + C^1) e-t
7. x = C1 (3) + c2 (1)e-2t
9. x = c. I ^ + c2( 1 ) e2t
2. x = c1 (1) e-t + ^(T) e-2t
c1 14
4. x = c, I "le3' + c2 ( 1 ) e2t
6. x = c1 (-1) eU1 + c2 (j) e2t
8. x = c1 (-2) +c2 (-i) et
10.x=cA2+1i)et+e-it
3
2
2
710
See SSM for detailed solutions to 16
20, 25
31c
'1'
1'
4
5
-7
1N
4
3-1,
11. x = C1 1 e + c2\ -2 e + c3 | 0 I e
12. x = c1 I - 4 I e + c2 I 0 I e + c3 I 1 I e
13. x = c1 I -5 I e 2t + c2 I -4 I e c + c3 I 1 I e
14. x = c1 I -4 I et + c2 I -1 I e 2t + c3 I 2 I eit
1
i
3
4 -2
-1 3-1,
15 x=-f(0 ^+KD *4t
16. x =
2 \1
e- +
25
17. x = - 02 et + 2 11 e2t
18. x = 6 I 2 j et + 3| -2 j e-t -
1 e4
20.x=s (1) t+cJ3)11
21. x = c1 (3) t2 + c
4
22. x = s?j + c41) t-2
29. (a) X = x2, X2 = -(c/a)x1 - (b/a)x2
30. (a) x = -55 (2) e-«20 + 29 (_2) e-/4
(c) T == 74.39
31. (a) x = cj-^ e(-2+72)t/2 + r (e(-2-7~2)t/2-
23. x = d 2 t-1 + c2^\ t2
+ c2 1 e
r12 = (-2 ± v))/2; node
(b) x = d (7) e(-1^'?)t + cA 7^j e(-1-^ ; r1 2 = _1 ±72; saddle point
(c) r12 = -1 ± V® ; a = 1
32.(a) (0=c^3)e-2t+^(1)e-t 33.(a) (cR - R
CL
> 0
Section 7.6, page 390
1. x c et
,et( cos2t D + ceA sin2t A
1 \vcos2t + sin2ty 2 y- cos2f + sin2ty
, _f (2cos2f\ , _t
2. x = c,e 1 I „ , I + cne 1
1^1 sin2^ + c2^l ?s^)
3 _ I 5 cos t \ / 5 sin t \
. x cM2cost + sin t) ^l-cos t + 2 sin t)
2
4
711
See SSM for detailed solutions to 7, 9
16abc, 18abc
21, 23abc, 29a
29bce
4 x = ce^i 5cos3* \ + ce‘/^ 5sin2t ,
1 l3(cos31 + sin3 t)J 2 \3(— cos 21 + sin31)
t cos t t
x = c,e + c, e
1 ' 2 cos t + sin t/ 2
2
sint — cos t + 2 sin f
6 X— c I — 2cos3f I _i_ c I — 2sin3t . X cMcos3t + 3sin3t/ cMsin3t— 3cos3t
xc
2
0
7. x = c1 I —3 I et + c2e^ I cos2f I + c3et
, sin2t )
0
sin2t cos 2t
2 e-2t \ c e-t
^ —72 sin 72 t \
cos 72 t v— cos 72 t — 72 sin 72 t y
(72cos72 t ^
sin 72 f 72 cos 721 — sin 72 t y
9. x e-
t I cos t — 3 sin t cos t sint
10. x e-
cos t - 5 sint —2cos t — 3 sin t
11.
13.
14.
15.
16.
17.
18.
19.
20.
(a)
(a)
(a)
(a)
(a)
(a)
(a)
(a)
(a)
12. (a) r = 5 ± i
r = -4 ± i
r = a ± i (b) a = 0
r = (a ±7a2 - 20)/2 (b) a = -72?, 0, 720
r = ±74 — 5a (b) a = 4/5
r = 4 ± 1v/3a (b) a = 0,25/12
r = — 1 ± 7—a (b) a = — 1, 0
r = -2 ± ^49 - 24a (b) a = 2,49/24
r = 1 a - 2 ± ya2 + 8a - 24 (b) a = -4 - 27^, -4 + 2710, 5/2
r = -1 ± 725 + 8a
- 24
(b) a = -25/8, -3
21. x = c1 t
cos(72ln t) 72 sin(72 in r)
+ c2t
sin ^72 in t) -72cos(72int)
22 I 5cos(ln t) \ + r( 5sin(ln t)
1 \2 cos(ln t) + sin(ln t) J 2 ^ — cos(ln t) + 2 sin(ln t)
)r = -:1 ± i, -4
23. (a)
24. (a)
r = -1± i 1
1 4 ± *•> 10
25. (b)
(c)
(d)
26. (b)
(c)
(d)
28. (b)
29. (a)
(b)
(c)
(d)
t/2 1 cos(t/2)
+ c2e~
4 sin(t/2)
Use c1 = 2, c2 = — 4 in answer to part (b). lim I(t) = lim V(t) = 0; no
t^TO t^TO
t/2 sin(t/2)
—4 cos(t/2)
I) = ce-t( cos t ) + ce-7 sin t
Vy 1 y— cos t — sin tj 2 y— sin t + cos t)
Use c1 = 2 and c2 = 3 in answer to part (b). lim I(t) = lim V(t) = 0; no
t^TO t^TO
r = ±i\/k/m (d) |r | is the natural frequency.
y1 = Ó2, y2 = -2y1 + Óç, y3 = y4 = y1 - 2y3 r = ±i, ±73 i
y1 = y3 = sin t + 2 cos t, y2 = y4 = —2 sin t + cos t
y1 = —y3 = sin 73 t + 2 cos 73 t, y2 = —y4 = —273 sin 73 t +73 cos 73 t
712
See SSM for detailed solutions to 2, 4
6, 10
11
1
3, 5
6. ®(t) =
7. ®(t) =
8. ®(t) =
9. ®(t) = 10. ®(t) =
11. x = -
2
Section 7.7, page 400
1. ®(t) =
2. ®(t) =
3. =
4. O(t) =
5. O(t) =
- 3 e-t I 4 2t 3e 2 e-t - 2 e2t\
2 e-t I 2 e2t 3e 4e-t - 1 e2tj
1 e~ t/2 I 1 e-t 2e e-t/2 - e-
4e~ t/2 - 1 e-t 4e 2 e-t/2 + 1e-
§ e - 2e -t - 2 et + 1 e-t \
§ e‘ - 2e -t - 2 et +3 e-7
1 e-3t + 4 e-‘ -1
5 ^ I 5
- 4 e-3t + 4 e2t cos t + 2 sin t
1 e-3t + 1 e2t
e-3t + 1 e2t
sint
- 5 sin t cost 2 sint
/ e ‘ cos2t -2e t sin2 A e - sin2f e - cos2ty
- 2 e2t + f e4t
- 2 e2t + f e4t
1 ^ _ 1 e4t
2 2
f e2t 1 e4t
e-t cos t I 2e-t sin t 5e-t sint -2e-2t I 3e-t
5 e-2t - 4e-t + f e2t \7 e-2t - 2e-t - 2 e2t
2
e-t sin t e-t cos t - 2e-t sin t.
~2t + e-t
-e
5 e-2t — 4 e-^ 4- 13 e2t
1 C7 C7 1 12
? + e-t
5 e-2t 4 e-t , 2.e2t
1 O -1 C7 1 i 'S C7
7 e-2t 2 e-t _ 13 e2t
4e 3e 12
1 A 4- 1
? C ^
e 2t + 1 e3
- f et - 1 e
\-1 ? - 1
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