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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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399 k=1 15
24. (b) y = Jgg U(2k-1)n(t)e-[t-(2k-1)n]/2°sm{V399[t - (2k - 1)n]/20}
Section 6.6, page 335
3. sin t * sin t = 2 (sin t — t cos t) is negative when t = 2n, for example.
4. F (s) = 2/s2(s2 + 4) 5. F (s) = 1/(s + 1)(s2 + 1)
6. F (s) = 1/s2(s - 1) 7. F (s) = s/(s2 + 1)2
8. f(t) = 1f0(t - T)3sint dT 9. f(t) = f e-(t-T) cos2r dr
10. f( t) = 1 f (t — r)e-(t-r) sin2r dr 11. f( t) = f sin(t — r)g(r) dr
2 0 s s 0
1 1 ft t* t (t )
12. y =— sin at +------------ sin a(t — r)g(r) d r 13. y = e-( T) sin(t — r) sin ar dr
o) a J0 -J0
14. y = e-(t-T)/2 sin2(t - r )g(r) dr
15. y = e-‘ /2cos t - 1 e-S/2sin t + ? e-(t-T)/2 sin(t - r)[1 - un(r)] dr
16. y = 2e-2t + te-2t + (t — r)e-2(t-T)g(r) dr
17. y = 2e-t - e-2t ^ [e
-(t-T) - e-2(t-T)] cos ar dr 18. y = [sinh(t — r) — sin(f — r)]g(r) dr
19. y = 4 cos t — 1 cos2t + 1 Jo [2 sin(t — r) — sin2(t — r)]g(r) dr
F(s)
20. ®(s) = E—
1 + K (s)
21. (c) 0(t) = 1 (4sin2t — 2sin t)
(d) u(t) = 1 (2 sin t — sin2t)
CHAPTER 7 Section 7.1, page 344
x2 = —2x1 — 0.5x2
2, 4, 5
2. xj = x2, x2 = —2x1 — 0.5x2 + 3 sin t
3. xj = x2, x2 = —(1 — 0.25t-2)x1 — t-1 x2
4. xj =
xj
x3,
x3j = x4,
x4j = x1
5. xj = x2, x2 = —q(t)x1 - p(t)x2 + g(t); x1(0) = u0, x2(0) = «0
6. y1 = Ó2, J-2 = -(k1 + k2) y1/m1 + k2 y2/ m 1 + Fl(t)/ml,
y3 = Ó4, y4 = k2y1 /m2 - (k2 + k3)y3/m2 + F2(t)/m2
2
Answers to Problems
707
See SSM for detailed solutions to 8, 9
12, 14, 19
21abc
— t — 3t
x2 = c^e - c2e
7. (a) xj = c1e-t + c2e—
(b) Cj = 5/2, c2 = —1/2 in solution in (a)
(c) Graph approaches origin in the first quadrant tangent to the line xj = x2.
8. x,
= 11 e2t 2 e— t
x - 11 e2t 4 -t
2 = 6 e 3 e
Graph is asymptotic to the line x1 = 2x2 in the first quadrant.
x - _3et/2 _ 1 e2t yl1 = 2 2 ’
2
x - 3 et/2 _ 1 > 2 = 2 2
Graph is asymptotic to the line Xj = x2 in the third quadrant.
10. x1 = - 7e-1 + 6e-2t
x2 = - 7e t + 9e 2t
Graph approaches the origin in the third quadrant tangent to the line xj = x2.
11. xj = 3cos2t + 4sin2t, x2 = —3sin2t + 4cos2t
Graph is a circle, center at origin, radius 5, traversed clockwise.
12. x1 = -2e-t/2cos2t + 2e-t/2sin2t, x2 = 2e-t/2cos2t + 2e-t/2sin2t
Graph is a clockwise spiral, approaching the origin.
13. LRCI" + LI' + RI = 0
21. (a) ^ Q1 + 40 Q2, Q1 (0) = 25
Q2 = 3 + 10 Q1 - 5 Q2- Q2(0) = 15
(b) Qf = 42, Qf = 36
(c) x'l = — It) x1 + 40 x2- x1(0) = -17 x2 = 10 x1 - 1 x2 - x2(0) = “21
22. (a) Q1 = 391 - 1^ Q1 + ^ Q2- Q1 (0) = Q0
Q2 = q2 + 30 Q1 — ^ Q2- Q2(0) = Q2
(b) Qf = 6(9q1 + q2)- Qf = 20(3q1 + 2q2)
(c) No
(d) 192 < Qf / Qf < 20
Section 7.2, page 355
1ac
/6 -6 3>
1. (a) 5 9 -2
\2 3 8 /
/-15 6 -121
(b) 7 -18 -1
26 3 5
<6 -12
(c) 4
12
89
(d^ 14
2 (a)(-1+2/ ~2+31) (b)
, , /-3 + 5/ 7 + 5A
(c)( 2 + 1 7 + Hi)
(d)
1T
12 -5
5 -8 5/
3 + 4i 61
11 + 61 6 - 51
8 + 71 4 - 41
2
1
'1
3. (a) 1 0 -1 (b) 2 -1
23
1,
i3 -1
4. (a)
3 21
21
'10
0
4
v1 + 1 -2 + 31
6 -4\
4 10
4 6
(b)
6 41
4
3 -21
(c), (d) ( 3
4
-1
4
3 + 21
1 - 1
2 + 1 -2 - 31
(c)
'3 + 21 2 + 1
A- 1 -2- 31
708
Answers to Problems
7 -11 -3 5 0 -1
See SSM for 6. (a) 11 20 17 (b) 2 7 4
detailed solutions -4 3 -12 -1 1 4
to 6, 10 8. (a) 41 (b) 12 - 81 (c) 2 + 21 (d) 16
12
14
22, 25
10.
3 4 \
11 11 1
2 J_
11 11/
-3 2
-3 3 -1
2 1 0
14. Singular
16.
18.
/ 1 3 _1_ \
10 10 10
2 4 2
10 10 10
7 1 3
\- 10 10 10/
/ 1 1 0 1 \
1 0 1 1
1 1 1 1
V 0 1 0 1 /
7et 5e- t 10e2t
(a) et 7e- t 2e2t
\ 8et 0 — e2t
(c)
6 -8 -111
9 15 6
-5 -1 5)
11.
13.
15.
12 1
2 4
1 3 1 3
\_3
-
0
0
17. Singular /
19.
6 5 0
V -2
1 3 0
1 - 3 1 3
0 1 3/
1 !\
4 8 1 i
1 1 2 - 4
0 2/
13 8
5 5
11 6
5 5 i
1 - 5 5
4 4
5 5
(b)
(c)
' 2e2t - 2 + 3e3t 4e2t - 1 - 3e3t ,-2e2t - 3 + 6e3t -1 + 6e-2t - 2et -3e3t + 3et - 2e4
1 + 4e-2t - et
2 + 2e-2t + et
3e3t + 2et - e4tN\
6e3t + ^ + e4t
- 5 /
t
—2e-
e
2et -e-t
K-et —3e-t
2e2 -2e2 4e2
(d) (e - 1)
J
2e-
e-
3e~
2 (e + m - 2(e+1)
e + 1 /
Section 7.3, page 366
1
2, 3 6, 8
14
15
1 x - -1
i. x1 — 3,
3
4
x*. = 7, x, = —1 2. No solution
2 — 3 3 —-3
X —- c, x2 — c + 1, x3 — c, where c is arbitrary x2 — - c, x3 — -c, where c is arbitrary
1
x1 — c,
x3 — 0
5. x1 — 0, x2 — 0,
7. x11) - 5x(2) + 2x(3) — 0 9. Linearly independent
12. 3x(1)(t) - 6x(2)(t) + x(3)(t) — 0
1 ' ? — 4, x(2) —
15. X1 — 2, x(1) —
16. X1 — 1 + 21, x(1) —
17. X1 — -3, x(1) —
2 1
11
6. Linearly independent 8. 2x(1) - 3x(2) + 4x(3) - x(4) — 0 10. x(1) + x(2) - x(4) — 0 13. Linearly independent
1
1+1
k2 — 1 - 21, x(2) —
> — -1, x(2) —
Answers to Problems
709
See SSM for detailed solutions to 18, 21
24
27
1, 2a
2bc, 6abcd
1
5
6, 7, 9
18. X, = 0- x(1) = I1. I ; X2 = 2, x(2) = ^
; A = -2, X(2) =
19. A = 2, x(1) = ) - _o x(2) - f 1
-V3
20. A.1 = -1/2, x(1) = (130
21. A = 1, x(1) = -3
X2 = -3/2, x® = ( 2y 0
X2 = 1 + 2i, x(2) = | 1
X3 = 1 - 2i, x(3) = 1
22. X1 = 1, x(1) =
A = 2, x(2) =
23. A = 1, x(1) = -2 ; X2 = 2, x(2) = Ml ; X3 = -1
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