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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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-f
y  c1 cos t I c2 sin i I f (°3 cos 2
y  c1et + c2e-t + c3e2t + c4e
2tef + c3t2et I c4f I c5
y  c1 + c2f + c3ef + c4e-t + c5 cos f + c6 sin f
f 2f
y  c1 + c2e + c3^ + c4 cos f + c5 sin f y  c1 + c2e2t + e-t (c3 cos \/3 f + c4 sin \/3 f)
y  ef[(c1 + c2f) cos f + (c3 + c4f) sin f] + e-t[(c5 + c6f) cos f + (c7 + c8f) sin f]
22. y  (c1 + c2f) cos f + (c3 + c4t) sin f 23. y  c1et + c2e'
24. y  c1e-t + c2e(-2+F2)t + c3e(-2-'^2)t
25. y  c1e-t/2 + c2e-t/3cos(f/V3) + c3e-t/3 sin(f/V3)
26. y  cje3t + c2e-2t + c3e(3+e3)t + c4e(3-e3)t
,(2^V5)t + (2-V5)f
u
6
e
694
See SSM for detailed solutions to 27 ,29 ,30, 31
34, 37, 38a, 39ac 1
5, 9, 13
17, 20, 22a 22be
1, 4, 5, 7
27. y = c1e t/2 + c2e t/4 + c3e 1 cos2t + c4e 1 sin2t
28. y = c1 e-t cos t + c2e-t sin t + c3e2t cos(\/3 t) + c4e2t sin(\/3 t)
29. y = 2  2cost + sint
30. y = 2 et/v^ sin (t/v'?)  2et/v^sin(t/V?)
31. y = 2t  3 32. y = 2cost  sin t
33. y = 3el  110e2t  6e2t  if e1/2 34. y = 123et + ff et/2cos t + j3
35. y = 8  18e1/3 + 8e1/2
36. y = 23 e t cos t  if et sin t  13e2t cos(V3t) + e2t sin(V3 0
37. y = 2 (cosh t  cos t) + 2 (sinh t  sin t)
38. (a) W = c, a constant (b) W =8 (c) W = 4
39. (b) u1 = c1 cos t + c2 sin t + c3 cosV6 f + c4 sin-/6 t
Section 4.3, page 224
1. y = c1et + c2tet + c3et + 1 tet + 3
2. y = c1 et + c2e t + c3 cos t + c4 sin t  3t  4 t sin t
3. y = c1e t + c2 cos t + c3 sin t + i tet + 4(t  1)
4. y = c1 + c2et + c3et + cos t
5. y = c1 + c2t + c3e2t + c4e2t  1 et  48t4  ^t2
6. y = c1 cos t + c2 sin f + c3t cos t + c4t sin t + 3 + 9 cos2t
7. y = c1 + c2t + c3t2 + c4et + et/2[cf cos(v/3 t/2) + c6 sin^V^ t/2)] + 23t4
8. y = c1 + c2t + c3t2 + c4e t + 2O sin2t + 410 cos2t
9. y = 13(1  cos2t) + 1t2
10. y = (t  4) cos t  (2t + 4) sin t + 3t + 4
11. y = 1 + 1 (t2 + 3t)  tet
12. y =f cos t  f sin t + 25et + ijOet + 520e3t + 6f cos2t  j490 sin2t
13. 7 (t) = t ( A0t3 + A1t2 + A2t + A3) + Bt2et
14. Y (t) = t ( A0t + A1)et + B cos t + C sin t
15. Y (t) = At2et + Bcos t + C sin t
16. Y (t) = At2 + ( B0t + B1)et + t (C cos2t + D sin2t)
17. Y (t) = t ( A0t2 + A1t + A2) + ( B0t + B1) cos t + (C0t + C1) sin t
18. Y(t) = Aet + (B0t + B1)et + te ((Ccos t + D sin t)
19. *0 = kn = ^0an + a an 1 +  + an1a + an
Section 4.4, page 229
1. y = c1 + c2 cos t + c3 sin t  ln cos t  (sin t) ln(sec t + tan t)
2. y = c1 + c2e + c3e   212
3. y = c1et + c2e 1 + c3e2t + 30e4t
4. y = c1 + c2 cos t + c3 sin t + ln(sec t + tan t)  t cos t + (sin t) ln cos t
5. y = c1et + c2 cos t + c3 sin t  1 e ( cos t
6. y = c1 cos t + c2 sin f + c31 cos t + c41 sin t  112 sin t
7. y = c1 et + c2 cos t + c3 sin t  2 (cos t) ln cos t + 1 (sin t) ln cos t  2 t cos t 
+ 2 e\ft (es/ cos ds
et/2 sin t
2 t sin t
695
See SSM for detailed solutions to 11, 14
16
8. y = Cj + c2el + c3e 1  lnsin t + ln(cos t + 1) + 2el ? ^e s/ sins^ ds
+ I et ? ^es/ sin s^ ds
9. Cj = 0, c2 = 2, c3 = 1 in answer to Problem 4
10. cj = 2, c2 = 7, c3 = 7, c4 = I in answer to Problem 6
11. cj = |, c2 = 2, c3 = 5, t0 = 0 in answer to Problem 7
12. c1 = 3, c2 = 0, c3 = en/2, t0 = n/2 in answer to Problem 8
13. Y (x) = x4/15
14. Y(t) = 1 f [ets  sin(t  s)  cos(t  s)]er(s) ds
Jt0
15. Y (t) = if [sinh(t  s)  sin(t  s)]er(s) ds
Jt0
16. Y(t) = i f e(ts')(t  s)2er(s) ds; Y(t) = tel ln |t|
2 Jt0
17. Y (x) = if X [(x/t2)  2(x2/t3) + (x3/t4)]g(t) dt
2 J xn
CHAPTER 5 Section 5.1, page 237
2, 5, 9, 12, 13
18, 19, 23, 25, 28
1. p = 1
3. p = to
5. p = 2
7. p = 3
^ (- 1)nx2n+1
9. >-----------------------------------------, p = oo
(2n + 1)!
11. 1 + (x - 1), p = to
V+ (x - 1)n, n
13. (-1)n
p =1
15. ? xn, p = 1
n=o
2. p = 2
4. p = 2
6. p = 1
8. p = e xn
1oi:
n=o
n!
p = to
12. 1 - 2(x + 1) + (x + 1)2, p = to
to
14. (-1)nxn, p = 1
n=o
TO
16. (-1)n+1(x - 2)n, p = 1
n=o
17. / = 1 + 22x-
+ (n + 1)2 xn + 
/ = 22 + 32  2x + 42  3x2 + 52  4x3 +  + (n + 2)2(n + 1)xn + 
18. / = a1 + 2a2x + 3a3x + 4a4^ +  + (n + 1)an+1 x + 
TOTO
= 1] nanxn-1 = 1](n + 1)a
n+1
xn
n= 1
n=o
y = 2a2 + 6a3 x + 12a4 x2 + 2oa5 x3 +  + (n + 2)(n + 1)an+2 xn + 
TOTO
= n(n - 1)anxn-2 = (n + 2)(n + 1)an+2xn
21. ^2(n+2)(n+1)an+2 xn n=o 22
23. TO !> + 1)anxn n=o 24
25. TO J2 [(n + 2)(n + 1)an+2 + nan]xn n=o 26
27. TO J2[(n + 1)nan+1 + an]xn 28
an-1J
n= 1
n = 1, 2,...;
n=o
a
2
aoe
o
n
696
See SSM for detailed solutions to 2
3
Section 5.2, page 247
1 3n+2 = 3n/(n + 2)(n + 1)
x2 x4 x6
7l(x) = 1 +  +  +  + 
2!
x3
4!
x5
6!
=
x2n
= coshx
y2(x) = x +-------\------\ + 
J2 3! 5! 7!
2 an+2 = an/(n + 2)
x2 x4 x6
yi (x) = 1 +------+-----------------------------+-
^ 2 2  4 2  4  6
x3 x5 x7
y2(x) = x +-------\---------------------\--------------------
J2 3 3 ? 5 3-5-7
n=0 (2n)!
to x2 n+1
= 'y------------------- = sinh x
n=0 (2n + 1)!
\
\
=
n=0
to
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