Page 362 - Schaum's Outline of Differential Equations

ANSWERS TO SUPPLEMENTARY PROBLEMS 345

2x
8.55. [-6]sin;c+[-l](2cosjc) 8.56. y = c 1e + c 2e- 2x
+ [2] (3 sin x + cos x) = 0
2
8.57. y = c^e " + c 2e^ x 8.58. y = c 1 sin 4x + c 2 cos 4x

8.59. y = c^* + c 2
8.60. Since y± and y 2 are linearly dependent, there is not enough information provided to exhibit the general solution.

8.61. y = CiX + c^e* + C 3y 3 where y 3 is a third particular solution, linearly independent from the other two.

8.62. Since the given set is linearly dependent, not enough information is provided to exhibit the general solution.

8.63. y = c^ + c 2e* + C 3e 2j:

2
4
8.64. y = CiX 2 + c^x" + c 3x + C 4y 4 + c sy s, where y 4 and y s are two other solutions having the property that the set (x , x',
4
x , y 4, y$} is linearly independent.
2
8.65. y = GI sin x + c 2 cos x + x — 2
8.66. Since e* and 3e" are linearly dependent, there is not enough information given to find the general solution.
x
8.67. y = c^ + c 2e~ + Cjxe* + 5
8.68. Theorem 8.1 does not apply, since a Q(x) = —(2lx) is not continuous about X Q = 0.

8.69. Yes; a Q(x) is continuous aboutx Q =l.

8.70. Theorem 8.1 does not apply, since bi(x) is zero at the origin.

CHAPTER 9

5x
x
9.17. y = c 1e + c 2e~ x 9.18. y = c le- + c 2e 6x
x
9.19. y = c 1e + c 2xe x 9.20. y = c 1 cos x + c 2 sin x

x
x
9.21. y = Cie~ cos x + c 2e~ sin x 9.22. y = cf^ x + c 2e~^ x
x
9.23. y = c^ + c 2xe- 3x 9.24. y = cf~ x cos -fix + c 2e~ sin J2x
9.25. y = c, e K3 + ^ m:t + c 2e K3 -^ mx 9.26. y = c le- (l!2)x + c 2xe- (l!2)x

4
16
50t
9.27. x = cje ' + c 2e ' 9.28. x = c ie- + c 2e- wt
) 2
5
<3 rs) 2
st
9.29. x = c 1e <3 + ^ " +c 2e -' " 9.30. x = c 1e +c 2te '
9.31. x = Ci cos 5t + c, sin 5t 9.32. x = c 1 + c 2e- 2St