26: Section 7.3 Answers

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1. \(y=c_1+c_2x+c_3e^{-x}+{2\over 3}x^4-{8\over 3}x^3+8x^2\)

2. \(y=c_1+c_2x+c_3e^{-8x}+{11\over 256}x^2+{7\over 32}x^3-{1\over 16}x^4\)

3. \(y=c_1e^x+c_2\cos x+c_3\sin x-7+{1\over 4}e^{-x}-{1\over 2}xe^x+{1\over 4}x^2e^x\)

4. \(y=c_1e^x+c_2xe^x+c_3x^2e^x-13+x+{1\over 6}x^3e^x\)

5. \(y=c_1e^{-x}+c_2e^{2x}+c_3e^{x/2}+1+{1\over 9}xe^{2x}-{1\over 20}e^{-2x}\)

6. \(y=c_1+c_2x+c_3e^x+c_4xe^x+{1\over 2}x^2+{1\over 2}x^2e^x\)

7. \(y=c_1+c_2x+c_3e^{2x}+c_4e^{-2x}-{5\over 16}x^2-{5\over 48}x^4-{1\over 16}xe^{2x}\)

8. \(y=c_1+c_2x+c_3e^{6x}-{1\over 4}x^2-{6\over 37}\cos x+{1\over 37}\sin x\)

9. \(y=c_1e^{2x}+c_2xe^{2x}+c_3e^{-2x}+{1\over 4}x^3e^{2x}-{3\over 16}x^2e^{2x}\)

10. \(y=c_1e^x+c_2xe^x+c_3x^2e^x-x-3-{2\over 3}x^3e^x\)

11. \(y=c_1e^{x}+c_2e^{2x}+c_3e^{-2x}+{5\over 4}+{1\over 3}xe^{x}+{1\over 4}xe^{2x}\)

12. \(y=c_1\cos x+c_2\sin x+c_3x\cos x+c_4x\sin x+x^2-2x-3\)

13. \(y=c_1+c_2x+c_3e^x+c_4e^{-x}-{2\over 3}x^3-{1\over 2}x^2e^{-x}-{5\over 2}xe^{-x}\)

14. \(y=c_1e^x+c_2e^{2x}+c_3e^{3x}+e^{-x}(2+x-x^{2})\)

15. \(y=c_1e^x+c_2e^{3x}+c_3e^{-2x}-\frac{e^{-3x}}{4}(3-x+x^{2})\)

16. \(y=c_1e^{-2x}+c_2e^{-x/2}+c_3e^{x/2}+e^{x}(1+x-x^{2})\)

17. \(y=c_1e^x+c_2e^{-x}+c_3e^{-3x}+e^{-2x}(1-5x+x^{2})\)

18. \(y=c_1e^x+c_2e^{-x}+c_3e^{-3x}-\frac{xe^{x}}{2}(1-x+x^{2}-x^{3})\)

19. \(y=c_1e^x+c_2e^{-x}\cos x+c_3e^{-x}\cos x+x^{2}e^{x}(1+x)\)

20. \(y=c_1e^{-2x}+c_2e^{-x/2}+c_3e^{x/2}+\frac{xe^{-2x}}{2}(2+x)\)

21. \(y=c_1e^x+c_2xe^x+c_3e^{-x}+\frac{x^{2}e^{x}}{2}(2+x)\)

22. \(y=c_1e^{2x}+c_2xe^{2x}+c_3e^{-x/2}+\frac{x^{2}e^{2x}}{2}(1+2x)\)

23. \(y=c_1e^{3x}+c_2xe^{3x}+c_3e^{-x}+x^{2}e^{3x}(2+x-x^{2})\)

24. \(y=c_1e^{4x}+c_2xe^{4x}+c_3e^{-x}+x^{2}e^{4x}(2+x)\)

25. \(y=c_1e^{x/2}+c_2xe^{x/2}+c_3x^2e^{x/2}+\frac{x^{3}e^{x/2}}{48}(1+x)\)

26. \(y=c_1e^{x}+c_2xe^{x}+c_3e^{-2x}+c_4e^{-3x}+e^{-x}(1-2x+x^{2})\)

27. \(y=c_1e^{-x}+c_2xe^{-x}+c_3e^{x}+c_4e^{-2x}+e^{2x}(1-x)\)

28. \(y=c_1+c_2e^{-4x}+c_3e^{-2x}\cos 2x+c_4e^{-2x}\sin 2x+e^{-2x}(1+x+x^{2}-x^{3})\)

29. \(y=c_1e^{-x/2}+c_2xe^{-x/2}+c_3e^{-x}+c_4e^{2x}+\frac{e^{x}}{3}(1-x)\)

30. \(y=c_1e^{-x/2}+c_2e^{-x}+c_3e^{-2x}+c_4e^{x}+xe^{x}(2+x)\)

31. \(y=c_1e^{-x}+c_2e^{x}+c_3e^{-2x}+c_4e^{2x}+\frac{xe^{x}}{6}(1+x^{2})\)

32. \(y=c_1e^{-x}+c_2xe^{-x}+c_3e^{2x}+c_4xe^{2x}+\frac{x^{2}e^{2x}}{6}(1+x+x^{2})\)

33. \(y=c_1+c_2e^{-x}+c_3e^{2x}+c_4xe^{2x}+\frac{x^{2}e^{2x}}{6}(3+x+x^{2})\)

34. \(y=c_1e^{-x}+c_2e^{x}+c_3xe^{x}+c_4x^2e^{x}+\frac{x^{3}e^{x}}{48}(2+x)\)

35. \(y=c_1e^{-x/2}+c_2e^{x}+c_3xe^{x}+c_4x^2e^{x}+\frac{x^{3}e^{x}}{6}(1+x)\)

36. \(y=c_1+c_2e^{-x}+c_3xe^{-x}+c_4x^2e^{-x}-\frac{x^{3}e^{-x}}{6}(1-x+x^{2})\)

37. \(y= c_1e^{x}+c_2e^{2x}+c_3xe^{2x}+c_4x^2e^{2x}+ \frac{x^{3}e^{2x}}{12}(2+x-x^{2})\)

38. \(y=c_1e^x+c_2e^{2x}+c_3e^{-2x}+ e^{−x} \left[ (1 + x) \cos x + (2 − x) \sin x\right] \)

39. \(y=c_1e^{-x}+c_2e^{2x}+c_3e^{-2x}+e^{-x}\left[ (1-x)\cos 2x+(1+x)\sin 2x\right] \)

40. \(y=c_1e^x+c_2\cos (\sqrt{2} x)+c_3\sin (\sqrt{2} x)+e^{2x}\left[ (1+x-x^{2})\cos x +(1+2x)\sin x\right]\)

41. \(y=c_1e^{2x}+c_2\cos x+c_3\sin x+\frac{e^{x}}{2}\left[ (1+x)\cos 2x+(1-x+x^{2})\sin 2x\right] \)

42. \(y=c_1e^{-3x}+c_2\cos 2x+c_3\sin 2x+\frac{x}{13}(8\cos 2x+14\sin 2x)\)

43. \(y=c_1e^{-x}+c_2e^x\cos x+c_3e^x\sin x+xe^{x}\left[ (1+x)\cos x+(3+x)\sin x\right]\)

44. \(y=c_1e^{3x}+c_2e^{2x}\cos 2x+c_3e^{2x}\sin 2x+\frac{xe^{2x}}{2}\left[(3-x)\cos 2x+\sin 2x\right]\)

45. \(y=c_1+c_2e^{3x}\cos 3x+c_3e^{3x}\sin 3x-\frac{xe^{3x}}{12}(x\cos 3x+\sin 3x)\)

46. \(y=c_1e^{-2x}+c_2e^{2x}+c_3e^{-x}\cos x+c_4e^{-x}\sin x-\frac{e^{x}}{10}(\cos x+7\sin x)\)

47. \(y= c_1e^{-x}+c_2e^{2x}+c_3e^{x}\cos x+c_4e^{x}\sin x +\frac{e^{x}}{12}(\cos 2x-\sin 2x)\)

48. \(y= c_1e^{x}+c_2e^{3x}+c_3e^{2x}\cos x+c_4e^{2x}\sin x +xe^{2x}\cos 2x\)

49. \(y= c_1e^{-x}+c_2xe^{-x}+c_3e^{-2x}+c_4xe^{-2x} -\frac{e^{-x}}{2}\left[ (1+x)\cos x+(2-x)\sin x\right]\)

50. \(y= c_1e^{-2x}+c_2e^{x}+c_3e^{-x}\cos x+c_4e^{-x}\sin x +\frac{xe^{-x}}{10}(\cos x+2\sin x)\)

51. \(y= c_1e^{x}+c_2e^{2x}+c_3e^{x}\cos 2x+c_4e^{x}\sin 2x +\frac{xe^{x}}{4-}(3\cos 2x-\sin 2x)\)

52. \(y= c_1e^{-x}+c_2e^{-3x}+c_3e^{-2x}\cos 3x+c_4e^{-2x}\sin 3x +\frac{xe^{-2x}}{8}\left[(1-x)\cos 3x+(1+x)\sin 3x\right]\)

53. \(y= c_1e^{x}+c_2e^{2x}+c_3e^{x}\cos 2x+c_4e^{x}\sin 2x -\frac{xe^{x}}{4}(1+x)\sin 2x\)

54. \(y= c_1e^{-x}\cos x+c_2e^{-x}\sin x+c_3xe^{-x}\cos x+c_4xe^{-x}\sin x+ \frac{x^{2}e^{-x}}{4}(\cos x-2\sin x)\)

55. \(y=c_1e^{2x}\cos 2x+c_2e^{2x}\sin 2x+c_3xe^{2x}\cos 2x+c_4xe^{2x}\sin 2x-\frac{x^{2}e^{2x}}{32}(\cos 2x-\sin 2x)\)

56. \(y=c_1e^{2x}\cos x+c_2e^{2x}\sin x+c_3xe^{2x}\cos x+c_4xe^{2x}\sin x+\frac{x^{2}e^{2x}}{8}(1+x)\sin x\)

57. \(y= c_1e^{x}+c_2xe^{x}+c_3e^{2x}+2x^{2}e^{x}+xe^{2x}-\cos x\)

58. \(y=c_1e^x+c_2\cos x+c_3\sin x+e^{2x}+xe^{x}+2x\cos x\)

59. \(y=c_1+c_2e^x+c_3e^{-x}+2x+x^{2}+2xe^{x}-3xe^{-x}+4e^{3x}\)

60. \(y=c_1e^{2x}+c_2e^x\cos x+c_3e^x\sin x+xe^{x}(\cos 2x-2\sin 2x)+2xe^{2x}+1\)

61. \(y=c_1e^{-x}+c_2xe^{-x}+c_3x^2e^{-x}+x^{2}e^{-2x}(1+2x)-\cos 2x+\sin 2x\)

62. \(y=c_1e^{-x}+c_2xe^{-x}+c_3e^x+2x^{2}(1+x)e^{-x}+x\cos x-2\sin x\)

63. \(y= c_1e^{-x}+c_2e^x+c_3e^{-2x}+c_4e^{2x}+2xe^{x}+xe^{2x}+\cos x\)

64. \(y=c_1e^x\cos x+c_2e^x\sin x+c_3e^x\cos 2x+c_4e^x\sin 2x+\frac{xe^{x}}{6}(\cos x+\sin 2x)\)

65. \(y=c_1e^x+c_2xe^x+c_3e^{-2x}+c_4xe^{-2x}+\frac{x^{2}}{54}\left[(2+2x)e^{x}+3e^{-2x}\right]\)

66. \(y=c_1e^{-x}+c_2xe^{-x}+c_3x^2e^{-x}+c_4e^{2x}+x^{3}(1+x)e^{-x}+xe^{-2x}\)

67. \(y=c_1e^x+c_2xe^x+c_3e^x\cos x+c_4e^x\sin x+xe^{x}(2x^{2}+\cos x+\sin x)\)

68. \(y=e^{2x}(1+x)+c_{1}e^{-x}+e^{x}(c_{2}+c_{3}x)\)

69. \(y=e^{3x}\left( 1-x-\frac{x^{2}}{2}\right)+c_{1}e^{x}+e^{-x}(c_{2}\cos x+c_{3}\sin x)\)

70. \(y=xe^{2x}(1+x)^{2}+c_{1}e^{x}+c_{2}e^{2x}+c_{3}e^{3x}\)

71. \(y=x^{2}e^{-x}(1-x)^{2}+c_{1}+e^{-x}(c_{2}+c_{3}x)\)

72. \(y=\frac{x^{3}e^{x}}{24}(4+x)+e^{x}(c_{1}+c_{2}x+c_{3}x^{2})\)

73. \(y=\frac{x^{2}e^{-x}}{16}(1+2x-x^{2})+e^{x}(c_{1}+c_{2}x)+e^{-x}(c_{3}+c_{4}x)\)

74. \(y=e^{-2x}\left[\left(1+\frac{x}{2}\right)\cos x+\left(\frac{3}{2}-2x\right)\sin x\right] +c_{1}e^{x}+c_{2}e^{-x}+c_{3}e^{-2x}\)

75. \(y=-xe^{x}\sin 2x+c_{1}+c_{2}e^{x}+e^{x}(c_{3}\cos x+c_{4}\sin x)\)

76. \(y=-\frac{x^{2}e^{x}}{16}(1+x)\cos 2x+e^{x}\left[ (c_{1}+c_{2}x)\cos 2x+(c_{3}+c_{4}x)\sin 2x\right]\)

77. \(y=8-6e^x+3xe^x+5x-{1\over 2}x^2e^x+{1\over 6}x^3e^x\)

78. \(y=2+x-2e^x-{1\over 6}x^3-{1\over 24}x^4+xe^x\)

79. \(y=11-11e^x+9xe^x+2x-12x^2e^x+{1\over 2}e^{5x}\)

80. \(y=-{23\over 12}e^{-2x}-{59\over 24}e^x\cos (\sqrt{3} x)+{17\over 72}\sqrt{3} e^{x}\sin (\sqrt{3} x)+{1\over 4}x-{5\over 8}+{2\over 3}xe^{-2x}\)

81. \(y=(x^{2}+2)e^{x}-e^{-2x}+e^{3x}\)

82. \(y=e^{-x}(1+x+x^{2})+(1-x)e^{x}\)

83. \(y=\left(\frac{x^{2}}{12}+16\right)xe^{-x/2}-e^{x}\)

84. \(y=(2-x)(x^{2}+1)e^{-x}+\cos x-\sin x\)

85. \(y=(2-x)\cos x-(1-7x)\sin x+e^{-2x}\)

86. \(y=2+e^{x}\left[ (1+x)\cos x-\sin x-1\right]\)

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