Section 5.5

1. $y_{p} (x) = \frac{1}{25} e^{3 x}$ $y_{p} (x) = \frac{1}{25} e^{3 x}$
2. $y_{p} (x) = - \frac{1}{4} (5 + 6 x)$ $y_{p} (x) = - \frac{1}{4} (5 + 6 x)$
3. $y_{p} (x) = \frac{1}{39} (\cos 3 x - 5 \sin 3 x)$ $y_{p} (x) = \frac{1}{39} (\cos 3 x - 5 \sin 3 x)$
4. $y_{p} (x) = \frac{1}{9} (- 4 e^{x} + 3 {x e}^{x})$ $y_{p} (x) = \frac{1}{9} (- 4 e^{x} + 3 {x e}^{x})$
5. $y_{p} (x) = \frac{1}{26} (13 + 3 \cos 2 x - 2 \sin 2 x)$ $y_{p} (x) = \frac{1}{26} (13 + 3 \cos 2 x - 2 \sin 2 x)$
6. $y_{p} (x) = \frac{1}{343} (4 - 56 x + 49 x^{2})$ $y_{p} (x) = \frac{1}{343} (4 - 56 x + 49 x^{2})$
7. $y_{p} (x) = - \frac{1}{6} (e^{x} - e^{- x}) = - \frac{1}{3} \sinh x$ $y_{p} (x) = - \frac{1}{6} (e^{x} - e^{- x}) = - \frac{1}{3} \sinh x$
8. $y_{p} (x) = \frac{1}{4} x \sinh 2 x$ $y_{p} (x) = \frac{1}{4} x \sinh 2 x$
9. $y_{p} (x) = - \frac{1}{3} + \frac{1}{16} (2 x^{2} - x) e^{x}$ $y_{p} (x) = - \frac{1}{3} + \frac{1}{16} (2 x^{2} - x) e^{x}$
10. $y_{p} (x) = \frac{1}{6} (2 x \sin 3 x - 3 x \cos 3 x)$ $y_{p} (x) = \frac{1}{6} (2 x \sin 3 x - 3 x \cos 3 x)$
11. $y_{p} (x) = \frac{1}{8} (3 x^{2} - 2 x)$ $y_{p} (x) = \frac{1}{8} (3 x^{2} - 2 x)$
12. $y_{p} (x) = 2 x + \frac{1}{2} x \sin x$ $y_{p} (x) = 2 x + \frac{1}{2} x \sin x$
13. $y_{p} (x) = \frac{1}{65} e^{x} (7 \sin x - 4 \cos x)$ $y_{p} (x) = \frac{1}{65} e^{x} (7 \sin x - 4 \cos x)$
14. $y_{p} (x) = \frac{1}{24} (- 3 x^{2} e^{x} + x^{3} e^{x})$ $y_{p} (x) = \frac{1}{24} (- 3 x^{2} e^{x} + x^{3} e^{x})$
15. $y_{p} (x) \equiv - 17$ $y_{p} (x) \equiv - 17$
16. $y_{p} (x) = \frac{1}{81} (45 + e^{3 x} - 6 {x e}^{3 x} + 9 x^{2} e^{3 x})$ $y_{p} (x) = \frac{1}{81} (45 + e^{3 x} - 6 {x e}^{3 x} + 9 x^{2} e^{3 x})$
17. $y_{p} (x) = \frac{1}{4} (x^{2} \sin x - x \cos x)$ $y_{p} (x) = \frac{1}{4} (x^{2} \sin x - x \cos x)$
18. $y_{p} (x) = - \frac{1}{144} (24 {x e}^{x} - 19 {x e}^{2 x} + 6 x^{2} e^{2 x})$ $y_{p} (x) = - \frac{1}{144} (24 {x e}^{x} - 19 {x e}^{2 x} + 6 x^{2} e^{2 x})$
19. $y_{p} (x) = \frac{1}{8} (10 x^{2} - 4 x^{3} + x^{4})$ $y_{p} (x) = \frac{1}{8} (10 x^{2} - 4 x^{3} + x^{4})$
20. $y_{p} (x) = - 7 + \frac{1}{3} {x e}^{x}$ $y_{p} (x) = - 7 + \frac{1}{3} {x e}^{x}$
21. $y_{p} (x) = x e^{x} (A \cos x + B \sin x)$ $y_{p} (x) = x e^{x} (A \cos x + B \sin x)$
22. $y_{p} (x) = A x^{3} + B x^{4} + C x^{5} + D x e^{x}$ $y_{p} (x) = A x^{3} + B x^{4} + C x^{5} + D x e^{x}$
23. $y_{p} (x) = A x \cos 2 x + B x \sin 2 x + C x^{2} \cos 2 x + D x^{2} \sin 2 x$ $y_{p} (x) = A x \cos 2 x + B x \sin 2 x + C x^{2} \cos 2 x + D x^{2} \sin 2 x$
24. $y_{p} (x) = A x + B x^{2} + (C x + D x^{2}) e^{- 3 x}$ $y_{p} (x) = A x + B x^{2} + (C x + D x^{2}) e^{- 3 x}$
25. $y_{p} (x) = A x e^{- x} + B x^{2} e^{- x} + C x e^{- 2 x} + D x^{2} e^{- 2 x}$ $y_{p} (x) = A x e^{- x} + B x^{2} e^{- x} + C x e^{- 2 x} + D x^{2} e^{- 2 x}$
26. $y_{p} (x) = (A x + B x^{2}) e^{3 x} \cos 2 x + (C x + D x^{2}) e^{3 x} \sin 2 x$ $y_{p} (x) = (A x + B x^{2}) e^{3 x} \cos 2 x + (C x + D x^{2}) e^{3 x} \sin 2 x$
27. $y_{p} (x) = A x \cos x + B x \sin x + C x \cos 2 x + D x \sin 2 x$ $y_{p} (x) = A x \cos x + B x \sin x + C x \cos 2 x + D x \sin 2 x$
28. $y_{p} (x) = (Ax + {Bx}^{2} + {Cx}^{3}) \cos 3 x + (Dx + {Ex}^{2} + {Fx}^{3}) \sin 3 x$ $y_{p} (x) = (Ax + {Bx}^{2} + {Cx}^{3}) \cos 3 x + (Dx + {Ex}^{2} + {Fx}^{3}) \sin 3 x$
29. $y_{p} (x) = A x^{3} e^{x} + B x^{4} e^{x} + C x e^{2 x} + D x e^{- 2 x}$ $y_{p} (x) = A x^{3} e^{x} + B x^{4} e^{x} + C x e^{2 x} + D x e^{- 2 x}$
30. $y_{p} (x) = (A + B x + C x^{2}) \cos x + (D + E x + F x^{2}) \sin x$ $y_{p} (x) = (A + B x + C x^{2}) \cos x + (D + E x + F x^{2}) \sin x$
31. $y (x) = \cos 2 x + \frac{3}{4} \sin 2 x + \frac{1}{2} x$ $y (x) = \cos 2 x + \frac{3}{4} \sin 2 x + \frac{1}{2} x$
32. $y (x) = \frac{1}{6} (15 e^{- x} - 16 e^{- 2 x} + e^{x})$ $y (x) = \frac{1}{6} (15 e^{- x} - 16 e^{- 2 x} + e^{x})$
33. $y (x) = \cos 3 x - \frac{2}{15} \sin 3 x + \frac{1}{5} \sin 2 x$ $y (x) = \cos 3 x - \frac{2}{15} \sin 3 x + \frac{1}{5} \sin 2 x$
34. $y (x) = \cos x - \sin x + \frac{1}{2} x \sin x$ $y (x) = \cos x - \sin x + \frac{1}{2} x \sin x$
35. $y (x) = e^{x} (2 \cos x - \frac{5}{2} \sin x) + \frac{1}{2} x + 1$ $y (x) = e^{x} (2 \cos x - \frac{5}{2} \sin x) + \frac{1}{2} x + 1$
36. $y (x) = \frac{1}{192} (234 + 240 x - 9 e^{- 2 x} - 33 e^{2 x} - 12 x^{2} - 4 x^{4})$ $y (x) = \frac{1}{192} (234 + 240 x - 9 e^{- 2 x} - 33 e^{2 x} - 12 x^{2} - 4 x^{4})$
37. $y (x) = 4 - 4 e^{x} + 3 x e^{x} + x - \frac{1}{2} x^{2} e^{x} + \frac{1}{6} x^{3} e^{x}$ $y (x) = 4 - 4 e^{x} + 3 x e^{x} + x - \frac{1}{2} x^{2} e^{x} + \frac{1}{6} x^{3} e^{x}$
38. $y (x) = \frac{1}{85} [e^{- x} (176 \cos x + 197 \sin x) - (6 \cos 3 x + 7 \sin 3 x)]$ $y (x) = \frac{1}{85} [e^{- x} (176 \cos x + 197 \sin x) - (6 \cos 3 x + 7 \sin 3 x)]$
39. $y (x) = - 3 + 3 x - \frac{1}{2} x^{2} + \frac{1}{6} x^{3} + 4 e^{- x} + x e^{- x}$ $y (x) = - 3 + 3 x - \frac{1}{2} x^{2} + \frac{1}{6} x^{3} + 4 e^{- x} + x e^{- x}$
40. $y (x) = \frac{1}{4} (5 e^{- x} + 5 e^{x} + 10 \cos x - 20)$ $y (x) = \frac{1}{4} (5 e^{- x} + 5 e^{x} + 10 \cos x - 20)$
41. $y_{p} (x) = 255 - 450 x + 30 x^{2} + 20 x^{3} + 10 x^{4} - 4 x^{5}$ $y_{p} (x) = 255 - 450 x + 30 x^{2} + 20 x^{3} + 10 x^{4} - 4 x^{5}$
42. $y (x) = 10 e^{- x} + 35 e^{2 x} + 210 \cos x + 390 \sin x + y_{p} (x)$ $y (x) = 10 e^{- x} + 35 e^{2 x} + 210 \cos x + 390 \sin x + y_{p} (x)$ where $y_{p} (x)$ $y_{p} (x)$ is the particular solution of Problem 41 .
43. (b) $y (x) = c_{1} \cos 2 x + c_{2} \sin 2 x + \frac{1}{4} \cos x - \frac{1}{20} \cos 3 x$ $y (x) = c_{1} \cos 2 x + c_{2} \sin 2 x + \frac{1}{4} \cos x - \frac{1}{20} \cos 3 x$
44. $y (x) = e^{- x / 2} [c_{1} \cos (\frac{1}{2} x \sqrt{3}) + c_{2} \sin (\frac{1}{2} x \sqrt{3})] + \frac{1}{26} (- 3 \cos 2 x + 2 \sin 2 x) + \frac{1}{482} (15 \cos 4 x + 4 \sin 4 x)$ $y (x) = e^{- x / 2} [c_{1} \cos (\frac{1}{2} x \sqrt{3}) + c_{2} \sin (\frac{1}{2} x \sqrt{3})] + \frac{1}{26} (- 3 \cos 2 x + 2 \sin 2 x) + \frac{1}{482} (15 \cos 4 x + 4 \sin 4 x)$
45. $y (x) = c_{1} \cos 3 x + c_{2} \sin 3 x + \frac{1}{24} - \frac{1}{10} \cos 2 x - \frac{1}{56} \cos 4 x$ $y (x) = c_{1} \cos 3 x + c_{2} \sin 3 x + \frac{1}{24} - \frac{1}{10} \cos 2 x - \frac{1}{56} \cos 4 x$
46. $y (x) = c_{1} \cos x + c_{2} \sin x + \frac{1}{16} (3 x \cos x + 3 x^{2} \sin x) + \frac{1}{128} (3 \sin 3 x - 4 x \cos 3 x)$ $y (x) = c_{1} \cos x + c_{2} \sin x + \frac{1}{16} (3 x \cos x + 3 x^{2} \sin x) + \frac{1}{128} (3 \sin 3 x - 4 x \cos 3 x)$
47. $y_{p} (x) = \frac{2}{3} e^{x}$ $y_{p} (x) = \frac{2}{3} e^{x}$
48. $y_{p} (x) = - \frac{1}{12} (6 x + 1) e^{- 2 x}$ $y_{p} (x) = - \frac{1}{12} (6 x + 1) e^{- 2 x}$
49. $y_{p} (x) = x^{2} e^{2 x}$ $y_{p} (x) = x^{2} e^{2 x}$
50. $y_{p} (x) = \frac{1}{16} (4 x \cosh 2 x - \sinh 2 x)$ $y_{p} (x) = \frac{1}{16} (4 x \cosh 2 x - \sinh 2 x)$
51. $y_{p} (x) = - \frac{1}{4} (\cos 2 x \cos x - \sin 2 x \sin x) + \frac{1}{20} (\cos 5 x \cos 2 x + \sin 5 x \sin 2 x) = - \frac{1}{5} \cos 3 x$ $y_{p} (x) = - \frac{1}{4} (\cos 2 x \cos x - \sin 2 x \sin x) + \frac{1}{20} (\cos 5 x \cos 2 x + \sin 5 x \sin 2 x) = - \frac{1}{5} \cos 3 x$ (!)
52. $y_{p} (x) = - \frac{1}{6} x \cos 3 x$ $y_{p} (x) = - \frac{1}{6} x \cos 3 x$
53. $y_{p} (x) = \frac{2}{3} x \sin 3 x + \frac{2}{9} (\cos 3 x) \ln | \cos 3 x |$ $y_{p} (x) = \frac{2}{3} x \sin 3 x + \frac{2}{9} (\cos 3 x) \ln | \cos 3 x |$
54. $y_{p} (x) = - 1 - (\cos x) \ln | \csc x - \cot x |$ $y_{p} (x) = - 1 - (\cos x) \ln | \csc x - \cot x |$
55. $y_{p} (x) = - \frac{1}{8} (1 - x \sin 2 x)$ $y_{p} (x) = - \frac{1}{8} (1 - x \sin 2 x)$
56. $y_{p} (x) = - \frac{1}{9} e^{x} (3 x + 2)$ $y_{p} (x) = - \frac{1}{9} e^{x} (3 x + 2)$
58. $y_{p} (x) = x^{3} (\ln x - 1)$ $y_{p} (x) = x^{3} (\ln x - 1)$
59. $y_{p} (x) = \frac{1}{4} x^{4}$ $y_{p} (x) = \frac{1}{4} x^{4}$
60. $y_{p} (x) = - \frac{72}{5} x^{4 / 3}$ $y_{p} (x) = - \frac{72}{5} x^{4 / 3}$
61. $y_{p} (x) = \ln x$ $y_{p} (x) = \ln x$
62. $y_{p} (x) = - x^{2} + x \ln | \frac{1 + x}{1 - x} | + \frac{1}{2} (1 + x^{2}) \ln | 1 - x^{2} |$ $y_{p} (x) = - x^{2} + x \ln | \frac{1 + x}{1 - x} | + \frac{1}{2} (1 + x^{2}) \ln | 1 - x^{2} |$

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Table of Contents for Section 5.5

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Table of Contents for
Section 5.5