Section 5.3

1. $y (x) = c_{1} e^{2 x} + c_{2} e^{- 2 x}$ $y (x) = c_{1} e^{2 x} + c_{2} e^{- 2 x}$
2. $y (x) = c_{1} + c_{2} e^{3 x / 2}$ $y (x) = c_{1} + c_{2} e^{3 x / 2}$
3. $y (x) = c_{1} e^{2 x} + c_{2} e^{- 5 x}$ $y (x) = c_{1} e^{2 x} + c_{2} e^{- 5 x}$
4. $y (x) = c_{1} e^{x / 2} + c_{2} e^{3 x}$ $y (x) = c_{1} e^{x / 2} + c_{2} e^{3 x}$
5. $y (x) = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x}$ $y (x) = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x}$
6. $y (x) = e^{- 5 x / 2} [c_{1} \exp (\frac{1}{2} x \sqrt{5}) + c_{2} \exp (- \frac{1}{2} x \sqrt{5})]$ $y (x) = e^{- 5 x / 2} [c_{1} \exp (\frac{1}{2} x \sqrt{5}) + c_{2} \exp (- \frac{1}{2} x \sqrt{5})]$
7. $y (x) = c_{1} e^{3 x / 2} + c_{2} x e^{3 x / 2}$ $y (x) = c_{1} e^{3 x / 2} + c_{2} x e^{3 x / 2}$
8. $y (x) = e^{3 x} (c_{1} \cos 2 x + c_{2} \sin 2 x)$ $y (x) = e^{3 x} (c_{1} \cos 2 x + c_{2} \sin 2 x)$
9. $y (x) = e^{- 4 x} (c_{1} \cos 3 x + c_{2} \sin 3 x)$ $y (x) = e^{- 4 x} (c_{1} \cos 3 x + c_{2} \sin 3 x)$
10. $y (x) = c_{1} + c_{2} x + c_{3} x^{2} + c_{4} e^{- 3 x / 5}$ $y (x) = c_{1} + c_{2} x + c_{3} x^{2} + c_{4} e^{- 3 x / 5}$
11. $y (x) = c_{1} + c_{2} x + c_{3} e^{4 x} + c_{4} x e^{4 x}$ $y (x) = c_{1} + c_{2} x + c_{3} e^{4 x} + c_{4} x e^{4 x}$
12. $y (x) = c_{1} + c_{2} e^{x} + c_{3} {x e}^{x} + c_{4} x^{2} e^{x}$ $y (x) = c_{1} + c_{2} e^{x} + c_{3} {x e}^{x} + c_{4} x^{2} e^{x}$
13. $y (x) = c_{1} + c_{2} e^{- 2 x / 3} + c_{3} x e^{- 2 x / 3}$ $y (x) = c_{1} + c_{2} e^{- 2 x / 3} + c_{3} x e^{- 2 x / 3}$
14. $y (x) = c_{1} e^{x} + c_{2} e^{- x} + c_{3} \cos 2 x + c_{4} \sin 2 x$ $y (x) = c_{1} e^{x} + c_{2} e^{- x} + c_{3} \cos 2 x + c_{4} \sin 2 x$
15. $y (x) = c_{1} e^{2 x} + c_{2} x e^{2 x} + c_{3} e^{- 2 x} + c_{4} x e^{- 2 x}$ $y (x) = c_{1} e^{2 x} + c_{2} x e^{2 x} + c_{3} e^{- 2 x} + c_{4} x e^{- 2 x}$
16. $y (x) = (c_{1} + c_{2} x) \cos 3 x + (c_{3} + c_{4} x) \sin 3 x$ $y (x) = (c_{1} + c_{2} x) \cos 3 x + (c_{3} + c_{4} x) \sin 3 x$
17. $y (x) = c_{1} \cos (x / \sqrt{2}) + c_{2} \sin (x / \sqrt{2}) + c_{3} \cos (2 x / \sqrt{3}) + c_{4} \sin (2 x / \sqrt{3})$ $y (x) = c_{1} \cos (x / \sqrt{2}) + c_{2} \sin (x / \sqrt{2}) + c_{3} \cos (2 x / \sqrt{3}) + c_{4} \sin (2 x / \sqrt{3})$
18. $y (x) = c_{1} e^{2 x} + c_{2} e^{- 2 x} + c_{3} \cos 2 x + c_{4} \sin 2 x$ $y (x) = c_{1} e^{2 x} + c_{2} e^{- 2 x} + c_{3} \cos 2 x + c_{4} \sin 2 x$
19. $y (x) = c_{1} e^{x} + c_{2} e^{- x} + c_{3} x e^{- x}$ $y (x) = c_{1} e^{x} + c_{2} e^{- x} + c_{3} x e^{- x}$
20. $y (x) = e^{- x / 2} [(c_{1} + c_{2} x) \cos (\frac{1}{2} x \sqrt{3}) + (c_{3} + c_{3} x) \sin (\frac{1}{2} x \sqrt{3})]$ $y (x) = e^{- x / 2} [(c_{1} + c_{2} x) \cos (\frac{1}{2} x \sqrt{3}) + (c_{3} + c_{3} x) \sin (\frac{1}{2} x \sqrt{3})]$
21. $y (x) = 5 e^{x} + 2 e^{3 x}$ $y (x) = 5 e^{x} + 2 e^{3 x}$
22. $y (x) = e^{- x / 3} [3 \cos (x / \sqrt{3}) + 5 \sqrt{3} \sin (x / \sqrt{3})]$ $y (x) = e^{- x / 3} [3 \cos (x / \sqrt{3}) + 5 \sqrt{3} \sin (x / \sqrt{3})]$
23. $y (x) = e^{3 x} (3 \cos 4 x - 2 \sin 4 x)$ $y (x) = e^{3 x} (3 \cos 4 x - 2 \sin 4 x)$
24. $y (x) = \frac{1}{2} (- 7 + e^{2 x} + 8 e^{- x / 2})$ $y (x) = \frac{1}{2} (- 7 + e^{2 x} + 8 e^{- x / 2})$
25. $y (x) = \frac{1}{4} (- 13 + 6 x + 9 e^{- 2 x / 3})$ $y (x) = \frac{1}{4} (- 13 + 6 x + 9 e^{- 2 x / 3})$
26. $y (x) = \frac{1}{5} (24 - 9 e^{- 5 x} - 25 {x e}^{- 5 x})$ $y (x) = \frac{1}{5} (24 - 9 e^{- 5 x} - 25 {x e}^{- 5 x})$
27. $y (x) = c_{1} e^{x} + c_{2} e^{- 2 x} + c_{3} x e^{- 2 x}$ $y (x) = c_{1} e^{x} + c_{2} e^{- 2 x} + c_{3} x e^{- 2 x}$
28. $y (x) = c_{1} e^{2 x} + c_{2} e^{- x} + c_{3} e^{- x / 2}$ $y (x) = c_{1} e^{2 x} + c_{2} e^{- x} + c_{3} e^{- x / 2}$
29. $y (x) = c_{1} e^{- 3 x} + e^{3 x / 2} [c_{2} \cos (\frac{3}{2} x \sqrt{3}) + c_{3} \sin (\frac{3}{2} x \sqrt{3})]$ $y (x) = c_{1} e^{- 3 x} + e^{3 x / 2} [c_{2} \cos (\frac{3}{2} x \sqrt{3}) + c_{3} \sin (\frac{3}{2} x \sqrt{3})]$
30. $y (x) = c_{1} e^{- x} + c_{2} e^{2 x} + c_{3} \cos (x \sqrt{3}) + c_{4} \sin (x \sqrt{3})$ $y (x) = c_{1} e^{- x} + c_{2} e^{2 x} + c_{3} \cos (x \sqrt{3}) + c_{4} \sin (x \sqrt{3})$
31. $y (x) = c_{1} e^{x} + e^{- 2 x} (c_{2} \cos 2 x + c_{3} \sin 2 x)$ $y (x) = c_{1} e^{x} + e^{- 2 x} (c_{2} \cos 2 x + c_{3} \sin 2 x)$
32. $y (x) = c_{1} e^{2 x} + (c_{2} + c_{3} x + c_{4} x^{2}) e^{- x}$ $y (x) = c_{1} e^{2 x} + (c_{2} + c_{3} x + c_{4} x^{2}) e^{- x}$
33. $y (x) = c_{1} e^{3 x} + e^{- 3 x} (c_{2} \cos 3 x + c_{3} \sin 3 x)$ $y (x) = c_{1} e^{3 x} + e^{- 3 x} (c_{2} \cos 3 x + c_{3} \sin 3 x)$
34. $y (x) = c_{1} e^{2 x / 3} + c_{2} \cos 2 x + c_{3} \sin 2 x$ $y (x) = c_{1} e^{2 x / 3} + c_{2} \cos 2 x + c_{3} \sin 2 x$
35. $y (x) = c_{1} e^{- x / 2} + c_{2} e^{- x / 3} + c_{3} \cos 2 x + c_{4} \sin 2 x$ $y (x) = c_{1} e^{- x / 2} + c_{2} e^{- x / 3} + c_{3} \cos 2 x + c_{4} \sin 2 x$
36. $y (x) = c_{1} e^{7 x / 9} + e^{- x} (c_{2} \cos x + c_{3} \sin x)$ $y (x) = c_{1} e^{7 x / 9} + e^{- x} (c_{2} \cos x + c_{3} \sin x)$
37. $y (x) = 11 + 5 x + 3 x^{2} + 7 e^{x}$ $y (x) = 11 + 5 x + 3 x^{2} + 7 e^{x}$
38. $y (x) = 2 e^{5 x} - 2 \cos 10 x$ $y (x) = 2 e^{5 x} - 2 \cos 10 x$
39. $y^{(3)} - 6 y ″ + 12 y ′ - 8 y = 0$ $y^{(3)} - 6 y ″ + 12 y ′ - 8 y = 0$
40. $y^{(3)} - 2 y ″ + 4 y ′ - 8 y = 0$ $y^{(3)} - 2 y ″ + 4 y ′ - 8 y = 0$
41. $y^{(4)} - 16 y = 0$ $y^{(4)} - 16 y = 0$
42. $y^{(6)} + 12 y^{(4)} + 48 y ″ + 64 y = 0$ $y^{(6)} + 12 y^{(4)} + 48 y ″ + 64 y = 0$
44. (a) $x = i, - 2 i (b) x = - i, 3 i$ $x = i, - 2 i (b) x = - i, 3 i$
45. $y (x) = c_{1} e^{- i x} + c_{2} e^{3 i x}$ $y (x) = c_{1} e^{- i x} + c_{2} e^{3 i x}$
46. $y (x) = c_{1} e^{3 i x} + c_{2} e^{- 2 i x}$ $y (x) = c_{1} e^{3 i x} + c_{2} e^{- 2 i x}$
47. $y (x) = c_{1} \exp ([1 + i \sqrt{3}] x) + c_{2} \exp (- [1 + i \sqrt{3}] x)$ $y (x) = c_{1} \exp ([1 + i \sqrt{3}] x) + c_{2} \exp (- [1 + i \sqrt{3}] x)$
48. $y (x) = \frac{1}{3} (e^{x} + \exp [\frac{1}{2} (- 1 + i \sqrt{3}) x] + \exp [\frac{1}{2} (- 1 - i \sqrt{3}) x])$ $y (x) = \frac{1}{3} (e^{x} + \exp [\frac{1}{2} (- 1 + i \sqrt{3}) x] + \exp [\frac{1}{2} (- 1 - i \sqrt{3}) x])$
49. $y (x) = 2 e^{2 x} - 5 e^{- x} + 3 \cos x - 9 \sin x$ $y (x) = 2 e^{2 x} - 5 e^{- x} + 3 \cos x - 9 \sin x$
52. $y (x) = c_{1} \cos (3 \ln x) + c_{2} \sin (3 \ln x)$ $y (x) = c_{1} \cos (3 \ln x) + c_{2} \sin (3 \ln x)$
53. $y (x) = x^{- 3} [c_{1} \cos (4 \ln x) + c_{2} \sin (4 \ln x)]$ $y (x) = x^{- 3} [c_{1} \cos (4 \ln x) + c_{2} \sin (4 \ln x)]$
54. $y (x) = c_{1} + c_{2} \ln x + c_{3} x^{- 3}$ $y (x) = c_{1} + c_{2} \ln x + c_{3} x^{- 3}$
55. $y (x) = c_{1} + x^{2} (c_{2} + c_{3} \ln x)$ $y (x) = c_{1} + x^{2} (c_{2} + c_{3} \ln x)$
56. $y (x) = c_{1} + c_{2} \ln x + c_{3} {(\ln x)}^{2}$ $y (x) = c_{1} + c_{2} \ln x + c_{3} {(\ln x)}^{2}$
57. $y (x) = c_{1} + x^{3} (c_{2} x^{- \sqrt{3}} + c_{3} x^{+ \sqrt{3}})$ $y (x) = c_{1} + x^{3} (c_{2} x^{- \sqrt{3}} + c_{3} x^{+ \sqrt{3}})$
58. $y (x) = x^{- 1} [c_{1} + c_{2} \ln x + c_{3} {(\ln x)}^{2}]$ $y (x) = x^{- 1} [c_{1} + c_{2} \ln x + c_{3} {(\ln x)}^{2}]$

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Table of Contents for Section 5.3

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Table of Contents for
Section 5.3