Section 5.1

1. $y (x) = \frac{5}{2} e^{x} - \frac{5}{2} e^{- x}$ $y (x) = \frac{5}{2} e^{x} - \frac{5}{2} e^{- x}$
2. $y (x) = 2 e^{3 x} - 3 e^{- 3 x}$ $y (x) = 2 e^{3 x} - 3 e^{- 3 x}$
3. $y (x) = 3 \cos 2 x + 4 \sin 2 x$ $y (x) = 3 \cos 2 x + 4 \sin 2 x$
4. $y (x) = 10 \cos 5 x - 2 \sin 5 x$ $y (x) = 10 \cos 5 x - 2 \sin 5 x$
5. $y (x) = 2 e^{x} - e^{2 x}$ $y (x) = 2 e^{x} - e^{2 x}$
6. $y (x) = 4 e^{2 x} + 3 e^{- 3 x}$ $y (x) = 4 e^{2 x} + 3 e^{- 3 x}$
7. $y (x) = 6 - 8 e^{- x}$ $y (x) = 6 - 8 e^{- x}$
8. $y (x) = \frac{1}{3} (14 - 2 e^{3 x})$ $y (x) = \frac{1}{3} (14 - 2 e^{3 x})$
9. $y (x) = 2 e^{- x} + x e^{- x}$ $y (x) = 2 e^{- x} + x e^{- x}$
10. $y (x) = 3 e^{5 x} - 2 {x e}^{5 x}$ $y (x) = 3 e^{5 x} - 2 {x e}^{5 x}$
11. $y (x) = 5 e^{x} \sin x$ $y (x) = 5 e^{x} \sin x$
12. $y (x) = e^{- 3 x} (2 \cos 2 x + 3 \sin 2 x)$ $y (x) = e^{- 3 x} (2 \cos 2 x + 3 \sin 2 x)$
13. $y (x) = 5 x - 2 x^{2}$ $y (x) = 5 x - 2 x^{2}$
14. $y (x) = 3 x^{2} - 16 / x^{3}$ $y (x) = 3 x^{2} - 16 / x^{3}$
15. $y (x) = 7 x - 5 x \ln x$ $y (x) = 7 x - 5 x \ln x$
16. $y (x) = 2 \cos (\ln x) + 3 \sin (\ln x)$ $y (x) = 2 \cos (\ln x) + 3 \sin (\ln x)$
21. Linearly independent
22. Linearly independent
23. Linearly independent
25. Linearly dependent
25. Linearly independent
26. Linearly independent
28. $y (x) = 1 - 2 \cos x - \sin x$ $y (x) = 1 - 2 \cos x - \sin x$
29. There is no contradiction because if the given differential equation is divided by $x^{2}$ $x^{2}$ to get the form in Eq. (8), then theresulting coefficient functions $p (x) = - 4 / x$ $p (x) = - 4 / x$ and $q (x) = 6 / x^{2}$ $q (x) = 6 / x^{2}$ are not continuous at $x = 0$ $x = 0$ .
33. $y (x) = c_{1} e^{x} + c_{2} e^{2 x}$ $y (x) = c_{1} e^{x} + c_{2} e^{2 x}$
34. $y (x) = c_{1} e^{- 5 x} + c_{2} e^{3 x}$ $y (x) = c_{1} e^{- 5 x} + c_{2} e^{3 x}$
35. $y (x) = c_{1} + c_{2} e^{- 5 x}$ $y (x) = c_{1} + c_{2} e^{- 5 x}$
36. $y (x) = c_{1} + c_{2} e^{- 3 x / 2}$ $y (x) = c_{1} + c_{2} e^{- 3 x / 2}$
37. $y (x) = c_{1} e^{- x / 2} + c_{2} e^{x}$ $y (x) = c_{1} e^{- x / 2} + c_{2} e^{x}$
38. $y (x) = c_{1} e^{- x / 2} + c_{2} e^{- 3 x / 2}$ $y (x) = c_{1} e^{- x / 2} + c_{2} e^{- 3 x / 2}$
39. $y (x) = (c_{1} + c_{2} x) e^{- x / 2}$ $y (x) = (c_{1} + c_{2} x) e^{- x / 2}$
40. $y (x) = (c_{1} + c_{2} x) e^{2 x / 3}$ $y (x) = (c_{1} + c_{2} x) e^{2 x / 3}$
41. $y (x) = c_{1} e^{- 4 x / 3} + c_{2} e^{5 x / 2}$ $y (x) = c_{1} e^{- 4 x / 3} + c_{2} e^{5 x / 2}$
42. $y (x) = c_{1} e^{- 4 x / 7} + c_{2} e^{3 x / 5}$ $y (x) = c_{1} e^{- 4 x / 7} + c_{2} e^{3 x / 5}$
43. $y ″ + 10 y ′ = 0$ $y ″ + 10 y ′ = 0$
44. $y ″ - 100 y = 0$ $y ″ - 100 y = 0$
45. $y ″ + 20 y ′ + 100 y = 0$ $y ″ + 20 y ′ + 100 y = 0$
46. $y ″ - 110 y ′ + 1000 y = 0$ $y ″ - 110 y ′ + 1000 y = 0$
47. $y ″ = 0$ $y ″ = 0$
48. $y ″ - 2 y ′ - y = 0$ $y ″ - 2 y ′ - y = 0$
49. The high point is $(\ln \frac{7}{4}, \frac{16}{7})$ $(\ln \frac{7}{4}, \frac{16}{7})$ .
50. $(- \ln 2, - 2)$ $(- \ln 2, - 2)$
52. $y (x) = c_{1} x + c_{2} / x$ $y (x) = c_{1} x + c_{2} / x$
53. $y (x) = c_{1} x^{- 4} + c_{2} x^{3}$ $y (x) = c_{1} x^{- 4} + c_{2} x^{3}$
54. $y (x) = c_{1} x^{- 3 / 2} + c_{2} x^{1 / 2}$ $y (x) = c_{1} x^{- 3 / 2} + c_{2} x^{1 / 2}$
55. $y (x) = c_{1} + c_{2} \ln x$ $y (x) = c_{1} + c_{2} \ln x$
56. $y (x) = x^{2} (c_{1} + c_{2} \ln x)$ $y (x) = x^{2} (c_{1} + c_{2} \ln x)$

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Table of Contents for Section 5.1

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Table of Contents for
Section 5.1