Section 1.5

1. $y (x) = 2 (1 - e^{- x})$ $y (x) = 2 (1 - e^{- x})$
2. $y (x) = (3 x + C) e^{2 x}$ $y (x) = (3 x + C) e^{2 x}$
3. $y (x) = e^{- 3 x} (x^{2} + C)$ $y (x) = e^{- 3 x} (x^{2} + C)$
4. $y (x) = (x + C) e^{x^{2}}$ $y (x) = (x + C) e^{x^{2}}$
5. $y (x) = x + 4 x^{- 2}$ $y (x) = x + 4 x^{- 2}$
6. $y (x) = x^{2} + 32 / x^{5}$ $y (x) = x^{2} + 32 / x^{5}$
7. $y (x) = 5 x^{1/2} + C x^{- 1/2}$ $y (x) = 5 x^{1/2} + C x^{- 1/2}$
8. $y (x) = 3 x + {Cx}^{- 1/3}$ $y (x) = 3 x + {Cx}^{- 1/3}$
9. $y (x) = x (7 + \ln x)$ $y (x) = x (7 + \ln x)$
10. $y (x) = 3 x^{3} + {Cx}^{3/2}$ $y (x) = 3 x^{3} + {Cx}^{3/2}$
11. $y (x) \equiv 0$ $y (x) \equiv 0$
12. $y (x) = \frac{1}{4} x^{5} - 56 x^{- 3}$ $y (x) = \frac{1}{4} x^{5} - 56 x^{- 3}$
13. $y (x) = (e^{x} + e^{- x}) / 2$ $y (x) = (e^{x} + e^{- x}) / 2$
14. $y (x) = x^{3} \ln x + 10 x^{3}$ $y (x) = x^{3} \ln x + 10 x^{3}$
15. $y (x) = [1 - 5 \exp (- x^{2})] / 2$ $y (x) = [1 - 5 \exp (- x^{2})] / 2$
16. $y (x) = 1 + e^{- \sin x}$ $y (x) = 1 + e^{- \sin x}$
17. $y (x) = (1 + \sin x) / (1 + x)$ $y (x) = (1 + \sin x) / (1 + x)$
18. $y (x) = x^{2} (\sin x + C)$ $y (x) = x^{2} (\sin x + C)$
19. $y (x) = \frac{1}{2} \sin x + C \csc x$ $y (x) = \frac{1}{2} \sin x + C \csc x$
20. $y (x) = - 1 + \exp (x + \frac{1}{2} x^{2})$ $y (x) = - 1 + \exp (x + \frac{1}{2} x^{2})$
21. $y (x) = x^{3} \sin x$ $y (x) = x^{3} \sin x$
22. $y (x) = (x^{3} + 5) e^{x^{2}}$ $y (x) = (x^{3} + 5) e^{x^{2}}$
23. $y (x) = x^{3} (2 + C e^{- 2 x})$ $y (x) = x^{3} (2 + C e^{- 2 x})$
24. $y (x) = \frac{1}{3} [1 + 16 {(x^{2} + 4)}^{- 3/2}]$ $y (x) = \frac{1}{3} [1 + 16 {(x^{2} + 4)}^{- 3/2}]$
25. $y (x) = [\exp (- \frac{3}{2} x^{2})] [3 {(x^{2} + 1)}^{3/2} - 2]$ $y (x) = [\exp (- \frac{3}{2} x^{2})] [3 {(x^{2} + 1)}^{3/2} - 2]$
26. $x (y) = 1/2 y^{2} + C / y^{4}$ $x (y) = 1/2 y^{2} + C / y^{4}$
27. $x (y) = e^{y} (C + \frac{1}{2} y^{2})$ $x (y) = e^{y} (C + \frac{1}{2} y^{2})$
28. $x (y) = \frac{1}{2} [y + (1 + y^{2}) (\tan^{- 1} y + C)]$ $x (y) = \frac{1}{2} [y + (1 + y^{2}) (\tan^{- 1} y + C)]$
30. $y (x) = x^{1/2} \int_{1}^{x} t^{- 1/2} \cos t d t$ $y (x) = x^{1/2} \int_{1}^{x} t^{- 1/2} \cos t d t$
29. $y (x) = [\exp (x^{2})] [C + \frac{1}{2} \sqrt{π} erf (x)]$ $y (x) = [\exp (x^{2})] [C + \frac{1}{2} \sqrt{π} erf (x)]$
32. (a) $y (x) = \sin x - \cos x$ $y (x) = \sin x - \cos x$ ; (b) $y (x) = {C e}^{- x} + \sin x - \cos x$ $y (x) = {C e}^{- x} + \sin x - \cos x$ ; (c) $y (x) = 2 e^{- x} + \sin x - \cos x$ $y (x) = 2 e^{- x} + \sin x - \cos x$
33. After about 7 min 41 s
34. About 22.2 days
35. About 5.5452 years
36. (a) $x (t) = (60 - t) - {(60 - t)}^{3} / 3600$ $x (t) = (60 - t) - {(60 - t)}^{3} / 3600$ ; (b) About 23.09 lb
37. 393:75 lb
38. (a) $x (t) = 50 e^{- t / 20}$ $x (t) = 50 e^{- t / 20}$ ; (b) $y (t) = 150 e^{- t / 40} - 100 e^{- t / 20}$ $y (t) = 150 e^{- t / 40} - 100 e^{- t / 20}$ ; (c) 56.25 lb
39. (b) $y_{max} = 100 e^{- 1} \approx 36.79 (gal)$ $y_{max} = 100 e^{- 1} \approx 36.79 (gal)$
41. (b) Approximately $1,308,283
43. $- 50.0529, - 28.0265, - 6.0000$ $- 50.0529, - 28.0265, - 6.0000$ , 16.0265, 38.0529
44. 3.99982, 4.00005, 4.00027, 4.00050, 4.00073
45. $x (t) = 20 (1 - e^{- t / 10})$ $x (t) = 20 (1 - e^{- t / 10})$ ; $x = 10$ $x = 10$ after $t = 10 \ln 2 \approx 6.93$ $t = 10 \ln 2 \approx 6.93$ months.
46. $x (t) = \frac{20}{101} (101 - 102 e^{- t / 10} + \cos t + 10 \sin t)$ $x (t) = \frac{20}{101} (101 - 102 e^{- t / 10} + \cos t + 10 \sin t)$ ; $x = 10$ $x = 10$ after $t = 6.47$ $t = 6.47$ months.

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Table of Contents for Section 1.5

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Table of Contents for
Section 1.5