1. $y\left(x\right)=C\exp \left(-x^2\right)$

2. $y\left(x\right)=1/\left(x^2+C\right)$

3. $y\left(x\right)=C\exp \left(-\cos x\right)$

4. $y\left(x\right)=C{\left(1+x\right)}^4$

5. $y\left(x\right)=\sin \left(C+\sqrt{x}\right)$

6. $y\left(x\right)={\left(x^{\mathrm{3/2}}+C\right)}^2$

7. $y\left(x\right)={\left(2x^{\mathrm{4/3}}+C\right)}^{\mathrm{3/2}}$

8. $y\left(x\right)={\sin }^{-1}\left(x^2+C\right)$

9. $y(x)=C(1+x)/(1-x)$

10. $y(x)=(1+x)/[1+C(1+x)]-1$

11. $y(x)={(C-x^2)}^{-\mathrm{1/2}}$

12. $y^2+1={\mathrm{Ce}}^{x^2}$

13. $\ln (y^4+1)=C+4\sin x$

14. $3y+2y^{\mathrm{3/2}}=3x+2x^{\mathrm{3/2}}+C$

15. $1/(3y^3)-2/y=1/x+\ln |x|+C$

16. $y(x)={\sec }^{-1}(C\sqrt{1+x^2})$

17. $\ln |1+y|=x+\frac{1}{2}{x}^2+C$

18. $y(x)=\tan (C-\frac{1}{x}-x)$

19. $y\left(x\right)=2\exp \left(e^x\right)$

20. $y\left(x\right)=\tan \left(x^3+\pi /4\right)$

21. $y^2=1+\sqrt{x^2-16}$

22. $y\left(x\right)=-3\exp \left(x^4-x\right)$

23. $y\left(x\right)=\frac{1}{2}\left(1+e^{2x-2}\right)$

24. $y\left(x\right)=\frac{\pi }{2}\sin x$

25. $y\left(x\right)=x\exp \left(x^2-1\right)$

26. $y\left(x\right)=1/\left(1-x^2-x^3\right)$

27. $y=\ln \left(3e^{2x}-2\right)$

28. $y\left(x\right)={\tan }^{-1}\left(\sqrt{x}-1\right)$

29. (a) General solution $y(x)=-1/(x-C)$; (b) The singular solution $y(x)\equiv 0$. (c) In the following figure we see that there is a unique solution through every point of the $xy$-plane.

30. General solution $y(x)={(x-C)}^2$; singular solution $y(x)\equiv 0$. (a) No solution if $b<0$; (b) Infinitely many solutions (for all x) if $b\geqq 0;(\text{c})$ Two solutions near (a, b) if $b>0$.

31. Separation of variables gives the same general solution $y={(x-C)}^2$ as in Problem 30 , but the restriction that $y\text{′}=2\sqrt{y}\geqq 0$ implies that only the right halves of the parabolas qualify as solution curves. In the figure below we see that through the point (a, b) there passes (a) No solution curve if $b<0$, (b) a unique solution curve if $b>0$, (c) Infinitely many solution curves if $b=0$.

32. General solution $y(x)=\pm \sec (x-C)$; singular solutions $y(x)\equiv \pm 1$.

(a) No solution if $|b|<1$; (b) A unique solution if $|b|>1$; (c) Infinitely many solutions if $b=\pm 1$.

33. About 51840 persons

34. $t\approx 3.87$ hr

35. About 14735 years

36. Age about 686 years

37. $21103:48

38. $44.52

39. 2585 mg

40. About 35 years

41. About $4.86\times {10}^9$ years ago

42. About 1.25 billion years

43. After a total of about 63 min have elapsed

44. About 2.41 minutes

45. (a) 0.495 m; (b $(8.32\times {10}^{-7})I_0$; (c) 3.29 m

46. (a) About 9.60 inches; (b) About 18,200 ft

47. After about 46 days

48. About 6 billion years

49. After about 66 min 40 s

50. (a) $A(t)=10·3^{2t/15}$; (b) About 20.80 pu; (c) About 15.72 years

51. (a) $A(t)=15·{\left(\frac{2}{3}\right)}^{t/5}$; (b) approximately 7.84 su; (c) After about 33.4 months

52. About 120 thousand years ago

53. About 74 thousand years ago

54. 3 hours

55. 972 s

56. At time $t=\mathrm{2048/1562}\approx 1.31$ (in hours)

58. 1:20 p.m.

59. (a) $y(t)={(8-7t)}^{\mathrm{2/3}}$; (b) at 1:08:34 p.m.; (c) $r=\frac{1}{60}\sqrt{\frac{7}{12}}\approx 0.15$ (in.)

60. About 6 min 3 sec

61. Approximately 14 min 29 s

62. The tank is empty about 14 seconds after 2:00 p.m.

63. (a) 1:53:34 p.m.; (b) $r\approx 0.04442\text{ ft}\approx 0.53\text{ in.}$

64. $r=\frac{1}{720}\sqrt{3}\text{ ft}$, about $\frac{1}{35}\text{ in.}$

65. At approximately 10:29 a.m.