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14.7: Section 2.4 Answers

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Aug 11, 2020
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- 14.6: Section 2.3 Answers
- 14.8: Section 2.5 Answers

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1. $y=\frac{1}{1-ce^{x}}$

2. $y=x^{2/7}(c-\ln |x|)^{1/7}$

3. $y=e^{2/x}(c-1/x)^{2}$

4. $y=\pm\frac{\sqrt{2x+c}}{1+x^{2}}$

5. $y=\pm (1-x^{2}+ce^{-x^{2}})^{-1/2}$

6. $y=\left[\frac{x}{3(1-x)+ce^{-x}} \right] ^{1/3}$

7. $y=\frac{2\sqrt{2}}{\sqrt{1-4x}}$

8. $y=\left[1-\frac{3}{2}e^{-(x^{2}-1)/4} \right]^{-2}$

9. $y=\frac{1}{x(11-3x)^{1/3}}$

10. $y=(2e^{x}-1)^{2}$

11. $y=(2e^{12x}-1-12x)^{1/3}$

12. $y=\left[\frac{5x}{2(1+4x^{5})} \right]^{1/2}$

13. $y=(4e^{x/2}-x-2)^{2}$

14. $P=\frac{P_{0}e^{at}}{1+aP_{0}\int_{0}^{t}\alpha (\tau )e^{a\tau }d\tau };\quad\lim_{t\to\infty }P(t)=\left\{\begin{array}{cc}{\infty }&{\text{if }L=0,}\\[4pt]{0}&{\text{if }L=\infty ,}\\[4pt]{1/aL}&{\text{if }0<L<\infty }\end{array} \right.$

15. $y=x(\ln |x|+c)$

16. $y=\frac{cx^{2}}{1-cx}\quad y=-x$

17. $y=\pm x(4\ln |x|+c)^{1/4}$

18. $y=x\sin ^{-1}(\ln |x|+c)$

19. $y=x\tan (\ln |x|+c)$

20. $y=\pm x\sqrt{cx^{2}-1}$

21. $y=\pm x\ln (\ln |x|+c)$

22. $y=-\frac{2x}{2\ln |x|+1}$

23. $y=x(3\ln x+27)^{1/3}$

24. $y=\frac{1}{x}\left(\frac{9-x^{4}}{2} \right)^{1/2}$

25. $y=-x$

26. $y=-\frac{x(4x-3)}{(2x-3)}$

27. $y=x\sqrt{4x^{6}-1}$

28. $\tan ^{-1}\frac{y}{x}-\frac{1}{2}\ln (x^{2}+y^{2})=c$

29. $(x+y)\ln |x|+y(1-\ln |y|)+cx=0$

30. $(y+x)^{3}=3x^{3}(\ln |x|+c)$

31. $(y+x)=c(y-x)^{3};\quad y=x;\quad y=-x$

32. $y^{2}(y-3x)=c;\quad y≡0;\quad y=3x$

33. $(x-y)^{3}(x+y)=cy^{2}x^{4};\quad y=0;\quad y=x;\quad y=-x$

34. $\frac{y}{x}+\frac{y^{3}}{x^{3}}=\ln |x|+c$

40. Choose $X_{0}$ and $Y_{0}$ so that

$aX_{0}+bY_{0}=\alpha\nonumber$ $cX_{0}+dY_{0}=\beta\nonumber$

41. $(y+2x+1)^{4})2y-6x-3)=c;\quad y=3x+3/2;\quad y=-2x-1$

42. $(y+x-1)(y-x-5)^{3}=c;\quad y=x+5;\quad y=-x+1$

43. $\ln |y-x-6|-\frac{2(x+2)}{y-x-6}=c;\quad y=x+6$

44. $(y_{1}=x^{1/3}y=x^{1/3}(\ln |x|+c)^{1/3}$

45. $y_{1}=x^{3};\quad y=\pm x^{3}\sqrt{cx^{6}-1}$

46. $y_{1}=x^{2};\quad y=\frac{x^{2}(1+cx^{4})}{1-cx^{4}} y=-x^{2}$

47. $y_{1}=e^{x};\quad y=-\frac{e^{x}(1-2ce^{x}}{1-ce^{x}};\quad y=-2e^{x}$

48. $y_{1}=\tan x; y=\tan x\tan (\ln |\tan x|+c)$

49. $y_{1}=\ln x;\quad y=\frac{2\ln x(1+c(\ln x)^{4})}{1-c(\ln x)^{4}};\quad y=-2\ln x$

50. $y_{1}=x^{1/2};\quad y=x^{1/2}(-2\pm\sqrt{\ln |x|+c})$

51. $y_{1}=e^{x^{2}};\quad y=e^{x^{2}}(-1\pm\sqrt{2x^{2}+c})$

52. $y=\frac{-3+\sqrt{1+60x}}{2x}$

53. $y=\frac{-5+\sqrt{1+48x}}{2x^{2}}$

56. $y=1+\frac{1}{x+1+ce^{x}}$

57. $y=e^{x}-\frac{1}{1+ce^{-x}}$

58. $y=1-\frac{1}{x(1-cx)}$

59. $y=x-\frac{2x}{x^{2}+c}$

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