# A.2.4: Section 2.4 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. $$P=\frac{P_{0}e^{at}}{1+aP_{0}\int_{0}^{t}\alpha (\tau )e^{a\tau }d\tau };\quad\lim_{t\to\infty }P(t)=\left\{\begin{array}{cc}{\infty }&{\text{if }L=0,}\\{0}&{\text{if }L=\infty ,}\\{1/aL}&{\text{if }0<L<\infty }\end{array} \right.$$

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

This page titled A.2.4: Section 2.4 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.