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Mathematics LibreTexts

Section 2.4 Answers

  • Page ID
    28698
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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. \(y=x-\frac{2x}{x^{2}+c}\)