Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

Section 4.5 Answers

  • Page ID
    28905
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    1. \(y'=\frac{2xy}{x^{2}+3y^{2}}\)

    2. \(y'=-\frac{y^{2}}{(xy-1)}\)

    3. \(y'=-\frac{y(x^{2}+y^{2}-2x^{2}\ln |xy|)}{x(x^{2}+y^{2}-2y^{2}\ln |xy|)}\)

    4. \(xy'-y=-\frac{x^{1/2}}{2}\)

    5. \(y'+2xy=4xe^{x^{2}}\)

    6. \(xy'+y=4x^{3}\)

    7. \(y' − y = \cos x − \sin x\)

    8. \((1+x^{2})y'-2xy=(1-x)^{2}e^{x}\)

    10. \(y'g-yg'=f'g-fg'\)

    11. \((x-x_{0})y'=y-y_{0}\)

    12. \(y'(y^{2}-x^{2}+1)+2xy=0\)

    13. \(2x(y-1)y'-y^{2}+x^{2}+2y=0\)

    14. 

    1. \(y=-81+18x,\: (9,81)\quad y=-1+2x,\: (1,1)\)
    2. \(y = −121 + 22x,\: (11, 121)\quad y = −1 + 2x,\: (1, 1)\)
    3. \(y = −100 − 20x,\: (−10, 100)\quad y = −4 − 4x,\: (−2, 4)\)
    4. \(y = −25 − 10x,\: (−5, 25)\quad y = −1 − 2x,\: (−1, 1)\)

    15. (e) \(y=\frac{5+3x}{4},\:(-3/5,4/5)\quad y=-\frac{5-4x}{3},\: (4/5,-3/5)\)

    17. 

    1. \(y=-\frac{1}{2}(1+x),\: (1,-1);\quad y=\frac{5}{2}+\frac{x}{10},\: (25,5)\)
    2. \(y=\frac{1}{4}(4+x),\: (4,2)\quad y=-\frac{1}{4}(4+x),\: (4,-2)\)
    3. \(y = \frac{1}{2}(1 + x),\: (1, 1)\quad y = \frac{7}{2} + \frac{x}{14},\: (49, 7)\)
    4. \(y = −\frac{1}{2}(1 + x),\: (1, −1)\quad y = −\frac{5}{2} −\frac{x}{10},\: (25, −5)\)

    18. \(y=2x^{2}\)

    19. \(y=\frac{cx}{\sqrt{|x^{2}=1|}}\)

    20. \(y=y_{1}+c(x-x_{1})\)

    21. \(y=-\frac{x^{3}}{2}-\frac{x}{2}\)

    22. \(y=-x\ln |x|+cx\)

    23. \(y=\sqrt{2x+4}\)

    24. \(y=\sqrt{x^{2}-3}\)

    25. \(y=kx^{2}\)

    26. \((y-x)^{3}(y+x)=k\)

    27. \(y^{2}=-x+k\)

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

    29. \(y^{2}=-2x-\ln (x-1)^{2}+k\)

    30. \(y=1+\sqrt{\frac{9-x^{2}}{2}};\text{ those with }c>0\)

    33. \(\tan ^{-1}\frac{y}{x}-\frac{1}{2}\ln (x^{2}+y^{2})=k\)

    34. \(\frac{1}{2}\ln (x^{2}+y^{2})+(\tan\alpha )\tan ^{-1}\frac{y}{x}=k\)