Skip to main content
Mathematics LibreTexts

14.23: Section 5.6 Answers

  1. Last updated
  2. Save as PDF
  • Page ID
    45912

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    1. \(y = 1 − 2x + c_{1}e^{−x} + c_{2}xe^{x} ;\: \{e^{−x}, xe^{x} \}\)

    2. \(y=\frac{4}{3x^{2}}+c_{1}x+\frac{c_{2}}{x};\: \{x, 1/x\}\)

    3. \(y=\frac{x(\ln |x|)^{2}}{2}+c_{1}x+c_{2}x\ln |x|;\: \{x, x\ln |x|\}\)

    4. \(y = (e^{2x} + e^{x} ) \ln(1 + e^{−x}) + c_{1}e^{2x} + c_{2}e^{x};\: \{e^{2x}, e^{x}\}\)

    5. \(y=e^{x}\left(\frac{4}{5}x^{7/2}+c_{1}+c_{2}x \right);\: \{e^{x}, xe^{x}\}\)

    6. \(y = e^{x} (2x^{3/2} + x^{1/2} \ln x + c_{1}x^{1/2} + c_{2}x^{−1/2} );\: \{x^{1/2} e^{x}, x^{−1/2}e^{−x}\}\)

    7. \(y = e^{x} (x \sin x + \cos x \ln | \cos x| + c_{1} \cos x + c_{2} \sin x);\: \{e^{x} \cos x, e^{x} \sin x\}\)

    8. \(y = e ^{−x ^{2}} (2e^{−2x} + c_{1} + c_{2}x);\: \{e^{ −x^{2}}, xe^{−x ^{2}} \}\)

    9. \(y = 2x + 1 + c_{1}x^{2} + \frac{c_{2}}{x^{2}};\: \{x^{2}, 1/x^{2} \}\)

    10. \(y=\frac{xe^{2x}}{9}+xe^{-x}(c_{1}+c_{2}x);\: \{xe^{-x}, x^{2}e^{-x}\}\)

    11. \(y=xe^{x}\left(\frac{x}{3}+c_{1}+\frac{c_{2}}{x^{2}} \right);\: \{xe^{x}, e^{x}/x\}\)

    12. \(y=-\frac{(2x-1)^{2}e^{x}}{8}+c_{1}e^{x}+c_{2}xe^{-x};\: \{e^{x}, xe^{-x}\}\)

    13. \(y=x^{4}+c_{1}x^{2}+c_{2}x^{2}\ln |x|; \{x^{2}, x^{2}\ln |x|\}\)

    14. \(y = e^{−x} (x^{3/2} + c_{1} + c_{2}x ^{1/2} );\: \{e^{−x}, x^{1/2} e^{−x} \}\)

    15. \(y = e^{x} (x+c_{1}+c_{2}x^{2});\: \{e^{x}, x^{2}e^{x} \}\)

    16. \(y=x^{1/2}\left(\frac{e^{2x}}{2}+c_{1}+c_{2}e^{x} \right);\:\{x^{1/2}, x^{1/2}e^{x}\}\)

    17. \(y = −2x^{2} \ln x + c_{1}x^{2} + c_{2}x^{4} ;\: \{x^{2}, x^{4} \}\)

    18. \(\{e^{x}, e^{x}/x\}\)

    19. \(\{x^{2}, x^{3}\}\)

    20. \(\{\ln |x|, x\ln |x|\}\)

    21. \(\{\sin\sqrt{x},\cos\sqrt{x}\}\)

    22. \(\{e^{x}, x^{3}e^{x}\}\)

    23. \(\{x^{a}, x^{a}\ln |x|\}\)

    24. \(\{x\sin x,x\cos x\}\)

    25. \(\{e^{2x}, x^{2}e^{2x}\}\)

    26. \(\{x^{1/2}, x^{1/2}\cos x\}\)

    27. \(\{x^{1/2}e^{2x}, x^{1/2}e^{-2x}\}\)

    28. \(\{1/x, e^{2x}\}\)

    29. \(\{e^{x}, x^{2}\}\)

    30. \(\{e^{2x}, x^{2}e^{2x}\}\)

    31. \(y=x^{4}=6x^{2}-8x^{2}\ln |x|\)

    32. \(y=2e^{2x}-xe^{-x}\)

    33. \(y=\frac{(x+1)}{4}[-e^{x}(3-2x)+7e^{-x}]\)

    34. \(y=\frac{x^{2}}{4}+x\)

    35. \(y=\frac{(x+2)^{2}}{6(x-2)}+\frac{2x}{x^{2}-4}\)

    38.

    1. \(y=\frac{-kc_{1}\sin kx+kc_{2}\cos kx}{c_{1}\cos kx+c_{2}\sin kx}\)
    2. \(y=\frac{c_{1}+2c_{2}e^{x}}{c_{1}+c_{2}e^{x}}\)
    3. \(y=\frac{-6c_{1}+c_{2}e^{7x}}{c_{1}+c_{2}e^{7x}}\)
    4. \(y=-\frac{7c_{1}+c_{2}e^{6x}}{c_{1}+c_{2}e^{6x}}\)
    5. \(y=-\frac{(7c_{1}-c_{2})\cos x+(c_{1}+7c_{2})\sin x}{c_{1}\cos x+c_{2}\sin x}\)
    6. \(y=\frac{-2c_{1}+3c_{2}e^{5x/6}}{6(c_{1}+c_{2}e^{5x/6})}\)
    7. \(y=\frac{c_{1}+c_{2}(x+6)}{6(c_{1}+c_{2}x)}\)

    39.

    1. \(y=\frac{c_{1}+c_{2}e^{x}(1+x)}{x(c_{1}+c_{2}e^{x})}\)
    2. \(y=\frac{-2c_{1}x+c_{2}(1-2x^{2})}{c_{1}+c_{2}x}\)
    3. \(y=\frac{-c_{1}+c_{2}e^{2x}(x+1)}{c_{1}+c_{2}xe^{2x}}\)
    4. \(y=\frac{2c_{1}+c_{2}e^{-3x}(1-x)}{c_{1}+c_{2}xe^{-3x}}\)
    5. \(y=\frac{(2c_{2}x-c_{1})\cos x-(2c_{1}x+c_{2})\sin x}{2x(c_{1}\cos x+c_{2}\sin x)}\)
    6. \(y=\frac{c_{1}+7c_{2}x^{6}}{x(c_{1}+c_{2}x^{6})}\)

    14.23: Section 5.6 Answers is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      Transcluded
      yes
    2. Tags
      1. source[1]-math-43770