Skip to main content
Mathematics LibreTexts

18: Section 5.5 Answers

  1. Last updated
  2. Save as PDF
  • Page ID
    103637

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

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

    \( \newcommand{\dsum}{\displaystyle\sum\limits} \)

    \( \newcommand{\dint}{\displaystyle\int\limits} \)

    \( \newcommand{\dlim}{\displaystyle\lim\limits} \)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    \(\newcommand{\longvect}{\overrightarrow}\)

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

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

    1. \(y=c_1e^{-3x}+c_2e^{3x}-6\)

    2. \(y=c_1+c_2e^{-x}+3x\)

    3. \(y=c_1e^{-2x}+c_2xe^{-2x}+{1\over 2}x+1\)

    4. \(y=c_1e^{-3x}+c_2e^{4x}+{1\over 7}xe^{4x}\)

    5. \(y=c_1e^{-x}+c_2e^{3x}-e^x+3\)

    6. \(y=c_1\cos 5x+c_2\sin 5x+{1\over 4}\sin x\)

    7. \(y=c_1e^{-3x}+c_2xe^{-3x}-{1\over 49}xe^{4x}+{2\over 343}e^{4x}\)

    8. \(y=c_1e^{-x}+c_2e^{x}+{1\over 6}x^3e^x-{1\over 4}x^2e^x+{1\over 4}xe^x-5\)

    9. \(y=c_1e^x\cos 2x+c_2e^x\sin 2x+{1\over 3}e^x\sin x\)

    10. \(y=c_1\cos 5x+c_2\sin 5x-2x\cos 5x\)

    11. \(y=c_1e^{-x/2}\cos {\sqrt 3\over 2}x+c_2e^{-x/2}\sin {\sqrt 3\over 2}x+\sin x+2\cos x-x\cos x\)

    12. \(y=c_1e^x+c_2e^{2x}+e^{3x}\left(-\frac{1}{4}+\frac{x}{2} \right)\)

    13. \(y=c_1e^x+c_2e^{5x}+e^{-3x}\left(1-\frac{x}{4}\right)\)

    14. \(y=c_1e^{3x}+c_2e^{-x}+e^{x}\left(2-\frac{3x}{4}\right)\)

    15. \(y=c_1e^{-x}+c_2xe^{-x}+e^{2x} (1−3x+x^{2})\)

    16. \(y =c_1\cos 2x+c_2\sin 2x+e^{−x} (1+x^{2} )\)

    17. \(y=c_1e^{2x}+c_2e^{-x}+e^{x} (−2+x+ 2x^{2} )\)

    18. \(y=c_1e^{5x}+c_2e^{-x}+xe^{-x}\left(\frac{1}{6}+\frac{x}{2} \right)\)

    19. \(y=c_1e^x+c_2e^{2x}+xe^{x} (1 + 2x)\)

    20. \(y=c_1e^{-4x}+c_2e^{3x}+xe^{3x}\left(-1+\frac{x}{2} \right)\)

    21. \(y= c_1e^{-x/2}+c_2e^{2x}+xe^{2x} (−2+x)\)

    22. \(y= c_1e^{-x}+c_2xe^{-x}+x^{2}e^{-x}\left(1+\frac{x}{2} \right)\)

    23. \(y= c_1e^x+c_2xe^{x}+x^{2}e^{x}\left(\frac{1}{2}-x \right)\)

    24. \(y= c_1e^{2x}+c_2xe^{2x}+\frac{x^{2}e^{2x}}{2}(1-x+x^{2})\)

    25. \(y= c_1e^{-x/3}+c_2xe^{-x/3}+\frac{x^{2}e^{-x/3}}{27}(3-2x+x^{2})\)

    26. \(y=\frac{e^{3x}}{4}(-1+2x)+c_{1}e^{x}+c_{2}e^{2x}\)

    27. \(y=e^{x}(1-2x)+c_{1}e^{2x}+c_{2}e^{4x}\)

    28. \(y=\frac{e^{2x}}{5}(1-x)+e^{-3x}(c_{1}+c_{2}x)\)

    29. \(y = xe^{x} (1 − 2x) + c_{1}e^{x} + c_{2}e^{−3x}\)

    30. \(y = e^{x} \left[ x^{2} (1 − 2x) + c_{1} + c_{2}x\right ]\)

    31. \(y=c_1e^{-x}+c_2e^{-2x}+\cos x+2\sin x\)

    32. \(y=c_1e^{(-3+\sqrt 5)x/2}+c_2e^{(-3-\sqrt 5)x/2}+\cos x+(2-2x)\sin x\)

    33. \(y=c_1e^{-x}+c_2xe^{-x}+e^{x}(-2\cos x+3\sin x)\)

    34. \(y=c_1e^{(-3+\sqrt 17)x/2}+c_2e^{(-3-\sqrt 17)x/2}+\frac{e^{2x}}{2}(\cos 2x-\sin 2x)\)

    35. \(y= c_1e^{x/2}\cos {\sqrt 3\over 2}x+c_2e^{x/2}\sin {\sqrt 3\over 2}x-e^{x}(x\cos x-\sin x)\)

    36. \(y =c_1e^{(-3+\sqrt 17)x/2}+c_2e^{(-3-\sqrt 17)x/2}+ e^{−2x} (1 − 2x)(\cos 3x − \sin 3x)\)

    37. \(y = c_1\cos 2x+c_2\sin 2x+x(\cos 2x − 3 \sin 2x)\)

    38. \(y =c_1e^{x}+c_2xe^{x} + −x [(2 − x) \cos x + (3 − 2x) \sin x]\)

    39. \(y=c_1\cos {x\over 2}+c_2\sin {x\over 2}+ x\left[x\cos\left(\frac{x}{2}\right)-3\sin\left(\frac{x}{2}\right) \right]\)

    40. \(y = c_1e^{-x}\cos x+c_2e^{-x}\sin x+ xe^{−x} (3 \cos x + 4 \sin x)\)

    41. \(y = c_1e^{x}\cos 2x+c_2e^{x}\sin 2x+xe^{x} [(−1 + x) \cos 2x + (1 + x) \sin 2x]\)

    42. \(y =c_1e^{-x}+c_2xe^{-x} −(14 − 10x) \cos x − (2 + 8x − 4x^{2} ) \sin x\)

    43. \(y = c_1e^{-x}+c_2e^{-2x} +(1 + 2x + x^{2}) \cos x + (1 + 3x^{2}) \sin x\)

    44. \(y= c_1e^{-x}+c_2e^{-2x} +\frac{x^{2}}{2}(\cos 2x-\sin 2x)\)

    45. \(y = c_1e^{2x}+c_2e^{3x} e^{x} (x^{2} \cos x + 2 \sin x)\)

    46. \(y = c_1e^{x}+c_2xe^{x} +e^{x} (1 − x^{2} )(\cos x + \sin x)\)

    47. \(y = c_1e^{x}\cos x+c_2e^{x}\sin x+e^{x} (x^{2} − x^{3})(\cos x + \sin x)\)

    48. \(y =c_1e^{-x}+c_2xe^{-x} + e^{−x} [(1 + 2x) \cos x − (1 − 3x) \sin x]\)

    49. \(y =c_1\cos 3x+c_2\sin 3x + x(2 \cos 3x − \sin 3x)\)

    50. \(y= c_1e^{-x}+c_2e^{-2x}−x^{3} \cos x + (x + 2x^{2} ) \sin x\)

    51. \(y = c_1e^{-x}+c_2e^{-3x}−e^{−x}[ (x + x^{2} ) \cos x − (1 + 2x) \sin x]\)

    52. \(y={5\over 8}e^{-8x}+{5\over 8}e^{8x}-{1\over 4}\)

    53. \(y=-{41\over 125}+{41\over 125}e^{5x}-{1\over 10}x^2+{9\over 25}x\)

    54. \(y=-\pi\cos x-{11\over 3}\sin x-{8\over 3}\cos 2x+2x\cos x\)

    55. \(y=2e^{2x}\cos 2x-{3\over 64}e^{2x}\sin 2x+{1\over 8}x^3+{3\over 16}x^2+{3\over 32}x\)

    56. \(y = −e^{2x} (1 + x) + 2e^{−x} − e^{5x}\)

    57. \(y = xe^{2x} + 3e^{x} − e^{−4x}\)

    58. \(y = e ^{-x} (2 + x − 2x^{2}) − e^{−3x}\)

    59. \(y = e ^{-2x} (3 − x) − 2e^{5x}\)

    60. \(y = e^{x} (2 \cos x + 3 \sin x) + 3e^{x} − e^{6x}\)

    61. \(y = e^{x} [(1 + 2x) \cos x + (1 − 3x) \sin x]\)

    62. \(y = e^{x} (\cos x−2 \sin x)+e^{−3x} (\cos x+\sin x)\)

    63. \(y = e^{3x} [(2 + 2x) \cos x − (1 + 3x) \sin x]\)

    64. \(y = e^{3x} [(2 + 3x) \cos x + (4 − x) \sin x]+3e^{x}−5e^{2x}\)

    65. \(y = xe^{−2x} \cos x + 3 \cos 2x\)

    66. \(y=-\frac{3}{8}\cos 2x+\frac{1}{4}\sin 2x+e^{-x}-\frac{13}{8}e^{-2x}-\frac{3}{4}xe^{-2x}\)

    67. a. \(y=c_1e^{-x/2}\cos {\sqrt 3\over 2}x+c_2e^{-x/2}\cos {\sqrt 3\over 2}x-\frac{e^{x}}{3}(1-x)\)
    b. \(y=c_1e^{-x/2}\cos {\sqrt 3\over 2}x+c_2e^{-x/2}\cos {\sqrt 3\over 2}x+e^{-x}(3+2x)\)
    c. \(y=c_1e^{-x/2}\cos {\sqrt 3\over 2}x+c_2e^{-x/2}\cos {\sqrt 3\over 2}x-\frac{e^{x}}{3}(1-x)+e^{-x}(3+2x)\)

    68. a. \(y=c_1e^{4x}+c_2e^{3x}+e^{x} (3 + 7x)\)
    b. \(y=c_1e^{4x}+c_2e^{3x}+xe^{3x}\)
    c. \(y=c_1e^{4x}+c_2e^{3x}+e^{x} (3 + 7x)+xe^{3x}\)

    69. a. \(y=c_1e^{4x}+c_2xe^{4x}+x^{3} e^{4x}\)
    b. \(y=c_1e^{4x}+c_2xe^{4x}+ 1 + 2x + x^{2}\)
    c. \(y=c_1e^{4x}+c_2xe^{4x}+ x^{3} e^{4x} + 1 + 2x + x^{2}\)

    70. a. \(y=c_1e^{x}+c_2e^{2x}+ xe^{2x} (1 − 2x)\)
    b. \(y=c_1e^{x}+c_2e^{2x}+ xe^{x}\)
    c. \(y=c_1e^{x}+c_2e^{2x}+ xe^{2x} (1 − 2x) + xe^{x}\)

    71. a. \(y = c_1e^x\cos x+c_2e^x\sin x+ e^{x} (1 + x)\)
    b. \(y = c_1e^x\cos x+c_2e^x\sin x+ x^{2} e^{−x}\)
    c. \(y = c_1e^x\cos x+c_2e^x\sin x+ e^{x} (1 + x) + x^{2} e^{−x}\)

    72. a. \(y=c_1\cos x+c_2\sin x+ x^{2} e^{−x}\)
    b. \(y=c_1\cos x+c_2\sin x+ e^{3x} (1 − x^{2} )\)
    c. \(y=c_1\cos x+c_2\sin x+ x^{2} e^{−x} + e^{3x} (1 − x^{2} )\)

    This page titled 18: Section 5.5 Answers is shared under a not declared license and was authored, remixed, and/or curated by William F. Trench.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      Author
      William F. Trench
      License Version
      2.0
      Show Page TOC
      no
    2. Tags
      1. source[1]-math-43769