A.8.2: Section 8.2 Answers
- Page ID
- 43784
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.
- \(\frac{t^{3}e^{7t}}{2}\)
- \(2e^{2t}\cos 3t\)
- \(\frac{e^{-2t}}{4}\sin 4t\)
- \(\frac{2}{3}\sin 3t\)
- \(t\cos t\)
- \(\frac{e^{2t}}{2}\sinh 2t\)
- \(\frac{2te^{2t}}{3}\sin 9t\)
- \(\frac{2e^{3t}}{3}\sinh 3t\)
- \(e^{2t}t\cos t\)
2.
- \(t^{2}e^{7t}+\frac{17}{6}t^{3}e^{7t}\)
- \(e^{2t}\left(\frac{1}{6}t^{3}+\frac{1}{6}t^{4}+\frac{1}{40}t^{5}\right)\)
- \(e^{-3t}\left(\cos 3t+\frac{2}{3}\sin 3t\right)\)
- \(2\cos 3t+\frac{1}{3}\sin 3t\)
- \((1-t)e^{-t}\)
- \(\cosh 3t+\frac{1}{3}\sinh 3t\)
- \(\left( 1-t-t^{2}-\frac{1}{6}t^{3}\right)e^{-t}\)
- \(e^{t}\left(2\cos 2t+\frac{5}{2}\sin 2t\right)\)
- \(1-\cos t\)
- \(3\cosh t+4\sinh t\)
- \(3e^{t}+4\cos 3t+\frac{1}{3}\sin 3t\)
- \(3te^{-2t}-2\cos 2t-3\sin 2t\)
3.
- \(\frac{1}{4}e^{2t}-\frac{1}{4}e^{-2t}-e^{-t}\)
- \(\frac{1}{5}e^{-4t}-\frac{41}{5}e^{t}+5e^{3t}\)
- \(-\frac{1}{2}e^{2t}-\frac{13}{10}e^{-2t}-\frac{1}{5}e^{3t}\)
- \(-\frac{2}{5}e^{-4t}-\frac{3}{5}e^{t}\)
- \(\frac{3}{20}e^{2t}-\frac{37}{12}e^{-2t}+\frac{1}{3}e^{t}+\frac{8}{5}e^{-3t}\)
- \(\frac{39}{10}e^{t}+\frac{3}{14}e^{3t}+\frac{23}{105}e^{-4t}-\frac{7}{3}e^{2t}\)
4.
- \(\frac{4}{5}e^{-2t}-\frac{1}{2}e^{-t}-\frac{3}{10}\cos t+\frac{11}{10}\sin t\)
- \(\frac{2}{5}\sin t+\frac{6}{5}\cos t+\frac{7}{5}e^{-t}\sin t-\frac{6}{5}e^{-t}\cos t\)
- \(\frac{8}{13}e^{2t}-\frac{8}{13}e^{-t}\cos 2t+\frac{15}{26}e^{-t}\sin 2t\)
- \(\frac{1}{2}te^{t}+\frac{3}{8}e^{t}+e^{-2t}-\frac{11}{8}e^{-3t}\)
- \(\frac{2}{3}te^{t}+\frac{1}{9}e^{t}+te^{-2t}-\frac{1}{9}e^{-2t}\)
- \(-e^{t}+\frac{5}{2}te^{t}+\cos t-\frac{3}{2}\sin t\)
5.
- \(\frac{3}{5}\cos 2t+\frac{1}{5}\sin 2t-\frac{3}{5}\cos 3t-\frac{2}{15}\sin 3t\)
- \(-\frac{4}{15}\cos t+\frac{1}{15}\sin t+\frac{4}{15}\cos 4t-\frac{1}{60}\sin 4t\)
- \(\frac{5}{3}\cos t+\sin t-\frac{5}{3}\cos 2t-\frac{1}{2}\sin 2t\)
- \(-\frac{1}{3}\cos \frac{t}{2}+\frac{2}{3}\sin\frac{t}{2}+\frac{1}{3}\cos t-\frac{1}{3}\sin t\)
- \(\frac{1}{15}\cos\frac{t}{4}-\frac{8}{15}\sin\frac{t}{4}-\frac{1}{15}\cos 4t+\frac{1}{30}\sin 4t\)
- \(\frac{2}{5}\cos\frac{t}{3}-\frac{3}{5}\sin\frac{t}{3}-\frac{2}{5}\cos\frac{t}{2}+\frac{2}{5}\sin\frac{t}{2}\)
6.
- \(e^{t}(\cos 2t+\sin 2t)-e^{-t}\left( \cos 3t+\frac{4}{3}\sin 3t\right)\)
- \(e^{3t}\left(-\cos 2t+\frac{3}{2}\sin 2t\right)+e^{-t}\left(\cos 2t+\frac{1}{2}\sin 2t\right)\)
- \(e^{-2t}\left(\frac{1}{8}\cos t+\frac{1}{4}\sin t\right)-e^{2t}\left(\frac{1}{8}\cos 3t-\frac{1}{12}\sin 3t\right)\)
- \(e^{2t}\left(\cos t+\frac{1}{2}\sin t\right)-e^{3t}\left(\cos 2t-\frac{1}{4}\sin 2t\right)\)
- \(e^{t}\left(\frac{1}{5}\cos t+\frac{2}{5}\sin t\right)-e^{-t}\left(\frac{1}{5}\cos 2t+\frac{2}{5}\sin 2t\right)\)
- \(e^{t/2}\left(-\cos t+\frac{9}{8}\sin t\right)+e^{-t/2}\left(\cos t-\frac{1}{8}\sin t\right)\)
7.
- \(1-\cos t\)
- \(\frac{e^{t}}{16}(1-\cos 4t)\)
- \(\frac{4}{9}e^{2t}+\frac{5}{9}e^{-t}\sin 3t-\frac{4}{9}e^{-t}\cos 3t\)
- \(3e^{t/2}-\frac{7}{2}e^{t}\sin 2t-3e^{t}\cos 2t\)
- \(\frac{1}{4}e^{3t}-\frac{1}{4}e^{-t}\cos 2t\)
- \(\frac{1}{9}e^{2t}-\frac{1}{9}e^{-t}\cos 3t+\frac{5}{9}e^{-t}\sin 3t\)
8.
- \(-\frac{3}{10}\sin t+\frac{2}{5}\cos t-\frac{3}{4}e^{t}+\frac{7}{20}e^{3t}\)
- \(-\frac{3}{5}e^{-t}\sin t+\frac{1}{5}e^{-t}\cos t-\frac{1}{2}e^{-t}+\frac{3}{10}e^{t}\)
- \(-\frac{1}{10}e^{t}\sin t-\frac{7}{10}e^{t}\cos t+\frac{1}{5}e^{-t}+\frac{1}{2}e^{2t}\)
- \(-\frac{1}{2}e^{t}+\frac{7}{10}e^{-t}-\frac{1}{5}\cos 2t+\frac{3}{5}\sin 2t\)
- \(\frac{3}{10}+\frac{1}{10}e^{2t}+\frac{1}{10}e^{t}\sin 2t-\frac{2}{5}e^{t}\cos 2t\)
- \(-\frac{4}{9}e^{2t}\cos 3t+\frac{1}{3}e^{2t}\sin 3t-\frac{5}{9}e^{2t}+e^{t}\)
9. \(\frac{1}{a}e^{\frac{b}{a}t}f\left(\frac{t}{a}\right)\)