42: Section 10.6 Answers
- Page ID
- 103608
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \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. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]+c_{2}\left[\begin{array}{c}{3}\\{2}\end{array}\right]e^{t}-\left[\begin{array}{c}{11}\\{11}\end{array}\right]t-\left[\begin{array}{c}{15}\\{10}\end{array}\right]\)
2. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^t+c_{2}\left[\begin{array}{c}{1}\\{3}\end{array}\right]e^{-t}+\left[\begin{array}{c}{4}\\{8}\end{array}\right]t+\left[\begin{array}{c}{0}\\{-4}\end{array}\right]\)
3. \({\bf y}=c_{1}\left[\begin{array}{c}{10}\\{3}\end{array}\right]e^{3t/2}+c_{2}\left[\begin{array}{c}{2}\\{1}\end{array}\right]e^{t/2}+\left[\begin{array}{c}{-13/2}\\{-13/4}\end{array}\right]te^{t/2}+\left[\begin{array}{c}{-15/2}\\{-9/4}\end{array}\right]e^{t/2}\)
4. \({\bf y}=c_{1}\left[\begin{array}{c}{-\sin 2t}\\{2\cos 2t}\end{array}\right]e^{2t}+c_{2}\left[\begin{array}{c}{\cos 2t}\\{2\sin 2t}\end{array}\right]e^{2t}+\left[\begin{array}{c}{-{1\over 8}\sin 2t\cos 4t-{1\over 8}\cos 2t\cos 4t}\\{{1\over 4}\cos 2t\sin 4t-{1\over 4}\sin 2t\cos 4t}\end{array}\right]e^{2t}\)
5. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{1}\end{array}\right]e^t+c_{2}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{2t}+\left[\begin{array}{c}{4}\\{2}\end{array}\right]te^t+\left[\begin{array}{c}{3}\\{3}\end{array}\right]e^t\)
6. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\{1}\end{array}\right]e^t+c_{2}\left[\begin{array}{c}{1}\\{1}\end{array}\right]e^{2t}+\left[\begin{array}{c}{1/10}\\{-3/20}\end{array}\right]e^{-3t}-\left[\begin{array}{c}{3}\\{1}\end{array}\right]\)
7. \({\bf y}=c_{1}\left[\begin{array}{c}{4}\\{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{-2}\\{1}\end{array}\right]e^{-3t}+\left[\begin{array}{c}{-12}\\{0}\end{array}\right]t-\left[\begin{array}{c}{4/3}\\{4/3}\end{array}\right]\)
8. \({\bf y}=c_{1}\left[\begin{array}{c}{4}\\{1}\end{array}\right]e^{3t}+c_{2}\left[\begin{array}{c}{-2}\\{1}\end{array}\right]e^{-3t}+\left[\begin{array}{c}{-te^t-{1\over 4}e^t}\\{-{1\over 8}e^{-t}-{1\over 8}e^t}\end{array}\right]\)
9. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{-1}\end{array}\right]e^t+c_{2}\left(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]te^{t}+\left[\begin{array}{c}{0}\\{1/2}\end{array}\right]e^t\right)+\left[\begin{array}{c}{1/2}\\{-2}\end{array}\right]e^{-t}\)
10. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{-1}\end{array}\right]e^t+c_{2}\left(\left[\begin{array}{c}{1}\\{-1}\end{array}\right]te^{t}+\left[\begin{array}{c}{0}\\{1/2}\end{array}\right]e^t\right)+\left[\begin{array}{c}{3}\\{-5}\end{array}\right]\)
11. \({\bf y}=c_{1}\left[\begin{array}{c}{\cos t}\\{\sin t}\end{array}\right]+c_{2}\left[\begin{array}{c}{\sin t}\\{-\cos t}\end{array}\right]+\left[\begin{array}{c}{t\cos t+\sin t\ln|\sec t|}\\{t\sin t-\cos t\ln|\sec t|}\end{array}\right]\)
12. \({\bf y}=c_{1}\left[\begin{array}{c}{-\sin t}\\{\cos t}\end{array}\right]e^t+c_{2}\left[\begin{array}{c}{\cos t}\\{\sin t}\end{array}\right]e^{t}+\left[\begin{array}{c}{-3}\\{3}\end{array}\right]e^t\)
13. \({\bf y}=c_{1}\left[\begin{array}{c}{-\sin t}\\{\cos t}\end{array}\right]e^t+c_{2}\left[\begin{array}{c}{\cos t}\\{\sin t}\end{array}\right]e^{t}+\left[\begin{array}{c}{\cos t}\\{\sin t}\end{array}\right]te^t\)
14. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{2}\end{array}\right]e^{-2t}+c_{2}\left(\left[\begin{array}{c}{1}\\{2}\end{array}\right]te^{-2t}+\left[\begin{array}{c}{1/2}\\{1/2}\end{array}\right]e^{-2t}\right)+\left[\begin{array}{c}{2t+\ln t-2t\ln t}\\{4t+3\ln t-4t\ln t}\end{array}\right]e^{-2t}\)
15. \({\bf y}=c_{1}\left[\begin{array}{c}{\cos t}\\{-\sin t}\end{array}\right]+c_{2}\left[\begin{array}{c}{\sin t}\\{\cos t}\end{array}\right]+\left[\begin{array}{c}{\cos t}\\{-\sin t}\end{array}\right]t+\left[\begin{array}{c}{-\sin t}\\{\sin t\tan t}\end{array}\right]+\left[\begin{array}{c}{\sin t}\\{\cos t}\end{array}\right]\ln|\sec t|\)
16. \({\bf y}=c_{1}\left[\begin{array}{c}{\cos t}\\{-\sin t}\end{array}\right]+c_{2}\left[\begin{array}{c}{\sin t}\\{\cos t}\end{array}\right]+\left[\begin{array}{c}{\sin t\ln|\csc t-\cot t|}\\{\cos t\ln|\csc t-\cot t|}\end{array}\right]\)
17. \({\bf y}=c_{1}\left[\begin{array}{c}{2\sin t}\\{\cos t}\end{array}\right]e^t+c_{2}\left[\begin{array}{c}{2\cos t}\\{-\sin t}\end{array}\right]e^t+\left[\begin{array}{c}{3\sin t}\\{{3\over 2}\cos t}\end{array}\right]te^t+\left[\begin{array}{c}{\cos t}\\{-{1\over 2}\sin t}\end{array}\right]e^t\ln|\sin t|+\left[\begin{array}{c}{-2\cos t}\\{\sin t}\end{array}\right]e^t\ln|\sec t|\)
18. \({\bf y}=c_{1}\left[\begin{array}{c}{\cos t-\sin t}\\{\cos t}\end{array}\right]+c_{2}\left[\begin{array}{c}{\cos t+\sin t}\\{\sin t}\end{array}\right]+\left[\begin{array}{c}{3\sin t\cos t-\cos^2 t-2\sin^2 t+(\sin t-\cos t)\ln|\sec t+\tan t|}\\{\sin^2 t-\cos^2 t-\cos t\ln|\sec t+\tan t|}\end{array}\right]\)
19. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{-1}\\{0}\end{array}\right]+c_{2}\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{2t}+c_{3}\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]e^{3t}+\left[\begin{array}{c}{-{1\over 4}e^{2t}+{1\over 2}te^{2t}}\\{-e^{t}+{1\over 4}e^{2t}+{1\over 2}te^{2t}}\\{\phantom{-}{1\over 2}t^2e^{3t}}\end{array}\right]\)
20. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]e^{t}+c_{2}\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right]e^{2t}+c_{3}\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right]e^{2t}+\left[\begin{array}{c}{-1/2}\\{-1}\\{-1/2}\end{array}\right]t+\left[\begin{array}{c}{-3/4}\\{-1}\\{-3/4}\end{array}\right]+\left[\begin{array}{c}{2}\\{2}\\{0}\end{array}\right]e^t+\left[\begin{array}{c}{2}\\{2}\\{2}\end{array}\right]te^t\)
21. \({\bf y}=\left[\begin{array}{c}{2}\\{2}\end{array}\right]te^{2t}+\left[\begin{array}{c}{-1}\\{1}\end{array}\right]e^{2t}+\left[\begin{array}{c}{-2}\\{2}\end{array}\right]te^{4t}+\left[\begin{array}{c}{2}\\{0}\end{array}\right]e^{4t}\)
22. \({\bf y}=\left[\begin{array}{c}{3}\\{3}\end{array}\right]t-\left[\begin{array}{c}{1}\\{4}\end{array}\right]+\left[\begin{array}{c}{1}\\{1}\end{array}\right]\ln t\)