A.9.1: Section 9.1 Answers
- Page ID
- 43790
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)2. \(y=2x^{2}-3x^{3}+\frac{1}{x}\)
3. \(y=2e^{x}+3e^{-x}-e^{2x}+e^{-3x}\)
4. \(y_{i}=\frac{(x-x_{0})^{i-1}}{(i-1)!},\quad 1\leq i\leq n\)
5.
b. \(y_{1}=-\frac{1}{2}x^{3}+x^{2}+\frac{1}{2x},\quad y_{2}=\frac{1}{3}x^{2}-\frac{1}{3x},\quad y_{3}=\frac{1}{4}x^{3}-\frac{1}{3}x^{2}+\frac{1}{12x}\)
c. \(y=k_{0}y_{1}+k_{1}y_{2}+k_{2}y_{3}\)
7. \(2e^{-x^{2}}\)
8. \(\sqrt{2}K\cos x\)
9.
a. \(W(x)=2e^{3x}\)
d. \(y=e^{x}(c_{1}+c_{2}x+c_{3}x^{2})\)
10.
- \(2\)
- \(-e^{3x}\)
- \(4\)
- \(4/x^{2}\)
- \(1\)
- \(2x\)
- \(2/x^{2}\)
- \(e^{x}(x^{2}-2x+2)\)
- \(-240/x^{5}\)
- \(6e^{2x}(2x-1)\)
- \(-128x\)
24.
- \(y'''=0\)
- \(xy'''-y''-xy'+y=0\)
- \((2x-3)y'''-2y''-(2x-5)y'=0\)
- \((x^{2}-2x+2)y'''-x^{2}y''+2xy'-2y=0\)
- \(x^{3}y'''+x^{2}y''-2xy'+2y=9\)
- \((3x-1)y'''-(12x-1)y''+9(x+1)y'-9y=0\)
- \(x^{4}y^{(4)}+5x^{3}y'''-3x^{2}y''-6xy'+6y=0\)
- \(x^{4}y^{(4)}+3x^{2}y'''-x^{2}y''+2xy'-2y=0\)
- \((2x-1)y^{(4)}-4xy'''+(5-2x)y''+4xy'-4y=0\)
- \(xy^{(4)}-y'''-4xy''+4y'=0\)