A.13.2: Section 13.2 Answers
- Page ID
- 43750
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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. \((e^{bx}y')'+ce^{bx}y=0\)
2. \((xy')'+\left(x-\frac{\nu ^{2}}{x}\right)y=0\)
3. \((\sqrt{1-x^{2}}y')'+\frac{\alpha ^{2}}{\sqrt{1-x^{2}}}y=0\)
4. \((x^{b}y')'+cx^{b-2}y=0\)
5. \((e^{-x^{2}}y')'+2\alpha e^{-x^{2}}y=0\)
6. \((xe^{-x}y')'+\alpha e^{-x}y=0\)
7. \(((1-x^{2})y')'+\alpha (\alpha +1)y=0\)
9. \(\lambda_{n}=n^{2}\pi ^{2},\quad y_{n}=e^{-x}\sin n\pi x\: (n=\) positive integer)
10. \(\lambda_{0}=-1,\quad y_{0}=1\quad\lambda_{n}=n^{2}\pi ^{2},\quad y_{n}=e^{-x}(n\pi\cos n\pi x+\sin n\pi x)\: (n=\) positive integer)
11.
- \(\lambda =0\) is an eigenvalue \(y_{0}=2-x\)
- none
- \(5.0476821,\: 14.9198790,\: 29.7249673,\: 49.4644528\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)
12.
- \(\lambda =0\) isn't an eigenvalue
- \(−0.5955245\quad y=\cosh\sqrt{-\lambda }x\)
- \(8.8511386,\: 38.4741053,\: 87.8245457,\: 156.9126094\quad y=\cos\sqrt{\lambda }x\)
13.
- \(\lambda =0\) isn't an eigenvalue
- none
- \(0.1470328,\: 1.4852833,\: 4.5761411,\: 9.6059439\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x+\sin\sqrt{\lambda }x\)
14.
- \(\lambda =0\) isn't an eigenvalue
- \(−0.1945921\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x\)
- \(1.9323619,\: 5.9318981,\: 11.9317920,\: 19.9317507\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)
15.
- \(\lambda =0\) isn't an eigenvalue
- \(−1.0664054\quad y=\cosh\sqrt{-\lambda }x\)
- \(1.5113188,\: 8.8785880,\: 21.2104662,\: 38.4805610\quad y=\cos\sqrt{\lambda }x\)
16.
- \(\lambda =0\) isn't an eigenvalue
- \(−1.0239346\quad y=\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x\)
- \(2.0565705,\: 9.3927144,\: 21.7169130,\: 38.9842177\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)
17.
- \(\lambda =0\) isn't an eigenvalue
- \(−0.4357577\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x\)
- \(0.3171423,\: 3.7055350,\: 9.1970150,\: 16.8760401\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)
18.
- \(\lambda =0\) isn't an eigenvalue
- \(−2.1790546,\: −9.0006633\quad y=\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-3\sinh\sqrt{-\lambda }x\)
- \(5.8453181,\: 17.9260967,\: 35.1038567,\: 57.2659330\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x-3\sin\sqrt{\lambda }x\)
19.
- \(\lambda =0\) is an eigenvalue \(y_{0}=2-x\)
- \(−1.0273046\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x\)
- \(8.8694608,\: 16.5459202,\: 26.4155505,\: 38.4784094\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)
20.
- \(\lambda =0\) isn't an eigenvalue
- \(−7.9394171,\: −3.1542806\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-5\sinh\sqrt{-\lambda }x\)
- \(29.3617465,\: 78.777456,\: 147.8866417,\: 236.7229622\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-5\sin\sqrt{\lambda }x\)
21. \(λ = 0,\quad y = xe^{−x}\quad 20.1907286,\: 118.8998692,\: 296.5544121,\: 553.1646458\quad y = e^{−x} \sin \sqrt{\lambda }x\)
22. \(λ_{n} = n^{2}\pi ^{2},\quad y_{n} = x \sin n\pi (x − 2)\quad (n =\) positive integer)
23. \(λ = 0,\quad y = x(2 − x)\quad 20.1907286,\: 118.8998692,\: 296.5544121\: 553.1646458,\quad y = x \sin\sqrt{\lambda } (x − 2)\)
24. \(3.3730893,\: 23.1923372,\: 62.6797232,\: 121.8999231,\: 200.8578309\quad y = x \sin\sqrt{\lambda } (x − 1)\)
25.
- \(-L<\delta <0\)
- \(\delta =-L\)
26. \(\lambda _{0}=-1\alpha ^{2}\quad y_{0}=e^{-x/a}\quad\lambda _{n}=n^{2},\quad y_{n}=n\alpha\cos nx-\sin nx,\quad n=1,2,\ldots\)
27.
- \(y=x-\alpha \)
- \(y=\alpha k\cosh kx-\sin kx\)
- \(y=\alpha k\cos kx-\sin kx\)
29. b. \(\lambda =-\alpha ^{2}/\beta ^{2}\quad y=e^{-\alpha x/\beta }\)