A.4.4: Section 4.4 Answers
- Page ID
- 43763
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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. \(\overline{y}=0\) 0 is a stable equilibrium; trajectories are \(v^{2}+\frac{y^{2}}{4}=c\)
2. \(\overline{y}=0\) 0 is an unstable equilibrium; trajectories are \(v^{2}+\frac{2y^{3}}{3}=c\)
3. \(\overline{y}=0\) 0 is a stable equilibrium; trajectories are \(v^{2}+\frac{2|y|^{3}}{3}=c\)
4. \(\overline{y}=0\) 0 is a stable equilibrium; trajectories are \(v^{2}-e^{-y}(y+1)=c\)
5. equilibria: \(0\) (stable) and \(−2, 2\) (unstable); trajectories: \(2v^{2} − y^{4} + 8y^{2} = c\); separatrix: \(2v^{2} − y^{4} + 8y^{2} = 16\)
6. equilibria: \(0\) (unstable) and \(−2, 2\) (stable); trajectories: \(2v^{2} + y^{4} − 8y^{2} = c\); separatrix: \(2v^{2} + y^{4} − 8y^{2} =0\)
7. equilibria: \(0, −2, 2\) (stable), \(−1, 1\) (unstable); trajectories: \(6v^{2} + y^{2}(2y^{4} − 15y^{2} + 24) = c\); separatrix: \(6v^{2} + y^{2} (2y^{4} − 15y^{2} + 24) = 11\)
8. equilibria: \(0, 2\) (stable) and \(−2, 1\) (unstable); trajectories: \(30v^{2} + y^{2}(12y^{3} − 15y^{2} − 80y + 120) = c\); separatrices: \(30v^{2} + y^{2} (12y^{3} − 15y^{2} − 80y + 120) = 496\) and \(30v^{2} + y^{2} (12y^{3} − 15y^{2} − 80y + 120) = 37\)
9. No equilibria if \(a < 0; 0\) is unstable if \(a = 0\); \(\sqrt{a}\) is stable and \(−\sqrt{a}\) is unstable if \(a > 0\).
10. \(0\) is a stable equilibrium if \(a ≤ 0\); \(−\sqrt{a}\) and \(\sqrt{a}\) are stable and \(0\) is unstable if \(a > 0\).
11. \(0\) is unstable if \(a ≤ 0\); \(−\sqrt{a}\) and \(\sqrt{a}\) are unstable and \(0\) is stable if \(a > 0\).
12. \(0\) is stable if \(a ≤ 0; 0\) is stable and \(−\sqrt{a}\) and \(\sqrt{a}\) are unstable if \(a ≤ 0\).
22. An equilibrium solution \(\overline{y}\) of \(y'' + p(y) = 0\) is unstable if there’s an \(€> 0\) such that, for every \(δ > 0\), there’s a solution of (A) with \(\sqrt{(y(0)-\overline{y})^{2}+v^{2}(0)}<\delta \), but \(\sqrt{(y(t)-\overline{y})^{2}+v^{2}(t)}\geq €\) for some \(t>0\).