14.32: Sección 4.4 Respuestas
- Page ID
- 115222
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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 es un equilibrio estable; las trayectorias son\(v^{2}+\frac{y^{2}}{4}=c\)
2. \(\overline{y}=0\)0 es un equilibrio inestable; las trayectorias son\(v^{2}+\frac{2y^{3}}{3}=c\)
3. \(\overline{y}=0\)0 es un equilibrio estable; las trayectorias son\(v^{2}+\frac{2|y|^{3}}{3}=c\)
4. \(\overline{y}=0\)0 es un equilibrio estable; las trayectorias son\(v^{2}-e^{-y}(y+1)=c\)
5. equilibrios:\(0\) (estable) e\(−2, 2\) (inestable); trayectorias:\(2v^{2} − y^{4} + 8y^{2} = c\); separatriz:\(2v^{2} − y^{4} + 8y^{2} = 16\)
6. equilibrios:\(0\) (inestable) y\(−2, 2\) (estable); trayectorias:\(2v^{2} + y^{4} − 8y^{2} = c\); separatriz:\(2v^{2} + y^{4} − 8y^{2} =0\)
7. equilibrios:\(0, −2, 2\) (estable),\(−1, 1\) (inestable); trayectorias:\(6v^{2} + y^{2}(2y^{4} − 15y^{2} + 24) = c\); separatriz:\(6v^{2} + y^{2} (2y^{4} − 15y^{2} + 24) = 11\)
8. equilibrios:\(0, 2\) (estable) e\(−2, 1\) (inestable); trayectorias:\(30v^{2} + y^{2}(12y^{3} − 15y^{2} − 80y + 120) = c\); separatrices:\(30v^{2} + y^{2} (12y^{3} − 15y^{2} − 80y + 120) = 496\) y\(30v^{2} + y^{2} (12y^{3} − 15y^{2} − 80y + 120) = 37\)
9. No hay equilibrios si\(a < 0; 0\) es inestable si\(a = 0\);\(\sqrt{a}\) es estable y\(−\sqrt{a}\) es inestable si\(a > 0\).
10. \(0\)es un equilibrio estable si\(a ≤ 0\);\(−\sqrt{a}\) y\(\sqrt{a}\) son estables y\(0\) es inestable si\(a > 0\).
11. \(0\)es inestable si\(a ≤ 0\);\(−\sqrt{a}\) y\(\sqrt{a}\) son inestables y\(0\) es estable si\(a > 0\).
12. \(0\)es estable si\(a ≤ 0; 0\) es estable y\(−\sqrt{a}\) y\(\sqrt{a}\) son inestables si\(a ≤ 0\).
22. Una solución\(\overline{y}\) de equilibrio de\(y'' + p(y) = 0\) es inestable si hay\(€> 0\) tal que, para cada\(δ > 0\), hay una solución de (A) con\(\sqrt{(y(0)-\overline{y})^{2}+v^{2}(0)}<\delta \), pero\(\sqrt{(y(t)-\overline{y})^{2}+v^{2}(t)}\geq €\) para algunos\(t>0\).