10.5E: Sistemas Homogéneos de Coeficiente Constante II (Ejercicios)
- Page ID
- 114734
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Q10.5.1
En Ejercicios 10.5.1-10.5.12 encuentra la solución general.
1. \({\bf y}'=\left[\begin{array}{cc}{3}&{4}\\{-1}&{7}\end{array}\right]{\bf y}\)
2. \({\bf y}'=\left[\begin{array}{cc}{0}&{-1}\\{1}&{-2}\end{array}\right]{\bf y}\)
3. \({\bf y}'=\left[\begin{array}{cc}{-7}&{4}\\{-1}&{-11}\end{array}\right]{\bf y}\)
4. \({\bf y}'=\left[\begin{array}{cc}{3}&{1}\\{-1}&{1}\end{array}\right]{\bf y}\)
5. \({\bf y}'=\left[\begin{array}{cc}{4}&{12}\\{-3}&{-8}\end{array}\right]{\bf y}\)
6. \({\bf y}'=\left[\begin{array}{cc}{-10}&{9}\\{-4}&{2}\end{array}\right]{\bf y}\)
7. \({\bf y}'=\left[\begin{array}{cc}{-13}&{16}\\{-9}&{11}\end{array}\right]{\bf y}\)
8. \({\bf y}'=\left[\begin{array}{ccc}{0}&{2}&{1}\\{-4}&{6}&{1}\\{0}&{4}&{2}\end{array}\right]{\bf y}\)
9. \({\bf y}'=\frac{1}{3}\left[\begin{array}{ccc}{1}&{1}&{-3}\\{-4}&{-4}&{3}\\{-2}&{1}&{0}\end{array}\right]{\bf y}\)
10. \({\bf y}'=\left[\begin{array}{ccc}{-1}&{1}&{-1}\\{-2}&{0}&{2}\\{-1}&{3}&{-1}\end{array}\right]{\bf y}\)
11. \({\bf y}'=\left[\begin{array}{ccc}{4}&{-2}&{-2}\\{-2}&{3}&{-1}\\{2}&{-1}&{3}\end{array}\right]{\bf y}\)
12. \({\bf y}'=\left[\begin{array}{ccc}{6}&{-5}&{3}\\{2}&{-1}&{3}\\{2}&{1}&{1}\end{array}\right]{\bf y}\)
Q10.5.2
En Ejercicios 10.5.13-10.5.23 resolver el problema de valor inicial.
13. \({\bf y}'=\left[\begin{array}{cc}{-11}&{8}\\{-2}&{-3}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{6}\\{2}\end{array}\right]\)
14. \({\bf y}'=\left[\begin{array}{cc}{15}&{-9}\\{16}&{-9}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{5}\\{8}\end{array}\right]\)
15. \({\bf y}'=\left[\begin{array}{cc}{-3}&{-4}\\{1}&{-7}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{2}\\{3}\end{array}\right]\)
16. \({\bf y}'=\left[\begin{array}{cc}{-7}&{24}\\{-6}&{17}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{3}\\{1}\end{array}\right]\)
17. \({\bf y}'=\left[\begin{array}{cc}{-7}&{3}\\{-3}&{-1}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{0}\\{2}\end{array}\right]\)
18. \({\bf y}'=\left[\begin{array}{ccc}{-1}&{1}&{0}\\{1}&{-1}&{-2}\\{-1}&{-1}&{-1}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{6}\\{5}\\{-7}\end{array}\right]\)
19. \({\bf y}'=\left[\begin{array}{ccc}{-2}&{2}&{1}\\{-2}&{2}&{1}\\{-3}&{3}&{2}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{-6}\\{-2}\\{0}\end{array}\right]\)
20. \({\bf y}'=\left[\begin{array}{ccc}{-7}&{-4}&{4}\\{-1}&{0}&{1}\\{-9}&{-5}&{6}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{-6}\\{9}\\{-1}\end{array}\right]\)
21. \({\bf y}'=\left[\begin{array}{ccc}{-1}&{-4}&{-1}\\{3}&{6}&{1}\\{-3}&{-2}&{3}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{-2}\\{1}\\{3}\end{array}\right]\)
22. \({\bf y}'=\left[\begin{array}{ccc}{4}&{-8}&{-4}\\{-3}&{-1}&{-3}\\{1}&{-1}&{9}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{-4}\\{1}\\{-3}\end{array}\right]\)
23. \({\bf y}'=\left[\begin{array}{ccc}{-5}&{-1}&{11}\\{-7}&{1}&{13}\\{-4}&{0}&{8}\end{array}\right]{\bf y},\quad {\bf y}(0)=\left[\begin{array}{c}{0}\\{2}\\{2}\end{array}\right]\)
Q10.5.3
Las matrices de coeficientes en los Ejercicios 10.5.24-10.5.32 tienen valores propios de multiplicidad\(3\). Encuentra la solución general.
24. \({\bf y}'=\left[\begin{array}{ccc}{5}&{-1}&{1}\\{-1}&{9}&{-3}\\{-2}&{2}&{4}\end{array}\right]{\bf y}\)
25. \({\bf y}'=\left[\begin{array}{ccc}{1}&{10}&{-12}\\{2}&{2}&{3}\\{2}&{-1}&{6}\end{array}\right]{\bf y}\)
26. \({\bf y}'=\left[\begin{array}{ccc}{-6}&{-4}&{-4}\\{2}&{-1}&{1}\\{2}&{3}&{1}\end{array}\right]{\bf y}\)
27. \({\bf y}'=\left[\begin{array}{ccc}{0}&{2}&{-2}\\{-1}&{5}&{-3}\\{1}&{1}&{1}\end{array}\right]{\bf y}\)
28. \({\bf y}'=\left[\begin{array}{ccc}{-2}&{-12}&{10}\\{2}&{-24}&{11}\\{2}&{-24}&{8}\end{array}\right]{\bf y}\)
29. \({\bf y}'=\left[\begin{array}{ccc}{-1}&{-12}&{8}\\{1}&{-9}&{4}\\{1}&{-6}&{1}\end{array}\right]{\bf y}\)
30. \({\bf y}'=\left[\begin{array}{ccc}{-4}&{0}&{-1}\\{-1}&{-3}&{-1}\\{1}&{0}&{-2}\end{array}\right]{\bf y}\)
31. \({\bf y}'=\left[\begin{array}{ccc}{-3}&{-3}&{4}\\{4}&{5}&{-8}\\{2}&{3}&{-5}\end{array}\right]{\bf y}\)
32. \({\bf y}'=\left[\begin{array}{ccc}{-3}&{-1}&{0}\\{1}&{-1}&{0}\\{-1}&{-1}&{-2}\end{array}\right]{\bf y}\)
Q10.5.4
33. Bajo los supuestos del Teorema 10.5.1, supongamos\({\bf u}\) y\(\hat{\bf u}\) son vectores tales que
\[(A-\lambda_1I){\bf u}={\bf x}\quad\mbox{and }\quad (A-\lambda_1I)\hat{\bf u}={\bf x},\nonumber \]
y dejar
\[{\bf y}_2={\bf u}e^{\lambda_1t}+{\bf x}te^{\lambda_1t} \quad\mbox{and }\quad \hat{\bf y}_2=\hat{\bf u}e^{\lambda_1t}+{\bf x}te^{\lambda_1t}.\nonumber \]
Demostrar que\({\bf y}_2-\hat{\bf y}_2\) es un múltiplo escalar de\({\bf y}_1={\bf x}e^{\lambda_1t}\).
34. Bajo los supuestos del Teorema 10.5.2, vamos
\[\begin{aligned} {\bf y}_1 &={\bf x} e^{\lambda_1t},\\ {\bf y}_2&={\bf u}e^{\lambda_1t}+{\bf x} te^{\lambda_1t},\mbox{ and }\\ {\bf y}_3&={\bf v}e^{\lambda_1t}+{\bf u}te^{\lambda_1t}+{\bf x} {t^2e^{\lambda_1t}\over2}.\end{aligned}\nonumber \]
Completar la prueba del Teorema 10.5.2 mostrando que\({\bf y}_3\) es una solución de\({\bf y}'=A{\bf y}\) y que\(\{{\bf y}_1,{\bf y}_2,{\bf y}_3\}\) es linealmente independiente.
35. Supongamos que la matriz\[A=\left[\begin{array}{cc}{a_{11}}&{a_{12}}\\{a_{21}}&{a_{22}}\end{array}\right]\nonumber\] tiene un valor propio repetido\(\lambda_1\) y el espacio propio asociado es unidimensional. Dejar\({\bf x}\) ser un\(\lambda_1\) -eigenvector de\(A\). Demostrar que si\((A-\lambda_1I){\bf u}_1={\bf x}\) y\((A-\lambda_1I){\bf u}_2={\bf x}\), entonces\({\bf u}_2-{\bf u}_1\) es paralelo a\({\bf x}\). Concluimos de esto que todos los vectores\({\bf u}\) tales que\((A-\lambda_1I){\bf u}={\bf x}\) definen los mismos semiplanos positivos y negativos con respecto a la línea\(L\) a través del origen paralelos a\({\bf x}\).
Q10.5.5
En Ejercicios 10.5.36-10.5.45 trazar trayectorias del sistema dado.
36. \({\bf y}'=\left[\begin{array}{cc}{-3}&{-1}\\{4}&{1}\end{array}\right]{\bf y}\)
37. \({\bf y}'=\left[\begin{array}{cc}{2}&{-1}\\{1}&{0}\end{array}\right]{\bf y}\)
38. \({\bf y}'=\left[\begin{array}{cc}{-1}&{-3}\\{3}&{5}\end{array}\right]{\bf y}\)
39. \({\bf y}'=\left[\begin{array}{cc}{-5}&{3}\\{-3}&{1}\end{array}\right]{\bf y}\)
40. \({\bf y}'=\left[\begin{array}{cc}{-2}&{-3}\\{3}&{4}\end{array}\right]{\bf y}\)
41. \({\bf y}'=\left[\begin{array}{cc}{-4}&{-3}\\{3}&{2}\end{array}\right]{\bf y}\)
42. \({\bf y}'=\left[\begin{array}{cc}{0}&{-1}\\{1}&{-2}\end{array}\right]{\bf y}\)
43. \({\bf y}'=\left[\begin{array}{cc}{0}&{1}\\{-1}&{2}\end{array}\right]{\bf y}\)
44. \({\bf y}'=\left[\begin{array}{cc}{-2}&{1}\\{-1}&{0}\end{array}\right]{\bf y}\)
45. \({\bf y}'=\left[\begin{array}{cc}{0}&{-4}\\{1}&{-4}\end{array}\right]{\bf y}\)