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10.5E: Constant Coefficient Homogeneous Systems II (Exercises)

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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$$\newcommand{\place}{\bigskip\hrule\bigskip\noindent} \newcommand{\threecol}{\left[\begin{array}{r}#1\\#2\\#3\end{array}\right]} \newcommand{\threecolj}{\left[\begin{array}{r}#1\$1\jot]#2\\[1\jot]#3\end{array}\right]} \newcommand{\lims}{\,\bigg|_{#1}^{#2}} \newcommand{\twocol}{\left[\begin{array}{l}#1\\#2\end{array}\right]} \newcommand{\ctwocol}{\left[\begin{array}{c}#1\\#2\end{array}\right]} \newcommand{\cthreecol}{\left[\begin{array}{c}#1\\#2\\#3\end{array}\right]} \newcommand{\eqline}{\centerline{\hfill\displaystyle#1\hfill}} \newcommand{\twochar}{\left|\begin{array}{cc} #1-\lambda\\#3-\lambda\end{array}\right|} \newcommand{\twobytwo}{\left[\begin{array}{rr} #1\\#3\end{array}\right]} \newcommand{\threechar}{\left[\begin{array}{ccc} #1-\lambda\\#4-\lambda\\#7 -\lambda\end{array}\right]} \newcommand{\threebythree}{\left[\begin{array}{rrr} #1\\#4\\#7 \end{array}\right]} \newcommand{\solutionpart}{\vskip10pt\noindent\underbar{\color{blue}\sc Solution({\bf #1})\ }} \newcommand{\Cex}{\fbox{\textcolor{red}{C}}\, } \newcommand{\CGex}{\fbox{\textcolor{red}{C/G}}\, } \newcommand{\Lex}{\fbox{\textcolor{red}{L}}\, } \newcommand{\matfunc}{\left[\begin{array}{cccc}#1_{11}(t)_{12}(t)&\cdots _{1#3}(t)\\#1_{21}(t)_{22}(t)&\cdots_{2#3}(t)\\\vdots& \vdots&\ddots&\vdots\\#1_{#21}(t)_{#22}(t)&\cdots_{#2#3}(t) \end{array}\right]} \newcommand{\col}{\left[\begin{array}{c}#1_1\\#1_2\\\vdots\\#1_#2\end{array}\right]} \newcommand{\colfunc}{\left[\begin{array}{c}#1_1(t)\\#1_2(t)\\\vdots\\#1_#2(t)\end{array}\right]} \newcommand{\cthreebythree}{\left[\begin{array}{ccc} #1\\#4\\#7 \end{array}\right]} 1 \ newcommand {\ dy} {\ ,\ mathrm {d}y} \ newcommand {\ dx} {\ ,\ mathrm {d}x} \ newcommand {\ dyx} {\ ,\ frac {\ mathrm {d}y}{\ mathrm {d}x}} \ newcommand {\ ds} {\ ,\ mathrm {d}s} \ newcommand {\ dt }{\ ,\ mathrm {d}t} \ newcommand {\dst} {\ ,\ frac {\ mathrm {d}s}{\ mathrm {d}t}}$$ In Exercises [exer:10.5.1}– [exer:10.5.12} find the general solution. [exer:10.5.1] $$\ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$ [exer:10.5.3] $$\ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$ [exer:10.5.5] $$\ ParseError: invalid DekiScript (click for details) Callstack: at 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(Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$ [exer:10.5.19] $$\ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$ [exer:10.5.21] $$\ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$ [exer:10.5.23] $$\ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$& [exer:10.5.25] $$\ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$& [exer:10.5.27] $$\ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$& [exer:10.5.29] $$\ ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Differential_Equations/Book:_Elementary_Differential_Equations_with_Boundary_Values_Problems_(Trench)/10:_Linear_Systems_of_Differential_Equations/10.5:_Constant_Coefficient_Homogeneous_Systems_II/10.5E:_Constant_Coefficient_Homogeneous_Systems_II_(Exercises)), /content/body/p/span, line 1, column 1$$& [exer:10.5.31] $$\{{\bf y}' =\threebythree{-3}{-3}445{-8}23{-5}\bf y}$$ [exer:10.5.32] $${\bf y}'={\threebythree{-3}{-1}01{-1}0{-1}{-1}{-2}}{\bf y}$$ [exer:10.5.33] Under the assumptions of Theorem [thmtype:10.5.1}, suppose $${\bf u}$$ and $$\hat{\bf u}$$ are vectors such that \[(A-\lambda_1I){\bf u}={\bf x}\quad\mbox{and }\quad (A-\lambda_1I)\hat{\bf u}={\bf x},$

and let

${\bf y}_2={\bf u}e^{\lambda_1t}+{\bf x}te^{\lambda_1t} \quad\mbox{and }\quad \hat{\bf y}_2=\hat{\bf u}e^{\lambda_1t}+{\bf x}te^{\lambda_1t}.$

Show that $${\bf y}_2-\hat{\bf y}_2$$ is a scalar multiple of $${\bf y}_1={\bf x}e^{\lambda_1t}$$. [exer:10.5.34] Under the assumptions of Theorem [thmtype:10.5.2}, let

\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}

Complete the proof of Theorem [thmtype:10.5.2} by showing that $${\bf y}_3$$ is a solution of $${\bf y}'=A{\bf y}$$ and that $$\{{\bf y}_1,{\bf y}_2,{\bf y}_3\}$$ is linearly independent. [exer:10.5.35] Suppose the matrix ?has a repeated eigenvalue $$\lambda_1$$ and the associated eigenspace is one-dimensional. Let $${\bf x}$$ be a $$\lambda_1$$-eigenvector of $$A$$. Show that if $$(A-\lambda_1I){\bf u}_1={\bf x}$$ and $$(A-\lambda_1I){\bf u}_2={\bf x}$$, then $${\bf u}_2-{\bf u}_1$$ is parallel to $${\bf x}$$. Conclude from this that all vectors $${\bf u}$$ such that $$(A-\lambda_1I){\bf u}={\bf x}$$ define the same positive and negative half-planes with respect to the line $$L$$ through the origin parallel to $${\bf x}$$. [exer:10.5.36] $${\bf y}'=\{\twobytwo{-3}{-1}41}{\bf y}$$& [exer:10.5.37]   $${\bf y}'=\{\twobytwo2{-1}10}{\bf y}$$ [exer:10.5.38]   $${\bf y}'=\{\twobytwo{-1}{-3}35}{\bf y}$$& [exer:10.5.39]   $${\bf y}'=\{\twobytwo{-5}3{-3}1}{\bf y}$$ [exer:10.5.40]   $${\bf y}'=\{\twobytwo{-2}{-3}34}{\bf y}$$& [exer:10.5.41]   $${\bf y}'=\{\twobytwo{-4}{-3}32}{\bf y}$$ [exer:10.5.42]   $${\bf y}'=\{\twobytwo0{-1}1{-2}}{\bf y}$$&

[exer:10.5.43] $${\bf y}'=\{\twobytwo01{-1}2}{\bf y}$$

[exer:10.5.44]   $${\bf y}'=\{\twobytwo{-2}1{-1}0}{\bf y}$$&

[exer:10.5.45]   $${\bf y}'=\{\twobytwo0{-4}1{-4}}{\bf y}$$