
% extra credit ch6, 2280 S2019
\documentclass[leqno]{article}
\input{linalg.sty}
\def\rf#1{(\ref{#1})}
\def\Prob#1#2{\par\bigskip\par\noindent
{\large\sf Problem #1}. {\bf #2}\par}
\parskip=3pt
\def\MAPLE{{\tt maple}}
% \begin{topics}{template} ... \end{topics}
% used like enumerate, itemize, description environments
% The item[] is never optional!
%
\newcommand{\topicslabel}[1]{\mbox{#1}\hfil}
\newenvironment{topics}[1]{
    \begin{list}{}{%
      \let\makelabel\topicslabel
      \settowidth{\labelwidth}{\topicslabel{#1}}
      \setlength{\leftmargin}{\labelwidth}
      \advance\leftmargin by 1.5em}
}{\end{list}}
\pagestyle{plain}
\setlength{\textheight}{9.8in}
\begin{document}
\thispagestyle{empty}
 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2280 Extra Credit Problems \\
\bf Chapter 6 \\
{\small\bf S2019}
\end{center}

{\bf Submitted work}. Please submit one stapled package per chapter.
Kindly label problems \fbox{\bf Extra Credit}. Label each problem with
its corresponding problem number, e.g., \fbox{Xc6.1-4}. Please attach
this printed sheet to simplify your work.

%09 Dec, 9.1: \quad \Y{4}, \Y{8}, \Y{18} \\
%10 Dec, 9.2: \quad  \Y{2}, \Y{12}, \Y{22} \\
%12 Dec, 9.3: \quad \Y{8}, \Y{9}, \Y{10} \\
%15 Dec, 9.4: \quad  \Y{4}, \Y{8}
\Prob{Xc6.1-4}{(Phase Portraits)}
Find the equilibrium points for the system. Plot a phase diagram using the {\tt maple} code below.
$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=& x - 2y +3, \\[2ex]
 \dd\frac{dy}{dt} &=& x-y+2.\end{array}\right.
$$

\BF{Example}: Plot the phase diagram of $\.u'=\twobytwo{1 & 2}{0 & 3}\.u+\colvector{4 \\ 5}$ using {\tt maple}.
\begin{verbatim}
 with(DEtools):
 equilEQ:=[0=x+2*y+4,0=3*y+5];
 solve(equilEQ,{x,y});# find diagram center (a,b)
 a:=-2/3;b:=-5/3;
 de:=[diff(x(t),t)=x(t)+2*y(t)+4,diff(y(t),t)=3*y(t)+5];
 ic:=[[x(0)=0,y(0)=-1],[x(0)=-1,y(0)=-1.5],[x(0)=0.5,y(0)=-2],
 [x(0)=0.5,y(0)=-1.5],[x(0)=-0.7,y(0)=-1.7]];
 DEplot(de,[x(t),y(t)],t=-10..10,ic,x=a-2..a+2,y=b-2..b+2,stepsize=0.05);
\end{verbatim}

\Prob{Xc6.1-8}{(Equilibrium Points)}
Find the equilibrium points for the system. Plot a phase diagram. The graph window should include the three equilibrium points.
$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=& x - 2y, \\[2ex]
 \dd\frac{dy}{dt} &=& 4x-x^3.\end{array}\right.
$$
% with(DEtools):
% equilEQ:=[0=x-2*y,0=4*x-x^3];
% solve(equilEQ,{x,y});
% de:=[diff(x(t),t)=x(t)-2*y(t),diff(y(t),t)=x(t)-x(t)^3];
% ic:=[[x(0)=0,y(0)=-1],[x(0)=0,y(0)=1],[x(0)=2,y(0)=-1.5],[x(0)=-2,y(0)=-1.5],
% [x(0)=2,y(0)=1.5],[x(0)=-2,y(0)=1.5]];
% DEplot(de,[x(t),y(t)],t=-10..10,ic,x=-3..3,y=-3..3,stepsize=0.05);

\Prob{Xc6.1-18}{(Stability)}
Determine if the equilibrium point $(0,0)$ is stable, asymptotically stable, or unstable.
Identify the equilibrium point as a node, saddle, center or spiral by examination of its
computer-generated direction field.

\begin{itemize}
\item[\BF{(a)}] $x'=y$, $y'=-x$
\item[\BF{(b)}] $x'=y$, $y'= -5x -4y$
\item[\BF{(c)}] $x=-2x$, $y'=-2y$
\item[\BF{(d)}] $x=y$, $y'=x$
\end{itemize}
% ans: stable center, stable spiral, stable node, unstable saddle

\Prob{Xc6.2-2}{(Classification by Eigenvalues)}
Compute the eigenvalues of $A$. Determine stability of equilibrium $(0,0)$ and classify as node (proper/improper), saddle, spiral, center.

\begin{itemize}
\item[\BF{(a)}] $A\twobytwo{0 & 1}{-1 & 0}$
\item[\BF{(b)}] $A\twobytwo{1 & 2}{2 & 1}$
\item[\BF{(c)}] $A\twobytwo{3 & -2}{4 & -1}$
\item[\BF{(c)}] $A\twobytwo{1 & -2}{2 & -3}$
\end{itemize}

\Prob{Xc6.2-12}{(Phase Portrait)}
Find the equilibrium point (it is unique) and plot by computer a phase diagram.
$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=& x +y-7, \\[2ex]
 \dd\frac{dy}{dt} &=& 3x-y-5.\end{array}\right.
$$
\Prob{Xc6.2-22}{(Almost Linear System)}
Linearize the system at its equilibria and determine the stability and type of each.
Plot a phase diagram by computer to verify the claims made.

$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=& 2x -5y+x^3, \\[2ex]
 \dd\frac{dy}{dt} &=& 4x-6y+y^4.\end{array}\right.
$$
\Prob{Xc6.3-8}{(Predator-Prey System)}
Linearize the system at equilibrium point $(0,0)$. Verify that
the phase diagram of the nonlinear system at $(0,0)$ is a saddle.
$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=& x(5-x-y), \\[2ex]
 \dd\frac{dy}{dt} &=& y(-2+x).\end{array}\right.
$$

\Prob{Xc6.3-9}{(Predator-Prey System)}
Linearize the system at equilibrium point $(5,0)$. Verify that
the phase diagram of the nonlinear system at $(5,0)$ is a saddle.
$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=& x(5-x-y), \\[2ex]
 \dd\frac{dy}{dt} &=& y(-2+x).\end{array}\right.
$$
\Prob{Xc6.3-10}{(Predator-Prey System)}
Linearize the system at equilibrium point $(2,3)$. Verify that
the phase diagram of the nonlinear system at $(2,3)$ is an asymptotically stable spiral.
$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=& x(5-x-y), \\[2ex]
 \dd\frac{dy}{dt} &=& y(-2+x).\end{array}\right.
$$

\Prob{Xc6.4-4}{(Almost Linear System)}
Linearize at $(0,0)$ and classify the equilibrium point $(0,0)$ of the nonlinear system,
using a phase diagram to verify the conclusion.
$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=& 2\sin x + \sin y, \\[2ex]
 \dd\frac{dy}{dt} &=& \sin x + 2\sin y.\end{array}\right.
$$
\Prob{Xc6.4-8}{(Almost Linear System)}
Linearize at all equilibria and classify the equilibrium points of the nonlinear system.
Use a phase diagram to verify the conclusions.
$$\left\{\begin{array}{lcl}
 \dd\frac{dx}{dt} &=&  y, \\[2ex]
 \dd\frac{dy}{dt} &=& \sin \pi x -y.\end{array}\right.
$$
{\bf End of extra credit problems chapter 6.}

\end{document}
