\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\F{{\boldsymbol F}}
\def\v{{\boldsymbol v}}
\def\x{{\boldsymbol x}}
\def\y{{\boldsymbol y}}
\def\A{{\boldsymbol A}}
\def\B{{\boldsymbol B}}
\def\I{{\boldsymbol I}}
\def\rank{\operatorname{rank}}

\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Homework 6}
\date{Due in class Tuesday, March 19, 2019}
\maketitle

\begin{enumerate}

\item For each of the following matrices,
\begin{itemize}
\item find the eigenvalues and eigenvectors, if necessary a
generalized eigenvector,
\item write out the general solution for the homogeneous linear
differential equation
\[
  \x' = \A \x \,,
\]
\item sketch the phase portrait of the differential equation.
\end{itemize}
\begin{enumerate}
\item $\displaystyle A =
\begin{pmatrix}
  3 & -2 \\ 2 & -2
\end{pmatrix}$

\item $\displaystyle A =
\begin{pmatrix}
  3 & -2 \\ 4 & -1
\end{pmatrix}$

\item $\displaystyle A =
\begin{pmatrix}
  3 & -4 \\ 1 & -1
\end{pmatrix}$

\end{enumerate}

\item For each of the following problems, find the solution of the
given initial value problem.  Draw the trajectory of the solution in
the $x_1$-$x_2$ plane and also draw the component plots of $x_1$
versus $t$ and of $x_2$ versus $t$.
\begin{enumerate}
\item $\displaystyle
\x' =
\begin{pmatrix}
  1 & -5 \\ 1 & -3
\end{pmatrix} \x \,,
\qquad
\x(0) =
\begin{pmatrix}
  1 \\ 1
\end{pmatrix}$

\item $\displaystyle
\x' =
\begin{pmatrix}
  1 & -4 \\ 4 & -7
\end{pmatrix} \x \,,
\qquad
\x(0) =
\begin{pmatrix}
  3 \\ 2
\end{pmatrix}$

\end{enumerate}
\item For each of the following systems of differential equations,
\begin{itemize}
\item find the equilibrium solution,
\item describe how the system behaves in the vicinity of the
equilibrium solution,
\item sketch the phase portrait.
\end{itemize}
\begin{enumerate}
\item $\displaystyle
\begin{aligned}[t]
  & x' = - x + y + 1 \,, \\
  & y' = x + y - 3
\end{aligned}$

\item $\displaystyle
\begin{aligned}[t]
  & x' = - x -4 \, y -4 \,, \\
  & y' = x - y - 6
\end{aligned}$

\end{enumerate}

\item Describe how the nature of the eigenvalues depends on the
parameter $\alpha$ in the matrix
\[
  A =
\begin{pmatrix}
  2 & \alpha \\ 1 & -3
\end{pmatrix} \,.
\]

\end{enumerate}
\end{document}
