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

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\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 7}
\date{Due in class Tuesday, March 26, 2019}
\maketitle

\begin{enumerate}

\item For each of the following systems of nonlinear differential equations,
\begin{itemize}
\item find all equilibrium points,
\item for each equilibrium point, write out the linear system
describing the evolution of small perturbations about the equilibrium
points, compute its eigenvalues and, if applicable, eigenvectors, and
\item sketch the phase portrait to the extent possible.
\end{itemize}

\begin{enumerate}
\item $\displaystyle
\begin{aligned}[t]
  x' &= x - x \, y \\ y' &= y + 2 \, x \, y
\end{aligned}
$

\item $\displaystyle
\begin{aligned}[t]
  x' &= 2 - x \\ y' &= y - x^2
\end{aligned}
$

\item $\displaystyle
\begin{aligned}[t]
  x' &= -(x-y) \, (1-x-y) \\ y' &= x \, (2+y) 
\end{aligned}
$

\end{enumerate}

\item Consider a general homogeneous constant-coefficient second-order
linear differential equation, i.e.,
\[
  a \, y'' + b \, y' + c \, y = 0 \,.
\]
\begin{enumerate}
\item Convert this equation into a system of linear first-order
equations with matrix $A$.

\item Show that
\begin{itemize}
\item when $b^2-4 ac>0$, the eigenvalues of $A$ are real and distinct,
\item when $b^2-4 ac=0$, $A$ has one eigenvalue of multiplicity $2$,
\item when $b^2-4 ac<0$, $A$ has a pair of complex conjugate
eigenvalues.
\end{itemize}
\item Find the general solution to the equation
\[
  y'' - 2 \, y' + 2 \, y = 0
\]
in terms of real-valued functions.
\end{enumerate}

\end{enumerate}
\end{document}
