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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\x{\boldsymbol{x}}

\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Homework 5}
\date{Due in class Thursday, March 22, 2018}
\maketitle

\begin{enumerate}

\item Find the (single) eigenvalue, eigenvector $\boldsymbol v$, and
generalized eigenvector $\boldsymbol w$ for each of the following
matrices. 

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

\item $\displaystyle A =
\begin{pmatrix}
  -\tfrac32 & 1 \\[3pt] -\tfrac14 & -\tfrac12
\end{pmatrix}$

\end{enumerate}

\item For the matrices from Problem 1, write out the general solution
ot the system $\dot \x = A \x$.  Describe how the solutions behave as
$t \to \infty$.

\item For the matrices from Problem 1, find the solution to $\dot \x =
A \x$ which corresponds to the initial condition $x(0) = (1,1)$.

\item Consider linear equations of the form 
\[
  \dot \x = A \x \,,
\]
where $A$ is a $2 \times 2$ matrix with \emph{real} coefficient.
Is the following true or false?  If a short argument for each
case.
\begin{enumerate}
\item $A$ can have repeated complex eigenvalues (i.e., eigenvalues
with a non-zero imaginary part).
\item If none of the eigenvalues of $A$ has a strictly positive real
part, then there is no solution which grows in time (i.e., then  the
trivial solution $\x(t) = \boldsymbol 0$ is stable). 

\end{enumerate}





\end{enumerate}
\end{document}
