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

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\Re{\operatorname{Re}}

\begin{document}

\title{Differential Equations}
\author{Homework 2}
\date{Due in class Tuesday, February 20, 2018}
\maketitle

\begin{enumerate}

\item 
\begin{enumerate}
\item Prove that for an $n \times n$ matrix $A$,
\[
   \frac{d}{dt} \e^{At} = A \e^{At}
\]
and conclude that the solution $x = x(t)$ to
\begin{equation*}
  \dot x = Ax \,,  \qquad x(0) = x_0 \,,
\end{equation*}
is given by 
\[
  x(t) = \e^{At} x_0 \,. 
\]

\item Show that if two $n \times n$ matrices $A$ and $B$ commute,
i.e., if $AB=BA$, then
\[
  B \e^{At} = \e^{At} B \,.
\]
\emph{Hint:} Verify that $B \e^{At} x_0$ and $\e^{At} Bx_0$ solve
$\dot x = Ax$, then use uniqueness.

\item Show that if $A$ and $B$ commute, then
\[
  \e^{(A+B)t} = \e^{At} \, \e^{Bt} \,.
\]
\emph{Hint:} Use the result from part (b) to show that both sides are
one parameter groups of transformations that solve the same
differential equation.

\item Use (c) to solve the differential equation
\[
  \dot x = Mx \qquad \mbox{where } M =
  \begin{pmatrix}
    2 & 1 & 0 \\
    0 & 2 & 1 \\
    0 & 0 & 2
  \end{pmatrix}.
\]
\end{enumerate}


\item Let $A \in \R^{n \times n}$ such that $\Re \lambda \leq \alpha$
for every eigenvalue $\lambda$ of $A$.  Let $k$ be the dimension of
the biggest Jordan block in the Jordan canonical form of $A$.  Show
that there exists a constant $C$ such that
\[
  \lVert \e^{At} \rVert
  \leq C \, \e^{\alpha t} \,
  \biggl(
    1 + t + \dots + \frac{t^{k-1}}{(k-1)!} 
  \biggr) \,.
\]
\emph{Hint:} First prove the result for a single Jordan block, using
the ideas from parts (c) and (d) of the previous question.  Then argue
for general $A$.

\item Consider the damped harmonic oscillator in the form
\[
  \dot x = A x
\]
with
\[
  A =
  \begin{pmatrix}
  0 & 1 \\ -1 & -1
  \end{pmatrix} \,.
\]
\begin{enumerate}
\item Show that $A$ satisfies the assumption of Problem~2 above.
\item Show that $A$ is \emph{not} strictly negative definite.  (A
matrix $A \in \R^{n \times n}$ is strictly negative definite if
$x^TAx<0$ for all $x \in \R^n$.)
\item Show that rate of change of the energy $E = x^Tx$ is not
strictly negative at all times.
\item Give a physical interpretation of this fact.  Moreover, argue
that it is nonetheless consistent with the conclusion of Problem~2.
\end{enumerate}
\item Find all equilibrium points, determine their stability
properties, and try to sketch the phase portraits of the following
differential equations:
\begin{enumerate}
\item $\begin{aligned}[t]
         \dot x_1 & = 2 x_1 - x_1^2 - x_1 x_2  \\
         \dot x_2 & = -x_2 + x_1 x_2
       \end{aligned}$
\item $\ddot y + \dot y + y^3 = 0$
\end{enumerate}
\end{enumerate}


\end{document}
