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\begin{document}

\title{Differential Equations}
\author{Summer Semester 2024, Exercise 2}
\date{Due Monday, April 29, 2024}
\maketitle

\begin{enumerate}

\item Consider a scalar second-order differential equation of the form
\[
  \ddot x + f(x) = 0 
\]
and suppose that $F$ is an anti-derivative of $f$, i.e., that
$F'(x)=f(x)$.  Show that the ``energy''
\[
  E = \tfrac12 \, \dot x^2 + F(x)
\]
is a constant of the motion, i.e., that $\dot E = 0$.


\item Recall the Volterra--Lotka system from Exercise~1, again with
all coefficients set to one,
\begin{gather*}
  \dot x = x - x \, y \,, \\
  \dot y = x \, y - y \,.
\end{gather*}
\begin{enumerate}
\item Show that $(x^*, y^*) = (1,1)$ is the only equilibrium point in
the open first quadrant.

\item Letting $x(t)=x^*+\xi(t)$ and $y(t)=y^*+\eta(t)$, show that for
small perturbations $\xi$ and $\eta$, the equations take the form
\begin{equation*}
  \begin{pmatrix}
    \dot \xi \\ \dot \eta
  \end{pmatrix}
  = A \begin{pmatrix}
    \xi \\ \eta
  \end{pmatrix} + \text{higher order terms}
\end{equation*}
where
\begin{equation*}
  A =
  \begin{pmatrix}
    0 & -1 \\ 1 & 0
  \end{pmatrix} \,.
\end{equation*}

\item Show that for the linearized system, i.e., ignoring the
higher-order terms in the expression from (b), the amplitude of the
perturbation $\xi$ and $\eta$ does not grow as $t \to \infty$.

\emph{Hint:} There are different ways to do this.  One option is to
observe that the linearized system is in fact a particular case of
Problem~1 by an appropriate change of notation (cf.\ Problem~1 of
Exercise 1).  What is $f$, what is $F$ in this case?  What can you
conclude from the the statement that $E$ is a constant of the motion?


\end{enumerate}

\end{enumerate}
\end{document}
