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\title{Differential Equations}
\author{Summer Semester 2024, Exercise 1}
\date{Due Wednesday, April 24, 2024}
\maketitle

\begin{enumerate}

\item Recall the equation for the mathematical pendulum,
\[
  \ddot \theta = - \sin \theta \,.
\]
Convert this equation into a first-order system of the form
\[
  \dot x = f(x) \,.
\]

\item Find the solution of the given initial value problem.
\begin{enumerate}
\item $x' - x = 2t \e^{2t}$, \quad $x(0)=1$
\item $t x' + 2x = t^2 - t + 1$, \quad $x(1) = \tfrac12$, \quad $t>0$
\item $x' = (\e^{-t} - \e^t)/(3+4x)$, \quad $x(0)=1$
\end{enumerate}


\item Consider the  Volterra--Lotka system, here 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 when $x>0$ and $y>0$ at time $t=0$, the solutions
remain strictly positive for as long as they exist.

\emph{Hint:} What equation is satisfied by $u=xy$?

\item Show that positive solutions exist for all times.

\emph{Hint:} What equation is satisfied by $v=x+y$?

\item What can you say when you drop the condition of positivity of
the initial values?

\emph{Hint:} Suppose $x>0$ and $y<0$.  First, set $w=-xy$.  Show that
\[
  2 \sqrt w \leq x-y \,.
\]
What equation is satisfied by $w$?

\end{enumerate}

\end{enumerate}
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