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\title{Differential Equations}
\author{Homework 1}
\date{Due in class Tuesday, February 13, 2018}
\maketitle

\begin{enumerate}

\item Consider the differential equation
\begin{gather*}
  \dot y = \sqrt y \,, \\
  y(0) = 0 \,.
\end{gather*}
\begin{enumerate}
\item Show that the solution is not unique.

\item Show that the vector field does not satisfy the assumptions of
the Picard-Lindel\"of theorem.
\end{enumerate}

\item Suppose that $f \in C(\R^n \times \R_+)$ satisfies a
\emph{quasi-Lipshitz condition} in the sense that there exists $L>0$
such that
\[
  \lVert f(x,t) - f(y,t) \rVert
  \leq L \, \lVert x-y \rVert \, 
       \bigl\vert \ln \lVert x-y \rVert \bigr\vert 
\]
for all $t \geq 0$ and $x,y \in \R^n$ with
$\lVert x - y \rVert \leq \tfrac12$. Write $x(t)$ and $y(t)$ to denote
two solutions of the differential equation
\[
  \dot x = f(x,t) \,.
\]
Show that
\[
  \lVert x(t) - y(t) \rVert^2 
  \leq \lVert x(0) - y(0) \rVert^2 \,
       \exp (\exp (ct)) \,.
\]
(What is $c$?)

\emph{Remark:} Local existence of solutions under the quasi-Lipshitz
condition can be shown by a modification of the Picard--Lindel\"of
argument which is slightly more involved than the argument required
here.  I pose this as a challenge problem.

\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.
\item Show that positive solutions exist for all times.
\item What can you say when you drop the condition of positivity of
the initial values?
\end{enumerate}

\item (From Verhulst, p.\ 23.) Consider the equation
\begin{equation*}
\ddot x - \lambda \, \dot x - (\lambda - 1) \, (\lambda - 2) \, x = 0
\end{equation*}
with $\lambda$ a real parameter.  Find the critical points and
characterize these points. Sketch the flow in the phase plane and
indicate the direction of the flow.

\end{enumerate}

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