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{\LARGE {Mathematics 118A, Winter 1998}}
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\begin{center}
\large Homework Assignment 4, due Friday, February 6
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\bigskip

\begin{enumerate}
\item Certain solutions to the (scalar) equation
\begin{equation}
  \tag{*}
  \dot x = \sqrt x
\end{equation}
fail to be unique.
\begin{enumerate}
\item Give two solutions which are equal at $t=0$, but different for
every $t>0$.
\item Revisit the proof of local existence and uniqueness of solutions
given in class.  How does the right side of (*) fail to satisfy the
assumptions made in that proof?
\end{enumerate}

\item Consider the system
\begin{align*}
  \dot x & = -y - x \, \sqrt{x^2 + y^2} \, \exp \sqrt{x^2 + y^2} ,
  \\
  \dot y & = x - y \, \sqrt{x^2 + y^2} \, \exp \sqrt{x^2 + y^2} .
\end{align*}
\begin{enumerate}
\item Determine the critical points and characterize them by
linearization.
\item Is $(0,0)$ an attractor?
\end{enumerate}

\item Verhulst, Exercise 3.5.  Use \emph{Mathematica} as much as
possible for doing the calculation, and graph the phase space for
different values of $c$ to illustrate your argument.

\item Use \emph{Mathematica} to study the damped oscillator equation
\[
  \ddot y = -y - k \, \dot y .
\]
\begin{enumerate}
\item Visualize the behavior of the eigenvalues of the system as a
function of $k$.

\item Solve the equation and plot the phase portrait near the origin
for each of the three distinct cases (a pair of complex conjugate
eigenvalues, one real eigenvalue, two distinct real eigenvalues).

Do not use the \emph{Mathematica} functions \texttt{DSolve} or
\texttt{MatrixExponential}.  Instead, carry out the matrix
computations (change of basis!) explicitly.  You may use the functions
\texttt{Eigenvalues}, \texttt{Eigenvectors}, and
\texttt{JordanDecomposition}---consult the online documentation for
details. 

\end{enumerate}


\end{enumerate}
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