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\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/11/07}\\
{\it Closed book \hfill Van Dommelen \hfill 10:00-12:00 EDT}
\end{center}

Solutions should be fully {\em derived} showing all intermediate
results, using class procedures.  Show all reasoning.  Bare answers
are absolutely not acceptable, because I will assume they come from
your calculator (or the math handbook, sometimes,) instead of from
you.  You must state what result answers what part of the question if
there is any ambiguity.  Answer what is asked; you do not get any
credit for making up your own questions and answering those.  Use the
stated procedures.

One book of mathematical tables, such as Schaum's Mathematical
Handbook, may be used, as well as a calculator, and a handwritten
letter-size formula sheet.

\begin{enumerate}

\item {\bf Background:} The following ODE arises in the analysis of
  hedging of variable annuities:
  \begin{displaymath}
    x y' - p y = 1  \qquad x\ge 0 \qquad p > 0
  \end{displaymath}

  {\bf Question:} 
  \begin{enumerate}
  \item Derive the general solution of the homogeneous equation.
  \item Derive the general solution of the inhomogeneous equation.
  \item Very neatly draw an exhaustive set of solution curves, and
    then some more, of the inhomogeneous equation for the case that
    $p=\frac12$.  (Note that $x\ge 0$.)
  \item Find the solution when $p=\frac12$ and $y(1)=0$.
  \item Comment on the initial value problem in which $p=\frac12$ and
    $y(0)=0$.
  \end{enumerate}

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Vibrations of simple systems are governed by
  the ODE:
  \begin{displaymath}
    y'' + c y' + y = F(t)
  \end{displaymath}
  where $F(t)$ is some given time-varying force acting on the system.
  
  {\bf Question:} Derive the solution of this equation if $c=1$,
  $y(0)=1$, and $y'(0)=2$, by means of Laplace transformation.  Work
  out completely; there may not be a $*$ or $H$ in the answer.  You
  must use the completion of the square method in this problem.

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} The following nonlinear system describes
  a biased Van der Pol oscillator:
  \begin{displaymath}
    x' = y \qquad y' = - 4(x^2-2x)y-x+1
  \end{displaymath}

  {\bf Question:}
  \begin{enumerate}
  \item Explain why we can learn information about the solutions
    of this system by studying the following system:
    \begin{displaymath}
    \left( \begin{array}{c} x_1' \\ x_2' \end{array} \right)
    =
    \left( \begin{array}{cc} 0 & 1 \\ -1 & 4 \end{array} \right)
    \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right)
    \end{displaymath}
    In particular, identify $x_1$ and $x_2$.
  \item Write the general solution for $x_1$ and $x_2$.
  \item Very neatly and quantitatively correct, draw an exhaustive set
    of solution curves in the $x_1,x_2$-plane.  Make sure all
    qualitative features are clearly visible.
  \item Classify the point precisely.
  \item Explain whether or not these solution curves are relevant
    to the solution curves of the original problem.  If they
    are, in what sense they are relevant.  If the relevance is not
    quite certain, indicate so.
  \item What can be said about the long-time behavior of typical
    solutions of the original system based on the obtained results?
  \end{enumerate}

\begin{rawhtml}
<A HREF="figures/3.jpg">Solution.</A>
\end{rawhtml}

\end{enumerate}

\end{document}

