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\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/10/10}\\
{\it Closed book \hfill Van Dommelen \hfill 7:30-9:30 am}
\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 exactly what is asked; you do not get
any credit for making up your own questions and answering those.  Use
the stated procedures.  Give exact, cleaned-up, answers where
possible.

One book of mathematical tables, such as Schaum{\rq}s Mathematical
Handbook, may be used, as well as a calculator, and a handwritten
letter-size formula sheet.  Table 3.1 of the book is attached.  In the
Laplace transform question, everything not in Table 3.1 must be
derived.  The convolution theorem is an exception, but this theorem
should only be used if absolutely unavoidable.

\begin{enumerate}
  
\item {\bf Background:} Damped spring mass systems are the simplest
  model for the typical dynamical problems in mechanical engineering.

  {\bf Question:} Solve
  \begin{displaymath}
    y'' + 4 y' + 8 y = 16 t + 13 B e^t \qquad y(0)=y_0 \quad y'(0)=y_0'
  \end{displaymath}
  where $B$, $y_0$, and $y_0'$ are given constants.  Use the method of
  undetermined coefficients.

  Be sure to show where every intermediate result comes from.

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Laplace transforms are a good way to solve
  dynamical systems, especially when their large-time behavior or
  stability is of interest.

  {\bf Question:} Solve
  \begin{displaymath}
    y'' + 4 y' + 8 y = 16 t \qquad y(0)=0 \quad y'(0)=0
  \end{displaymath}
  using the class Laplace transform procedures.

  You can only use the transforms in the supplied Table 3.1.  Anything
  else must be derived.

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Systems of first order ordinary differential
  equations can describe the dynamics of any system governed by ordinary
  differential equations.

  {\bf Question:} Solve the first order system
  \begin{displaymath}
    \dot x = x + y \quad \dot y = y + z \quad \dot z = z
    \qquad
    x(0)=1 \quad y(0)=2 \quad z(0) = 3
  \end{displaymath}
  using the class procedures for first order systems.  (Yes, it can
  easily be solved directly, but that gets no credit.)  Use the matrix
  exponential method.  Write the fundamental matrix $\Omega$ of the
  system, the matrix $e^{At}$, and the vector $(x,y,z)$.

  Be sure to show where every intermediate result comes from.

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

\end{enumerate}

\end{document}
