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\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/15/06}\\
{\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.

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:} Because Newton's second law is second order in
  time, simple dynamical systems are described by second order
  equations.  This looks at an example of such a system under
  exponential forcing.

  {\bf Question:} Derive the solution of
  \begin{displaymath}
    \ddot x - 4 \dot x + 3 x = e^t  \qquad x(0)=0 \quad \dot x(0)=1
  \end{displaymath}
  using the class procedures for guessing solutions for scalar
  equations of this type.  This also includes using the method of
  undetermined coefficients.

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} To study stability, Laplace transform solution
  is often more convenient.

  {\bf Question:} Solve the same problem as question 1, but this time
  do it using the method of Laplace transform and decomposition in
  partial fractions.

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} When we go to nonlinear, and more complicated
  dynamical systems, it is usually more convenient to use first order
  systems instead.
  
  {\bf Question:} Convert the {\em homogeneous} equation of the
  previous questions into a first order system and solve using vector
  procedures only.  Ignoring the initial conditions, very carefully
  draw all possible shapes of solution curves in the phase plane.  I
  need to see where curves are parallel to each other accurately, as
  well as the angles they make with the axes.  Finally, sketch the
  solution curve in the phase plane that satisfies the given initial
  conditions.

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

\end{enumerate}

\end{document}

