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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 11/3/96 }\\
{\it Closed book \hfill Van Dommelen \hfill 9:00-9:50am }
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used.

\begin{enumerate}

\item
In a model for turbulent flows, you need to solve the following system
of two ordinary differential equations:
\[ {{\rm d} x\over {\rm d}t} = x y \qquad
{{\rm d} y\over {\rm d}t} = 2 y^2 - x^2 \]
Eliminate the time $t$ between these two equations, then find $y$ as a
function of $x$.
\begin{rawhtml}
  <a href="figures/1.gif">Solution</a>
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\item
The potential $V$ across the condensor of a serial CL circuit
satisfies the differential equation:
\[ \ddot V + 4 V = e(t). \]
Find $V(t)$ if the applied potential takes the form $e(t)=1/\cos(2t)$.
Assume $V(0) = \dot V(0) = 0$.
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  <a href="figures/2.gif">Solution</a>
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\item
In a two degree-of-freedom dynamic system, one of the degrees of
freedom can be eliminated to produce the following fourth order
equation for the remaining unknown $x(t)$:
\[ \stackrel{\textstyle{....}}x + 13 \stackrel{\textstyle{..}}x + 36 x = 0 .\]
Solve this equation subject to the initial conditions:
\[ x(0) = 0 \quad \stackrel{\textstyle{.}}x(0) = 2
\quad \stackrel{\textstyle{..}}x(0) = 0 \quad
\stackrel{\textstyle{...}}x(0) = -13. \]
\begin{rawhtml}
  <a href="figures/3.gif">Solution</a>
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\end{enumerate}

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