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Midterm 3 2280 4:30}}}
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\begin{center}\Large\bf
Differential Equations 2280\\
Sample Midterm Exam 3\\
Thursday, 27 April 2006
\end{center}

{\bf Instructions}: This in-class exam is 50 minutes. No
calculators, notes, tables or books.
No answer check is expected.
Details count $75$\%. The answer counts $25$\%.

\EJECT

\begin{exercises}
\prob{1.}{(ch6)} ~\newline
 (a) Define {\em asymptotically stable equilibrium} for $\.u'=\.f(\.u)$, a 2-dimensional system.
\newline
 (b) Give examples of 2-dimensional systems of type saddle, spiral, center and node.
 \newline
 (c) Give a $2$-dimensional predator-prey example $\.u'=\.f(\.u)$ and
 explain the meaning of the variables in the model.

\ifsolutions
\begin{solution}{1} ~\newline

\end{solution}
\fi
\end{exercises}


 \EJECT
\begin{exercises}
\prob{2.}{(ch6)} ~\newline
  Find the equilibrium points of $x'=14x-x^2/2-xy$,
 $y'=16y-y^2/2-xy$ and classify the linearizations as node,
 spiral, center, saddle. What classifications can be deduced for
 the nonlinear system?

\ifsolutions
\begin{solution}{2} ~\newline

\end{solution}
\fi
\end{exercises}

 \EJECT
\begin{exercises}
\prob{3.}{(ch7)} ~\newline
 (a) Define the direct Laplace Transform. \newline
 (b) Define Heaviside's unit step function. \newline
 (c) Derive a Laplace integral formula for Heaviside's unit step
 function. \newline
 (d) Explain Laplace's Method, as applied to the differential
 equation $x'(t) + 2x(t) = e^t$, $x(0)=1$.

\ifsolutions
\begin{solution}{3} ~\newline

\end{solution}
\fi
\end{exercises}
 \EJECT
\begin{exercises}
\prob{4.}{(ch7)} ~\newline
  (a) Solve $\L(f(t))=\dd\frac{100}{s^2+1)(s^2+4)}$ for $f(t)$.
  \newline
  (b) Solve for $f(t)$ in the equation
  $\L(f(t))=\dd\frac{1}{s^2(s-3)}$.
  \newline
  (c) Find $\L(f)$ given $f(t)=(-t)e^{2t}\sin(3t)$.
  \newline
  (d) Find $\L(f)$ where $f(t)$ is the periodic sawtooth wave function of period
  $2$ equal to $t/2$ on $0\le t \le 2$.


\ifsolutions
\begin{solution}{4} ~\newline

\end{solution}
\fi
\end{exercises}

 \EJECT
\begin{exercises}
\prob{5.}{(ch7)} ~\newline
 (a) Solve $y''+4y'+4y=t^2$, $y(0)=y'(0)=0$ by Laplace's Method.
  \newline
 (b) Solve $x'''+x''-6x'=0$, $x(0)=x'(0)=0$, $x''(0)=1$ by
 Laplace's Method.
  \newline
 (c) Solve the system $x'=x+y$, $y'=x-y+e^t$, $x(0)=0$, $y(0)=0$
 by Laplace's Method.


\ifsolutions
\begin{solution}{5} ~\newline


\end{solution}
\fi
\end{exercises}


 \EJECTEND

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