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Midterm 3 2280 4:30}}}
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\begin{center}\Large\bf
Differential Equations 2280\\
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 $25$\%. Each problem is
scored 100.

\vfill {\sf Please discard this sheet after reading it.}\eject
\thispagestyle{empty} {\large\bf Name}.
\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} \hfill{
Midterm 3 2280 4:30}

\begin{exercises}
\prob{1.}{(ch6)} ~\newline
 (a) [30\%] Define {\em stable equilibrium} for $\.u'=\.f(\.u)$, a nonlinear 2-dimensional
 system in which $\.f$ is continuously differentiable.
\newline
 (b) [40\%] Give an example of a linear 2-dimensional system with a stable spiral at equilibrium point $x=y=0$.
 Draw a representative phase diagram about $x=y=0$.
 \newline
 (c) [30\%] Assume a $2$-dimensional predator-prey system $\.u'=\.f(\.u)$
 has equilibrium points $(0,0)$, $(160,0)$, $(0,180)$ and $(100,90)$.
 Explain the possible physical meanings of the equilibria, e.g., extinction,
 explosion, carrying capacity.

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

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


 \EJECT
\begin{exercises}
\prob{2.}{(ch6)} ~\newline
  (a) [40\%] Classify the linearization of the nonlinear system $x'=x+4y-xy^2$,
 $y'=2x-y+x^2y$ at equilibrium $x=y=0$ as node,
 spiral, center or saddle.
 \newline
 (b) [30\%] Is there a two-dimensional nonlinear system with a
 spiral at equilibrium $x=y=0$ while the linearization at $x=y=0$ is a
 center? Cite theorems or give an example.
 \newline
 (c) [30\%] If a two-dimensional system has a saddle at equilibrium
 $x=y=0$, then what are the possible classifications for the
 linearized system at $x=y=0$? Supply reasons and cite theorems.


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

\end{solution}
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\end{exercises}

 \EJECT
\begin{exercises}
\prob{3.}{(ch7)} ~\newline
 (a) [25\%] Display the integral formula for the direct Laplace Transform of $e^{-t^2}$.
 Cite theorems to explain why the integral exists.
 \newline
 (b) [25\%] Derive from the definition of the direct Laplace Transform a
  Laplace integral formula for the function $f(t)=t^2$. \newline
 (c) [25\%] Explain Laplace's Method, as applied to the differential
 equation $x''(t) = t$, $x(0)=x'(0)=0$.
 \newline
  (d) [25\%] Find $\L(f)$ given $f(t)=(-t)e^{2t}\sin(3t)$.
  \newline


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

\end{solution}
\fi
\end{exercises}
 \EJECT
\begin{exercises}
\prob{4.}{(ch7)} ~\newline
  (a) [30\%] Solve $\L(f(t))=\dd\frac{10}{s^2+8)(s^2+4)}$ for $f(t)$.
  \newline
  (b) [30\%] Solve for $f(t)$ in the equation
  $\L(f(t))=\dd\frac{s+1}{s^2(s-3)(s+2)}$.
  \newline
   (c) [40\%] Solve for $f(t)$ in the relation
  $$\L(f)=\dd\frac{d}{ds}\L(t^2\sin 3t) + \frac{2}{s^3} +
  \frac{s-1}{s^2+2s+5} +
  \left.\left(\L\left(t^3e^{9t}\cos 8t\right)\right)\right|_{s\to
  s+3}.$$


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

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

 \EJECT
\begin{exercises}
\prob{5.}{(ch7)} ~\newline
 (a) [50\%] Solve $x'''+3x''+2x'=0$, $x(0)=x'(0)=0$, $x''(0)=1$ by
 Laplace's Method.
  \newline
 (b) [50\%] Solve the system $x'=x+y$, $y'=x$, $x(0)=1$, $y(0)=0$
 by Laplace's Method.

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


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


 \EJECTEND

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