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\fancyhead[L]{\footnotesize MTH 1733}
\fancyhead[C]{\footnotesize Quiz 4}
\fancyhead[R]{\footnotesize Fall 1997}
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\begin{document}
%\author{Alexander I.~Suciu}
\thispagestyle{plain}

\begin{center}
\bfseries 
{Prof. A. Suciu\hspace*{\fill} Name: \Blank5}\\[1pc]
{MTH 1733 % Honors Calculus 1 
\hspace*{\fill} {\Large QUIZ 4} \hspace*{\fill} Fall 1997}\\[1.5pc]
\end{center}


\begin{questions}

\item \boxed{6\text{ points}}\:  
Solve the following initial value problem:
\[
y'-3y=3x+5,\quad y(0)=4.
\]

\vs

\item \boxed{6\text{ points}}\: 
Find the general solution of the following system of differential equations:
\[
\begin{array}{@{}r@{\:}c@{\:}r@{\:}c@{\:}r@{\;}c@{\;}l@{}}
x'&=&2x&+&y \\
y'&=&-x&+&4y
\end{array}
\]

\vs

\newpage

\item \boxed{6\text{ points}}\: 
A tank initially contains $20$~kg of salt dissolved in $200$~liters of water.  
A 	brine solution containing $3$~kg of salt/liter flows into the tank at the 
rate of $2$~liters/minute.  The mixture, kept uniform by stirring, flows out 
at the rate of $4$~liters/minute.  Set up an appropriate one-compartment model 
and write down the corresponding initial value problem (do not solve).

\vs\vs

\item \boxed{6\text{ points}}\: 
A certain bacteria culture grows at a rate proportional to its size, 
doubling every hour.  The culture contains 5~million bacteria at time 
$t=0$ (with time in hours).  

\begin{parts}
\item \label{one} Write down an initial value problem that models 
the growth of the bacteria culture.
\vs
\item Solve the intial value problem from part \eqref{one}.
\vs
\item At what time will there be $200$~million bacteria present?
\vs
\end{parts}

\newpage

\item \boxed{6\text{ points}}\:  
Consider the following autonomous differential equation:
\[ y'=y^2 - 2y -3 \]
\begin{parts}
\item Find the equilibrium solutions, specifying whether they are stable 
or unstable.
\vs\vs
\item \label{two}
Sketch the solution curves corresponding to the initial 
values $y(0)=-2$, $y(0)=2$, $y(0)=4$.  
\vs\vs\vs
\item  For each of the three curves $y=y(t)$ in part \eqref{two}, 
decide whether an inflection point occurs, and, if it does, at what 
value of $y$.
\vs
\end{parts}

\vs

\end{questions}
\end{document}