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Math 305 \hfill Homework Set 3  \hfill Spring 2017

\begin{center}Due Monday, Febuary 6\end{center}

{\bf I.} Find the general solution to each of the following 
differential equations. If an initial condition is given, also 
find the particular solution. 

\begin{enumerate}

\item $y' = y^3 \sin x$. 

\item $xy' + 2y = \sin x$, assume $x>0$.

\item $xy^2y' + y^3 = 1$,  assume $x>0$.

\item $ty' + y = e^t$, assume $t>0$.

\item $y' = e^x \cos^2 y$.

\item $\dis y' + \frac{y}{t-1} = \frac{1}{t-3}$, assume $1< t < 3$.

\item $x^2 y' - 3y = 2$, assume $x>0$.

\item $y' = \dis \frac{y^2 + 2xy}{x^2}$.

\item $y' = xy + 2y + 3x + 6$.

\item $x^2y' + 2xy = y^3$,  assume $x>0$.

\item $y' = xy$, $y(1) = 1$.

\item $y' = \dis \frac{x^2 + y^2}{xy}$.

\item $z' - 3 z = e^x$, with $z(0) = 2$.

\item $\dis y' + \frac{y}{2} = \sin \frac{x}{2}$, with $y(\pi) = 6$.

\end{enumerate}


{\bf II.} Use a computer to plot the direction field for each differential 
equation below. Use $-5 \leq x \leq 5$ and  $-5 \leq y \leq 5$. Also plot two solution curves. 


\begin{enumerate}
\item $y' = x- 3y$.

\item $y' = x - y^2$. 

\item  $y' = \sin (\pi xy)$.

\end{enumerate}

\pagebreak
{\bf III.} Do the following problems. Show all steps. Explain what you are doing. 
\begin{enumerate}

\item A storage tank contains 2000 gal of gasoline that initially
has 100 lb of an additive dissolved in it. At $t=0$ gasoline 
containing 2 lb of the additive per gallon is pumped into the tank 
at 40 gal/min. The galosine in the tank is kept well mixed. It is 
drained out at 40 gal/min. How many pounds of the additive 
are dissolved in the remaining gasoline after 30 minutes? 

\item A chemical spill has polluted a pond. Your company,
Clean Ponds Inc., has been contracted to clean the 
pond. Federal regulations require that 90\% of the 
pollutant be removed within one month (31 days). 
The pond has 600,000 gallons of water in it.

The following pump/filter systems are available:
\vspace{.14in}

\begin{center}
\begin{tabular}{|r||c|c|c|}
\hline
name & Econo-Pump & Super Filter     & Pump Master          \\
     & and Filter &   System         & with Filter          \\
\hline
\hline
cost &          \$150,000   &         \$250,000        &       \$350,000          \\
\hline
pump rate &   4000 gal/hr   &       3000 gal/hr        &        5000 gal/hr      \\
\hline
efficiency  &   40 \%       &        75 \%             &        65 \%            \\
\hline
\end{tabular}
\end{center}
\begin{quote}{\small
Notes to Table: The cost listed is the minimum rental charge per month.
Filter efficiency is the percentage of pollutant removed on each 
pass through the filter.
}\end{quote}


A) Which pump/filter system should you get?

Note: Since the pump sends the water back into the pond after it is filtered,
the water coming in is part new water and part filtered water. Assume the 
water in the pond is well mixed. (See figure.)



% @@@@@@@@@@@@@@@@@@@@@@@@@@@@ FIGURE @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
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B) Becauise of a lawsiut by an environmental group you must
get 98\%  of the polluntant out in one month (31 days). Now what 
do you do?


\pagebreak

\item You have a motor boat. Assume that its velicity is modeled by
\[
v' = F_m - kv,
\]
where $F_m$ is the force of the motor in pounds, $k$ is a constant, 
and $v$ is velocity in feet/second. As an experiment you run your 
boat with various motor forcings and measure the terminial velocity. 
The data is in the table below. 


\begin{center}
\begin{tabular}{| c | c | c |}
\hline 
$F_m$ & $v$ & \hspace*{.3in}$k$\hspace*{.3in} \\
\hline 
2 & 4.641589 & \\
\hline
4 & 7.368063 & \\
\hline
6 & 9.654894 & \\
\hline
8 & 11.690710 & \\
\hline
10& 13.572088 & \\
\hline
\end{tabular}
\end{center}


For each experimental run compute the value of $k$. Is the assumed 
model a good one? Explain. If it is not, find a better one. 

\end{enumerate}

\end{document}

