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\begin{document}
{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Math 226: Differential Equations and Linear Algebra }
\\
{\bf Fall Semester 1998}
\\
{\bf Exam 1}
\\
{\bf September 28, 1998}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination contains 8 problems on 7 sheets 
of paper including the front cover. 
Do all your work in this booklet and show your computations.
Calculators, books, and notes are not allowed.
\\[1.5ex]
\hline
\end{tabular}

\begin{center} {\bf Scores}

\begin{tabular}{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
  1      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &   {}\ 5 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  7      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  8      &   10 &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
Total  & 100\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular} 
\end{center}  

\vspace*{1cm}
{\bf Sign the pledge:}
\\ 
``On my honor, I have neither given nor received unauthorized aid on this Exam.''

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}

\newpage
\begin{problems}
\item 
Solve the following initial value problem and determine the interval 
where the solution is defined.
\[
     (t^2-9)\frac{dy}{dt}+2ty=\frac{2t^2-18}{t^2},\quad y(1)=-1.
\]
\vspace*{20cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item 
Find the value of $a$, for which the equation is exact, and then solve it 
using that value of $a$. 
\[
    (3x^2-axy^2+2y)dx+(3y^2-2x^2y+2x)dy=0.
\] 
You may leave your answer in implicit form. 
\vspace*{20cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item
Find the general solution of the equation 
\[
    2x\frac{dy}{dx}=2y+x e^{\frac{y}{x}}.
\]
You may leave your answer in implicit form. 
\vspace*{20cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item
(5 points, no partial credit)
Circle the differential equation whose direction field is shown in the
following picture.
\\[2ex]
\begin{tabular}{lll}
{\bf A.} $y'=y(x-y)$     & {\bf B.} $y'=(x-y)(x+y)$\hspace*{.6cm} & 
{\bf C.} $y'=(1-y)(x-y)$\\[1.5ex]
{\bf D.} $y'=(y-1)(x-y)$\hspace*{.6cm} & {\bf E.} $y'=(x-1)(x-y)$    &  
\end{tabular}
\vspace*{7cm}

\item 
A tank initially contains 300 gallons of pure water.
A mixture containing 1 lb of salt per gallon enters the tank at a rate   of
3 gal/min. 
The well-stirred mixture leaves the tank at a rate of 5 gal/min.
\\
Write the initial value problem needed to find the amount of salt $S(t)$ 
in the tank at time $t>0$ prior to the instant when the tank is empty. 
{\bf Do not solve it!}
\\ \relax
[Hint: First you should figure out how many gallons of mixture are in the 
tank at time $t$.] 

\vspace*{9cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item 
Given the differential equation
\[
  \frac{dy}{dt}=2y(y-1)(y-3).
\]
{\bf Do not attempt to solve it!}
\begin{subproblem}
\item Find the (constant) equilibrium solutions and classify each one
          as asymptotically stable, unstable,  or semistable.
\vspace*{9cm}

\item Sketch the graphs of the four solutions $y_1,\;y_2,\;y_3$, and $y_4$
          which have initial condition
          $y_1(0)=1,\;y_2(0)=2,\;y_3(0)=3$, and $y_4(0)=0.5$. 
          Neglect concavity.
\vspace*{9cm}

\end{subproblem}

\item 
Find the solution of the initial value problem 
\[
  6y''-7y'+y=0,\quad y(0)=0,\quad y'(0)=5.
\]
\vspace*{8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\item
Determine the largest interval in which the following initial value 
problem is certain to have a unique twice differentiable solution. 
\[
    (t-3)(t-1)ty''+t^2y'+(t-1)y=\sin t,\quad y(2)=-3,\quad y'(2)=4. 
\]
{\bf Do not attempt to find the solution!}
\vspace*{8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}

\end{problems}
\end{document}