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\begin{document}


\begin{center}
{\bf Math 230: Some Practice Problems for the First Exam }
\\
{\bf Fall Semester 1998}
\end{center}
%\vspace*{.4cm}

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

\item
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}

\begin{center}
\epsfig{file=prac1fig1.eps,height=6cm,width=6cm}
\end{center}


\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 $t$.]

\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.

\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.
\end{subproblem}

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


\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!}

\item
Use Euler's method with a step size of $.01$ to estimate $y(2.01)$
where $y$ solves the initial value problem
$$
y' + \cos(\pi ty) = 0.
$$
Estimate the difference between the approximate and actual values
of $y$ at $t=2.01$.

\item The textbook is a good source of other problems to work.  Note
in particular the miscellaneous problems on page 94.  These will be
a good source of first order ODE problems.  Be careful, though--we
didn't cover techniques for solving non-linear ODE's that aren't
separable, so ignore any problem involving these kinds of ODE's.

\end{problems}
\end{document}