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\title{Applied Differential Equations and Modeling}
\author{Midterm Exam}
\date{April 5, 2018}
\maketitle

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\begin{enumerate}



\item Solve the differential equation
\[
  t \, y' + 2 \, y = \sin t \,, \qquad y(\pi/2) = 1 \,.
\]
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\item Consider the differential equation
\[
  y' = 2 \, t \, y^2 \,.
\]
\begin{enumerate}
\item Solve the equation with initial condition $y(0) = a$.  (Note:
the case $a=0$ is special and requires separate discussion.)
\item For which values of $a$ does the solution exist for all $t \geq
0$?  If the solution fails to exist, state the interval of existence.
\end{enumerate}
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\item Consider the differential equation
\[
  y' = y \, (1 - y^2) \,.
\]

\begin{enumerate}
\item Find all equilibrium points of the equation.
\item Classify each equilibrium point as stable or unstable.
\item Indicate the equilibrium points in a $t$-$y$ graph and sketch
several other solutions \emph{without} solving the equation.
\end{enumerate}
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\item Let $P(t)$ denote the number of fish in a lake at time $t$, and
let $C$ denote the ``carrying capacity'' of the lake. Suppose further
that fishermen catch fish at a constant ``harvest rate'' $h$.  (Thus,
different from the homework problem on the same topic, the fishermen
increase their effort if the number of fish in the lake gets small!)
Then the population of fish satisfies the equation
\[
  \frac{dP}{dt} = \Bigl( 1 - \frac{P}{C} \Bigr) \, P - h
  \qquad \text{with} \qquad
  P(0) = P_0 \,.
\]
In the following, we assume that $h < C/4$.
\begin{enumerate}
\item  Show that the equation has two positive
equilibrium points, the larger one stable, the smaller one unstable.

\item Indicate the equilibrium points in a $t$-$P$ graph and sketch
several other solutions \emph{without} solving the equation.

\item How does the population of fish behave long-time?  Discuss, in
particular, the possibility of extinction.  Write a recommendation to
the fishermen.

\item \emph{Extra credit:} Show that, if $h>C/4$, the population will
become extinct in finite time no matter how large the initial
population of fish.
\end{enumerate}
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\item Consider the second order differential equation
\[
  y'' + y' - 2 \, y = 0 \,.
\]
\begin{enumerate}
\item Write this equation as a system of two first-order equations in
matrix form with matrix $A$.
\item Compute the eigenvalues of $A$.
\item Compute the eigenvectors of $A$.
\item Write out the general solution $y(t)$ for the second order
equation.
\item Write out the solution with initial condition $y(0) = 0$ and
$y'(0) = 1$.
\item Sketch the qualitative behavior of the equation in the $y$-$y'$
phase plane.
\end{enumerate}
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\end{enumerate}

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