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\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Midterm Exam}
\date{April 3, 2019}
\maketitle

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\begin{enumerate}

\item Solve the initial value problem
\begin{gather*}
  t \, y' + (t+1) \, y = 2 \, t \, \e^{-t} \,, \\
  y(1) = 1 \,.
\end{gather*}
On which interval of time does the solution exist?
\hfill(10)

\item Consider the differential equation
\[
  y' + y^3 = 0 \,.
\]
\begin{enumerate}
\item Solve the equation with initial condition $y(0) = a$.  
\item Determine how the interval of existence depends on the initial
value $a$.
\end{enumerate}
\hfill(10)


\item Consider the \emph{Gompertz growth model}
\[
  \frac{\d y}{\d t} = r \, y \, \ln \frac{K}{y} \,,
\]
where $r$ and $K$ are positive constants.
\begin{enumerate}
\item Find the equilibrium points and classify each as stable or
unstable.  Sketch a few solution curves in the $t$-$y$ plane.
\item Find the general solution to the differential equation and use
the formula to confirm your conclusion from part (a).
\item Give an interpretation of the constant $K$ when the model is
used to describe population growth.
\end{enumerate}
\hfill(5+5+5)


\item Find the general solution to the linear system of equations
\begin{gather*}
  x_1 + x_2 + 2 \, x_3 = 3 \,, \\
  x_1 + 2 \, x_2 + x_3 - x_4 = 4 \,, \\
  -x_1 - 3 \, x_3 - x_4 = -2 \,.
\end{gather*}
\hfill(10)


\item Consider the system of linear differential equations
\begin{gather*}
  \x' =
  \begin{pmatrix}
    -2 & -2 \\ 2 & 1
  \end{pmatrix} \, \x
\end{gather*}
\begin{enumerate}
\item Write out the general solution,
\item Find the solution with
\begin{gather*}
  \x(0) =
  \begin{pmatrix}
    0 \\ 1
  \end{pmatrix} \,.
\end{gather*}
\end{enumerate}
\hfill(5+5)

\item Consider the system of nonlinear differential equations
\begin{gather*}
  x' = 5 \, x - x \, y \,, \\
  y' = x \, y - y \,.
\end{gather*}

\begin{enumerate}
\item Find all equilibrium points,
\item for each equilibrium point, write out the linear system
describing the evolution of small perturbations about the equilibrium
points, compute its eigenvalues and, if real-valued, eigenvectors, and
\item determine the stability of each equilibrium point and sketch the
phase portrait to the extent possible.
\end{enumerate}
\hfill(5+5+5)

\end{enumerate}



\end{document}
