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\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Problem Set 8}
\date{Review -- Not for Credit}
\maketitle

\begin{enumerate}

\item Solve the initial value problem
\begin{gather*}
  t^3 \, y' + 4 \, t^2 \, y = \e^{-t} \,, \\
  y(-1) = 0 \,.
\end{gather*}
On which interval of time does the solution exist?

\item Consider the differential equation
\[
  y' = \frac{t}{y} \,.
\]
\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}

\item Consider a cylindrical water tank with cross sectional area $A$
filled with water up to height $h$.  There is a constant inflow into
the tank at rate $k$.  At the bottom, the tank has a hole with
effective cross sectional area $a$.  By Torricelli's principle, the
leakage rate is $a \sqrt{2gh}$, where $g$ is the constant of gravity.
\begin{enumerate}
\item Argue that the height $h$ as a function of time satisfies the
differential equation
\[
  \frac{\d y}{\d t} = \frac{k - a\sqrt{2gh}}A \,.
\]
\item Without solving the equation, determine how $h$ behaves as $t
\to \infty$.  Does the tank empty out, reach a stable equilibrium, or
overflow any finite height of the tank?

\item Find the general solution to the differential equation and use
the formula to confirm your conclusion from part (b).
\end{enumerate}

\item Find the general solution to the linear system of equations
\begin{gather*}
  x_1 - 2 \, x_2 + 4 \, x_3 = 2 \,, \\
  2 \, x_1 - x_2 - 2 \, x_3 = -1 \,, \\
  3 \, x_1 - x_2 + 2 \, x_3 = 1 \,, \\
  2 \, x_1 + 6 \, x_2 - 12 \, x_3 = -6 \,.
\end{gather*}

\item Consider the system of linear differential equations
\begin{gather*}
  \x' =
  \begin{pmatrix}
    8 & -4 \\ 1 & 4
  \end{pmatrix} \, \x
\end{gather*}
\begin{enumerate}
\item Write out the general solution,
\item Find the solution with
\begin{gather*}
  \x(0) =
  \begin{pmatrix}
    2 \\ -1
  \end{pmatrix} \,.
\end{gather*}
\end{enumerate}

\item Consider the system of nonlinear differential equations
\begin{gather*}
  x' = y \,, \\
  y' = -x + \tfrac16 \, x^3 - 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}

\end{enumerate}



\end{document}
