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\title{Applied Differential Equations and Modeling}
\author{Homework 2}
\date{Due in class Tuesday, February 20, 2018}
\maketitle

\begin{enumerate}

\item Consider the differential equation
\begin{equation*}
  y' = (1-2x)/y \,, \quad y(1) = -2 \,.
\end{equation*}
Solve this initial value problem, plot the graph of the solution, and
state on which interval the solution is defined.

\item Consider the differential equation
\begin{equation*}
  \frac{\d y}{\d x} = \frac{y-4x}{x-y} \,.
\end{equation*}
\begin{enumerate}
\item Show that, setting $v=x/y$, that this differential equation can
be written in the form
\begin{equation*}
  v + x \, \frac{\d v}{\d x} = \frac{v-4}{1-v} \,.
\end{equation*}

\item Show that this new form of the equation is separable, and solve
the equation.  Write, as your final answer, $y$ as a function of $x$.

\item Draw the direction field in the $x$-$y$-plane, and a few
integral curves (i.e., solution curves to the differential equation
which you found in part b).  

\emph{Remark:} Note that the computation above showed that the
direction field depends only on $y/x$; this should be clearly seen in
your answer.  Such equations are sometimes called \emph{homogeneous}.
\end{enumerate}

\item Suppose that a tank containing a certain liquid has an outlet
near the bottom.  Let $h(t)$ be the height of the liquid surface above
the outlet at time $t$.  Torricelli's principle states that the
outflow velocity $v$ at the outlet is equal to the velocity of a
particle falling freely (with no drag) from height $h$.

\begin{enumerate}
\item Show that $v = \sqrt{2 g h}$, where $g$ is the constant of
gravitational acceleration.
\item By equating the rate of outflow to the rate of change of volume
in the tank, show that $h(t)$ satisfies the differential equation
\[
  A(h) \, \frac{\d h}{\d t} = - \alpha \, a \, \sqrt{2gh} \,,
\]
where $A(h)$ is the area of the cross section of the tank at height
$h$ and $a$ is the area of the outlet.  The constant $\alpha$ is a
contraction coefficient that accounts for the observed fact that the
cross section of the (smooth) outflow stream is smaller that $a$.  The
value of $\alpha$ for water is about $0.6$.

\item Consider a water tank in the form of right circular cylinder
that is $3$\,m high above the outlet.  The radius of the tank is
$1$\,m and the radius of the circular outlet is $\tfrac1{10}$\,m.  If
the tank is initially full of water, determine how long it takes to
drain the tank down to the level of the outlet.

\end{enumerate}

\item A ball with mass $0.15$\,kg is thrown upward with initial
velocity $20$\,m/s from a position $30$\,m above ground.  The ball is
subject to gravitational force and a force due to to air resistance of
magnitude $k\, \lvert v \rvert$ with $k = 1/30$\,kg/s.

\begin{enumerate}
\item Find the maximum height above the ground that the ball reaches.
\item Find the time that the ball hits the ground.
\item Plot the graphs of velocity and position versus time.
\end{enumerate}

\item Solve the initial value problem
\[
  y' = 2 t y^2 \,, \quad y(0)=y_0
\]
and determine how the interval on which the solution exists depends on
$y_0$. 
\end{enumerate}
\end{document}
