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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due January 27 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Spring 2022 $|$ Dr.~Macauley}}


\begin{document}

%$\;$

\begin{enumerate}


  % Polking 2.4.18 (modififed)
\item In our first model of air resistance, the resistance force
  depends only on the velocity. This leads to the differential
  equation $mv'=-mg-rv$, where $-mg$ is the gravitation force and
  $R(v)=-rv$ is the air resistance force. For an object that drops a
  considerable distance, such as a parachutist, there is a dependence
  on the altitude as well. It is reasonable to assume that the
  resistance force is proportional to air pressure, as well as
  velocity. Furthermore, to a first-order approximation, the air
  pressure varies exponentially with the altitude (i.e., it is
  proportional to $e^{-ax}$, where $a$ is a constant and $x$ is the
  altitude). Propose and justify (but do not solve) a differential
  equation model (in $x$ instead of $v$) for the velocity of a falling
  object subject to such a resistance force. Recall that $x'=v$. \\



\item The population of snakes on a plane is believed to be growing
  according to the logistic equation:
  \[
  y'=ry\left(1-y/M\right)\,\qquad\qquad \text{Solution }
  y(t)=\frac{M}{1+Ce^{-rt}}\,.
  \]
  The maximum number of snakes that can live on the plane is
  $1000$. Initially, the population is $500$, and at this time, the
  rate of increase of snakes is $100$ per month.
  \begin{enumerate}
  \item How many months until the population reaches 90\% of the maximum?
  \item Sketch this solution curve in the $(t,y)$-plane, and the two
    steady-state solutions. \\
  \end{enumerate}



  % Modified version of Polking 2.4.43
\item Let $T(t)$ be the temperature of a cup of water at time $t$, in
  hours. Newton's law of cooling says that at any $t$, the rate of
  change $T'(t)$ is proportional to the difference in ambient
  temperature and $T(t)$, which can be described by the differential
  equation
  \[
  T'=k(A-T).
  \]
  The ambient temperature $A$ need not be constant; suppose it varies
  sinusoidally with time with a period of 24 hours. At 6am, the
  ambient temperature is at its minimum of $40^\circ$ and at 6pm, its
  maximum of $60^\circ$.
  \begin{enumerate}
  \item Write down a differential equation (that is, find $A(t)$) that
    models the temperature of the cup of water. Let $t=0$ be noon.
  \item Give a physical \emph{and} mathematical explanation why
    there is no steady-state solution.
  \item The long-term behavior of a solution to this equation is a
    sinusoid; explain why. What do you expect the period, amplitude,
    and phase shift of this solution to be (qualitatively) compared to
    $A(t)$?
  \item Without actually solving anything, sketch a graph of several
    solutions to this equation corresponding to different initial
    temperatures.
  \item Suppose that instead of measuring the temperature of a cup of
    water, $T(t)$ measures the temperature of a pond. What parameters
    would this change in the differential equation? What would your
    graphs in Part~(d) look like? Sketch solutions corresponding to
    the same initial conditions. 
  \end{enumerate}


\end{enumerate}

\end{document}
