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\def\d{\mathrm{d}}
\def\e{\mathrm{e}}
\def\C{\mathbb{C}}
\def\N{\mathbb{N}}
\def\R{\mathbb{R}}

\begin{document}

\title{Calculus and Elements of Linear Algebra I}
\author{Homework 9}
\date{Due on Moodle, Monday, November 16, 2020}
\maketitle


\begin{enumerate}

\item Radiocarbon dating is a method to determine the age of objects
composed of organic material.  It is based on the fact that
atmospheric carbon is composed of a fixed ratio of two stable carbon
isotopes, ${}^{12}$C and ${}^{13}$C, and an unstable
isotope,${}^{14}$C, which undergoes radioactive decay with a half-life
of $5\,730$ years.  (This ratio is maintained by the interaction of
cosmic rays with atmospheric nitrogen.)  Living organisms exchange
carbon with the atmosphere sufficiently fast so that the carbon ratio
in their tissue is close to the atmospheric ratio.  When an organism
dies, the amount of stable carbon in its organic matter remains
constant while ${}^{14}$C decays, so that the age can be estimated by
measuring the ``carbon ratio'', the ratio of the number of ${}^{14}$C
to the number of stable carbon atoms.

An archaeological object is found to have a carbon ratio which is
$74\%$ of the atmospheric carbon ratio.  Determine the age of the
object.

\item Frictional forces are typically proportional to the velocity,
i.e.,
\[
  F(v) = -k \, v 
\]
where $k$ is a positive constant.  Now assume that a particle of mass
$m$ subject to a frictional force has an initial velocity $v_0$ and
initial position $x_0=0$.
\begin{enumerate}
\item How does the velocity of the particle change with time?  Does it
come to a stand-still in a finite time?

\emph{Hint:} Use Newton's second law $F=m\,a = m \, \d v/\d t$.

\item Does the particle travel a finite or an infinite distance?

\item Use the definition of work,
\[
  W = \int_{x_0}^{x_{\text{final}}} F(x) \, \d x \,,
\]
to compute the work done by frictional forces on the particle.
Interpret the result.
\end{enumerate}

\item Solve the differential equation
\begin{gather*}
  \frac{\d y}{\d x} = y \, (2x + 1) \,, \\
   y(0) = 2 \,.
\end{gather*}

\item Solve the differential equation
\begin{gather*}
  \e^x \, \e^y \, \frac{\d y}{\d x} + \e^x \, \e^y = x^2 \,, \\
   y(0) = 0 \,.
\end{gather*}
\emph{Hint:} Recognize the left hand side of the equation as the
derivative of some expression, then integrate.

\item For each of the following equations, determine the equilibrium
points where $y'(x)=0$ and classify each as stable ($y'$ changes sign
from positive to negative at $x$) or unstable ($y'$ changes sign from
negative to positive at $x$).  Sketch a few solution curves (without
trying to solve the equation) in the $x$-$y$ plane.
\begin{enumerate}
\item $\displaystyle y' = y - y^2$
\item $\displaystyle y' = y \, (y-1) \, (y-2)$
\item $\displaystyle y' = \e^y - 1$
\end{enumerate}

\end{enumerate}

\end{document}
