\documentclass{article}

\usepackage{fullpage}
\usepackage{amsmath}
\usepackage{color}
\newcommand{\gp}[1]{{\left({#1}\right)}}
\newcommand{\px}[2]{\frac{\partial #1}{\partial #2}}

\begin{document}

\title{Math 142-2, Homework 7}
\author{Your name here}
\date{\today}
\maketitle


\section*{Problem 72.5}
\textbf{Sketch $dq/d\rho$ as a function of $x$, for fixed $t > 0$ (after the light turns green).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 73.1}
\textbf{Assume that the traffic density is initially
$$ \rho(x,0) = \left\{
\begin{array}{ll}
\rho_{\max} & x < 0 \\
\rho_{\max}/2 & 0 < x < a \\
0 & a < x.
\end{array}
\right. $$
Sketch the initial density. Determine and sketch the density at all later
times.  Assume that $u = u_{\max}(1 - \rho/\rho_{\max})$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 73.2}
\textbf{Calculate the maximum acceleration of a car which starts approximately one
  car length behind the traffic light (i.e., $x(0) = -1/\rho_{\max}$).  Assume
  that $u = u_{\max}(1 - \rho/\rho_{\max})$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 73.3}
\textbf{Calculate the velocity of a car at the moment it starts moving behind a
  light.  Assume that $u = u_{\max}(1 - \rho/\rho_{\max})$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 73.6}
\textbf{Assume
$$ u = u_{\max}\gp{1 - \frac{\rho^2}{\rho_{\max}^2}}. $$
Determine the traffic density that results after an infinite line of stopped
traffic is started by a red traffic light turning green.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 73.9}
\textbf{At what velocity does the information that the traffic light changed
  from red to green travel?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 74.2}
\textbf{Assume $u(\rho) = u_{\max}(1 - \rho^2/\rho_{\max}^2)$ and
$$ \rho(x,0) = \left\{
\begin{array}{ll}
\rho_{\max} & x < 0 \\
\rho_{\max}(L-x)/L & 0 < x < L \\
0 & L < x.
\end{array}
\right. $$
Determine $\rho(x, t)$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 74.3}
\textbf{Consider the following partial differential equation:
$$\px{\rho}{t} - \rho^2 \px{\rho}{x} = 0 \qquad -\infty < x < \infty. $$}

\textbf{(a) Why can't this equation model a traffic flow problem?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Solve this partial differential equation by the method of characteristics,
subject to the initial conditions:
$$ \rho(x,0) = \left\{
\begin{array}{ll}
1 & x < 0 \\
1-x & 0 < x < 1 \\
0 & 1 < x.
\end{array}
\right. $$}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 74.4}
\textbf{Consider the example solved in this section. What traffic density should be
  approached as $L \to 0$? Verify that as $L \to 0$ equation 74.9 approaches the
  correct traffic density.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Additional Problem 1}
\textbf{Consider the model (in polar coordinates):
$$ \frac{dr}{dt} = \sin(r) \qquad \frac{d\theta}{dt} = \cos(r). $$}

\textbf{(a) Identify all equilibria and identify their stability (stable,
  neutrally stable, or unstable) and shape (node, saddle, spiral/loop).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Identify all limit cycles and identify them as stable or unstable.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) For each limit cycle identify whether the limit cycle trajectories
  are clockwise or counterclockwise.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(d) One of the tools we have used for sketching phase planes has been
  isoclines, or places where the slope is horizontal ($\frac{dy}{dt} = 0$) or vertical
  ($\frac{dx}{dt} = 0$).  In this case, however, it is more convenient to consider
  places where the slope is radial ($\frac{d\theta}{dt} = 0$) or tangential ($\frac{dr}{dt}
  = 0$), along with arrows as before.  Sketch the phase plane for this model
  (for $r \le 2\pi$) by sketching these polar isoclines, the equilibria, and the
  limit cycles.  Also sketch an estimate for what you think the trajectories
  might look like.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Additional Problem 2}
\textbf{Consider the model (for all $x$ and $y$):
$$ \frac{dx}{dt} = (x - y) (1 - x^2 - y^2) \qquad \frac{dy}{dt} = (x + y) (1 - x^2 - y^2). $$}

\textbf{(a) Identify all equilibria.  Identify the stability (stable, neutrally
  stable, or unstable) and shape (node, saddle, spiral/loop) for the simple
  equilibrium (it should be clear which one this is).  What can be said about
  the stability of the other equilibria?  [Hint: there is more than one
    equilibrium for this model.]}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Identify all limit cycles and identify them as stable or
  unstable. [Hint: be careful.]}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Sketch a phase plane for this model, which should include all
  equilibria, limit cycles, and sample trajectories.  (You may draw isoclines or
  other guides if this helps you with the trajectories.)}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Additional Problem 3}
\textbf{Let $N(a,t)$ be the number of individuals with age $a$ at time $t$,
  which is valid for $a \ge -g$ and $t \ge 0$, where $g$ is the gestation time
  (time between conception and birth) is $g$.  The conception rate $b(a)$ and
  death rate $c(a)$ are functions of age.  The initial age distribution is
  $f(a)$.  Derive a \textit{continuous} model for the evolution of the
  population and its age distribution over time.  Be sure to completely specify
  your model.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\end{document}
