\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\begin{document}

\title{General Mathematics and Computational Science I}
\author{Exercise 22}
\date{December 4, 2007}
\maketitle

\begin{enumerate}

\item Consider the Romeo--Juliet system
\begin{gather*}
  R_{n+1} = a_R \, R_n + p_R \, J_n \,, \\
  J_{n+1} = a_J \, J_n + p_J \, R_n 
\end{gather*}
with coefficients
\begin{gather*}
  a_R = \frac32 \,, \quad
  a_J = \frac12 \,, \quad
  p_R = - \frac12 \,, \quad
  p_J = \frac12 \,.
\end{gather*}
(I.e., Juliet is a \emph{cautious lover} and Romeo \emph{likes to
tease but not to please}.)
\begin{enumerate}
\item Solve the equation for $R_n$.

\emph{Hint:} As done in class, convert the model into a single,
second-order difference equation for $R_n$.

\item What is the corresponding solution for $J_n$?

\emph{Hint:} Solve the first equation of the model for $J_n$.

\item Describe the outcome of the affair as a function of the initial
feelings for each other.

\end{enumerate}
  
\item Run the Romeo--Juliet system on a computer (using
\emph{Mathematica} or any other programming environment) with
parameters 
\begin{gather*}
  a_R = 1.1 \,, \quad
  a_J = 0.1 \,, \quad
  p_R = - 1 \,, \quad
  p_J = 1 \,.
\end{gather*}
What is the outcome of the affair?  Describe what you see. 
\end{enumerate}
\end{document}
