%Separation of Variables %If file is modified, please note here and in the footnote below. %Math340 \documentclass[12pt]{article} \usepackage{times} \pagestyle{empty} \addtolength{\textwidth}{1.2in} \addtolength{\textheight}{1.2in} \addtolength{\oddsidemargin}{-.58in} \addtolength{\evensidemargin}{-.58in} \renewcommand{\baselinestretch}{1.0} \parindent = 0cm \parskip = .1cm \begin{document} \begin{center} {\Large Separation of Variables \footnote{Copyright \copyright 2002 Steve Chapin and Larry Snyder. All rights reserved. Please address comments to young@math.ohiou.edu.}} \end{center} \begin{itemize} \item Enter the following sequence of commands: \\ \verb& syms t& \\ \verb& f = sqrt(9-t^2)& \\ \verb& int(f)& \dotfill An antiderivative of \verb$f$.\\ \verb& int(f, -3, 3)& \dotfill The definite integral.\\ \verb& g = t*acos(t)& \dotfill acos(t) is \verb$arccos t$ \\ \verb& diff(g)& \\ \verb& pretty(ans)& \\ \verb& diff(g, 2)& \dotfill The second derivative of \verb$g$. \\ \verb& pretty(ans)& \\ \verb& diff(g, 3)& \dotfill The third derivative of \verb$g$. \\ \verb& pretty(ans)& \\ \verb& ezplot(f, [-3, 3])& \\ \verb& ezplot(g, [-1, 1])& \item {\bf Remarks.} The graph of \verb&f& should be the upper half of a circle. It will be distorted because of the default scale on the y-axis. Display the graph again and in the \verb&Figure window&, click on \fbox{Edit}. Pull down to \verb&Axes Properties&. Reset the y-limits to be \verb$-1.5$ and \verb$3.1$, click on \fbox{Apply} and then \fbox{OK}. The graph should now appear more like a semicircle. Type \verb& help sym/diff & or \verb& help int & in \textsc{MatLab} for more info on the use of \verb&diff& or \verb∫&. \end{itemize} Following the methodology above, using a separate piece of paper, do the following. \begin{enumerate} \item Find the particular solution to the ODE $y'' = \sec y'$ that is tangent to the \verb$t$-axis at the origin ($y'(0) = 0, y(0) = 0$). Use the method of separation of variables, and make sure to include all of the steps. Use \textsc{MatLab} to compute the appropriate integrals. (Hint. Let $u = y'$ and remember to add a constant of integration where appropriate.) \item Find the area under the graph of the solution of the IVP in part (a) on the interval $[-1, 1]$. Make sure to write the formula you use, not just the answer. Again, use \textsc{MatLab} to compute the appropriate integrals. \item Use \textsc{MatLab} to plot the solution of the IVP in part (a) on the interval $[-1, 1]$. Sketch the graph, \emph{by hand}. DO NOT HAND IN A PRINTOUT! \item Find the absolute maximum value of the solution on the interval $[-1, 1]$. \end{enumerate} \end{document}