%Separation of Variables
%If file is modified, please note here and in the footnote below.
%Math340

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\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&int&.

\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}

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