%Direction Fields
%If you modify this file, please indicate it here and in the footnote below.
%Math340
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{\Large
Direction Fields
\footnote{Copyright \copyright 2002 Steve Chapin and Larry Snyder.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
\end{center}
\begin{enumerate}
\item \verb&dfield6& is a \textsc{MatLab} program for \textsc{MatLab} Version 6 that
may be retrieved from the website at \verb&http://math.rice.edu/~dfield/&
and other versions are also available at this site. If you don't have
it, copy it into \verb&C:\Matlab\Work& (or \verb&C:\MatlabR12\Work&).
\item In \textsc{MatLab}, enter the command: \verb& dfield6&
\item A \verb&DFIELD Setup& window appears.
\item The differential equation $x' = x^2 - t$
appears in the boxes for \\ \verb&The differential equation&.
\item Using \textsc{MatLab} notation, change these entries to enter the
differential equation $y' = \sin y$.
\item The independent variable by default is t so leave that entry
unchanged.
\item For \verb&The display window& settings,
\begin{enumerate}
\item enter \verb& -5 & for the minimum value of \verb&t&
\item enter \verb& 5 & for the maximum value of \verb&t&
\item enter \verb& -2*pi & for the minimum value of \verb&y&
\item enter \verb& 2*pi & for the maximum value of \verb&y&.
\end{enumerate}
\item Click on the \verb&Proceed& button. The direction field for your
differential equation will appear in another window.
\item At the top of this window, you can click on \verb&Options& and pull
down to \verb&Window& settings. Here you can select \verb&Arrows& instead
of \verb&Lines& for your direction field plot.
\item If you click at any point in the direction field plot, a
solution curve through that point is plotted. Several solution curves
can be plotted by clicking on more than one point.
\end{enumerate}
Following the methodology above, do the following.
\medskip
(a) Print out or carefully
sketch by hand the direction field of the
differential equation
$$
y' = \frac{2y}{t} \quad (Choose -5 \le t \le 5, and -10 \le y \le 10.)
$$
\medskip
(b) Superimpose some solutions (say, two above the
t-axis and two below the t-axis) on the direction
field in part (a).
\medskip
(c) Use the information in parts~(a) and (b) to guess a
one-parameter family of solutions of the differential equation.
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