%Special Functions
%If file is modified, please note here and in the footnote below.
%Math340

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{\Large
Special Functions
\footnote{Copyright \copyright 2002 Steve Chapin and Larry Snyder.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
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The antiderivatives of many elementary functions are not, themselves,
elementary funtions.  Some of these antiderivatives arise frequently
in certain subjects and have been given names.  These are examples of
what are called special functions, and other antiderivatives can
sometimes be expressed in terms of these special functions.

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Try the following \textsc{MatLab} commands:\\
\verb&   syms t&  \\
\verb&   int(exp(-t^2))&  \\
\verb&   int(sin(t^2))&  \\

Look up the definition of the special functions involved using the
\verb& mhelp & command which calls up the help feature in the Maple kernel.

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For the following find the general solution on the indicated interval,
by hand, using the method of variation of parameters, except using
\textsc{MatLab} to integrate $u_1'$ and $u_2'$.

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\textbf{Make sure you show all your work!  Look up
and write down, in standard mathematical notation, any special
functions that occur and any unfamiliar expressions or constants that
appear in these definitions.}

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(a) $y'' - 3y' = \frac{1}{t}, \quad t \texttt{>} 0$

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(b) $y'' -2y' + 2y = \frac{e^t}{t}, \quad t \texttt{>} 0$

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