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\chapter[ODESOLVE: Ordinary differential eqns]%
        {ODESOLVE: \protect\\ Ordinary differential equations solver}
\label{ODESOLVE}
\typeout{[ODESOLVE: Ordinary differential equations solver]}

{\footnotesize
\begin{center}
Malcolm A.H. MacCallum \\
School of Mathematical Sciences, Queen Mary and Westfield College \\
University of London \\
Mile End Road \\
London E1 4NS, England \\[0.05in]
e--mail: mm@maths.qmw.ac.uk
\end{center}
}
\ttindex{ODESOLVE}

\index{ordinary differential equations}
The ODESOLVE package is a solver for ordinary differential equations.
At the present time it has very limited capabilities,

\begin{enumerate}
\item it can handle only a single scalar equation presented as an
algebraic expression or equation, and
\item it can solve only first-order equations of simple types,
linear equations with constant coefficients and Euler equations.
\end{enumerate}

\noindent These solvable types are exactly those for
which Lie symmetry techniques give no useful information.

\section{Use}
The only top-level function the user should normally invoke is:
\ttindex{ODESOLVE}

\vspace{.1in}
\begin{tabbing}
{\tt ODESOLVE}(\=EXPRN:{\em expression, equation}, \\
\>VAR1:{\em variable}, \\
\>VAR2:{\em variable}):{\em list-algebraic}
\end{tabbing}
\vspace{.1in}

\noindent {\tt ODESOLVE} returns a list containing an equation (like solve):

\begin{description}
\item[EXPRN] is a single scalar expression such that EXPRN = 0 is the
ordinary differential equation (ODE for short) to be solved,
or is an equivalent equation.
\item[VAR1] is the name of the dependent variable.
\item[VAR2] is the name of the independent variable
\end{description}

\noindent (For simplicity these will be called y and x in the sequel)
The returned value is a list containing the equation giving the
general solution of the ODE (for simultaneous equations this will be a
list of equations eventually). It will contain occurrences of the
\index{ARBCONST operator}
operator {\tt ARBCONST} for the arbitrary constants in the general
solution.  The arguments of {\tt ARBCONST} should be new, as with {\tt
ARBINT} etc. in SOLVE. A counter {\tt !!ARBCONST} is used to arrange
this (similar to the way {\tt ARBINT} is implemented).

Some other top-level functions may be of use elsewhere, especially:
\ttindex{SORTOUTODE}

\vspace{.1in}
\noindent{\tt SORTOUTODE}(EXPRN:{\em algebraic}, Y:{\em var}, X:{\em var}):
{\em expression}
\vspace{.1in}

\noindent which finds the order and degree of the EXPRN as a
differential equation for Y with respect to Y and sets the linearity
and highest derivative occurring in reserved variables ODEORDER,
ODEDEGREE,\ttindex{ODEORDER}\ttindex{ODEDEGREE}\ttindex{ODELINEARITY}\ttindex{HIGHESTDERIV}ODELINEARITY
and HIGHESTDERIV.  An expression equivalent to the ODE is
returned, or zero if EXPRN (equated to 0) is not an ODE in the
given variables.

\section{Commentary}

The methods used by this package are described in detail in the full
documentation, which should be inspected together with the examples
file.