@@ -1,159 +1,159 @@
-\documentstyle[11pt,reduce]{article}
-\date{}
-\title{ODESOLVE}
-\author{Malcolm A.H. MacCallum \\ Queen Mary and Westfield College, London \\
-Email: mm@maths.qmw.ac.uk \\[0.1in]
-Other contributors: Francis Wright, Alan Barnes}
-\begin{document}
-\maketitle
-
-\index{ODESOLVE package}
-\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 vars.
-
-"The use of the function\ttindex{COFACTOR}
-
-\vspace{.1in}
-\begin{tabbing}
-{\tt COFACTOR}(\=ROW:{\em integer}, \\
-\>COLUMN:{\em integer}, \\
-\>MATRIX:{\em matrix}):{\em algebraic}
-\end{tabbing}
-\vspace{.1in}
-
-\noindent which uses variation of parameters, returns the cofactor of the
-element in row ROW and column COLUMN of the matrix MATRIX.  Errors occur
-if ROW or COLUMN do not simplify to integer expressions or if MATRIX is
-not square.
-
-\section{Tracing}
-
-
-Some rudimentary tracing is provided and is activated by the switch TRODE
-\index{tracing ! ODESOLVE}
-\ttindex{TRODE}
-(analogous to TRFAC and TRINT)
-
-\section{Comments}
-
-The intention in the long run is to develop a rather general and
-powerful ordinary differential equation solver incorporating the
-methods detailed below.  At present the program has not been optimized
-for efficiency and much work remains to be done to convert algebraic
-mode procedures to more efficient symbolic mode replacements.
-
-No attempt is made to extend the REDUCE integrator, although this is
-in some sense a problem of ODEs.  Thus the equation $\frac{dy}{dx} = g(x)$ will
-be solved if and only if $\int g(x) dx$ succeeds.
-
-The available and planned coverage is as follows:
-
-\begin{itemize}
-\item First-order equations: (first degree unless otherwise stated)
-
-\begin{itemize}
-\item Quadrature of $\frac{df}{dx} = g(x)$
-\item Linear equations
-\item Separable equations
-\item (Algebraically) homogeneous equations
-\item Equations reducible to the previous case by linear transformations
-\item Exact equations
-\item Bernoulli equations
-\end{itemize}
-
-The above are already implemented. Further 1st order cases are not:
-\begin{itemize}
-\item Riccati equations using Schmidt's methods and other special cases
-\item Hypotheses on the integrating factor following Char (SYMSAC 81)
-or Shtokhamer, Glinos and Caviness.
-\item Higher degree cases
-\end{itemize}
-\item Linear equations of higher order
-\begin{itemize}
-\item Constant coefficients case for driving terms solvable by
-variation of parameters using the integrator
-(Choice of method is discussed in the source of module lccode).
-\end{itemize}
-The above is already implemented. Further higher order methods are not:
-\begin{itemize}
-\item More complex driving terms via Laplace transforms (?)
-\item  Variable coefficients: Watanabe (EUROSAM 84) methods
-including Kovacic's algorithm as extended by Singer
-\item  Factorization of operators as in Schwarz's ISSAC-89 paper or
-Berkovich's 1990 book
-\item  Other methods based on Galois theory (see Ulmer's preprints
-from Karlsruhe, 1989, 1990 and Singer's 1989 review) or
-other ways of hunting Liouvillian solutions (see Singer's
-review in J. Symb. Comp., 1990).
-\end{itemize}
-\item Non-linear equations of order 2 and higher
-\begin{itemize}
-\item Lie algebra of point symmetries e.g. using Wolf's CRACK now available
-in REDUCE
-\item  Other special ansatze (see Wolf. op. cit), in particular
-contact transformations for 2nd order cases
-\end{itemize}
-\item Possibly (?) exploitation of Cartan's methods for equivalence of
-differential equations.
-\end{itemize}
-\end{document}
+\documentstyle[11pt,reduce]{article}
+\date{}
+\title{ODESOLVE}
+\author{Malcolm A.H. MacCallum \\ Queen Mary and Westfield College, London \\
+Email: mm@maths.qmw.ac.uk \\[0.1in]
+Other contributors: Francis Wright, Alan Barnes}
+\begin{document}
+\maketitle
+
+\index{ODESOLVE package}
+\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 vars.
+
+"The use of the function\ttindex{COFACTOR}
+
+\vspace{.1in}
+\begin{tabbing}
+{\tt COFACTOR}(\=ROW:{\em integer}, \\
+\>COLUMN:{\em integer}, \\
+\>MATRIX:{\em matrix}):{\em algebraic}
+\end{tabbing}
+\vspace{.1in}
+
+\noindent which uses variation of parameters, returns the cofactor of the
+element in row ROW and column COLUMN of the matrix MATRIX.  Errors occur
+if ROW or COLUMN do not simplify to integer expressions or if MATRIX is
+not square.
+
+\section{Tracing}
+
+
+Some rudimentary tracing is provided and is activated by the switch TRODE
+\index{tracing ! ODESOLVE}
+\ttindex{TRODE}
+(analogous to TRFAC and TRINT)
+
+\section{Comments}
+
+The intention in the long run is to develop a rather general and
+powerful ordinary differential equation solver incorporating the
+methods detailed below.  At present the program has not been optimized
+for efficiency and much work remains to be done to convert algebraic
+mode procedures to more efficient symbolic mode replacements.
+
+No attempt is made to extend the REDUCE integrator, although this is
+in some sense a problem of ODEs.  Thus the equation $\frac{dy}{dx} = g(x)$ will
+be solved if and only if $\int g(x) dx$ succeeds.
+
+The available and planned coverage is as follows:
+
+\begin{itemize}
+\item First-order equations: (first degree unless otherwise stated)
+
+\begin{itemize}
+\item Quadrature of $\frac{df}{dx} = g(x)$
+\item Linear equations
+\item Separable equations
+\item (Algebraically) homogeneous equations
+\item Equations reducible to the previous case by linear transformations
+\item Exact equations
+\item Bernoulli equations
+\end{itemize}
+
+The above are already implemented. Further 1st order cases are not:
+\begin{itemize}
+\item Riccati equations using Schmidt's methods and other special cases
+\item Hypotheses on the integrating factor following Char (SYMSAC 81)
+or Shtokhamer, Glinos and Caviness.
+\item Higher degree cases
+\end{itemize}
+\item Linear equations of higher order
+\begin{itemize}
+\item Constant coefficients case for driving terms solvable by
+variation of parameters using the integrator
+(Choice of method is discussed in the source of module lccode).
+\end{itemize}
+The above is already implemented. Further higher order methods are not:
+\begin{itemize}
+\item More complex driving terms via Laplace transforms (?)
+\item  Variable coefficients: Watanabe (EUROSAM 84) methods
+including Kovacic's algorithm as extended by Singer
+\item  Factorization of operators as in Schwarz's ISSAC-89 paper or
+Berkovich's 1990 book
+\item  Other methods based on Galois theory (see Ulmer's preprints
+from Karlsruhe, 1989, 1990 and Singer's 1989 review) or
+other ways of hunting Liouvillian solutions (see Singer's
+review in J. Symb. Comp., 1990).
+\end{itemize}
+\item Non-linear equations of order 2 and higher
+\begin{itemize}
+\item Lie algebra of point symmetries e.g. using Wolf's CRACK now available
+in REDUCE
+\item  Other special ansatze (see Wolf. op. cit), in particular
+contact transformations for 2nd order cases
+\end{itemize}
+\item Possibly (?) exploitation of Cartan's methods for equivalence of
+differential equations.
+\end{itemize}
+\end{document}