%From berk@info.ssu.samara.ru Fri Apr 30 01:54 MDT 1999
%Date: Fri, 30 Apr 1999 13:04:19 +0500 (KSD)
%From: "Lev M. Berkovich" <berk@info.ssu.samara.ru>
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\begin{center}
{\large \bf SOLDE: A REDUCE Package for Solving  of Second Order
Linear Ordinary Differential Equations}\\
{\large Lev M. Berkovich\\
Department of Algebra \& Geometry of Samara State
University\\
443011,~ Samara,~ Russia,~ e-mail: berk@info.ssu.samara.ru}
\end{center}

This paper uses abilities of the language REDUCE and the graphic
package GNUPLOT for the representation of solutions of the second order
linear differential equations. The procedure of searching solutions in
the analitic form uses the package SOLDE [1, 2], and the graphic output
of the found solutions uses the program in [3].

The algorithmic procedure SOLDE differents from known algorithms
(J.~Kovacic,~ M.~Singer).

The procedure, named SOLDE, finds the Liouvillian solutions of the
nonhomogeneous second order linear ordinary differential equations
$$
Ly\equiv a_2(x)y''+a_1(x)y'+a_0(x)y=f(x),
$$
where $a_2(x), a_1(x), a_0(x)$ and $f(x)$ belong to some differential
field $(k(x), D)$,~ $k$ is number field of the characteristics
$0$,~$D=d/dx$ is a derivation on $k(x)$. A differential field extension
of $(k(x),D)$ is a Liouvillian extension $(K(x),D)$
such that $K(x)\supset\overline{k}(x)$, and $D$ is a derivation on
$K(x)$.

It provides a user with the following information on the equation
being investigated.


$\bullet$ The Kummer--Liouville transformation
$$
y=v(x)z(t),~~dt=u(x)dx,
$$
that reduces the homogeneous equation $Ly=0$ to one with constant
coefficients
$$
z''(t)+b_1z'(t)+b_0z(t)=0.
$$

$\bullet$ The factorization
$$
Ly\equiv (D-\alpha_2)(D-\alpha_1)y=0,~ \alpha_i=\alpha_i(x)\in
K(x),~i=1,2.
$$

$\bullet$ The fundamental set of solutions $y_1(x),~y_2(x)$ of $Ly=0$.

$\bullet$ The partial solution $y_*$ of $Ly=f(x)$.

$\bullet$ Ly=0 has no Liouvillian solution.
\vspace{3mm}

{\bf \large References}
\vspace{2mm}

[1] Berkovich, L.M.~
{\it Factorization and transformations of ordinary differential 
equations}, Saratov University Publ., 1989, 192 P.

[2] Berkovich, L.M. and Berkovich, F.L.~ {\it Transformation and
factorization of second order linear ordinary differential equations
and its implementation in REDUCE}, Publ. Elektr. Fak. Univ. Beograd,
Ser. Mat., 1995, N 6,~11--24.

[3] Berkovich L.M., Frolov I.S. {\it Representation of the second order
linear differential equations solutions using the language REDUCE and
graphic pacckage GNUPLOT}, Vestnik Samarskogo gos. universiteta,
1997 N 2(4), 109--114.

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