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% lodo.htex
\spadtype{LinearOrdinaryDifferentialOperator(A, diff)} is the domain of linear
\index{operator!linear ordinary differential}
ordinary differential operators with coefficients in a ring
\spad{A} with a given derivation.
%This includes the cases of operators which are polynomials in \spad{D}
%acting upon scalar or vector expressions of a single variable.
%The coefficients of the operator polynomials can be integers, rational
%functions, matrices or elements of other domains.
\showBlurb{LinearOrdinaryDifferentialOperator}

\head{subsection}{Differential Operators with Series Coefficients}{ugxLinearOrdinaryDifferentialOperatorSeries}
% ----------------------------------------------------------------------

\noindent
{\bf Problem:}
Find the first few coefficients of \spad{exp(x)/x^i} of \spad{Dop phi} where
\begin{verbatim}
Dop := D^3 + G/x^2 * D + H/x^3 - 1
phi := sum(s[i]*exp(x)/x^i, i = 0..)
\end{verbatim}

\noindent
{\bf Solution:}
\xtc{
Define the differential.
}{
\spadcommand{Dx: LODO(EXPR INT, f +-> D(f, x)) \bound{Dxd}}
}
\xtc{
}{
\spadcommand{Dx := D() \free{Dxd}\bound{Dx}}
}
\xtc{
Now define the differential operator \spad{Dop}.
}{
\spadcommand{Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1 \free{Dx}\bound{Dop}}
}
\xtc{
}{
\spadcommand{n == 3 \bound{n3}}
}
\xtc{
}{
\spadcommand{phi == reduce(+,[subscript(s,[i])*exp(x)/x^i for i in 0..n]) \bound{phi}}
}
\xtc{
}{
\spadcommand{phi1 ==  Dop(phi) / exp x \bound{phi1}\free{Dop phi}}
}
\xtc{
}{
\spadcommand{phi2 == phi1 *x^(n+3) \bound{phi2}\free{phi1}}
}
\xtc{
}{
\spadcommand{phi3 == retract(phi2)@(POLY INT) \bound{phi3}\free{phi2}}
}
\xtc{
}{
\spadcommand{pans == phi3 ::UP(x,POLY INT) \free{phi3}\bound{pans}}
}
\xtc{
}{
\spadcommand{pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] \bound{pans1}\free{pans}}
}
\xtc{
}{
\spadcommand{leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) \bound{leq}\free{pans1}}
}
\xtc{
Evaluate this for several values of \spad{n}.
}{
\spadcommand{leq \free{n3 leq}}
}
\xtc{
}{
\spadcommand{n==4 \bound{n4}}
}
\xtc{
}{
\spadcommand{leq \free{n4 leq}}
}
\xtc{
}{
\spadcommand{n==7 \bound{n7}}
}
\xtc{
}{
\spadcommand{leq \free{n7 leq}}
}

% ----------------------------------------------------------------------
