FirstOrderRightFactors

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    {\Large\bf UserFunction FirstOrderRightFactors} }}
\vspace*{5mm}

Description of the arguments and its types.

\vspace*{1mm}\noindent{\bf FirstOrderRightFactors$(u:LDO(C,vs,O),v,n)$}

The argument $u$ has the type $LDO$ which abbreviates
{\em LinearDifferentialOperator}. The meaning of its parameters is as follows.

$C$: Coefficient type.

$xs=\{x\}|\{x,y\}\}$: There may be one or two differential variables.

$O:LEX|GRLEX$: Term ordering; applies only if there are two variables.

\vspace*{1mm}\noindent{\bf Specification.}
The first-order right factors of $u$ are returned.

\vspace*{1mm}\noindent{\bf Examples.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each
minisession is returned in a separate window. The first example is a
second-order operator.


\hspace*{4mm}{\tt L:=D(x,2)-D(y,2)+4/(x+y)*D(x);}

\hspace*{4mm}{\tt T==|LDO(RATF Q,\{x,y\},GRLEX);}

\hspace*{4mm}{\tt FirstOrderRightFactors(L|T|);}

\vspace{3mm}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11.5cm}{
The given operator:

$\partial_{xx}-\partial_{yy}+\Frac{4}{x+y}\partial_x$

The first order right factors:

$\partial_x-\partial_y+\Frac{2}{x+y}$ }}

\hspace*{4mm}{\tt L:=D(x,y,2)+(x+1+1/y)*D(x,y)+y*D(y,2)+(x+1/y)*D(x)}

\hspace*{30mm}{\tt +(x*y+y+2)*D(y)+x*y+2;}

\hspace*{4mm}{\tt lr:=FirstOrderRightFactors(L|T|);}

The second-order left factor of {\tt L} is obtained by submitting

\hspace*{4mm}{\tt l:=first lr;}

\hspace*{4mm}{\tt ls:=ExactQuotient(L,l|T|;}

\hspace*{4mm}{\tt FirstOrderRightFactors(ls|T|);}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11.5cm}{
The given operator:

$\partial_{xy}+\Frac{xy+1}{y}\partial_x+y\partial_y+xx+2$

The first order right factors:

$\partial_x+y,\partial_y+\Frac{xy+1}{y}$ }}


\vspace*{1mm}\noindent{\bf Related Functions.}
ExactQuotient, LoewyDecomposition,

\vspace*{5mm}

{\bf References}
D.~Grigoriev, F. Schwarz, {\em Factoring and Solving
Linear Partial Differential Equations}, Computing~{bf 73}, 179-197(2004).


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