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\centerline{\fbox{\parbox{10.5cm}{\Large\bf UserFunction DarbouxPolynomials} }}

\vspace{5mm}\noindent{\bf DarbouxPolynomials}$(u:LDO(C,v,O),n:Z)$
The first argument $u$ has the type $LDO$ which abbreviates
{\em LinearDifferentialOperator}. The meaning of its parameters is as follows.

$C$: Coefficient type.

$v=\{x_1,x_2,\ldots \}$: Variables in decreasing order.

$O=LEX|GRLEX|GREVLEX$: Term ordering.

$n$: Positive integer of type $Z$.

\vspace*{1mm}\noindent{\bf Specification.}
Returns Darboux polynomials up to order $n$.

\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.

\4{\tt d:=\{d:=(2*x**2*y-x)*D(x)+(2*x*y**2+y)*D(y))|LDO(POLY Q,{x,y})|;}

\4{\tt DarbouxPolynomials(d,1);}




\vspace*{1mm}\noindent{\bf Related Functions.} {\em FirstIntegral}.


{\bf References}

M.~J.~Prelle, M.~Singer, {\em Elementary First Integrals of Differential Equations}, Transactions of the American Mathematical Society {\bf 279}, 215-229(1983).

Y.~Man, {\em Computing Closed Form Solutions of First Order ODEs Using the Prelle-Singer Procedure}, Journal of Symbolic Computation {\bf 16}, 423-443(1993).

Y.~Man, M.~A.~H. Maccallum, {\em A Rational Approach to the Prelle-Singer Algorithm},  Journal of Symbolic Computation {\bf 24}, 31-43(1997).



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