LoewyFactor

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\centerline{\fbox{\parbox{8cm}{\Large\bf UserFunction LoewyFactor} }}

\vspace*{1mm}\noindent{\bf LoewyFactor}$(u:LODE(RATF Q,y,x)$
The single argument $u$ is a linear homogeneous differential equation of any order. Its  coefficients are rational functions in the independent variable and may contain in addition one or more parameters. The meaning of the parameters is as follows.

$C$: Coefficient type.

$y$: Dependent variable.

$x$: Independent variable.

\noindent{\bf Specification.}
The irreducible of the largest completely reducible right factor is 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.

\5{\tt LoewyFactor x**2*Df(y,x,2)+x**2*Df(y,x)-2*y;}

 

\hspace*{10mm}\fbox{\parbox{12cm}{

  The differential equation:

  \vspace*{1mm}
  $y''+y'-\Frac{2}{x^{2}}y = 0$

  Equation is  completely reducible. The Loewy factor:

  \vspace{1mm}
  $Lclm(D-\Frac{1}{x-2}+\Frac{1}{x},\hspace*{1mm}
    D+1-\Frac{1}{x+2}+\Frac{1}{x})y=0$ }}


\vspace{5mm}\noindent{\bf LoewyFactor}$(u:LDFMOD(C,zs,xs,O))$
The single argument $u$ has the type $LDFMOD$ which abbreviates
{\em LinearDifferentialFormModule}, it represents a system of linear homogeneous partial differential equations. The meaning of its parameters is as follows.

$C$ determines the type of the coefficients.

$zs=\{z_1,z_2,\ldots \}$ denotes the dependent variables in decreasing order.

$xs=\{x_1,x_2,\ldots \}$ denotes the independent variables in decreasing order.

$O:LEX|GRLEX$ determines the applied term ordering.

\vspace*{1mm}\noindent{\bf Specification.}
The Loewy factor of $u$ is 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.


\5{\tt sys:=\{Df(sigma,x,2)+y*Df(sigma,y)+Df(sigma,x),

   \hspace*{15mm} Df(sigma(x,y)-Df(sigma,y),
 
   \hspace*{15mm} Df(sigma,y,2)+3/y*Df(sigma,y);\} }

\5{\tt T==LDFMOD(RATF Q,\{sigma\},\{y,x\},GRLEX)|;}

\5{\tt LoewyFactor(sys|T|);}


\hspace*{10mm}\fbox{\parbox{12cm}{
   The given Janet basis:
   
    \vspace{1mm}
   $<\sigma_{x,x}+y\sigma_y+\sigma_x,\sigma_{x,y}-\sigma_y,
     \sigma_{y,y}+\Frac{3}{y}\sigma_y>$

   The Loewy factor:

\vspace{1mm}   $Lclm(\partial_x+1,\partial_y>,\partial_x-1,\partial_y+\frac{2}{y}>,<\partial_x,\partial_y>)\sigma$. }}

\vspace*{3mm}
{\bf Related Functions} LoewyDecomposition.

\vspace*{5mm}

{\bf References}

A.~Loewy, {\em \"{U}ber vollst\"{a}ndig reduzible lineare homogene Differentialgleichungen}, Mathematische Annalen {\bf 56}, 89-117 (1906)

F. Schwarz, {\em Solving Second Order Linear Differential Equations}, CRC Press, 2007.

 

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