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\centerline{\fbox{\parbox{11cm}{\Large\bf UserFunction LoewyDecomposition} }}

\vspace*{5mm}\noindent{\bf LoewyDecomposition}$(u:LODO(RATF\1 Q,x))$
The single argument $u$ is an ordinary differential operator of order 2 or 3.
The meaning of the parameters is as follows.

$x:$ Variable.

\vspace*{1mm}\noindent{\bf Specification.}
The Loewy decomposition of $u$ into irredicible right factors is returned.


\vspace{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

\4{\tt do:=D(x,2)+(x+1)/x**2*D(x)-2/x**3;}

\4{\tt LoewyDecomposition(do|LODO(RATF Q,x)|);}


The Differential Operator:


Loewy Decomposition of Type ${\cal L}^2_{2}:\1$

$(D+\Frac{1}{x+1})(D+\Frac{1}{x}+\Frac{1}{x^{2}}-\Frac{1}{x+1})$  }}



{\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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