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Loewy


Usage

Loewy L


Parameter Type Description
L $\mathbbm{Q}[x,\frac d{dx}]$ A differential operator of order at most 3


Returns

Returns either L, in which case L is completely reducible, or a right Loewy decomposition $[L_1,\dots,L_n]$ where $L = L_1 \cdots L_n$ and each $L_i \in \mathbbm{Q}(x)[\frac d{dx}]$ is a completely reducible right-factor of maximal order of $L_1 \cdots L_{i-1}$.


Example

We compute the Loewy decomposition of the differential equation

\begin{displaymath}
\frac{d^3 y}{dx^3} - \frac{x^2+1}x \frac{d^2 y}{dx^2} -
\frac{2x^4-x^2-1}{x^2} \frac{dy}{dx} + (2x^3-4x) y = 0
\end{displaymath} (6)

as follows:
1 --> L := D^3-(x^2+1)/x*D^2-(-x^2-1+2*x^4)/x^2*D+2*x^3-4*x;
2 --> v := Loewy(L);
3 --> tex(v);

\begin{displaymath}
\left[ D-x, D^{2}-{{1} \over {x}}\,D-2\,x^{2} \right]
\end{displaymath}

This means that the operator of (6) factors as

\begin{displaymath}
{\left({\frac d{dx} - x}\right)}{\left({\frac{d^2}{dx^2} - \frac 1x \frac d{dx} - 2x^2}\right)}
\end{displaymath}

where the second-order factor is a completely reducible right-factor of maximal order. Note that is is reducible in this example:
4 --> w := decompose(element(v,2));
5 --> tex(w);

\begin{displaymath}
\left[ x^{2}-2 , x\,u , 1 \right]
\end{displaymath}

which means that

\begin{displaymath}
\frac{d^2}{dx^2} - \frac 1x \frac d{dx} - 2x^2
\end{displaymath}

is a least common left multiple of $d/dx - x \sqrt{2}$ and $d/dx + x \sqrt{2}$.


See Also

decompose,factor


next up previous contents index
Next: makeIntegral Up: Supported functions Previous: leftLcm   Contents   Index
Manuel Bronstein 2002-09-04