factor

Usage

factor L

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

Description

Returns one of the following possible results:

(i)

L, in which case L is irreducible in ${\overline \mathbbm{Q}}(x)[d/dx]$.

(ii)

$[m,L_1,\dots,L_n]$ where $m \ge 0$ and $L_i \in \mathbbm{Q}(x)[\frac d{dx}]$, in which case each $L_i$ is irreducible in ${\overline \mathbbm{Q}}(x)[d/dx]$, and L can be expressed in terms of the $L_i$'s depending on $m$ as in the following table:

m	L
0	LeftLcm($L_1,\dots,L_n$)
1	$L_1 \cdots L_n$
2	LeftLcm($L_1, L_2 L_3$)
3	$L_1$ LeftLcm($L_2, L_3$)
4	$L_1$ LeftLcm($L_2, L_3$) $L_4$

(iii)

$[m,h(x),a(x,u),b(x,u),f_1(x,u),\dots,f_n(x,u)]$, where $m < 0$, $h(x) \in \mathbbm{Q}[x]$ is irreducible and $a(x,u)$, $b(x,u)$, $f_1(x,u),\dots,f_n(x,u) \in \mathbbm{Q}(x)[u]$, in which case each $f_i(x,D)$ is irreducible in ${\overline \mathbbm{Q}}(x)[d/dx]$, and L can be expressed depending on $m$ as in the following table:

m	L
-1	LeftLcm( $D + a(\alpha, x)/b(\alpha, x), f_1(x,D),\dots,f_n(x,D)$)
-4	$f_1(x,D)$ LeftLcm( $D + a(\alpha, x) / b(\alpha, x)$)
-5	$f_1(x,D)$ LeftLcm( $D + a(\alpha, x) / b(\alpha, x)$) $f_2(x,D)$

where $\alpha$ ranges over all the roots of $h(x)$.

Example

We factor the differential equation

$$\frac{d^3 y}{dx^3} - x \frac{d^2 y}{dx^2} - (x^2-1) \frac{dy}{dx} + (x^3-3x) y = 0$$ (5)

as follows:

1 --> L := D^3-x*D^2+(-x^2+1)*D-3*x+x^3;
2 --> v := factor(L);
3 --> tex(v);

$$\left[ 2 , D-x , D-x , D+x \right]$$

This means that the operator of (5) is a least common left multiple of $d/dx - x$ and of the product $(d/dx - x)(d/dx + x)$, which itself is not a least common left multiple of irreducible operators.

Usage within MAPLE

• In order not to conflict with the factor function in MAPLE, factor is available under MAPLE under the name dfactor. In addition, when using dfactor from inside MAPLE, the result returned from BERNINA is further transformed as in the case of the decompose function. So the above example in MAPLE would be:

  > L := D^3-x*D^2+(-x^2+1)*D-3*x+x^3;
  > dfactor(L, D, x);
  

  
  
  $${\it LeftLcm}(D-x,\left[D-x,D+x\right])$$
  
  

  
  See Also

  decompose,Loewy