decompose

Usage

decompose 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 cannot be written as a least common left multiple of lower order operators (L can still be either reducible or irreducible in that case).

(ii)

$[L_1,\dots,L_n]$ where $L_i \in \mathbbm{Q}(x)[\frac d{dx}]$, in which case L is a least common left multiple of $L_1,\dots,L_n$.

(iii)

$[h(x),a(x,u),b(x,u),f_1(x,u),\dots,f_n(x,u)]$, where $h(x) \in \mathbbm{Q}[x]$ 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 L is a least common left multiple of $f_1(x,D),\dots,f_n(x,D)$ and of $D + a(\alpha, x) / b(\alpha, x)$ where $\alpha$ ranges over all the roots of $h(x)$.

Example

We decompose the differential equation

$$\frac{d^3 y}{dx^3} - \frac{x^2+3}x \frac{d^2 y}{dx^2} - \frac{2x^4-x^2-3}{x^2} \frac{dy}{dx} + 2x^3 y = 0$$ (2)

as follows:

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

$$\left[ x^{2}+4\,x+2 , \left(-x-2\right)\,u , 1 , u-x \right]$$

This means that the operator of (2) is a least common left multiple of $d/dx - x$, $d/dx - (\alpha+2) x$ and $d/dx - (\beta+2) x$ where $\alpha$ and $\beta$ are the two roots of $x^2 + 4 x + 2 = 0$. Since $\alpha = -2 - \sqrt 2$ and $\beta = -2 + \sqrt 2$, our operator is a least common left multiple of $d/dx - x$, $d/dx + x \sqrt 2$ and $d/dx - x \sqrt 2$.

Usage within MAPLE

• When using decompose from inside MAPLE, the result returned from BERNINA is further transformed into one of the following:

  (i)

  L, in which case L cannot be written as a least common left multiple of lower order operators.

  (ii)

  An object of the form LeftLcm $(L_1,\dots,L_n)$ where the $L_i$'s are differential operators, in which case L is a least common left multiple of $L_1,\dots,L_n$.

  (iii)

  An object of the form LeftLcm $(L_\alpha, and~conjugates)$ where $L_\alpha$ is a differential operator containing an algebraic number $\alpha$, in which case L is a least common left multiple of all the conjugates of $L_\alpha$.

  So the above example in MAPLE would be:

  > L := D^3-(x^2+3)/x*D^2-(2*x^4-x^2-3)/x^2*D + 2*x^3;
  > decompose(L, D, x);
  

  
  
  $${\it LeftLcm}(D-\sqrt {2}x,D+\sqrt {2}x,D-x)$$
  
  

  
  See Also

  factor,Loewy