The input is a differential operator L with rational function coefficients.

The output is a list of irreducible factors $\left[\mathrm{L1}\,\mathrm{...}\,\mathrm{Lr}\right]$ such that $L=\mathrm{mult}\left(\mathrm{L1}\,\mathrm{...}\,\mathrm{Lr}\right)$.

The optional argument one step causes the computation to stop whenever a factorization is found. In this case the output is $\left[L\right]$ if $L$ is irreducible, but it contains two (not necessarily irreducible) factors if $L$ is reducible.

The argument domain describes the differential algebra. If this argument is the list $\left[\mathrm{Dx}\,x\right]$ then the differential operators are notated with the symbols $\mathrm{Dx}$ and $x$. They are viewed as elements of the differential algebra $C\left(x\right)\mathrm{[Dx]}$ where $C$ is the field of constants.

If the argument domain is omitted then the differential specified by the environment variable _Envdiffopdomain will be used. If this environment variable is not set then the argument domain may not be omitted.

This function is part of the DEtools package, and so it can be used in the form DFactor(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[DFactor](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$a≔\mathrm{RootOf}\left(x^2-10\right)$

$a≔\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)$

$R≔\left(\frac{35}{4}-\frac{5}{2}a\right)x^2-\frac{55}{2}+\frac{11}{2}a$

$R≔\left(\frac{35}{4}-\frac{5\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)}{2}\right)x^2-\frac{55}{2}+\frac{11\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)}{2}$

$L≔{\mathrm{DF}}^2-R$

$L≔{\mathrm{DF}}^2-\left(\frac{35}{4}-\frac{5\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)}{2}\right)x^2+\frac{55}{2}-\frac{11\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)}{2}$

$N≔\mathrm{DFactor}\left(L\,\left[\mathrm{DF}\,x\right]\right)$

$N≔\left[\mathrm{DF}+\frac{\left(-5+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)\right)\left(45x^6-60x^4\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)-300x^4+525x^2+150x^2\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)-110-34\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)\right)}{30x\left(3x^4-2x^2\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)-10x^2+2\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)+7\right)}\,\mathrm{DF}-\frac{\left(-5+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)\right)\left(45x^6-60x^4\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)-300x^4+525x^2+150x^2\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)-110-34\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)\right)}{30x\left(3x^4-2x^2\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)-10x^2+2\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)+7\right)}\right]$

$\mathrm{mult}\left(\mathrm{op}\left(N\right)\,\left[\mathrm{DF}\,x\right]\right)$

${\mathrm{DF}}^2+\frac{\left(-7+2\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)\right)\left(15x^2-22\mathrm{RootOf}\left({\mathrm{\_Z}}^2-10\right)-110\right)}{12}$

$L≔\mathrm{mult}\left({\mathrm{DF}}^2+x^3+\frac{1}{x^3}\,{\mathrm{DF}}^2+x^2-\frac{1}{x^3}\,\left[\mathrm{DF}\,x\right]\right)$

$L≔{\mathrm{DF}}^4+x^2\left(x+1\right){\mathrm{DF}}^2+\frac{2\left(2x^5+3\right)\mathrm{DF}}{x^4}+\frac{x^{11}+x^6+x^5-12x-1}{x^6}$

$\mathrm{DFactor}\left(L\,\left[\mathrm{DF}\,x\right]\,\mathrm{`one\; step`}\right)$

$\left[{\mathrm{DF}}^2+\frac{x^6+1}{x^3}\,{\mathrm{DF}}^2+\frac{x^5-1}{x^3}\right]$

van der Put, M., and Singer, M. F. Galois Theory of Linear Differential Equations, Vol. 328. Springer: 2003. An electronic version of this book is available at http://www4.ncsu.edu/~singer/ms_papers.html.

van Hoeij, M. "Factorization of Differential Operators with Rational Functions Coefficients." J. Symb. Comput. Vol. 24. (1997): 537-561.