The least common left multiple F=LCLM(L1 .. Ln) of operators $\mathrm{L1}..\mathrm{Ln}$ is defined as the operator with minimal order such that all solutions $\mathrm{L1}..\mathrm{Ln}$ are solutions of $F$ as well.

If the optional argument groundfield=ext where ext is a list of $\mathrm{RootOf}$s is given, then LCLM( L1 .. Ln) and all their conjugates over the field $Q\left(\mathrm{ext}\right)$ is computed. This LCLM is an element of $Q\left(\mathrm{ext}\,x\right)$ $\left[\mathrm{Dx}\right]$.

The quotes for the name groundfield are only necessary if groundfield has been given a value; otherwise they may be omitted.

The argument domain describes the differential algebra. If this argument is the list $\left[\mathrm{Dt}\,t\right]$ then the differential operators are notated with the symbols $\mathrm{Dt}$ and $t$. They are viewed as elements of the differential algebra $C\left(t\right)$ $\left[\mathrm{Dt}\right]$ 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 LCLM(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[LCLM](..).

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

$a≔\mathrm{RootOf}\left(x^2-2\right)$

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

$b≔\mathrm{RootOf}\left(x^2-3\right)$

$b≔\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\right)$

$A≔\left[\mathrm{Dx}\,x\right]$

$A≔\left[\mathrm{Dx}\,x\right]$

$L≔\mathrm{LCLM}\left(\mathrm{Dx}+a\,{\mathrm{Dx}}^2+b\mathrm{Dx}+x\,A\right)\:$

$\mathrm{degree}\left(L\,\mathrm{Dx}\right)$

$3$

$L≔\mathrm{LCLM}\left(\mathrm{Dx}+a\,{\mathrm{Dx}}^2+b\mathrm{Dx}+x\,A\,'\mathrm{groundfield}'=\left[b\right]\right)\:$

$\mathrm{degree}\left(L\,\mathrm{Dx}\right)$

$4$

$L≔\mathrm{LCLM}\left(\mathrm{Dx}+a\,{\mathrm{Dx}}^2+b\mathrm{Dx}+x\,A\,'\mathrm{groundfield}'=\left[a\right]\right)\:$

$\mathrm{degree}\left(L\,\mathrm{Dx}\right)$

$5$

$L≔\mathrm{LCLM}\left(\mathrm{Dx}+a\,{\mathrm{Dx}}^2+b\mathrm{Dx}+x\,A\,'\mathrm{groundfield}'=\left[\right]\right)$

$L≔{\mathrm{Dx}}^6-\frac{\left(5x^4+32x^3+36x^2-40x-13\right){\mathrm{Dx}}^5}{x^5+8x^4+12x^3-20x^2-13x-8}+\frac{\left(2x^6+11x^5-16x^4-88x^3+146x^2+161x+24\right){\mathrm{Dx}}^4}{x^5+8x^4+12x^3-20x^2-13x-8}-\frac{\left(3x^5-9x^4-124x^3-60x^2+297x+233\right){\mathrm{Dx}}^3}{x^5+8x^4+12x^3-20x^2-13x-8}+\frac{\left(x^7+4x^6-14x^5-21x^4+87x^3-248x^2-214x+65\right){\mathrm{Dx}}^2}{x^5+8x^4+12x^3-20x^2-13x-8}+\frac{2\left(3x^5+x^4-60x^3+12x^2+217x+207\right)\mathrm{Dx}}{x^5+8x^4+12x^3-20x^2-13x-8}-\frac{2\left(x^7+8x^6+12x^5-21x^4-41x^3-36x^2+56x+81\right)}{x^5+8x^4+12x^3-20x^2-13x-8}$

$\mathrm{degree}\left(L\,\mathrm{Dx}\right)$

$6$