For operators a and b (as for polynomials, see quo) there exist operators $q$ and $r$ such that $a=qb+r$ and $\mathrm{order}\left(r\right)<\mathrm{order}\left(b\right)$. The output rightdivision is the list $\left[q\&comma;r\right]$.

The procedure leftdivision does precisely the same, only the order of multiplication $q$ $b$ is reversed: $b$ $q$.

The argument domain describes the differential algebra. If this argument is the list $\left[\mathrm{Dt}\&comma;t\right]$ 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.

These functions are part of the DEtools package, and so they can be used in the form rightdivision(..) and leftdivision(..) only after executing the command with(DEtools).  However, they can always be accessed through the long form of the command by using DEtools[rightdivision](..) or DEtools[leftdivision](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\&colon;$

$\mathrm{rightdivision}\left({\mathrm{Dt}}^4\&comma;{\mathrm{Dt}}^2-\frac{18t}{1+2t+3t^3}\&comma;\left[\mathrm{Dt}\&comma;t\right]\right)$

$\left[{\mathrm{Dt}}^2+\frac{18t}{3t^3+2t+1}\&comma;-\frac{36\left(6t^3-1\right)\mathrm{Dt}}{{\left(3t^3+2t+1\right)}^2}+\frac{36\left(54t^5+12t^3-9t^2-2\right)}{{\left(3t^3+2t+1\right)}^3}\right]$