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DEtools

 GCRD
 find the Greatest Common Right Divisor of 2 differential operators

 Calling Sequence GCRD( f, g, domain)

Parameters

 f, g - differential operators domain - list containing two names

Description

 • For operators f and g there exists an operator $R$ such that the solution space of $R$ is the intersection of the solution spaces of f and g. This procedure finds this operator $R$. Like the gcd for polynomials, this $R$ is a combination $R=af+bg$ for some operators a and b.
 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{GCRD}\left({\mathrm{Dx}}^{4},{\mathrm{Dx}}^{2}-\frac{18x}{1+2x+3{x}^{3}},\left[\mathrm{Dx},x\right]\right)$
 ${\mathrm{Dx}}{-}\frac{{9}{}{{x}}^{{2}}{+}{2}}{{3}{}{{x}}^{{3}}{+}{2}{}{x}{+}{1}}$ (1)