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 exterior_power
 return the exterior power of a differential operator

 Calling Sequence exterior_power(L, n, domain) exterior_power(eqn, n, dvar)

Parameters

 L - differential operator n - positive integer domain - list containing two names eqn - homogeneous linear differential equation dvar - dependent variable

Description

 • The input L is a differential operator. The output of this procedure is a linear differential operator $M$ of minimal order such that for all solutions $\mathrm{y1}..\mathrm{yn}$ of L, the determinant of the Wronskian $w=\mathrm{det}\left(\mathrm{Matrix}\left(n,n,[\mathrm{y1},\mathrm{y1}',\mathrm{y1}\text{'}\text{'},\mathrm{..},\mathrm{y2},\mathrm{y2}',\mathrm{y2}\text{'}\text{'},\mathrm{..},\mathrm{yn},\mathrm{yn}',\mathrm{yn}\text{'}\text{'},\mathrm{..}]\right)\right)$ is a solution of $M$.
 • An important property of the exterior power $M$ is the following: If L has rational functions coefficients and L has a right-hand factor of order n, then M has a right-hand factor of order $1$ (in other words: $M$ has an exponential solution ${ⅇ}^{\int R\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx}$ where $R$ is a rational function).
 • 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 $C\left(t\right)\mathrm{\left[Dt\right]}$ where $C$ is the field of constants.
 • If the argument domain is omitted then the differential specified by the environment variable _Envdiffopdomain is used. If this environment variable is not set then the argument domain may not be omitted.
 • Instead of a differential operator, the input can also be a linear homogeneous differential equation, eqn. In this case the third argument must be the dependent variable dvar.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $A≔\left[\mathrm{Dx},x\right]$
 ${A}{≔}\left[{\mathrm{Dx}}{,}{x}\right]$ (1)
 > $L≔{\mathrm{Dx}}^{4}-2\mathrm{Dx}-{x}^{2}$
 ${L}{≔}{{\mathrm{Dx}}}^{{4}}{-}{{x}}^{{2}}{-}{2}{}{\mathrm{Dx}}$ (2)
 > $M≔\mathrm{exterior_power}\left(L,2,A\right)$
 ${M}{≔}{{\mathrm{Dx}}}^{{6}}{+}{4}{}{{x}}^{{2}}{}{{\mathrm{Dx}}}^{{2}}{+}{12}{}{x}{}{\mathrm{Dx}}$ (3)
 > $\mathrm{exterior_power}\left(\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)-y\left(x\right),2,y\left(x\right)\right)$
 ${y}{}\left({x}\right){+}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (4)