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=\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 ${\ⅇ}^{\int R\left(x\right)\ⅆx}$ 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 $C\left(t\right)\mathrm{[Dt]}$ 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.

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

$A≔\left[\mathrm{Dx}\,x\right]$

$A≔\left[\mathrm{Dx}\,x\right]$

$L≔{\mathrm{Dx}}^4-2\mathrm{Dx}-x^2$

$L≔{\mathrm{Dx}}^4-x^2-2\mathrm{Dx}$

$M≔\mathrm{exterior\_power}\left(L\,2\,A\right)$

$M≔{\mathrm{Dx}}^6+4x^2{\mathrm{Dx}}^2+12x\mathrm{Dx}$

$\mathrm{exterior\_power}\left(\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)-y\left(x\right)\,2\,y\left(x\right)\right)$

$y\left(x\right)+\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)$