return the exterior power of a differential operator
exterior_power(L, n, domain)
exterior_power(eqn, n, dvar)
list containing two names
homogeneous linear differential equation
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 y1..yn of L, the determinant of the Wronskian w=det⁡Matrix⁡n,n,[y1,y1',y1'',..,y2,y2',y2'',..,yn,yn',yn'',..] 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∫R⁡xⅆx where R is a rational function).
The argument domain describes the differential algebra. If this argument is the list Dt,t, then the differential operators are notated with the symbols Dt and t. They are viewed as elements of the differential algebra C⁡t⁢[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.
A ≔ Dx,x
L ≔ Dx4−2⁢Dx−x2
L ≔ Dx4−x2−2⁢Dx
M ≔ exterior_power⁡L,2,A
M ≔ Dx6+4⁢x2⁢Dx2+12⁢x⁢Dx
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