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DEtools

  

exterior_power

  

return the exterior power of a differential operator

 

Calling Sequence

Parameters

Description

Examples

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 y1..yn of L, the determinant of the Wronskian w=detMatrixn,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 ⅇRxⅆx 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 Ct[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.

Examples

withDEtools:

ADx,x

ADx,x

(1)

LDx42Dxx2

LDx4x22Dx

(2)

Mexterior_powerL,2,A

MDx6+4x2Dx2+12xDx

(3)

exterior_powerⅆ3ⅆx3yxyx,2,yx

yx+ⅆ3ⅆx3yx

(4)

See Also

DEtools[expsols]

diffop