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symmetric_power

  

calculate the symmetric power of a differential equation or operator

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

symmetric_power(L, m, domain)

symmetric_power(eqn, m, dvar)

Parameters

L

-

differential operator

m

-

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 every set of m solutions y1,...,ym of L the product y1y2ym is a solution of M.

• 

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 having rational function coefficients. In this case the third argument must be the dependent variable.

• 

This function is part of the DEtools package, and so it can be used in the form symmetric_power(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[symmetric_power](..).

Examples

withDEtools:

_EnvdiffopdomainDx,x:

LDx2+axDx+bx

LDx2+axDx+bx

(1)

Msymmetric_powerL,2

MDx3+3axDx2+2ax2+ⅆⅆxax+4bxDx+4bxax+2ⅆⅆxbx

(2)

To illustrate formally the meaning of the output of this command, consider a general second order ODE

ODEⅆ2ⅆx2yx+axyx=0

ODEⅆ2ⅆx2yx+axyx=0

(3)

The nth symmetric_power of ODE is another ODE having for a solution the nth power of the solution of ODE. For example, the solution of ODE can be written - formally - using the Maple DESol command; dsolve represents it that way:

soldsolveODE

solyx=DESolax_Yx+ⅆ2ⅆx2_Yx,_Yx

(4)

where in the above DESol(...) represents any linear combination of two independent solutions of ODE. The first symmetric power of ODE is then ODE itself (has for solution sol^1) and, for instance, for the second and third symmetric powers of ODE we have

ODE_2symmetric_powerODE,2,yx

ODE_22ⅆⅆxaxyx+4axⅆⅆxyx+ⅆ3ⅆx3yx

(5)

dsolveODE_2,yx

yx=DESolax_Yx+ⅆ2ⅆx2_Yx,_Yx2

(6)

ODE_3symmetric_powerODE,3,yx

ODE_39ax2+3ⅆ2ⅆx2axyx+10ⅆⅆxaxⅆⅆxyx+10axⅆ2ⅆx2yx+ⅆ4ⅆx4yx

(7)

dsolveODE_3,yx

yx=DESolax_Yx+ⅆ2ⅆx2_Yx,_Yx3

(8)

See Also

DEtools/symmetric_product

diffop