convert a recurrence into a differential equation - Maple Help

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LREtools[REtoDE] - convert a recurrence into a differential equation

Calling Sequence

REtoDE(problem, f(z))

REtoDE(problem, f(z), output=val)

Parameters

problem

-

either a RESol recurrence structure, or a recurrence definition

f(z)

-

name and variable of the function

output=val

-

form of the output, where output=DESol or output=diff

Description

• 

This routine returns the differential operator or differential equation satisfied by the generating function associated to the recurrence. This operator is returned as either a DESol operator or a differential equation (both possibly with initial conditions). No attempt is made to explicitly solve the result.

• 

For the definition of the format of a problem, see the help page for LREtools[REcreate]. The problem can be an RESol structure resulting from a call to LREtools[REcreate].

• 

The output option val can be either DESol which requests operator output (the default) or diff which requests the output in terms of a differential system.

  

Note: In some cases initial conditions are also needed, so the output=diff result is returned as a set. If no initial conditions are needed, the output is a single ODE.

Examples

withLREtools:

REtoDEan+22an+1+an=0,an,,fz

DESol2f+4z4Df+z22z+1D2f,f

(1)

rec:=REcreatean+22an+1+an=0,an,

rec:=RESolan+22an+1+an=0,an,a0=a0,a1=a1,INFO

(2)

REtoDErec,fz,output=diff

2fz+4z4ⅆⅆzfz+z22z+1ⅆ2ⅆz2fz

(3)

An example with conditions:

REtoDE2nan+2+n2+1an+1n+12an=0,an,,fz

DESol8z+8f+40z28Df+28z3+8z2+8zD2f+4z4+4z3D3f,f,f0=Df0

(4)

REtoDE2nan+2+n2+1an+1n+12an=0,an,,fz,output=diff

8z+8fz+40z28ⅆⅆzfz+28z3+8z2+8zⅆ2ⅆz2fz+4z4+4z3ⅆ3ⅆz3fz,f0=Df0

(5)

See Also

DESol, dsolve, LREtools, LREtools[REcreate], RESol, rsolve


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