substitute an algebraic function into a holonomic one - Maple Help

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gfun[algebraicsubs] - substitute an algebraic function into a holonomic one

Calling Sequence

algebraicsubs(deq, eq, y(z), ini)

Parameters

deq

-

linear differential equation in y(z) with polynomial coefficients

eq

-

algebraic equation in y(z)

y

-

name; holonomic function name

z

-

name; variable of the holonomic function y

ini

-

(optional) set; specify computation of initial conditions for the resulting differential equation

Description

• 

The gfun[algebraicsubs](deq, eq, y(z)) command returns a differential equation satisfied by the composition f@g where f is the holonomic function defined by the equation deq and g is the algebraic equation defined by eq.  The composition is holonomic by closure properties of holonomic functions.

• 

Let d1 be the differential order of deq, and d2 be the degree of eq. If the equation deq is homogeneous, then the order of f@g is at most d1d2.  Otherwise, it is at most d1+1d2.

• 

If initial conditions are specified using ini, the algebraicsubs function attempts to compute initial conditions for the resulting differential equation.

  

Initial conditions can be present in the differential equation if a set is specified, in the same way as initial conditions are specified for dsolve. In the case of a polynomial equation, they are specified in the optional parameter ini as a set, using the same syntax as for dsolve.

Examples

The differential equation satisfied by cos(t).

withgfun:

deq:=D2ft+ft:

The algebraic equation satisfied by sqrt(1-4*t).

eq:=algfuntoalgeq14t,ft:

The differential equation satisfied by cos(sqrt(1-4*t)).

algebraicsubsdeq,eq,ft

4ft+2ⅆⅆtft+1+4tⅆ2ⅆt2ft

(1)

algebraicsubsD2yx+yx,y0=1,Dy0=0,2x4y2+2yxx2,yx,y0=0,Dy0=1,D2y0=22

2048x8+768x6+864x4236x2+87yx+320x3+280xⅆⅆxyx+512x6136x2+18ⅆ2ⅆx2yx+128x38xⅆ3ⅆx3yx+32x4+4x2+3ⅆ4ⅆx4yx,y0=1,Dy0=0,D2y0=1,D3y0=62

(2)

See Also

dsolve, gfun, gfun/`diffeq+diffeq`, gfun/`rec+rec`, gfun/parameters, gfun[algfuntoalgeq]


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