gfun[poltodiffeq] - determine the differential equation satisfied by a polynomial in holonomic functions

Calling Sequence

poltodiffeq(P, listdiffeq, list_unknowns, y(z), <iniconds>, <homogeneous=bool>, <operator=name>)

Parameters

P - polynomial in z and the (possibly differentiated) variables in list_unknowns

listdiffeq - list containing, for each of the variables in list_unknowns, either a linear differential equation or a set containing the equation together with initial conditions

list_unknowns - list of the unknowns [y1(z), y2(z),...], in the same order as in listdiffeq

y,z - the name of the resulting function and the generic variable

homogeneous - (optional) boolean requesting a homogeneous output equation; the default is false

operator - (optional) name used verbatim as the differential operator

Description

If y1(z), y2(z),... are power series solutions of listdiffeq[1], listdiffeq[2],..., poltodiffeq outputs a linear differential equation verified by P(z,y1(z),...).
When homogeneous=true, the constant coordinate is retained in the linear-dependence computation so that the returned equation is homogeneous.
With operator=name, input and output initial conditions are not computed. Homogeneous output is a polynomial in the supplied name and inhomogeneous output has the form L(name)=-q.

Examples

> with(gfun):
Sin:={diff(y1(z),z,z)=-y1(z),y1(0)=0,D(y1)(0)=1}:
Cos:={diff(y2(z),z,z)=-y2(z),y2(0)=1,D(y2)(0)=0}:
poltodiffeq(y1(z)^2+y2(z)^2,[Sin,Cos],[y1(z),y2(z)],y(z));

\[\{4 \frac{d}{d z}y \! \left(z \right)+\frac{d^{3}}{d z^{3}}y \! \left(z \right), y \! \left(0\right) = 1, D\! \left(y \right)\! \left(0\right) = 0, D^{\left(2\right)}\! \left(y \right)\! \left(0\right) = 0\}\]

> poltodiffeq(y1(z)^2+diff(y1(z),z)^2,[Sin],[y1(z)],y(z));

\[\{\frac{d}{d z}y \! \left(z \right), y \! \left(0\right) = 1\}\]

gfun[poltodiffeq] - determine the differential equation satisfied by a polynomial in holonomic functions

Calling Sequence

Parameters

Description

Examples

See Also