gfun[diffeqtohomdiffeq] - make a differential equation homogeneous

gfun[rectohomrec] - make a recurrence homogeneous

Calling Sequence

diffeqtohomdiffeq(deq,y(z), <operator=name>)

rectohomrec(rec,u(n), <operator=name>)

Parameters

deq - linear differential equation in y(z) with polynomial coefficients

rec - linear recurrence with polynomial coefficients

y,z - function and variable of the differential equation

u,n - variable and index of the recurrence

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

Description

If deq is not homogeneous, then diffeqtohomdiffeq produces a differential equation of order increased by one which is homogeneous and cancels all the solutions of the original equation. If deq is homogeneous, it is unchanged.
If rec is not homogeneous, then rectohomrec produces a recurrence of order increased by one which is homogeneous and cancels all the solutions of the original equation. If rec is homogeneous, it is unchanged.
With operator=name, initial conditions are not computed and the homogeneous result is returned as a polynomial in the supplied name.

Examples

> with(gfun):
deq:=diff(y(x),x)*(x-1)+2*y(x)-2*x-3:
diffeqtohomdiffeq(deq,y(x));

\[4 y \! \left(x \right)+\left(-4 x -11\right) \left(\frac{d}{d x}y \! \left(x \right)\right)+\left(-2 x^{2}-x +3\right) \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)\]

> diffeqtohomdiffeq({deq,y(0)=2},y(x));

\[\{4 y \! \left(x \right)+\left(-4 x -11\right) \left(\frac{d}{d x}y \! \left(x \right)\right)+\left(-2 x^{2}-x +3\right) \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right), y \! \left(0\right) = 2, D\! \left(y \right)\! \left(0\right) = 1\}\]

> rec:=u(n+1)=u(n)+n^2+1:
rectohomrec(rec,u(n));

\[\left(-n^{2}-2 n -2\right) u \! \left(n \right)+\left(2 n^{2}+2 n +3\right) u \! \left(n +1\right)+\left(-n^{2}-1\right) u \! \left(n +2\right)\]

> rectohomrec({rec,u(0)=1},u(n));