diffeqtohomdiffeq - Maple Help

gfun

 diffeqtohomdiffeq
 make a differential equation homogeneous
 rectohomrec
 make a recurrence homogeneous

 Calling Sequence diffeqtohomdiffeq(deq, y(z)) rectohomrec(rec, u(n))

Parameters

 deq - linear differential equation in y(z) with polynomial coefficients y - name; differential equation function z - name; variable of the differential equation function y rec - linear recurrence with polynomial coefficients u - name; recurrence name n - name; index of the recurrence u

Description

 • The diffeqtohomdiffeq(deq, y(z)) command makes the differential equation, deq homogeneous.
 If deq is not homogeneous, the diffeqtohomdiffeq function 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.
 • The rectohomrec(rec, u(n)) command makes the recurrence, rec, homogeneous.
 If rec is not homogeneous, the rectohomrec function 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.

Examples

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $\mathrm{deq}≔\mathrm{diff}\left(y\left(x\right),x\right)\left(x-1\right)+2y\left(x\right)-2x-3:$
 > $\mathrm{diffeqtohomdiffeq}\left(\mathrm{deq},y\left(x\right)\right)$
 ${4}{}{y}{}\left({x}\right){+}\left({-}{4}{}{x}{-}{11}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{2}{}{{x}}^{{2}}{-}{x}{+}{3}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (1)
 > $\mathrm{diffeqtohomdiffeq}\left(\left\{\mathrm{deq},y\left(0\right)=2\right\},y\left(x\right)\right)$
 $\left\{{4}{}{y}{}\left({x}\right){+}\left({-}{4}{}{x}{-}{11}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{2}{}{{x}}^{{2}}{-}{x}{+}{3}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){,}{y}{}\left({0}\right){=}{2}{,}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){=}{1}\right\}$ (2)
 > $\mathrm{rec}≔u\left(n+1\right)=u\left(n\right)+{n}^{2}+1:$
 > $\mathrm{rectohomrec}\left(\mathrm{rec},u\left(n\right)\right)$
 $\left({-}{{n}}^{{2}}{-}{2}{}{n}{-}{2}\right){}{u}{}\left({n}\right){+}\left({2}{}{{n}}^{{2}}{+}{2}{}{n}{+}{3}\right){}{u}{}\left({n}{+}{1}\right){+}\left({-}{{n}}^{{2}}{-}{1}\right){}{u}{}\left({n}{+}{2}\right)$ (3)
 > $\mathrm{rectohomrec}\left(\left\{\mathrm{rec},u\left(0\right)=1\right\},u\left(n\right)\right)$
 $\left\{\left({-}{{n}}^{{2}}{-}{2}{}{n}{-}{2}\right){}{u}{}\left({n}\right){+}\left({2}{}{{n}}^{{2}}{+}{2}{}{n}{+}{3}\right){}{u}{}\left({n}{+}{1}\right){+}\left({-}{{n}}^{{2}}{-}{1}\right){}{u}{}\left({n}{+}{2}\right){,}{u}{}\left({0}\right){=}{1}{,}{u}{}\left({1}\right){=}{2}\right\}$ (4)