 poltodiffeq - Maple Help

gfun

 poltodiffeq
 determine the differential equation satisfied by a polynomial in holonomic functions Calling Sequence poltodiffeq(P, listdiffeq, list_unknowns, y(z)) Parameters

 P - polynomial in z and y1(z), y2(z), ... and possibly their derivatives and repeated derivatives listdiffeq - list containing, for each of y1(z), y2(z), ..., either a linear differential equation it satisfies or a set containing the equation together with initial conditions list_unknowns - list of function names $[\mathrm{y1}\left(z\right),\mathrm{y2}\left(z\right),...]$ y - name; holonomic function name z - name; variable of the holonomic function y Description

 • The poltodiffeq(P, listdiffeq, list_unknowns, y(z)) command returns a linear differential equation satisfied by the polynomial P.
 If y1(z), y2(z), ... are holonomic function solutions of listdiffeq, listdiffeq, ..., the poltodiffeq function returns a linear differential equation satisfied by $P\left(z,\mathrm{y1}\left(z\right),...\right)$. Examples

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $\mathrm{Sin}≔\left\{\frac{{ⅆ}^{2}}{ⅆ{z}^{2}}\mathrm{y1}\left(z\right)=-\mathrm{y1}\left(z\right),\mathrm{y1}\left(0\right)=0,\mathrm{D}\left(\mathrm{y1}\right)\left(0\right)=1\right\}:$
 > $\mathrm{Cos}≔\left\{\frac{{ⅆ}^{2}}{ⅆ{z}^{2}}\mathrm{y2}\left(z\right)=-\mathrm{y2}\left(z\right),\mathrm{y2}\left(0\right)=1,\mathrm{D}\left(\mathrm{y2}\right)\left(0\right)=0\right\}:$
 > $\mathrm{poltodiffeq}\left({\mathrm{y1}\left(z\right)}^{2}+{\mathrm{y2}\left(z\right)}^{2},\left[\mathrm{Sin},\mathrm{Cos}\right],\left[\mathrm{y1}\left(z\right),\mathrm{y2}\left(z\right)\right],y\left(z\right)\right)$
 $\left\{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{z}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right){,}{y}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){=}{0}{,}{{\mathrm{D}}}^{\left({2}\right)}{}\left({y}\right){}\left({0}\right){=}{0}\right\}$ (1)
 > $\mathrm{poltodiffeq}\left({\mathrm{y1}\left(z\right)}^{2}+{\left(\frac{ⅆ}{ⅆz}\mathrm{y1}\left(z\right)\right)}^{2},\left[\mathrm{Sin}\right],\left[\mathrm{y1}\left(z\right)\right],y\left(z\right)\right)$
 $\left\{\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right){,}{y}{}\left({0}\right){=}{1}\right\}$ (2)