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[1], listdiffeq[2], ..., the poltodiffeq function returns a linear differential equation satisfied by $P\left(z,\mathrm{y1}\left(z\right),...\right)$.

$\mathrm{with}\left(\mathrm{gfun}\right)\:$

$\mathrm{Sin}≔\left\{\mathrm{diff}\left(\mathrm{y1}\left(z\right)\,z\,z\right)=-\mathrm{y1}\left(z\right)\,\mathrm{y1}\left(0\right)=0\,\mathrm{D}\left(\mathrm{y1}\right)\left(0\right)=1\right\}\:$

$\mathrm{Cos}≔\left\{\mathrm{diff}\left(\mathrm{y2}\left(z\right)\,z\,z\right)=-\mathrm{y2}\left(z\right)\,\mathrm{y2}\left(0\right)=1\,\mathrm{D}\left(\mathrm{y2}\right)\left(0\right)=0\right\}\:$

$\mathrm{poltodiffeq}\left({\mathrm{y1}\left(z\right)}^2+{\mathrm{y2}\left(z\right)}^2\,\left[\mathrm{Sin}\,\mathrm{Cos}\right]\,\left[\mathrm{y1}\left(z\right)\,\mathrm{y2}\left(z\right)\right]\,y\left(z\right)\right)$

$\left\{\frac{{\ⅆ}^3}{\ⅆz^3}y\left(z\right)+4\frac{\ⅆ}{\ⅆz}y\left(z\right)\,y\left(0\right)=1\,\mathrm{D}\left(y\right)\left(0\right)=0\,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)=0\right\}$

$\mathrm{poltodiffeq}\left({\mathrm{y1}\left(z\right)}^2+{\mathrm{diff}\left(\mathrm{y1}\left(z\right)\,z\right)}^2\,\left[\mathrm{Sin}\right]\,\left[\mathrm{y1}\left(z\right)\right]\,y\left(z\right)\right)$

$\left\{\frac{\ⅆ}{\ⅆz}y\left(z\right)\,y\left(0\right)=1\right\}$