The algeqtodiffeq(p, y(z)) command computes a linear differential equation with polynomial coefficients verified by the function y(z).  The polynomial p defines an algebraic function, $\mathrm{RootOf}\left(p\,y\right)$ in Maple terms.  The resulting equation is of order at most $\mathrm{degree}\left(p\,y\right)-1$.

The resulting linear differential equation contains initial conditions in zero ($y\left(0\right)$, $\mathrm{D}\left(y\right)\left(0\right)$, and so on), and can thus be passed directly to dsolve. In general, $y\left(0\right)$ is a RootOf a polynomial, $\mathrm{D}\left(y\right)\left(0\right)$ a rational expression in $y\left(0\right)$, ${\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)$ a rational expression in $y\left(0\right)$, $\mathrm{D}\left(y\right)\left(0\right)$, and so on.

If initial conditions are specified using ini, the algeqtodiffeq function attempts to compute initial conditions for the resulting differential equation.

If a particular solution of eq is selected by specifying initial terms of its power series expansion at the origin using the same syntax as that of initial conditions for dsolve, the algeqtodiffeq function returns the corresponding initial conditions together with the differential equation.

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

$\mathrm{algeqtodiffeq}\left(y=1+z{y}^2\,y\left(z\right)\right)$

$1+\left(-1+2z\right)y\left(z\right)+\left(4z^2-z\right)\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)$

$\mathrm{algeqtodiffeq}\left(56a^3+7a^3{y}^3-14yz\,y\left(z\right)\,\left\{y\left(0\right)=-2\right\}\right)$

$\left\{-y\left(z\right)z+3\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)z^2+\left(-108a^9+2z^3\right)\left(\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)\right)\,y\left(0\right)=\mathrm{-2}\,\mathrm{D}\left(y\right)\left(0\right)=-\frac{1}{3a^3}\right\}$

$p≔y=1+zy+z{y}^5$

$p≔y=z{y}^5+zy+1$

$\mathrm{deq}≔\mathrm{algeqtodiffeq}\left(p\,y\left(z\right)\right)$

$\mathrm{deq}≔\left\{\left(-147840z^3+169920z^2-22320z+240\right)y\left(z\right)+\left(393216z^8-294912z^7-466944z^6+92160z^5+447360z^4+65664z^3-265488z^2+29064z-120\right)\left(\frac{\ⅆ}{\ⅆz}y\left(z\right)\right)+\left(589824z^9-1130496z^8-30720z^7+878592z^6-132480z^5-444384z^4+203256z^3+67800z^2-1392z\right)\left(\frac{{\ⅆ}^2}{\ⅆz^2}y\left(z\right)\right)+\left(196608z^{10}-606208z^9+489472z^8+165888z^7+83200z^6+626112z^5+271848z^4+24488z^3-1408z^2\right)\left(\frac{{\ⅆ}^3}{\ⅆz^3}y\left(z\right)\right)+\left(16384z^{11}-69632z^{10}+105472z^9-56064z^8+188480z^7+166896z^6+38012z^5+1333z^4-256z^3\right)\left(\frac{{\ⅆ}^4}{\ⅆz^4}y\left(z\right)\right)\,y\left(0\right)=1\right\}$

$\mathrm{rec}≔\mathrm{diffeqtorec}\left(\mathrm{deq}\,y\left(z\right)\,u\left(n\right)\right)$

$\mathrm{rec}≔\left\{\left(16384n^4+98304n^3+180224n^2+98304n\right)u\left(n\right)+\left(-69632n^4-466944n^3-1060864n^2-958464n-294912\right)u\left(n+1\right)+\left(105472n^4+700416n^3+1332224n^2+208896n-995328\right)u\left(n+2\right)+\left(-56064n^4-170496n^3+1257216n^2+5973504n+6543360\right)u\left(n+3\right)+\left(188480n^4+1968000n^3+7213120n^2+11107200n+6572160\right)u\left(n+4\right)+\left(166896n^4+2962656n^3+18918576n^2+51195456n+49204800\right)u\left(n+5\right)+\left(38012n^4+956064n^3+8804404n^2+35087184n+50788512\right)u\left(n+6\right)+\left(1333n^4+53814n^3+747191n^2+4381134n+9313488\right)u\left(n+7\right)+\left(-256n^4-8064n^3-95216n^2-499464n-982080\right)u\left(n+8\right)\,u\left(0\right)=1\,u\left(1\right)=2\,u\left(2\right)=12\,u\left(3\right)=112\,u\left(4\right)=1232\,u\left(5\right)=14832\,u\left(6\right)=189184\,u\left(7\right)=2512064\right\}$

p_generator:=rectoproc(rec,u(n),list):

$\mathrm{p\_generator}\left(30\right)$

$\left[1\,2\,12\,112\,1232\,14832\,189184\,2512064\,34358784\,480745984\,6848734464\,99003237376\,1448575666176\,21411827808256\,319255531155456\,4796005997940736\,72520546008219648\,1102912584949792768\,16859182461720526848\,258886644574700699648\,3991711460817459806208\,61775021926688418365440\,959229931916911121530880\,14940323391360408046796800\,233352506098550016631111680\,3654109325605190169830359040\,57356169042767373344673103872\,902258550678887785413876908032\,14222150544386213214581667397632\,224605955315162319622246867402752\,3553395680818726488887774467325952\right]$