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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):

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

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