The gfun[algebraicsubs](deq, eq, y(z)) command returns a differential equation satisfied by the composition $f\circ g$ where f is the holonomic function defined by the equation deq and g is the algebraic equation defined by eq.  The composition is holonomic by closure properties of holonomic functions.

Let d1 be the differential order of deq, and d2 be the degree of eq. If the equation deq is homogeneous, then the order of $f\circ g$ is at most $\mathrm{d1}\mathrm{d2}$.  Otherwise, it is at most $\left(\mathrm{d1}+1\right)\mathrm{d2}$.

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

Initial conditions can be present in the differential equation if a set is specified, in the same way as initial conditions are specified for dsolve. In the case of a polynomial equation, they are specified in the optional parameter ini as a set, using the same syntax as for dsolve.

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

$\mathrm{deq}≔{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(t\right)+f\left(t\right)\:$

$\mathrm{eq}≔\mathrm{algfuntoalgeq}\left(\mathrm{sqrt}\left(1-4t\right)\,f\left(t\right)\right)\:$

$\mathrm{algebraicsubs}\left(\mathrm{deq}\,\mathrm{eq}\,f\left(t\right)\right)$

$-4f\left(t\right)+2\frac{\ⅆ}{\ⅆt}f\left(t\right)+\left(-1+4t\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}f\left(t\right)\right)$

$\mathrm{algebraicsubs}\left(\left\{{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(x\right)+y\left(x\right)\,y\left(0\right)=1\,\mathrm{D}\left(y\right)\left(0\right)=0\right\}\,2x^4-y^2+2yx-x^2\,y\left(x\right)\,\left\{y\left(0\right)=0\,\mathrm{D}\left(y\right)\left(0\right)=1\,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)=2\mathrm{sqrt}\left(2\right)\right\}\right)$

$\left\{\left(-2048x^8+768x^6+864x^4-236x^2+87\right)y\left(x\right)+\left(320x^3+280x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-512x^6-136x^2+18\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(128x^3-8x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+\left(-32x^4+4x^2+3\right)\left(\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)\right)\,y\left(0\right)=1\,\mathrm{D}\left(y\right)\left(0\right)=0\,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)=\mathrm{-1}\,{\mathrm{D}}^{\left(3\right)}\left(y\right)\left(0\right)=-6\sqrt{2}\right\}$