The gfun[holexprtodiffeq](expr, y(z)) command produces a differential equation satisfied by the holonomic expression expr.

If expr is a holonomic expression, then the gfun[holexprtodiffeq] function returns a differential equation in y and z that is satisfied by expr. Initial conditions are returned when possible.

Not all holonomic functions are recognized by holexprtodiffeq. It currently accepts the following functions.

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

$\mathrm{holexprtodiffeq}\left(\mathrm{BesselJ}\left(2\,x\right)\,y\left(x\right)\right)$

$\left\{x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^2-4\right)y\left(x\right)\,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)=\frac{1}{4}\right\}$

$\mathrm{holexprtodiffeq}\left(\mathrm{arcsec}\left(\frac{1}{x}\right)+{\sin \left(x\right)}^2\,y\left(x\right)\right)$

$\left\{\left(16x^5-8x^3+52x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(16x^6-40x^4+44x^2-20\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(4x^5-2x^3+13x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+\left(4x^6-10x^4+11x^2-5\right)\left(\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)\right)\,y\left(0\right)=\frac{\mathrm{\pi }}{2}\,\mathrm{D}\left(y\right)\left(0\right)=\mathrm{-1}\,{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(0\right)=2\,{\mathrm{D}}^{\left(3\right)}\left(y\right)\left(0\right)=\mathrm{-1}\right\}$