The gfun[`diffeq+diffeq`](eq1, eq2, y(z)) command determines the differential equation satisfied by the sum of two holonomic functions, eq1 and eq2.

If f and g are holonomic function solutions of eq1 and eq2, respectively, the gfun[`diffeq+diffeq`] function returns a linear differential equation verified by $f+g$.

The differential order of the returned equation is at most the sum of eq1 and eq2's differential orders for the gfun[`diffeq+diffeq`] function.

The gfun[`diffeq*diffeq`](eq1, eq2, y(z)) command determines the differential equation satisfied by the Cauchy product of two holonomic functions, eq1 and eq2.

If f and g are holonomic function solutions of eq1 and eq2), respectively, the gfun[`diffeq*diffeq`] function returns a linear differential equation verified by $fg$.

The differential order of the returned equation is at most the product of eq1 and eq2's differential orders for the gfun[`diffeq*diffeq`] function.

The gfun[hadamardproduct](eq1, eq2, y(z)) command determines the differential equation satisfied by the Hadamard product of two holonomic functions, eq1 and eq2.

If f and g are holonomic function solutions of eq1 and eq2, respectively, the gfun[hadamardproduct] function returns a linear differential equation verified by the Hadamard product of f and g.  The function whose coefficient of $z^n$ in the Taylor expansion around 0 is the product of the corresponding coefficients of f and g.

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

$\mathrm{eq1}≔\mathrm{D}\left(y\right)\left(x\right)-y\left(x\right)\:$

$\mathrm{eq2}≔\left(1+x\right){\mathrm{D}}^{\left(2\right)}\left(y\right)\left(x\right)+\mathrm{D}\left(y\right)\left(x\right)\:$

$\mathrm{`diffeq+diffeq`}\left(\mathrm{eq1}\,\mathrm{eq2}\,y\left(x\right)\right)$

$\left(-x-3\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-x^2-2x+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(x^2+3x+2\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)$

$\mathrm{`diffeq*diffeq`}\left(\mathrm{eq1}\,\mathrm{eq2}\,y\left(x\right)\right)$

$xy\left(x\right)+\left(-2x-1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(1+x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{hadamardproduct}\left(\mathrm{eq1}\,\mathrm{eq2}\,y\left(x\right)\right)$

$\left\{\left(1+x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)x-{\mathrm{\_t}}_2{\mathrm{\_C}}_1\,y\left(0\right)={\mathrm{\_t}}_2{\mathrm{\_C}}_0\right\}$