The diffeqtohomdiffeq(deq, y(z)) command makes the differential equation, deq homogeneous.

If deq is not homogeneous, the diffeqtohomdiffeq function produces a differential equation of order increased by one which is homogeneous and cancels all the solutions of the original equation.

If deq is homogeneous, it is unchanged.

The rectohomrec(rec, u(n)) command makes the recurrence, rec, homogeneous.

If rec is not homogeneous, the rectohomrec function produces a recurrence of order increased by one which is homogeneous and cancels all the solutions of the original equation.

If rec is homogeneous, it is unchanged.

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

$\mathrm{deq}≔\mathrm{diff}\left(y\left(x\right)\,x\right)\left(x-1\right)+2y\left(x\right)-2x-3\:$

$\mathrm{diffeqtohomdiffeq}\left(\mathrm{deq}\,y\left(x\right)\right)$

$4y\left(x\right)+\left(-4x-11\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-2x^2-x+3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{diffeqtohomdiffeq}\left(\left\{\mathrm{deq}\,y\left(0\right)=2\right\}\,y\left(x\right)\right)$

$\left\{4y\left(x\right)+\left(-4x-11\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-2x^2-x+3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\,y\left(0\right)=2\,\mathrm{D}\left(y\right)\left(0\right)=1\right\}$

$\mathrm{rec}≔u\left(n+1\right)=u\left(n\right)+n^2+1\:$

$\mathrm{rectohomrec}\left(\mathrm{rec}\,u\left(n\right)\right)$

$\left(-n^2-2n-2\right)u\left(n\right)+\left(2n^2+2n+3\right)u\left(n+1\right)+\left(-n^2-1\right)u\left(n+2\right)$

$\mathrm{rectohomrec}\left(\left\{\mathrm{rec}\,u\left(0\right)=1\right\}\,u\left(n\right)\right)$

$\left\{\left(-n^2-2n-2\right)u\left(n\right)+\left(2n^2+2n+3\right)u\left(n+1\right)+\left(-n^2-1\right)u\left(n+2\right)\,u\left(0\right)=1\,u\left(1\right)=2\right\}$