The SeparateVariables(ODE, y(x)) returns an equivalent ODE in a form where the dependent and independent variables are separated.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\right)\:$

$\mathrm{ode1}≔t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\mathrm{diff}\left(z\left(t\right)\,t\right)=0$

$\mathrm{ode1}≔t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)=0$

$\mathrm{SeparateVariables}\left(\mathrm{ode1}\,z\left(t\right)\right)$

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

The Student[ODEs][SeparateVariables] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.