The general form of the homogeneous equation of class G is given by the following:

homogeneousG_ode := diff(y(x),x) = y(x)/x*F(y(x)/x^alpha);

$\mathrm{homogeneousG\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\frac{y\left(x\right)F\left(\frac{y\left(x\right)}{x^{\mathrm{\alpha }}}\right)}{x}$

where F is an arbitrary functions of its argument. This type of ODE can be solved in a general manner by dsolve and the coefficients of the infinitesimal symmetry generator are also found by symgen.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\,\mathrm{symgen}\right)$

$\left[\mathrm{odeadvisor}\,\mathrm{symgen}\right]$

$\mathrm{odeadvisor}\left(\mathrm{homogeneousG\_ode}\right)$

$\left[\left[\mathrm{\_homogeneous}\,\mathrm{class\; G}\right]\right]$

$\mathrm{symgen}\left(\mathrm{homogeneousG\_ode}\right)$

$\left[\mathrm{\_\ξ}=x\,\mathrm{\_\η}=\mathrm{\alpha }y\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{homogeneousG\_ode}\right)$

$\mathrm{ans}≔y\left(x\right)=\mathrm{RootOf}\left(-\ln \left(x\right)+\mathrm{c\_\_1}+{\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{\mathrm{\_a}\left(-\mathrm{\alpha }+F\left(\mathrm{\_a}\right)\right)}\ⅆ\mathrm{\_a}\right)x^{\mathrm{\alpha }}$

$\mathrm{odetest}\left(\mathrm{ans}\,\mathrm{homogeneousG\_ode}\right)$

$0$