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

homogeneousB_ode := F(diff(y(x),x), y(x)/x);

$\mathrm{homogeneousB\_ode}≔F\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\,\frac{y\left(x\right)}{x}\right)$

where F is an arbitrary functions of its arguments. See Differentialgleichungen, by E. Kamke, p. 19. 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.

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

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

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

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

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

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

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

$0$