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

homogeneousC_ode := diff(y(x),x)=F((a*x+b*y(x)+c)/(r*x+s*y(x)+t));

$\mathrm{homogeneousC\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=F\left(\frac{ax+by\left(x\right)+c}{rx+sy\left(x\right)+t}\right)$

where F is an arbitrary function of its argument. 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.

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

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

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

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

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

$\left[\mathrm{\_\ξ}=\frac{asx-brx-bt+sc}{as-br}\,\mathrm{\_\η}=\frac{asy-bry+at-cr}{as-br}\right]$

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

$\mathrm{ans}≔y\left(x\right)=\frac{at-cr+\mathrm{RootOf}\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{F\left(\frac{\mathrm{\_a}b-a}{\mathrm{\_a}s-r}\right)+\mathrm{\_a}}\ⅆ\mathrm{\_a}+\ln \left(x\left(as-br\right)-bt+sc\right)+\mathrm{c\_\_1}\right)\left(x\left(as-br\right)-bt+sc\right)}{-as+br}$

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

$0$