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

homogeneousD_ode := diff(y(x),x)= y(x)/x+g(x)*f(y(x)/x);

$\mathrm{homogeneousD\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\frac{y\left(x\right)}{x}+g\left(x\right)f\left(\frac{y\left(x\right)}{x}\right)$

where f(y(x)/x) and g(x) are arbitrary functions of their arguments. See Differentialgleichungen, by E. Kamke, p. 20. 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.

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

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

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

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

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

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

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

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

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

$0$

$f≔u\mapsto u$

$f≔u\mapsto u$

$\mathrm{allvalues}\left(\mathrm{value}\left(\mathrm{ans}\right)\right)$

$y\left(x\right)={\ⅇ}^{\int \frac{g\left(x\right)}{x}\ⅆx+\mathrm{c\_\_1}}x$

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

$0$