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

homogeneousA_ode := diff(y(x),x)=f(y(x)/x);

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

where f(y(x)/x) is an arbitrary function. See Kamke's book, p. 19. This type of ODE can be solved in a general manner:

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

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

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

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

$\mathrm{dsolve}\left(\mathrm{homogeneousA\_ode}\right)$

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