Would the substitution $x=z-h$ and $y=w-k$ transform equation $\frac{dy}{dx}=f\left(\frac{ax+by+c}{dx+ey+f}\right)$ to a homogeneous one? I need help with this exercise, and I apologize for the English :)
If $ab \neq bd$ then, we can choose the appropriate constant $h$ and $k$ adequately so that the sustitutions $x=z-h$ and $y=w-k$ reduce the differential equation.
$$\frac{dy}{dx}=f\left(\frac{ax+by+c}{dx+ey+f}\right)$$
a homogeneous equation. How to proceed?

If $\frac{a}{d}=\frac{b}{e}$ then $u=ax+by$
If $\frac{a}{d}\ne\frac{b}{e}$ then $x\to x-\alpha$ and $y\to y-\beta$ where $\alpha$ and $\beta$ are solution of system equations is given by
$$\left\{ \begin{matrix}
   ax+by+c=0  \\
   dx+ey+f=0  \\
\end{matrix} \right.$$
By substitution we have
$$y'=f\left(\frac{aX+bY}{dX+eY}\right)$$
where
$X=x-\alpha$ and $Y=y-\beta$
