If $\frac{a}{a_1}=\frac{b}{b_1}$, then the substitution to be used to solve the differential equation…
If $\frac{a}{a_1}=\frac{b}{b_1}$, then the substitution to be used to solve the differential equation $\frac{d y}{d x}=\frac{a x+b y+c}{a_1 x+b_1 y+c_1}$ by using separation of variables is
$x=x+h, y=y+k$
$a x+b y=z$
$y=V(x) \cdot x$
$x=a t, y=b t$
Solution
If however $\frac{a}{a_1}=\frac{b}{b_1}=m$ (say), then the differential equation becomes of the form $\frac{d y}{d x}=\frac{m\left(a_1 x+b_1 y\right)+C}{a_1 x+b_1 y+C_1}$.
To solve such a differential equation put $u=a_1 x+b_1 y$, get rid of $y$ and then the transformed equation will be such that the variables are separable.