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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
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$a x+b y=z$
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.
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.
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