A differential equation of the form \(\frac{{dy}}{{dx}} = f(ax + by + c)\) can be reduced to variables separable form by substituting \(ax + by + c = z \Rightarrow a + b\frac{{dy}}{{dx}} = \frac{{dz}}{{dx}}\)

The equation, then becomes \(\frac{1}{b}\left( {\frac{{dz}}{{dx}} - a} \right) = f(z) \Rightarrow \frac{{dz}}{{dx}} = a + bf(z)\)

\( \Rightarrow \frac{{dz}}{{a + bf(z)}} = dx\)

Hence the variables are separated in terms of \(z\) and \(x\).