Answer
Verify (See below)
Work Step by Step
\[y'=F(ax+by+c)\]
\[\frac{dy}{dx}=F(ax+by+c)\;\;\;\ldots (1)\]
Substitute $\:V=ax+by+c$ _____(2)
Differentiate (2) with respect to $x$
\[\frac{dV}{dx}=a+b\frac{dy}{dx}\]
From (1) and (2)
\[\frac{dV}{dx}=a+bF(V)\]
Separating variables
\[\frac{1}{bF(V)+a}dV=dx\]
Hence change of variables $x,y$ to $x,V$ where $V=ax+by+c$ reduces (1) to the separable form \[\frac{1}{bF(V)+a}dV=dx\]
