Answer
Answer is written below.
Work Step by Step
\[\frac{dy}{dx}=\frac{y}{x}F(xy)\;\;\;\ldots (1)\]
Substitute $\: V=xy\;\;\;\ldots (2)$
Differentiate (2) with respect to $x$
\[\frac{dV}{dx}=y+x\frac{dy}{dx}\]
From (1)
\[\frac{dV}{dx}=y+x\left[\frac{y}{x}F(xy)\right]\]
\[\frac{dV}{dx}=y[1+F(xy)]\]
From (2)
\[\frac{dV}{dx}=\frac{V}{x}[1+F(V)]\]
\[\frac{1}{V[F(V)+1]}\frac{dV}{dx}=\frac{1}{x}\]
Hence change of variables $V=xy$ transforms the differential equation (1) into separable equation
\[\frac{1}{V[F(V)+1]}\frac{dV}{dx}=\frac{1}{x}\]
