(Steps Shown) Show that the change of variables V=xy transforms the DE (dy)/(dx)=y/xF(xy) into the separable DE (1)/(V[ F(V)+1 ])(dV)/(dx)=1/x (b) Use the


Question: Show that the change of variables \(V=xy\) transforms the DE

\[\frac{dy}{dx}=\frac{y}{x}F\left( xy \right)\]

into the separable DE

\[\frac{1}{V\left[ F\left( V \right)+1 \right]}\frac{dV}{dx}=\frac{1}{x}\]

(b) Use the above result to solve

\[\frac{dy}{dx}=\frac{y}{x}\left[ \ln \left( yx \right)-1 \right]\]

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