Solve `x((dy)/(dx))=y(logy-logx+1)`

Explanation

$$
\begin{aligned}
&\mathrm{x} \frac{d y}{d x}=y(\log y-\log x+1) \\
&\Rightarrow \frac{d y}{d x}=\frac{y}{x}\left(\log \frac{y}{x}+1\right) \\
&\text { Put } y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x} \\
&\Rightarrow v+x \frac{d v}{d x}=v(\log v+1) \\
&\Rightarrow \frac{d v}{v \log v}=\frac{d x}{x} \\
&\text { On integrating both sides, we get } \\
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