solve xdy/dx=y(log y - log x - 1)

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{x}.\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}\left(\mathrm{logy}-\mathrm{logx}-1\right) \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}\left\{\log \left(\frac{\mathrm{y}}{\mathrm{x}}\right)-1\right\} \left[\mathrm{As} \mathrm{logm}-\mathrm{logn}=\log \left(\frac{\mathrm{m}}{\mathrm{n}}\right)\right] \\ \mathrm{Let} \\ \frac{\mathrm{y}}{\mathrm{x}}=\mathrm{t} \\ \Rightarrow \mathrm{y}=\mathrm{xt} \\ \mathrm{Differentiating} \mathrm{with} \mathrm{respect} \mathrm{to} \mathrm{x}, \mathrm{we} \mathrm{get}: \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x}.\frac{\mathrm{dt}}{\mathrm{dx}}+\mathrm{t}.\frac{\mathrm{dx}}{\mathrm{dx}} \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x}.\frac{\mathrm{dt}}{\mathrm{dx}}+\mathrm{t} \\ \Rightarrow \mathrm{x}.\frac{\mathrm{dt}}{\mathrm{dx}}+\mathrm{t}=\mathrm{t}\left(\mathrm{logt}-1\right) \\ \Rightarrow \mathrm{x}.\frac{\mathrm{dt}}{\mathrm{dx}}+\mathrm{t}=\mathrm{t}.\mathrm{logt}-\mathrm{t} \\ \Rightarrow \mathrm{x}.\frac{\mathrm{dt}}{\mathrm{dx}}=\mathrm{t}.\mathrm{logt}-2\mathrm{t} \\ \Rightarrow \mathrm{x}.\frac{\mathrm{dt}}{\mathrm{dx}}=\mathrm{t}.\left(\mathrm{logt}-2\right) \\ \Rightarrow \frac{\mathrm{dt}}{\mathrm{t}\left(\mathrm{logt}-2\right)}=\frac{\mathrm{dx}}{\mathrm{x}} \\ \Rightarrow \int \frac{\mathrm{dt}}{\mathrm{t}\left(\mathrm{logt}-2\right)}=\int \frac{\mathrm{dx}}{\mathrm{x}} ;\left(\mathrm{i}\right) \\ \mathrm{In} \int \frac{\mathrm{dt}}{\mathrm{t}\left(\mathrm{logt}-2\right)} \mathrm{if} \mathrm{we} \mathrm{put} \mathrm{logt}-2=\mathrm{u}, \mathrm{then} \\ \frac{\mathrm{dt}}{\mathrm{t}}=\mathrm{du} \\ \Rightarrow \int \frac{\mathrm{dt}}{\mathrm{t}\left(\mathrm{logt}-2\right)}=\int \frac{\mathrm{du}}{\mathrm{u}}=\log \left|\mathrm{u}\right|=\log \left|\mathrm{logt}-2\right| \\ \mathrm{Hence} \left(\mathrm{i}\right) \mathrm{becomes}: \\ \log \left|\mathrm{logt}-2\right|=\log \left|\mathrm{x}\right|+\log \left|\mathrm{C}\right|, \mathrm{where} \log \left|\mathrm{C}\right| \mathrm{is} \mathrm{integration} \mathrm{constant}. \\ \Rightarrow \log \left|\log \left(\frac{\mathrm{y}}{\mathrm{x}}\right)-2\right|=\ln \left|\mathrm{Cx}\right| \\ \Rightarrow \boxed{\left|\mathbf{log}\left(\frac{\mathbf{y}}{\mathbf{x}}\right)\mathbf{-}\mathbf{2}\right|\mathbf{=}\left|\mathbf{Cx}\right|} \end{gathered}$$

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