Mathematics - Differential Equation Question with Solution | TestHub

$$\begin{array}{rl}& \frac{xdy}{dx}=y(\log y-\log x+1)\\ & \frac{dy}{dx}=\frac{y}{x}\left(\log \left(\frac{y}{x}\right)+1\right)\end{array}$$ Put $y=vx$ $\frac{dy}{dx}=v+\frac{xdv}{dx}$ $\Rightarrow v+\frac{xdv}{dx}=v(\log v+1)$ $\frac{xdv}{dx}=v\log v$ $\Rightarrow \frac{dv}{v\log v}=\frac{dx}{x}$ put $\log v=z$ $\frac{1}{v}dv=dz$ $\Rightarrow \frac{dz}{z}=\frac{dx}{x}$ $\ln z=\ln x+\ln c$ $z=cx$ $\log v=cx$ $\log \left(\frac{y}{x}\right)=cx$