If $x \frac{d y}{d x}=y(\log y-\log x+1)$ then the solution of the equation is :
Answer & explanation
Correct answer: option 4
$\frac{d y}{d x}=\frac{y}{x}\left(\log \frac{y}{x}+1\right)$
Put $y=vx \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$
∴ $v+x \frac{d v}{d x}=v(\log v+1)$
$\Rightarrow \frac{d v}{v \log v}=\frac{d x}{x}$
$\Rightarrow \log (\log v)=\log x+\log c=\log c x$
$\Rightarrow \log v=cx \Rightarrow \log \left(\frac{y}{x}\right)=c x$
Hence (4) is the correct answer.