A) \[\log \left( \frac{y}{x} \right)=cx\]
B) \[\frac{y}{x}=\log y+c\]
C) \[y=\log y+1\]
D) \[y=xy+c\]
Correct Answer: A
Solution :
Given \[\frac{dy}{dx}=\frac{y}{x}\left( \log \frac{y}{x}+1 \right)\]. Put \[y=vx\]Þ \[\frac{dy}{dx}=v+x.\frac{dv}{dx}\] \[\therefore v+x.\frac{dv}{dx}=v(\log v+1)\] \[v+x\frac{dv}{dx}=v\log v+v\] Þ \[x\frac{dv}{dx}=v\log v\]Þ \[\frac{dv}{v\log v}=\frac{dx}{x}\] Integrating both sides, \[\int_{{}}^{{}}{\frac{dv}{v\log v}}=\int_{{}}^{{}}{\frac{dx}{x}}\] \[\log \log v=\log x+\log c\]\[\Rightarrow \]\[\log v=xc\]Þ\[\log (y/x)=\,x\,c\].
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