A) \[y\log \left( \frac{x}{y} \right)=cx\]
B) \[x\log \left( \frac{y}{x} \right)=cy\]
C) \[\log \left( \frac{y}{x} \right)=cx\]
D) \[\log \left( \frac{x}{y} \right)=cy\]
Correct Answer: C
Solution :
\[x\frac{dy}{dx}=y(\log y-\log x+1)\] \[\therefore \] \[\frac{dy}{dx}=\frac{y}{x}[\log (y/x)+1]\] Put, \[\frac{y}{x}=t\] \[\Rightarrow \] \[y=xt\] \[\Rightarrow \] \[\frac{dy}{dx}=t+x\frac{dt}{dx}\] \[\therefore \] \[t+x\frac{dt}{dx}=t(\log t+1)\] \[\Rightarrow \] \[x\frac{dt}{dx}=t\log t\] \[\Rightarrow \] \[\frac{dt}{t\log t}=\frac{dx}{x}\] \[\Rightarrow \] \[\log (\log t)=\log x+\log c\] \[\Rightarrow \] \[\log t=xc\] \[\Rightarrow \] \[\log (y/x)=cx\]
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