Question

If $x \frac{dy}{dx} =y \left(\log \,y - \log \,x\right)$ then the solution of the equation is

• A. $\log (\frac {x} {y}) =cy$
• B. $\log (\frac {y} {x}) =cx$
• C. $x \log (\frac {y} {x}) =cy$
• D. $y \log (\frac {x} {y}) =cx$

Answer 1

Answer: (B)

$\log (\frac {y} {x}) =cx$

Answer 2

$x \frac{dy}{dx} =y \left(\log \,y - \log \,x+1\right)$ $\Rightarrow \frac{dy}{dx} = \frac{y}{x}\left(log \frac{y}{x}+1\right)$. Put $\frac{y}{x} = z$ $\therefore \frac{dy}{dx}=x\frac{dz}{dx}+z$ $\therefore z+x \frac{dz}{dx}= z\left(log\,z+1\right)$ $\Rightarrow x \frac{dz}{dx} \,z\,log\,z$ $\Rightarrow \frac{dz}{z\,log\,z} = \frac{dy}{x}$ $\Rightarrow log \,\left(log \,z\right) = log \,x + log\, C$ $\Rightarrow log \,z = Cx$ $\Rightarrow log \left(\frac{y}{x}\right) = Cx$