Question

Solution of $\frac{dy}{dx}=\frac{y(x\log y-y)}{x(y\log x-x)}$ is

• A. $x^y=c{y}^x$
• B. $xy=c$
• C. $(xy{)}^x=c$
• D. None of these

Answer 1

Answer: (A)

$\frac{dy}{dx}=\frac{y(x\log y-y)}{x(y\log x-x)}$
$\Rightarrow x(y\log x-x)\frac{dy}{dx}=y(x\log y-y)$
$\Rightarrow \left(\log x-\frac{x}{y}\right)\frac{dy}{dx}=\log y-\frac{y}{x}$
$\Rightarrow \frac{y}{x}+\log x\cdot \frac{dy}{dx}=\log y+\frac{x}{y}\frac{dy}{dx}$
$\Rightarrow \frac{d}{dx}(y\log x)=\frac{d}{dx}(x\log y)$
$\Rightarrow y\log y=x\log y+\log c$
$\Rightarrow \log x^y=\log y^x+\log c$
$\Rightarrow x^y=c{y}^x$