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. ${\left(xy\right)}^x=c$
• D. None of these

Answer 1

The correct option is A $x^y=c{y}^x$

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