Question

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

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

Answer 1

Answer: (A)

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