Question

The 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

$x^y=c{y}^x$

Explanation for the correct option:

Find the solution of given differential equation

Given differential equation: $\frac{dy}{dx}=\frac{y(x\log y-y)}{x(y\log x-x)}$

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

Hence, option (A) is the correct answer.