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If $c$ is any arbitrary constant, then the general solution of the differential equation $y d x-x d y=x y d x$ is given by
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Verified Answer
The correct answer is:
$y=c x e^{-x}$
Given $y d x-x d y=x y d x$
$\Rightarrow \frac{y d x-x d y}{x y}=d x \Rightarrow d\left[\ln \left(\frac{x}{y}\right)\right]=d x$
Integrating both sides, we get $\ln \frac{x}{y}+\ln c=x$
$y e^x=c x$.
$\Rightarrow \frac{y d x-x d y}{x y}=d x \Rightarrow d\left[\ln \left(\frac{x}{y}\right)\right]=d x$
Integrating both sides, we get $\ln \frac{x}{y}+\ln c=x$
$y e^x=c x$.
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