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Q.

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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a

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

b

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

c

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

d

$y e^{x} = c x$

answer is D.

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Detailed Solution

Given $y d x - x d y = x y d x$

$\begin{aligned} \Rightarrow \frac{y d x - x d y}{x y} = d x \\ \Rightarrow d [\ln (\frac{x}{y})] = d x \end{aligned}$

Integrating both sides, we get $\ln \frac{x}{y} + \ln c = x$

$\Rightarrow y e^{x} = c x .$

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