The solutions of differential equation xy1=2xe−yx+y is

xdydx=2xe−yx+ydydx=2e−yx+yxV+xdvdx=2e−V+V∫evdv=2∫1xdxev=2log⁡x+2log⁡cey/x=2(log⁡xc)

The solutions of differential equation xy1=2xe−yx+y is

xdydx=2xe−yx+ydydx=2e−yx+yxV+xdvdx=2e−V+V∫evdv=2∫1xdxev=2log⁡x+2log⁡cey/x=2(log⁡xc)

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The solutions of differential equation $x y^{1} = 2 x e^{- \frac{y}{x}} + y$ is

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$e^{- \frac{y}{x}} = x + c$

$e^{\frac{y}{x}} = \log | c x |$

$e^{y / x} = 2 \log | c x |$

$e^{\frac{y}{x}} + \log | c x | = 0$

answer is D.

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Detailed Solution

$\begin{array}{l} x \frac{d y}{d x} = 2 x e^{- \frac{y}{x}} + y \\ \frac{d y}{d x} = 2 e^{- \frac{y}{x}} + \frac{y}{x} \\ V + \frac{x d v}{d x} = 2 e^{- V} + V \\ \int e^{v} d v = 2 \int \frac{1}{x} d x \\ e^{v} = 2 \log x + 2 \log c \\ e^{y / x} = 2 (\log x c) \end{array}$