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Question Number 133508 by Ahmed1hamouda last updated on 22/Feb/21

Commented by Ahmed1hamouda last updated on 22/Feb/21

$S olve the differential equations$
	Answered by Olaf last updated on 22/Feb/21

	$y d x - 2 x d y = (y - 2) e^{y} d y$ $y \frac{d x}{d y} - 2 x = (y - 2) e^{y} (1)$ $Let x (y) = u (y) e^{y}$ $(1) : y [\frac{d u}{d y} e^{y} + {u e}^{y}] - 2 {u e}^{y} = (y - 2) e^{y}$ $y [\frac{d u}{d y} + u] - 2 u = y - 2$ $y \frac{d u}{d y} + u (y - 2) = y - 2$ $y \frac{d u}{d y} = (y - 2) (1 - u)$ $- \frac{d u}{1 - u} = - \frac{y - 2}{y}$ $\ln ∣ 1 - u ∣ = 2 \ln ∣ y ∣ - y + C_{1}$ $\ln ∣ 1 - u ∣ = \ln (C_{2} y^{2} e^{- y})$ $1 - u = C_{2} y^{2} e^{- y}$ $u = 1 - C_{2} y^{2} e^{- y}$ $x = {u e}^{y} = e^{y} - C_{2} y^{2}$
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