Solve this. ((x+y−a)/(x+y−b)) ∙ (dy/dx) = ((x+y+a)/(x+y+b))


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Question Number 204180 by naka3546 last updated on 08/Feb/24

$Solve this .$ $\frac{x + y - a}{x + y - b} \cdot \frac{d y}{d x} = \frac{x + y + a}{x + y + b}$
	Answered by mr W last updated on 08/Feb/24

	$u = x + y$ $\frac{d u}{d x} = 1 + \frac{d y}{d x}$ $\frac{u - a}{u - b} \times (\frac{d u}{d x} - 1) = \frac{u + a}{u + b}$ $\frac{d u}{d x} = \frac{u^{2} + (a - b) u - a b}{u^{2} - (a - b) u - a b} + 1$ $\frac{d u}{d x} = \frac{2 (u^{2} - a b)}{u^{2} - (a - b) u - a b}$ $\frac{u^{2} - (a - b) u - a b}{u^{2} - a b} d u = 2 d x$ $(1 - \frac{(a - b) u}{u^{2} - a b}) d u = 2 d x$ $u - \frac{(a - b) \ln (u^{2} - a b)}{2} = 2 x + C$ $2 u - (a - b) \ln (u^{2} - a b) = 4 x + C$ $\Rightarrow 2 (y - x) - (a - b) \ln [{(x + y)}^{2} - a b] = C$
	Commented by naka3546 last updated on 09/Feb/24

	$Thank you so much, sir .$
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