The solution of the differential equation $(x + y) d y - (x - y) d x = 0$ is

y^{2} + 2 x y + x^{2} = k

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y^{2} + 2 x y - x^{2} = k

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y^{2} + 2 x y + x^{2} = 0

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y^{2} - 2 x y - x^{2} = k

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Solution

The correct option is B $y^{2} + 2 x y - x^{2} = k$
$(x + y) d y - (x - y) d x = 0 \Rightarrow \frac{d y}{d x} = \frac{x - y}{x + y}$
Substitute $y = x v \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$
$∴ x \frac{d v}{d x} + v = \frac{x - x v}{x + x v} = \frac{- v + 1}{v + 1}$
$\Rightarrow \frac{d v}{d x} = \frac{- v^{2} - 2 v + 1}{x (v + 1)} \Rightarrow \frac{\frac{d v}{d x} (v + 1)}{- v^{2} - 2 v + 1} = \frac{1}{x}$
$\Rightarrow \int \frac{\frac{d v}{d x} (v + 1)}{- v^{2} - 2 v + 1} d x = \int \frac{d x}{x}$
$\Rightarrow - \frac{1}{2} log (- v^{2} - 2 v + 1) = log x + c \Rightarrow y^{2} + 2 x y - x^{2} = k$

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