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I dont know the best way of solving this. I am able to solve it in the following manner.

y - x y' = x + y y'

(x + y) y' = y - x --- equation (1)

Let y = g x => y' = g + x g'

Substitute in equation (1),

x g + x² g' + x g² + x² g g' = g x - x

x g' (1 + g) = - (g² + 1)

- 1/x = (g + 1) g' / (g² + 1)

-1/ x = g g' / (g² + 1) + g' / (g² + 1)

Integrating we get,

Ln 1/x + Ln K = 1/2 Ln (1 + g²) + tan⁻¹ g

Substitute g = y/x, and simplify.

You may verify the solution, by differentiating this equation and obtaining the expression for y'.

y - x y' = x + y y'

(x + y) y' = y - x --- equation (1)

Let y = g x => y' = g + x g'

Substitute in equation (1),

x g + x² g' + x g² + x² g g' = g x - x

x g' (1 + g) = - (g² + 1)

- 1/x = (g + 1) g' / (g² + 1)

-1/ x = g g' / (g² + 1) + g' / (g² + 1)

Integrating we get,

Ln 1/x + Ln K = 1/2 Ln (1 + g²) + tan⁻¹ g

Substitute g = y/x, and simplify.

You may verify the solution, by differentiating this equation and obtaining the expression for y'.