A curve amongst the family of curves represented by the differential equation, x 2 - y 2 d x + 2 x y d y = 0 which passes through 1 , 1 , is

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x 2 d x - y 2 d x + 2 x y d y = 0 x 2 d x = y 2 d x - 2 x y d y ⇒ - d x = d y 2 x (exact differential equation) ⇒ - x = y 2 x + C ⇒ x 2 + y 2 + C x = 0 ∵ It passes through 1 , 1 . Hence, C = - 2 ⇒ x 2 + y 2 - 2 x = 0

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