If a curve y = f x passes through the point ( 1 , - 1 ) and satisfies the differential equation, y 1 + x y d x = x d y , then f - 1 2 is equal to

Step-by-step solution

Given differential equation, y d x + x y 2 d x = x d y ⇒ x d y - y d x y 2 = x d x ⇒ - d x y = d x 2 2 On integrating we get, ⇒- x y = x 2 2 + C ∵ It passes through ( 1 , - 1 ) . ∴ 1 = 1 2 + C   ⇒     C = 1 2 ∴ x 2 + 1 + 2 x y = 0     ⇒     y = - 2 x x 2 + 1 ∴ f - 1 2 = 4 5