Let y = y x be a solution curve of the differential equation, 1 - x 2 y 2 d x = y d x + x d y , If the line x = 1 intersects the curve y = y x at y = 2 and the line x = 2 intersects the curve y = y x at y = α , then a value of α is

Step-by-step solution

Given, 1 - x 2 y 2 d x = y d x + x d y ⇒ d x = y d x + x d y 1 - x y 2 ⇒ d x = d y x 1 - x y 2 ⇒ d x = 1 2 d x y 1 - x y + d x y 1 + x y Now integrating both side we get, ⇒ 2 x + c = ln 1 + x y 1 - x y ⇒ x y + 1 x y - 1 = e c e 2 x Now given, y 1 = 2 so putting the value in above equation we get, 3 = e c e 2 ⇒ e c = 3 e 2 Hence, the equation becomes x y + 1 x y - 1 = 3 e 2 x - 2 Now finding y 2 so putting x = 2 in above equation we get, 2 y + 1 2 y - 1 = 3 e 2 ⇒ 2 y + 1 = 2