Let y = y ( x ) be the solution of the differential equation x + 2 e y + 1 x + 2 + y + 1 d x = x + 2 d y , y 1 = 1 . If the domain of y = y x is an open interval ( α , β ) , then | α + β | is equal to ___________.

Step-by-step solution

The given differential equation is x + 2 e y + 1 x + 2 + y + 1 d x = x + 2 d y ,   y 1 = 1 Let, y + 1 = Y ⇒ d y = d Y and x + 2 = X ⇒ d x = d X and at y = 1 ,   Y = 2 and at x = 1 ,   X = 3 . Thus, the differential equation becomes X e Y X + Y d X = X d Y ⇒ X d Y - Y d X = X e Y X d X ⇒ X d Y - Y d X X 2 = X e Y X X 2 d X ⇒ d d X Y X = X e Y X X 2 d X ⇒ d Y X e - Y X = d X X Integrating both sides w.r.to X , we get ∫ d Y X e - Y X = ∫ d X X ⇒ - e