Let the solution curve of the differential equation x d y = x 2 + y 2 + y d x , x > 0 , intersect the line x = 1 at y = 0 and the line x = 2 at y = α . Then the value of α is

Step-by-step solution

Given, x d y = x 2 + y 2 + y d x ⇒ x d y - y d x = x 2 + y 2 d x ⇒ x d y - y d x x 2 = 1 + y 2 x 2 · d x x ⇒ d y x 1 + y x 2 = d x x Now integrating both side we get, ⇒ ∫ d y x 1 + y x 2 = ∫ d x x ⇒ ln y x + y x 2 + 1 = ln x + ln c ⇒ y + y 2 + x 2 x = c x ⇒ y + y 2 + x 2 = c x 2 Now given when x = 1 , y = 0 ⇒ 0 + 1 = c ⇒ c = 1 So equation of curve is y + x 2 + y 2 = x 2 Now at x = 2 , y = α So putting the value in curve we get, 2 + 4 + &#94