Let y = y ( x ) be the solution of the differential equation x d y - y d x = x 2 - y 2 d x , x ≥ 1 , with y ( 1 ) = 0 . If the area bounded by the line x = 1 , x = e π , y = 0 and y = y ( x ) is α e 2 π + β , then the value of 10 ( α + β ) is equal to ___ .

Step-by-step solution

Given x d y - y d x = x 2 - y 2 d x ⇒ x d y - y d x x 2 = 1 x 1 - y 2 x 2 d x ⇒ ∫ d y x 1 - y x 2 = ∫ d x x ⇒ sin - 1 y x = l n x + c at x = 1 , y = 0 ⇒ c = 0 y = x sin ( l n x ) A = ∫ 1 e π x sin l n x d x x = e t , d x = e t d t ⇒ ∫ 0 π e 2 t sin t d t = A α e 2 π + β = e 2 t 5 ( 2 sin t - cos t ) 0 π = 1 + e 2 π 5 α = 1 5 ,   β = 1 5 So, 10 ( α + β ) = 4