Let y = y x be a solution of the differential equation ( x cos x ) d y + ( x y sin x + y cos x - 1 ) d x = 0 , 0 < x < π 2 . If π 3 y π 3 = 3 , then π 6 y " π 6 + 2 y ' π 6 is equal to _ _ _ _ _ _ _ .

Step-by-step solution

Given, x   cos x d y + x y   sin x + y   cos x - 1 d x = 0 ⇒ x   cos x d y d x + y x   sin x + y   cos x = 1 ⇒ d y d x + y x   sin x + cos x x   cos x = 1 x   cos x Which is a linear differential equation of form d y d x + P x y = Q x Now we know that integrating factor will be e ∫ P d x ⇒ I F = e ∫ tan x + 1 x d x = e ln sec x + ln x = x · sec x Hence, solution of linear differential equation is given by, ⇒ y · x   sec