Let y = y x be the solution of the differential equation x d y = y + x 3 cos x d x with y π = 0 , then y π 2 is equal to:

Step-by-step solution

We have, x d y = y + x 3 cos x d x ⇒ x d y = y d x + x 3 cos x d x ⇒ x d y - y d x x 2 = x 3 cos x d x x 2 ⇒ d d x y x = ∫ x cos x d x ⇒ y x = x sin x - ∫ 1 . sin x d x Therefore, y x = x sin x + cos x + C At x = π ,   y = 0 , 0 = - 1 + C ⇒ C = 1 , x = π , y = 0 So, y x = x sin x + cos x + 1 ⇒ y = x 2 sin x + x cos x + x Hence, y π 2 = π 2 4 + π 2 .