Let y = y x be the solution of the differential equation sin ⁡ x d y d x + y cos x ⁡ = 4 x , x ∈ 0 , π . If y π 2 = 0 , then y π 6 is equal to

Step-by-step solution

d y d x + y cot   x = 4 x sin x ,   x ∈ 0 ,   π I.F. = e ∫ cot x = e ln ⁡ sin x = sin x ∴ y   sin x = ∫ 4 x sin x . sin x   d x y sin x = 2 x 2 + c ∵ y π 2 = 0 ∴ 0 × 1 = 2 × π 2 2 + c ⇒ c = - π 2 2 y π 6 = 2 × π 6 2 - π 2 2 1 2 = π 2 18 - π 2 2 × 2 = π 2 9 - π 2 = - 8 π 2 9