Let y = y x be the solution of the differential equation d y d x + 2 y 2 cos 4 x - cos 2 x = x e tan - 1 2 cot 2 x , 0 < x < π 2 with y π 4 = π 2 32 . If y π 3 = π 2 18 e - tan - 1 α , then the value of 3 α 2 is equal to ______.

Step-by-step solution

Given d y d x + 2 2 cos 4 x - cos 2 x y = x e tan - 1 2 cot 2 x This is a linear differential equation. Here I = ∫ d x 2 cos 4 x - cos 2 x = ∫ d x 1 2 + cos 2 2 x 2 + cos 2 x - cos 2 x = 2 2 ∫ sec 2 2 x d x 2 + tan 2 2 x Put tan 2 x = t I = tan - 1 tan 2 x 2 ∴ I F = e tan - 1 tan 2 x 2 = e cot - 1 2 cot 2 x So general solution will be y e cot - 1 2 cot 2 x = ∫ x e tan - 1 2 cot 2 x e cot - 1 2 cot 2 x d x y e cot - 1 2 cot 2 x = e π 2 x 2 2 + c y π 4 = π 2 32 ⇒ c =