If y = y x , x ∈ 0 , π 2 be the solution curve of the differential equation sin 2 2 x d y d x + 8 sin 2 2 x + 2 sin 4 x y = 2 e - 4 x 2 sin 2 x + cos 2 x , with y π 4 = e - π , then y π 6 is equal to

Step-by-step solution

Given, sin 2 2 x d y d x + 8 sin 2 2 x + 2 sin 4 x y = 2 e - 4 x 2 sin 2 x + cos 2 x Now rewriting the given differential equation we get, ⇒ d y d x + 8 + 4 cot 2 x y = 2 e - 4 x sin 2 2 x 2 sin x + cos 2 x Which is a linear differential equation. Now calculating I F = e ∫ 8 + 4 cot 2 x dx = e 8 x + 2 ln sin 2 x = e 8 x · sin 2 2 x Now solution of differential equation is given by, y × I F = ∫ 2 e - 4 x 2 sin 2 x + cos 2 x sin 2 2 x × I F d x ⇒ y e 8 x · sin 2 2 x = &#874