The general solution of the differential equation d y d x = 2 y tan ⁡ x + tan 2 ⁡ x , ∀ x ∈ 0 , π 2 is yf x = x 2 - sin ⁡ 2 x 4 + C , (where, C is an arbitrary constant). If f π 4 = 1 2 , then the value of f π 3 is equal to

Step-by-step solution

Given equation is d y d x + y - 2 tan ⁡ x = tan 2 ⁡ x IF = e 2 ln ⁡ cos ⁡ x = cos 2 ⁡ x Thus, the solution is y ⋅ cos 2 ⁡ x = ∫ tan 2 ⁡ x · cos 2 ⁡ x d x ⇒ y ⋅ cos 2 ⁡ x = ∫ 1 - cos ⁡ 2 x 2 d x = x 2 - sin ⁡ 2 x 4 + C ⇒ f x = cos 2 ⁡ x ⇒ f π 3 = 1 2 2 = 1 4