If y = y x is the solution curve of the differential equation d y d x + y tan x = x sec x , 0 ≤ x ≤ π 3 , y 0 = 1 , then y π 6 is equal to

Step-by-step solution

Given: d y d x + y tan x = x sec x This is a linear differential equation. I . F . = e ∫ tan x d x = sec x Then solution of differential equation is y sec x = ∫ x sec 2 x d x ⇒ y sec x = x tan x - ∫ tan x d x ⇒ y sec x = x tan x - ln sec x + C Given: y 0 = 1 ⇒ c = 1 ∴ y sec x = x tan x - ln sec x + 1 At x = π 6 , we get y sec π 6 = π 6 tan π 6 - ln sec π 6 + 1 ⇒ y 2 3 = π 6 1 3 - ln 2 3 + 1 ⇒ y = π 12 - 3 2 ln 2 3 + 3 2 ⇒