Let S = 0 , 2 π - π 2 , 3 π 4 , 3 π 2 , 7 π 4 . Let y = y x , x ∈ S , be the solution curve of the differential equation d y d x = 1 1 + sin 2 x , y π 4 = 1 2 . If the sum of abscissas of all the points of intersection of the curve y = y x with the curve y = 2 sin x is k π 12 , then k is equal to _____.

Step-by-step solution

Given differential equation d y d x = 1 1 + sin 2 x ∫ d y = ∫ d x sin x + cos x 2 ∫ d y = ∫ sec 2 x 1 + tan x 2 y = - 1 1 + tan x + C So y π 4 = 1 2 So 1 2 = - 1 2 + C ⇒ C = 1 So y = - 1 1 + tan x + 1 y = - 1 + 1 + tan x 1 + tan x y = tan x 1 + tan x Solving this equation with y = 2 sin x tan x 1 + tan x = 2 sin x i.e. sin x = 0 ,    1 2 = sin x + cos x x = π or 1 2 = sin x + π 4 ⇒ sin π 6 = sin x + π 4 x + π 4 = π - π 6 , 2 &#960