If the solution curve, of the differential equation d y d x = x + y - 2 x - y passing through the point ( 2 , 1 ) is tan - 1 y - 1 x - 1 - 1 β log e α + y - 1 x - 1 2 = log e x - 1 , then 5 β + α is equal to

Step-by-step solution

Given: d y d x = x + y - 2 x - y Now, finding the intersection point of x + y - 2 = 0 and x - y = 0 . ⇒ x + x - 2 = 0 , x = y ⇒ x = 1 , y = 1 ⇒ x , y ≡ 1 , 1 Let, x = X + 1 , y = Y + 1 ⇒ d Y d X = X + Y X - Y Putting, Y = t X ⇒ d Y d X = t + X d t d X ⇒ 1 + t 1 - t = t + X d t d X ⇒ 1 + t - t + t 2 1 - t = X d t d X ⇒ 1 X d X = 1 - t 1 + t 2 d t ⇒ ∫ 1 X d X = ∫ 1 - t 1 + t 2 d t ⇒ log X = tan - 1 t - 1 2 log 1 + t 2 + C ⇒ log x - 1 = tan - 1 y - 1 x - 1 - 1 2 log 1 + y - 1 x - 1 2 + C It is given that the solution