If the solution curve of the differential equation ( y - 2 log e x ) d x + ( x log e x 2 ) d y = 0 , x > 1 passes through the points e , 4 3 and ( e 4 , α ) , then α is equal to _ _ _ _ _ _ _

Step-by-step solution

Given, ( y - 2 log e x ) d x + ( x log e x 2 ) d y = 0 ,   x > 1 ⇒ 2 x ln x d y d x + y = 2 ln x   where   log e x = ln x ⇒ d y d x + y 2 x ln x = 1 x Which is linear differential equation, So, I . F = e ∫ 1 2 x ln x = ln x Now, solution of the equation is given by, y · ln x = ∫ ln x x d x ⇒ y · ln x = 2 3 ln x 3 2 + C         . . . ( i ) Given, e q . ( i ) passes through point e ,   4 3 So, C = 2 3 Hence, solution of differential e