Let x = x y be the solution of the differential equation 2 y + 2 log e y + 2 d x + x + 4 - 2 log e y + 2 d y = 0 , y > - 1 with x e 4 - 2 = 1 . Then x e 9 - 2 is equal to

Step-by-step solution

Given, 2 y + 2 log e y + 2 d x + x + 4 - 2 log e y + 2 d y = 0 Let x + 4 = u ,   y + 2 = v d x = d u ,   d y = d v So, the equation becomes, 2 v   ln v du = - u - 2   ln v d v ⇒ 2 v   ln v d u d v + u = 2   ln v ⇒ d u d v + 1 2 v   ln v .   u = 1 v Which is a linear differential equation, So, I F = e 1 2 ∫ 1 v   ln v = e 1 2 ln ln v = ln v 1 2 Now solution of differential equation is given by, u · ln v 1 2 = ∫ 1 v · ln v 1 2 d v ⇒