In the paper we shall concentrate on the the uniqueness property of the solution of a specific type of differential equation as obtained from the conclusion of Brück Conjecture by radically improving, extending and generalizing a result of Bhoosnurmath-Kulkarni-Prabhu. Some examples have been given in the paper to show that one condition in our main result is sharp and a number of examples have been exhibited to show that one condition used in the paper is essential.