Abstract: A simplified method, Laplace transformation, is used to discuss the radial Schrodinger equation with the weakest bound electron potential model (WBEPM). Through using such method, the second-order differential equation is reduced to a first-order differential equation and the exact bound state solutions including energy spectrum and normalized wave functions are obtained by making use of the integral. The results agree with those obtained by Zheng. It is most important that the two kinds of new recursion relations of radial wave functions are derived by the same method. These new recursion relations are the relations between the effective principal and angular-momentum quantum numbers, and are comprehensive in application to the calculations of transition probabilities in atomic and molecular physics.