Wave function have a key role in providing the info about a physical system in quantum mechanics. Hence, herein, we present the Whittaker-function approach as a very useful method for finding the exact solutions of the Schrödinger equation for some quantum systems. Initially, the properties of the Whittaker function and its development in hypergeometric function was done, and then the discussion on the termination condition in this function. By using the Whittaker function and for the interaction potentials energy (Kratzer, Pseudo harmonic and Morse), exact solutions of the Schrödinger equation and the determination of the vibrational energy levels along with the associated wave functions for CO and NO diatomic molecules are analytically and numerically obtained. Furthermore, the comparison between results obtained from the Whittaker function approach and existing literature was carried out which shows a considerable agreement. Finally, we believe that the results presented here will be very useful for finding satisfactory analytic solutions and can provide very vital tools to check and improve numerical solutions for more complex physical systems.