  Exact bound state solutions and corresponding normalized eigenfunctions of
the radial Schr\"odinger equation are studied for the pseudoharmonic and
Mie-type potentials by using the Laplace transform approach. The analytical
results are obtained and seen that they are the same with the ones obtained
before. The energy eigenvalues of the inverse square plus square potential and
three-dimensional harmonic oscillator are given as special cases. It is shown
the variation of the first six normalized wavefunctions of the above
potentials. It is also given numerical results for the bound states of two
diatomic molecular potentials, and compared the results with the ones obtained
in literature.
