The time-dependent Schrodinger equation governing the time-dependent weakly coupling Bosonic gas is exactly solved by use of the Lewis-Riesenfeld invariant theory in the present paper. This enables us to treat the time evolution of the Bosonic gas. Since the Bogoliubov transformation formulation in the time-dependent case may yield a phase factor that cannot be determined by the theory itself, we calculate the transformation coefficients between the Bosonic occupation representation and the quasi-particle representation and, based on this, obtain the complete set of vector basis of the quasi-particle representation for the time-dependent weakly coupling Bosonic gas.