In this paper the generalized time-dependent quantum oscillator with a moving boundary is studied. Its Hamiltonian is of the non-homogeneous quadratic form of space-coordinate and momentum with time-dependent coefficients. We obtain an orthonormalized and complete set of rigorous evolving states of Exp-Sin type, as well as the necessary and sufficient condition for the existence of states of this type. Our results are of considerable generality, including as particular cases almost all the results given in literature. In addition, a misunderstanding of a few authors on the differential with respect to time is clarified and we point out that the differentiation with respect to either time or space-coordinate can be performed in the ordinary sense.