We perform an analytic investigation of the stability of line-driven disk winds, independent of hydrodynamic simulations. Our motive is to determine whether or not line-driven disk winds can account for the wide/broad UV resonance absorption lines seen in cataclysmic variables (CVs) and quasi-stellar objects (QSOs). In both CVs and QSOs observations generally indicate that the absorption arising in the outflowing winds has a steady velocity structure on time scales exceeding years (for CVs) and decades (for QSOs). However, published results from hydrodynamic simulations of line-driven disk winds are mixed, with some researchers claiming that the models are inherently unsteady, while other models produce steady winds. The analytic investigation presented here shows that if the accretion disk is steady, then the line-driven disk wind emanating from it can also be steady. In particular, we show that a gravitational force initially increasing along the wind streamline, which is characteristic of disk winds, does not imply an unsteady wind. The steady nature of line-driven disk winds is consistent with the 1D streamline disk-wind models of Murray and collaborators and the 2.5D time-dependent models of Pereyra and collaborators. This paper emphasizes the underlying physics behind the steady nature of line-driven disk winds using mathematically simple models that mimic the disk environment.