The most effective and widely used methods for integrating the Orr–Sommerfeld equation by shooting are the continuous orthogonalization method and the compound–matrix method. In this paper, we consider this problem from a differential–geometric point of view. A new definition of orthogonalization is presented: restriction of the Orr–Sommerfeld to a complex Stiefel manifold; and this definition leads to a new formulation of continuous orthogonalization, which differs in a precise and interesting geometric way from existing orthogonalization routines. Present orthogonalization methods based on Davey's algorithm are shown to have a different differential–geometric interpretation: restriction of the Orr–Sommerfeld equation to a complex Grassmanian manifold. This leads us to introduce the concept of a Grassmanian integrator, which preserves linear independence and not necessarily orthogonality. Using properties of Grassmanian manifolds and their tangent spaces, a new Grassmanian integrator is introduced that generalizes Davey's algorithm. Furthermore, it is shown that the compound–matrix method is a dual Grassmanian integrator: it uses Plucker coordinates for integrating on a Grassmanian manifold, and this characterization suggests a new algorithm for constructing the compound matrices. Extension of the differential–geometric framework to general systems of linear ordinary differential equations is also discussed.