The compound matrix method for ordinary differential systems

Abstract A generalization of the compound matrix method is presented to deal with eigenvalue and boundary-value problems involving unstable systems of ordinary differential equations. Details are given for fourth- and sixth-order problems. Next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints, and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed. Using the Orr-Sommerfeld problem as an example, it is also shown how the techniques used in the derivation of certain auxiliary systems, which play a crucial role in the generalization of the compound matrix method, can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions. Finally, there is a brief discussion of the relationship between the compound matrix method anf the Riccati method.