Operator factorization and the solution of second-order linear ordinary differential equations

Abstract

The theory and application of second-order linear ordinary differential equations is reviewed from the standpoint of the operator factorization approach to the solution of ordinary differential equations (ODE). Using the operator factorization approach, the general second-order linear ODE is solved, exactly, in quadratures and the resulting formulae used in the development of the standard theory of the general second-order linear ODE along with the derivation of further formulae for the exact solution of large classes of second-order linear ODE. Further, an iterative method is developed (involving the Green function) from the basic integral solution formulae which enables the procurement of solution in series of second-order linear ODE. The theory presented here, which depends implicitly on the solution of the Riccati equation, is restricted to initial value problems; the extension to boundary value (eigenvalue) problems is indicated in a conclusions and discussion section, along with the relevant hints as to how to extend the technique to higher-order linear ODE. The present discussion has obvious applications in the teaching of ODE, as well as being of more general interest to a wider audience of practitioners, who require a working knowledge of the present subject matter.