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.
1.Introduction
The factorization method for solving second-order linear ODE (ordinary differential equations) has always been reasonably well known. For example, for ODE with constant coefficients, we can refer to Boas [1], Margenau and Murphy [2] (both quoted by Powles [3]) Jeffrey [4], Hildebrand [5] and Ayres [6]. However, in the 'root papers' (for example, Polya [7], Mammana [8], Schro¨dinger [9] or Infeld and Hull [10]) the factorization method is developed to deal with the general problem of second-order linear ODE with variable coefficients. It is the aim of this paper to review and reappraise the factorization method for second-order linear ODE and show that this factorization method provides an effective unifying theme for the development of solution methods in the practice of second-order linear ODE. However, we do not discuss the problem of eigenvalue equations or general