Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 79, pp. 1-18.

Title: Analytical solution of linear ordinary differential equations by
       differential transfer matrix method

Authors: Sina Khorasani (Georgia Inst. of Technology, Atlanta, GA, USA)
 Ali Adibi (Georgia Inst. of Technology, Atlanta, GA, USA)

Abstract: 
 We report a new analytical method for finding the exact solution
 of homogeneous linear ordinary differential equations with
 arbitrary order and variable coefficients. The method is based on
 the definition of jump transfer matrices and their extension into
 limiting differential form. The approach reduces the $n$th-order
 differential equation to a system of $n$ linear differential
 equations with unity order. The full analytical solution is then
 found by the perturbation technique. The important feature of the
 presented method is that it deals with the evolution of
 independent solutions, rather than its derivatives. We prove the
 validity of method by direct substitution of the solution in the
 original differential equation. We discuss the general properties
 of differential transfer matrices and present several analytical
 examples, showing the applicability of the method.
 
Submitted June 2, 2003. Published August 5, 2003.
Math Subject Classifications: 34A05, 34A25, 34A30.
Key Words: Ordinary differential equations;
           differential transfer matrix method; matrix exponential.