Liouville-Green asymptotics for almost-diagonal second-order matrix differential equations

A WKB (or Liouville-Green) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y''= (D(t) + G(t)) Y, on [a, + infinity), where D(t) is a nonsingular diagonal matrix. A basis for the right-module of its solutions can be represented explicitly, and precise computable bounds for the error terms involved are given. The double asymptotic nature with respect to both, t and some parameters that might affect the matrix coefficient, is shown. Examples and applications are given.