On the asymptotic behavior of a fundamental set of solutions

Abstract

We consider nth order homogeneous linear ordinary differential equations whose coefficients have an asymptotic expansion as x → ∞ x \to \infty in terms of real powers of x and are analytic in sectors of the complex plane. In earlier work Bank (Funkcial. Ekvac. 11 (1968), 87-100) developed a method for reading off the asymptotic behavior of solutions directly from the equation, except in certain cases where roots asymptotically coalesce. For our results, we consider coefficients in a field of the type developed by Strodt (Trans. Amer. Math. Soc. 105 (1962), 229-250). By successive algebraic transforms, we show that an equation in the exceptional case can be reduced to the nonexceptional case and so the asymptotic behavior of the solutions can be read from the equation. This generalizes the classical results when ∞ \infty is a singular point and the coefficients are analytic in neighborhoods of ∞ \infty . The strength of our results is that the coefficients need not be defined in a full neighborhood of ∞ \infty , and that the asymptotic behavior can be read directly from the equation.