Asymptotic Integration of Linear Ordinary Differential Equations of Order N

This paper introduces the notion of scalar dichotomies, an analog of the Levinson dichotomic conditions for diagonal systems adapted to differential equations of order n. Using this concept, a general theory of asymptotic integration of the nonautonomous equation x(n) + (a1(t) + b1(t)) x(n-1) + ... + (an(t) + bn(t)) x = 0 is given, provided the solutions of the equation x(n) + a1(t)x(n-1) + ... + an(t)x = 0, up to the p-derivative, 0 p n - 1, are known. We study in details the constant coefficient case ai(t) = constant, where we obtain an extension of the Ghizzetti theorem. All results are obtained without using canonical forms.