Euler case for a general fourth-order differential equation

as x →∞, where x is the independent variable and the prime denotes d/dx. The functions pi(x) (0 ≤ i ≤ 2) and qi(x) (i = 1,2) are defined on an interval [a,∞), are not necessarily real-valued, and are all nowhere zero in this interval. Our aims are to identify relations between q0, q1, p0, p1, and p2 that represents an Euler case for (1.1) and to obtain the asymptotic forms of four linearly independent solutions under this case. Al-Hammadi [2] obtained an asymptotic formula of Liouville-Green type for (1.1) which extends those of Walker [9]. Also in [1], we consider (1.1) with p1 = q2 = 0 and we give a complete analysis for the case where