CLASSIFICATION OF THE SOLUTIONS OF A DIFFERENTIAL-EQUATION ACCORDING TO THEIR ASYMPTOTIC-BEHAVIOR

The solutions of the differential equation Lny + p(x)y = 0, where Lny = ρn(ρn-1… (ρ1(ρ0y)')'.)’ and p(x) is of one sign, are classified according to their behaviour as x→∞. The solution space is decomposed into disjoint, non-empty sets Sk, 0≦k≤n, such that (-1)n-kp(x)≤0. We study the growth properties and the density of the zeros of the solutions which belong to the different sets St, the structure of the sets and its connection with (k, n-k)-disfocality. © 1979, Royal Society of Edinburgh. All rights reserved.