v(x) a known function, and a;, a', any numbers (real or complex) such that ainn O and a' l/amn is either nonreal or positive. This equation is a particular case of the general linear equation (see Wright [7]), where the coefficient au,^x+ac4 in (1.1) is replaced by a known function A,,(x), and also of the linear equation with asymptotically constant coefficients (see Wright [8], Cooke [1]), where A;,,(x) tends to a finite limit as xoo. Reference was made to (1.1) in Wright and Yates [10], which deals with a generalization of the Bessel function integral involved in Lemma 4 of this paper, and the equation was previously considered by Hoheisel [3], who was chiefly concerned with the existence and uniqueness of its solutions. Mirolyubov [4] has also discussed (1.1), taking x to be complex, but merely stating in general terms (without proof) that a solution may be expressed as the sum of two contour integrals (1). We consider here the asymptotic expansion for large positive x of the solution of (1.1) satisfying certain boundary conditions. In all our results only the leading term of the expansion is given, but further terms are of course calculable. As a preliminary to more general considerations, Cooke [1, Theorem 1.1] has established, for the values m = 1 =n, a result similar to Case (a) of Theorem 1 of the present paper. His method of proof is essentially that used here, although the details are worked out somewhat differently. However, the conditions he imposed exclude the possibility of an asymptotic expansion of the form of Cases (b) and (c) of Theorem 1(2). I should like to express my gratitude to Professor E. M. Wright for sug-