Construction of an asymptotic solution of a system of third-order linear differential equations with a small parameter

Abstract

where ~ > 0 is a small real parameter. Systems of firstand second-order linear differential equations with a small parameter have been treated by many authors (for a bibliography see [1, 2], for example). We study system (I) for the first time. We consider the characteristic equation det [Ao (t) ~E] = 0, ( 3 ) where E is the identity matrix. We will suppose that the roots ~1(t), ~2(t),...,Xn(t) of this equation are simple, i.e., ;~,(t) = Xj(t), V/E[O; L] , i f i--C:]; i, ] = 1,n, (4) a n d ~ j ( t ) :~O, v tE[O; L], 1 = 1, n.