On an internal boundary layer problem

In this paper, we consider the boundary value problemεy″+a(x)y′+b(x)y=0, x∈[x−,x+], x−<0<x+,y(x−)=A, y(x+)=B,where A and B are two prescribed constants, and 0<ε⪡1 is a small positive parameter. As x→0, it is assumed that a(x)∼αx and b(x)∼β, where α>0 and β/α≠1,2,3,… . Under certain smoothness conditions on a(x) and b(x), an asymptotic solution is constructed, which holds uniformly for x∈[x−,x+]. This result is proved rigorously by using the method of successive approximation.