On a boundary-layer problem

This is a continuation of our earlier article concerning the boundary-value problem {εy″ + a(x)y′ + b(x) = 0, x ∈ [x _-, x ₊], x _- <0 <x ₊, y(x _-) = A, y(x ₊) = B, where A, B are prescribed constants, and 0 ≤ ε≪ 1 is a small positive parameter. In that article, we assumed the coefficients a(x) and b(x) are sufficiently smooth functions with the behavior given by a(x) ∼ αx and b(x) ∼ β as x → 0, where α > 0 and β/α ≠ 1, 2, 3,.... In the present article, we are concerned with the case α <0 and β/α ≠ 0, -1, -2,.... An asymptotic solution is obtained for the problem, which holds uniformly for all x in [x _-, x ₊]. Our result is proved rigorously, and shows that a previous result in the literature is incorrect.