On singularly perturbed systems of ODE with a multiple root of the degenerate equation

We consider a boundary-value problem for a system of two second-order ODE with distinct powers of a small parameter at the second derivative in the first and second equations. When one of the two equations of the degenerate system has a double root, the asymptotic behaviour of the boundary-layer solution of the boundary-value problem turns out to be qualitatively different from the known asymptotic behaviour in the case when those equations have simple roots. In particular, the scales of the boundary-layer variables and the very algorithm for constructing the boundary-layer series depend on the type of the boundary conditions for the unknown functions. We construct and justify asymptotic expansions of the boundary-layer solution for boundary conditions of a particular type. These expansions differ from those for other boundary conditions.