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.

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关于退化方程具有复数根的ODE的奇摄动系统

我们考虑一类具有两个二阶ODE的系统的边值问题，该二阶ODE在第一和第二个方程的二阶导数处具有小参数的不同幂。当退化系统的两​​个方程之一具有双根时，在这些方程具有简单根的情况下，边值问题的边界层解的渐近行为在质量上不同于已知的渐近行为。 。特别是，边界层变量的标度和构造边界层级数的算法取决于未知函数的边界条件的类型。我们针对特定类型的边界条件构造并证明边界层解的渐近展开。这些扩展不同于其他边界条件的扩展。