An intermediate boundary layer in singularly perturbed first-order equations

The Cauchy problem for a first-order ordinary differential equation with a small parameter at the derivative and a singular initial point is studied. A sufficient condition is found under which an intermediate boundary layer appears in a singularly perturbed problem described by first-order ordinary differential equations. A complete asymptotic expansion of the solution in the form of an asymptotic series in the sense of Erdélyi is constructed using a modified method of boundary functions. The obtained decomposition is justified; i.e. an estimate for the remainder term is obtained.