Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation

We consider the Cauchy problem for the nonlinear differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varepsilon \frac{{du}} {{dx}} = f(x,u),u(0,\varepsilon ) = R_0 , $$\end{document} where ɛ > 0 is a small parameter, f(x, u) ∈ C∞ ([0, d] × ℝ), R0 > 0, and the following conditions are satisfied: f(x, u) = x − up + O(x2 + |xu| + |u|p+1) as x, u → 0, where p ∈ ℕ \ {1} f(x, 0) > 0 for x > 0; fu2(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ0+∞fu2(x, u) du = −∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].