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 epsilon du/dx = f(x,u), u(0, epsilon) = R-0, where epsilon > 0 is a small parameter, f (x, u) is an element of C-infinity ([0, d] x R), R-0 > 0, and the following conditions are satisfied: f (x, u) = x - u(p) + O(x(2) + vertical bar xu vertical bar + vertical bar u vertical bar(p+1)) as x, u -> 0, where p is an element of N\{1}; f(x, 0) > 0 for x > 0; f(u)(2) (x, u) < 0 for (x, u) is an element of [0, d] x (0, + infinity); integral(+infinity)(0) f(u)(2)(x, u) du = -infinity. 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].