Asymptotics of the solution of a nonlinear Cauchy problem

The solution uniform asymptotics of the initial value problem for equation $\varepsilon u '= x^2-u^2 + \varepsilon f (x)$ singularly depending on small parameter $ \varepsilon $ is considered. The equation contains the unexplored case of the right-hand side, though equations of this type are well studied. The three-scale solution asymptotic expansion is constructed by the matching method, justificated by the upper and lower solutions method.