Asymptotics of solution of the Riccati equation

Uniform asymptotics is found for a solution of the initial value problem to the equation $ \varepsilon^2 u '= -u^2 + \varepsilon f (x) $, singularly depending on a small parameter $\varepsilon$. Equations of this type are already well studied, but this equation represents an unexplored case of the right-hand side behavior. By the method of asymptotics matching the three-scale asymptotic expansion for a solution is constructed and is justificated by the method of upper and lower solutions.