Uniform asymptotic expansions of solutions of an inhomogeneous equation

We treat a class of equations given by ε2u″(x)=u(x)(q(x,ε)−u(x)), u(−1)=α−, u(1)=α+, and obtain rigorous uniform asymptotic expansions of the solutions as ε→0. A key tool is a new formula of variation of constants that works for such quadratic equations. Included are solutions with one or more spikes. One example of this class of problems is a famous problem studied by Carrier and discussed formally by Bender and Orszag in the book “Advanced Mathematical Methods for Scientists and Engineers”. In this paper we give the first rigorous derivation for the well-known asymptotics for that problem. Another more applied example is also covered by our theory.