It's been a long time since I did any singular perturbations, but when I did, the text we used was Kevorkian and Cole; it definitely covers this type of problem. I think that Bender and Orszag's Multiple Scale and Singular Perturbation Methods (sorry, MO won't let me add another link) has a section on it also, but not in as much depth.

FWIW, I remember Bender and Orszag being a better book in general.

It's been a long time since I did any singular perturbations, but when I did, the text we used was Kevorkian and Cole; it definitely covers this type of problem. I think that Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers (sorry, MO won't let me add another link) has a section on it also, but not in as much depth.

FWIW, I remember Bender and Orszag being a better book in general.