It sounds like you are looking for a specific textbook or PDF related to Applied Asymptotic Analysis by Peter D. Miller (often referred to as the "Miller" text in applied math circles).
: The methods detailed in the book are essential for fields such as quantum mechanics applied asymptotic analysis miller pdf
Verdict: Miller is the best choice if you need to derive asymptotics for integrals in the complex plane or need a rigorous justification for a research paper. Bender & Orszag is better for a first, playful exposure. It sounds like you are looking for a
For ( y'' + (1 + \epsilon x) y = 0 ), substitute ( y = y_0 + \epsilon y_1 + \dots ) → secular terms appear. Read actively : As you read, take notes