Many different physics techniques for approximately solving differential equations seem to follow the same basic pattern. One starts with some differential equation $Df(x) = s(x)$ (or $s(x) f(x)$), where $D$ is some differential operator (not necessarily linear), $s$ is a known source function of one or several variables, and $f$ is an unknown function to be found in the same number of variables as $s$. One then solves the special case where $s(x) \equiv s$ is constant for the particular solution $g(s,x)$ such that $D\_x g(s,x) \equiv s$ (or $s\, g(s,x)$). Then one assumes that, as long as $s(x)$ is "slowly varying", we can just locally use that constant-$s$ solution to get an approximate solution for varying $s$: $f(x) \approx g(s(x), x)$.

The textbook case of this technique is the WKB approximation in nonrelativistic 1D quantum mechanics, but I believe that similar ideas are used in the slowly-varying-envelope approximation in wave mechanics, quasi-FLRW spacetimes in general relativity with slowly varying matter distributions, the Born series, etc.

Is there a general math terminology/theory for this trick? I've seen variations on this idea used many time in different physics contexts, but I'm not sure if I've seen a more abstract or general unifying mathematical discussion of these special instances.