For linear differential equations of any order, with non-constant coefficients (in general). See here and here.

As shown in the example in the notes, multiple scales fails when the frequency of the fast oscillation depends on the slow scale.

Then, one has to instead use the WKB ansatz:

$y=e^{i\phi(x)/\epsilon}A(x;\epsilon)$

in the dispersive case, or

$y=e^{\phi(x)/\epsilon}A(x;\epsilon)$

in the dissipative case.

When substituting this in an equation, given in a certain form (a form in which all second order ODEs can be expressed, see first lectures by Bender), one gets a series of equations for the term of increasing order in $\epsilon$

• Eikonal equation
• Transport equation

Turning points

Use Matched asymptotic expansions. In the turning point itself, the leading order solution is an Airy function