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==Example: undamped Duffing equation==
[[File:Duffing-equation-perturbative-approaches.svg|thumb|431x431px|Here the differences between <math display="inline">\mathcal{O}(\varepsilon)</math> approaches for both regular perturbation theory and multiple-scale analysis can be seen, and how they compare to the exact solution for <math display="inline">\varepsilon = \frac{1}{4}</math>]]
===Differential equation and energy conservation===
As an example for the method of multiple-scale analysis, consider the undamped and unforced [[Duffing equation]]:<ref>This example is treated in: Bender & Orszag (1999) pp. 545–551.</ref>
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