Method of multiple scales

Remarks

Example: $x'' + 2\varepsilon x' + x = 0$. Order $0$ gives $\partial_t^2 X_0 + X_0 = 0$, so $X_0 = A(T) \cos t + B(T) \sin t$. Order $1$ gives $\partial_t^2 X_1 + X_1 = -2 \partial_t \partial_T X_0 - 2 \partial_t X_0 = 2(A' + A) \sin t - 2(B' + B) \cos t$. Bounded $X_1$ requires $A' = -A$ and $B' = -B$, so $X_0 = e^{-T} (a \cos t + b \sin t)$. Every exact solution has the form $e^{-\varepsilon t} \bigl(c_1 \cos(\sqrt{1 - \varepsilon^2}\, t) + c_2 \sin(\sqrt{1 - \varepsilon^2}\, t)\bigr)$. Its frequency differs from $1$ by $O(\varepsilon^2)$, so the approximation has error $O(\varepsilon)$ for $0 \le t \le C / \varepsilon$.

Non-example: the regular expansion $x = x_0(t) + \varepsilon x_1(t)$ with one time gives $x_1$ containing $t \cos t$ and $t \sin t$ terms, so the error is not small when $t \ge 1/\varepsilon$.