 Method of multiple scales

The method of multiple scales describes motion with widely separated time or length scales by treating them as independent variables during an asymptotic expansion. With fast time $t$ and slow time $T=ϵt$, write $u(t;ϵ)=U(t,T;ϵ)$ and useAt each order, a solvability condition in the method of multiple scales removes resonant forcing forcing that would generate a secular term. The resulting amplitude-phase equations describe accumulated changes over many fast periods.^[ref]

For $u^{\prime \prime }+2ϵu^{\prime }+u=0$, put $U_0=A(T)\cos t+B(T)\sin t$. The first correction obeysIts cosine and sine coefficients vanish precisely when $A_T=-A$ and $B_T=-B$. For $u(0)=1$, $u^{\prime }(0)=0$, the leading result is $e^{-ϵt}\cos t$ with bounded corrections. A fixed-time expansion instead contains $-ϵt\cos t$, which becomes order one when $t=O({ϵ}^{-1})$. This explains the slow variable's purpose; eliminating resonances is a construction step, while an error estimate is needed to establish uniform accuracy on a stated time interval.