Multiple-Scale Analysis

We assume that

θ(x) = θ0(x) + θ1(x) + 2θ2(x) +

3θ3(x) + · · · .

But the initial conditions force θ = O(), so θ0 = 0. The equations for the other functions are

θ′′1 + θ1 = 0 θ(0) = 1, θ ′(0) = 0;

θ′′2 + θ2 = 0 θ(0) = 0, θ ′(0) = 0;

θ′′3 + θ3 = θ 3 1/6 θ(0) = 0, θ

′(0) = 0; · · · .