= Solution
{c}
Write $A=a e^{i\phi}$. The modulation equation gives $a_T=0$ and $\phi_T=-5a^2/3$. Matching the leading initial conditions gives $a=1$, $\phi(0)=0$. Therefore
$$
\boxed{U_0=2\cos(T_0-5T/3),\qquad
u(t)\sim2\cos\bigl[(1-5\epsilon^2/3)t\bigr].}
$$
The leading <amplitude> remains two and has no slow growth or decay. The <frequency> changes from one to $1-5\epsilon^2/3+O(\epsilon^3)$; expanding this shifted <cosine> recovers the second-order <secular term> from part (i).
There are bounded first-order waveform changes as well. If they are retained while matching $u(0)=2$, one may write $2\cos\tau+\epsilon[2-(4/3)\cos\tau-(2/3)\cos2\tau]+\cdots$, where $\tau=(1-5\epsilon^2/3)t$. Thus the first harmonic's coefficient is $2-4\epsilon/3$, with a mean shift and a second harmonic. These bounded shape corrections are distinct from the constant leading <amplitude> in the zeroth-order multiple-scales solution.
Back to article page
