Consider
$$
\ddot{u}+\omega_{0}^{2} u+\epsilon u^{4}=2 K \cos \Omega t
$$
Show that to first-order resonances exist when \(\Omega \approx 4 \omega_{0}, 2
\omega_{0}, \frac{1}{4} \omega_{0}, \frac{1}{7} \omega_{0}, 0\), \(\frac{2}{3}
\omega_{0}\), and \(\frac{3}{2} \omega_{0}\). Use the methods of multiple scales
and averaging to determine the equations describing the amplitude and phase
for each case.
