Consider the one dimensional harmonic oscillator with angular frequency \(\omega\) perturbed by the small non-linear potential \(\epsilon q^{4}\).

(a) Find the solution using the perturbation technique introduced in the text to first order in the small perturbation;

(b) Improve your solution from part (a) by implementing the technique outlined at the end of the perturbations section, writing a solution with angular frequency \(\Omega=\omega+\epsilon \omega_{1}\). That is, your solution now depends on \(s=\Omega t\) instead of \(s=\omega t\). Fix \(\omega_{1}\) so that you cancel a term in the solution that is not periodic.