# Consider the one dimensional harmonic oscillator with angular frequency (omega) perturbed by the small non-linear potential (epsilon

## Question:

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.

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