Inverted Pendulum Viewer

$$\begin{align*} &\text{Original: } \ddot{\theta} + \frac{g}{l} \sin (\theta) = \frac{ A \omega^2}{l} \left( \cos (\theta) \cos(\omega t) \right)\\ &\text{First-Order: } \ddot{\theta} + \frac{g}{l}(\pi - \theta) = -\frac{ A \omega^2}{l}\cos(\omega t) \\ &\text{Second-Order: } \ddot{\theta} + \frac{g}{l}\left((\pi - \theta) + \frac{(\theta - \pi)^3}{6}\right) = \frac{ A \omega^2}{l}\left(-1 + \frac{(x-\pi)^2}{2}\right)\cos(\omega t) \end{align*}$$