this equation does not have elementary closed-form solutions, but can be explored in a variety of ways. it is closely approximated by the mathieu equation, for instance, when the amplitude of oscillations are small. analyses show that the pendulum stays upright for fast oscillations. the first plot shows that when formula_78 is a slow oscillation, the pendulum quickly falls over when disturbed from the upright position. the angle formula_5 exceeds 90° after a short time, which means the pendulum has fallen on the ground. if formula_78 is a fast oscillation the pendulum can be kept stable around the vertical position. the second plot shows that when disturbed from the vertical position, the pendulum now starts an oscillation around the vertical position (formula_81). the deviation from the vertical position stays small, and the pendulum doesn't fall over.