The sinusoid and the phasor

TL;DR

This paper explores replacing the sinusoid in the Mathieu equation with a phasor, revealing that solutions transition from bounded or exponential growth to always unbounded linear growth, indicating a fundamental change in behavior.

Contribution

It introduces a novel variation of the Mathieu equation using a phasor, demonstrating a distinct solution behavior not observed in the traditional form.

Findings

Solutions grow linearly with time when using a phasor.

Traditional Mathieu solutions are either bounded or grow exponentially.

Replacing sinusoid with a phasor fundamentally alters solution dynamics.

Abstract

Mathieu equation is widely used to study several natural phenomenon. In this paper, we show that replacing the sinusoid in the Mathieu equation with a phasor can lead to solutions that behave in a totally different way. Solutions of Mathieu equation are either bounded or grow unboundedly at an exponential rate. Solutions of this new equation are always unbounded and grow linearly with time.