Analyzing Sound Wave Equations in a Synthesizer App

You are helping a friend develop a simple synthesizer app that generates different types of sound waves. The app models audio signals using second-order differential equations, where the solution [latex]y(t)[/latex] represents the amplitude of the sound wave at time [latex]t[/latex]. Different equation settings produce different audio behaviors: some create sustained tones, others create sounds that fade out, and some produce unusual effects. Your task is to analyze several differential equations to determine what type of sound each setting will produce. You will classify equations, find characteristic equations, solve for general solutions, and use initial conditions to find specific waveforms. Understanding these mathematical relationships will help your friend’s app generate the desired audio output.