Simple Harmonic Oscillator Equation

The universal differential equation x″ + ω²x = 0 describes any system with a linear restoring force. Its general solution is x(t) = A·cos(ωt + φ), with amplitude A and phase φ set by initial conditions.

Springs, small-angle pendulums, LC circuits, small-amplitude vibrations of any stable equilibrium. Whenever the restoring force is proportional to displacement, this equation governs the motion.

Breaks down when the restoring force is nonlinear — large-angle pendulums and non-Hookean springs are not true SHM. Adding damping changes the equation to x″ + γx′ + ω₀²x = 0.

Writing ω² = m/k instead of k/m. Forgetting that ω in the solution must match the ω in the differential equation. Assuming A and φ are always 1 and 0 — they depend on initial position and velocity.

• oscillators everywhere

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