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VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 2 Theory

Second-Order Linear ODEs (Harmonic Oscillators, Damping & Resonance)

Underdamped, critically damped, and overdamped dynamical responses

Adjust the damping ratio ζ and natural frequency ω₀ in y'' + 2ζω₀y' + ω₀²y = 0 to see the oscillation envelope ±e^(-ζω₀t) control decay.

1. Characteristic Roots & The Damping Envelope

Substituting y = e^(rt) into ay'' + by' + cy = 0 yields the characteristic polynomial ar² + br + c = 0. Complex conjugate roots r = -α ± iω_d produce oscillations trapped inside the exponential envelope ±A e^(-αt).

r2+2ζω0r+ω02=0  ⟹  r=−ζω0±ω0ζ2−1r^2 + 2\zeta\omega_0 r + \omega_0^2 = 0 \implies r = -\zeta\omega_0 \pm \omega_0\sqrt{\zeta^2 - 1}

Quick Reference & Cheat Sheet

Damping Regimes
ζ<1 (Underdamped),ζ=1 (Critically Damped),ζ>1 (Overdamped)\zeta < 1 \text{ (Underdamped)}, \quad \zeta = 1 \text{ (Critically Damped)}, \quad \zeta > 1 \text{ (Overdamped)}
Determines whether the system oscillates across equilibrium or decays monotonically.