Complex Roots and Oscillation

You will learn how to solve second-order linear homogeneous equations when the characteristic equation has complex roots, and interpret the solutions as oscillatory motion with or without damping.

When the characteristic equation $ar^2 + br + c = 0$ has complex roots $\alpha \pm \beta i$, the general solution is:

$$y = e^{\alpha x} (C_1 \cos \beta x + C_2 \sin \beta x)$$

• The term $e^{\alpha x}$ controls growth or decay (damping).
• The term $C_1 \cos \beta x + C_2 \sin \beta x$ produces oscillation with frequency $\beta$.

Special cases:

• $\alpha = 0$: undamped simple harmonic motion (pure oscillation forever)
• $\alpha \lt 0$: damped oscillation (most real systems)
• $\alpha \gt 0$: growing oscillation (unstable)

The visual shows three cases: undamped oscillation (constant amplitude), lightly damped (decaying envelope), and heavily damped (rapid decay with minimal oscillation).

Example 1: Undamped Oscillation

Solve $y'' + 4y = 0$.

Solution: Characteristic equation: $r^2 + 4 = 0 \implies r = \pm 2i$

General solution: $y = C_1 \cos 2x + C_2 \sin 2x$

Example 2: Damped Oscillation

Solve $y'' + 6y' + 13y = 0$.

Solution: Characteristic equation: $r^2 + 6r + 13 = 0 \implies r = -3 \pm 2i$

General solution: $y = e^{-3x} (C_1 \cos 2x + C_2 \sin 2x)$

Example 3: Initial Value Problem

Solve $y'' + 2y' + 2y = 0$ with $y(0) = 1$, $y'(0) = 0$.

Solution: Characteristic roots: $-1 \pm i$

General solution: $y = e^{-x} (C_1 \cos x + C_2 \sin x)$

Applying initial conditions: $C_1 = 1$, $C_2 = 1$

Final solution: $y = e^{-x} (\cos x + \sin x)$

This mathematics describes real oscillating systems everywhere: car suspensions absorbing bumps, electrical circuits in radios and phones, bridges swaying in the wind, and even the motion of tall buildings during earthquakes. Engineers use these equations to design systems that either resonate at specific frequencies (like musical instruments) or dampen vibrations quickly (like shock absorbers).