We "guess" y = eʳˣ because exponential functions reproduce themselves under differentiation. This converts the DE into the characteristic quadratic equation.
The Characteristic Equation
Three cases based on the discriminant b² − 4ac:
Case 1: Two distinct real roots r₁, r₂: y = C₁e^(r₁x) + C₂e^(r₂x) Case 2: Repeated root r: y = (C₁ + C₂x)eʳˣ Case 3: Complex roots α ± βi: y = eᵅˣ(C₁cos(βx) + C₂sin(βx))
Case 3 is the most physically interesting — it produces oscillation. The trigonometric functions appear through Euler's formula: eⁱᶿ = cos θ + i sin θ. The eigenvalue approach to systems x' = Ax yields the same three cases.
Critically damped (b² = 4mk): Fastest return without oscillation
Overdamped (b² > 4mk): Slow exponential decay
Resonance occurs when forcing frequency matches natural frequency — amplitude grows without bound (in the undamped case). This explains why soldiers break step on bridges and why opera singers can shatter glass. The Laplace transform method handles these problems elegantly.