Second-Order Differential Equations

Springs, circuits, and waves — second-order DEs describe oscillatory systems.

Homogeneous with Constant Coefficients

Form: ay'' + by' + cy = 0
Trial solution: y = eʳˣ → ar² + br + c = 0

We "guess" y = eʳˣ because exponential functions reproduce themselves under differentiation. This converts the DE into the characteristic quadratic equation.

The Characteristic Equation

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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.

Example: y'' + 4y = 0

Characteristic: r² + 4 = 0 → r = ±2i (complex roots, α = 0, β = 2)

y = C₁cos(2x) + C₂sin(2x) — pure oscillation!

Non-Homogeneous Equations

ay'' + by' + cy = g(x)
General solution: y = yₕ + yₚ
yₕ = homogeneous solution, yₚ = particular solution

Methods for finding yₚ:

  • Undetermined coefficients: Guess yₚ based on g(x). If g = polynomial, guess polynomial; if g = eᵃˣ, guess Aeᵃˣ; if g = sin/cos, guess A·cos + B·sin
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  • Variation of parameters: More general but requires integration. Works for any g(x)

Springs & Circuits

Spring-mass: my'' + by' + ky = F(t)
RLC circuit: LQ'' + RQ' + Q/C = E(t)
  • Undamped (b = 0): Pure oscillation at natural frequency ω₀ = √(k/m)
  • Underdamped (b² < 4mk): Decaying oscillation — connects to wave applications
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  • 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.
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