In This Lesson Homogeneous with Constant Coefficients 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 The Characteristic Equation Non-Homogeneous Equations Springs & Circuits 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 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 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!
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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 Variation of parameters: More general but requires integration . Works for any g(x) Springs & Circuits 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
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 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.