In This Lesson Homogeneous with Constant Coefficients The Characteristic Equation Non-Homogeneous Equations 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 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 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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·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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.