In This Lesson Homogeneous with Constant Coefficients The Characteristic Equation Non-Homogeneous Equations Springs & Circuits 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:
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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 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
ay'' + by' + cy = g(x)
General solution: y = yₕ + yₚ
yₕ = homogeneous solution, yₚ = particular solution
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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)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 Underdamped (b² < 4mk): Decaying oscillation — connects to wave applications Critically damped (b² = 4mk): Fastest return without oscillation Overdamped (b² > 4mk): Slow exponential decay 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 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.