In This Lesson Homogeneous with Constant Coefficients 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 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:
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:
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Spring-mass: my'' + by' + ky = F(t)
RLC circuit: LQ'' + RQ' + Q/C = E(t)
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Laplace transform method handles these problems elegantly.
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