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Ordinary differential equations

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Differential Equations & Dynamical SystemsGradoExplicación9 min

A Damped Spiral That Rings Down to One, Not to Zero

The characteristic roots of u'' + u' + u are the primitive cube roots of unity, so the homogeneous part decays with envelope e to the minus x over two and oscillates with period 4 pi over root 3, which is 7.2552. Substituting that homogeneous solution back into the equation leaves a residual of exactly minus one, and that residual is the whole distance between the common wrong answer and the right one. The constant u = 1 solves the equation by itself, so every solution settles on 1, and the constant trial works only because the coefficient on u is not zero.

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