SME
Differential Equations · Problem type
Saddle, node, spiral or center?
Use this for “What kind of equilibrium does x′ = Ax have at the origin when A = [[−1, −2], [2, −1]]?”
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Work out T and D for each A, then T² − 4D.
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Assumptions
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T = a + d is the sum of the eigenvalues and D = ad − bc their product.
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D < 0: a saddle. D > 0 and T² − 4D ≥ 0: a node. T² − 4D < 0: a spiral, or a center when T = 0.
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Stable when T < 0 (trajectories head in), unstable when T > 0.
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