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Second-Order ODE as a First-Order System

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A linear second-order ODE y'' + p y' + q y = 0 becomes a first-order system in two unknowns by introducing v = y'. The state vector (y, v) evolves under a single 2×2 matrix, opening up every first-order tool — Euler, RK4, eigenvector analysis.

second-order ODEfirst-order systemstate vectorcompanion matrixphase plane
ForutsetterEuler's Method: Tangent Steps Along a Slope Field
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