Systems of Differential Equations

Linear Algebra (Matrices) · Axiom Academy

LESSON Systems of Differential Equations Each eigenvector is a mode that evolves on its own as — so the whole system decouples into independent grow/decay pieces. 1. One Eigenvector = One Independent Mode Suppose . Try the guess : then and — they match. So every eigenpair gives a solution that lives on the eigenvector's line and just stretches by : it grows if , decays if . grow mode ( ): shoots out along decay mode ( ): collapses in along 2. The Eigenvalue Sets the Rate: Each mode's size over time is just the scalar . Watch the two amplitudes climb and fall: the mode explodes upward while the mode melts toward zero. The eigenvalue is literally the per-unit-time growth rate of its mode. . The mode blows up; bigger means faster blow-up. This direction dominates the long run. . The mode dies out; more negative means faster decay. Its contribution fades. . The mode neither grows nor shrinks — it holds a constant displacement along . gives e^ at scaling times rotation at rate b — a spiral. (Real distinct here.) 3. Add the Modes: the Phase-Plane Trajectory Because the equation is linear, any sum of solutions is a solution . So the general state is the superposition of the modes, weighted by constants set by the start point. Watch the two mode-vectors add tip-to-tail each instant — their sum's tip traces the actual trajectory through the phase plane. You've seen how eigenvalues and eigenvectors turn a tangled system into independent grow/decay modes you simply add together.

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