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Propositions from Linear ODE systems

All propositions from Linear ODE systems.

[PROPOSITION]

If $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$ solve $\mathbf{x}'=A\mathbf{x}$ , then every linear combination

\alpha\mathbf{x}_1(t)+\beta\mathbf{x}_2(t)

also solves the system.

Differentiation and multiplication by $A$ are both linear:

\frac{d}{dt}(\alpha\mathbf{x}_1+\beta\mathbf{x}_2) =\alpha A\mathbf{x}_1+\beta A\mathbf{x}_2 =A(\alpha\mathbf{x}_1+\beta\mathbf{x}_2).

Part of the thread Roots, modes and evolution, in 7 places.

A linear evolution splits into modes $r^n$ or $e^{rt}$ , and the roots $r$ decide growth, decay and oscillation.

THExponential modescoupled ODEs: eigenvectors as modesALG 4 / Here