Mr. Integral
Math notes/Differential Equations/Systems and Phase Planes

Forced Systems and Variation of Constants — Cheat sheet

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Core rule

x=eAtx0+∫0teA(t−τ)b(τ)dτ.\mathbf x=e^{At}\mathbf x_0+\int_0^t e^{A(t-\tau)}\mathbf b(\tau)d\tau.

Watch for

An equilibrium can exist even if it is unstable. Long-time convergence requires stability of the homogeneous dynamics, not only successful solution of Ax∗+b=0A\mathbf x_*+\mathbf b=0.