Exponential forcing of a constant-coefficient differential system
= Exponential forcing of a constant-coefficient differential system
{title2=$x_p(t)=e^{\lambda t}(\lambda I-A)^{-1}v$}
For $x'-Ax=e^{\lambda t}v$, substituting an exponential particular solution reduces the differential equation to $(\lambda I-A)u=v$. The displayed solution exists whenever $\lambda$ is not an <eigenvalue> of $A$. Resonance can require polynomial factors in time. The general solution adds $e^{tA}c$, expressed using the <matrix exponential>.