Exponential forcing of a constant-coefficient differential system ($x_p(t)=e^{\lambda t}(\lambda I-A{)}^{-1}v$)

For $x^{\prime }-Ax=e^{\lambda t}v$, 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.