For a first order inhomogenous system of linear differential equations, what is a good way of defining resonance? I apologize for the title being slightly unclear (at least to me it seems so), so if anyone has a better suggestion feel free to change it.
Anyways, for example, when dealing with a second order differential equation of the form:
$$ ay'' + by' + cy = f(x) $$
with solutions $y_1(x)$ and $y_2(x)$ we can say that there is resonance in the system provided that $f(x)$ is linearly dependant on $y_1(x)$ and/or $y_2(x)$.
Now consider the linear system 
$$\textbf{x}' (t)  + \textbf{P}\textbf{x}(t) = \textbf{z}(t)$$
where $\textbf{x}(t)$, $\textbf{z}(t)$ are n-vectors, $\textbf{P}$ is a n-by-n matrix and $\textbf{z}(t)$ is a forcing term of the system. As the solutions are then of the form
$$\textbf{x}_i(t)=a_i \textbf{v}_i e^{\lambda_i t} $$
where the $\lambda_i$ and $\textbf{v}_i$ are eigenvalues and eigenvectors of $\textbf{P}$ respectively (assuming the eigenvectors are orthogonal), can we say that resonance is occuring when the $\textbf{z}(t)$ is linearly dependant of the solutions $\textbf{x}_i(t)$?
 A: Suppose $\mathbf{z}(t)=\sum_i a_i\mathbf{v}_ie^{\lambda_i t}$ (for fixed $a_i$, not all zero), and based on that, we look for a particular solution to the nonhomogeneous problem $$\textbf{x}' (t)  + \textbf{P}\,\textbf{x}(t) = \textbf{z}(t),\tag{N}$$ of the form $$\mathbf{x}_p(t)=\sum_{i} b_i\mathbf{v}_ie^{\lambda_i t},$$ (where the $b_i$ are to be determined). Then we are destined to fail: upon substituting $\mathbf{x}_p$ back into (N), the left-hand side becomes zero (due to the fact that (N) is linear and $\mathbf{x}_p$ is a linear combination of solutions to the homogeneous problem), whereas the right-hand side is not.
Thus, the method of undetermined coefficients---whether vector or scalar form---requires multiplying the original "guess" for $\mathbf{x}_p$ by the smallest power of $t$ for which the resulting function is not in the span of the fundamental set of solutions (to the homogeneous problem), $\{\mathbf{v}_ie^{\lambda_i t}\}$.
(I presume this bold part above is what you mean by resonance.)
Hope that helps.
