117 Resonance

$\bullet$ Let $U_{0}\neq 0$ and $\nu=\omega$ , so that the input angular frequency equals the natural angular frequency. Then the general solution is

X=c_{1}\begin{bmatrix}\cos\nu t\\ -\sin\nu t\end{bmatrix}+c_{2}\begin{bmatrix}\sin\nu t\\ \nu\cos\nu t\end{bmatrix}+{{U_{0}}\over{2\nu}}\begin{bmatrix}t\sin\nu t\\ \sin\nu t+t\nu\cos\nu t\end{bmatrix}

where the solution oscillates unboundedly. Whereas the input is bounded, the output is unbounded. So the system is not BIBO stable.