Find the general solution to the following systems. Distinguish between the vibrational and unstable modes. What constraints on the initial conditions ensure that the unstable modes are not excited? (a) $\frac{d^2 u}{d t^2}=-4 u-2 v, \frac{d^2 v}{d t^2}=-2 u-v$.
(b) $\frac{d^2 u}{d t^2}=-u-3 v, \frac{d^2 v}{d t^2}=-3 u-9 v$.
(c) $\frac{d^2 u}{d t^2}=-2 u+v-2 w, \frac{d^2 v}{d t^2}=u-v$,
$\frac{d^2 w}{d t^2}=-2 u-4 w$.
(d) $\frac{d^2 u}{d t^2}=$
$\frac{d^2 v}{d t^2}=u-v+2 w, \frac{d^2 w}{d t^2}=-2 u+2 v-4 w$.