Solution
= Solution
The <characteristic equation> is $m^2+2m+2=0$, with roots $-1\pm i$. Therefore
$$
\boxed{u(t)=e^{-t}(A\cos t+B\sin t)}.
$$
If the second component is also wanted, the relation $v=u'+3u$ gives
$$
v(t)=e^{-t}\bigl[(2A+B)\cos t+(2B-A)\sin t\bigr].
$$
These are damped oscillations associated with the complex conjugate <eigenvalues> of the coefficient matrix.