Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 15 / b / ii / Solution

The characteristic equation is $m^2+2m+2=0$, with roots $-1\pm i$. ThereforeIf the second component is also wanted, the relation $v=u^{\prime }+3u$ givesThese are damped oscillations associated with the complex conjugate eigenvalues of the coefficient matrix.