Past exam of the mathematical methods II course of the University of Oxford / 2014 / 9 / b / Solution

Set $\mathbf{x}=\mathbf{a}\cos (\omega t+\phi )$. The two normal modes satisfyThe characteristic equation is $(2-s)(1-s)-1=s^2-3s+1=0$. HenceHere $a_{\pm }$ denotes the scalar coefficient requested in the source, whereas bold $\mathbf{a}$ denotes a mode-amplitude vector. Both roots are positive.