 Solution

The normal modes are the eigenvectors of the stiffness matrix. The in-phase mode has normalized vector $(1,1)/\sqrt{2}$; its central spring is unstretched. The opposite-phase mode has normalized vector $(1,-1)/\sqrt{2}$ and stretches the central spring. Their frequencies areEach mode has arbitrary amplitude and phase. Thus $x=A_{-}\cos ({\omega }_{-}t+{\varphi }_{-})(1,1)/\sqrt{2}+A_{+}\cos ({\omega }_{+}t+{\varphi }_{+})(1,-1)/\sqrt{2}$ is the general real motion. Equal coordinates belong to ${\omega }_{-}$ and opposite coordinates to ${\omega }_{+}$; the amplitudes are not fixed until initial data are supplied.