= Solution
The <normal modes> are the <eigenvectors> of the stiffness matrix. The in-phase mode has normalized vector $(1,1)/\sqrt2$; its central spring is unstretched. The opposite-phase mode has normalized vector $(1,-1)/\sqrt2$ and stretches the central spring. Their frequencies are
$$
\boxed{\omega_-^2=\frac km,\qquad\omega_+^2=\frac{k+2k'}m=\frac km(1+2\epsilon).}
$$
Each mode has arbitrary amplitude and phase. Thus $x=A_-\cos(\omega_-t+\phi_-)(1,1)/\sqrt2+A_+\cos(\omega_+t+\phi_+)(1,-1)/\sqrt2$ 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.
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