Solution
= Solution
Let $X_1,X_2$ be longitudinal displacements. <Hooke's law> gives
$$
m\begin{pmatrix}\ddot X_1\\\ddot X_2\end{pmatrix}
=-k\begin{pmatrix}2&-1\\-1&2\end{pmatrix}\begin{pmatrix}X_1\\X_2\end{pmatrix}.
$$
The symmetric and antisymmetric <eigenvectors> are respectively $(1,1)$ and $(1,-1)$, so
$$
\boxed{\omega_{x,+}=\sqrt{k/m},\qquad \omega_{x,-}=\sqrt{3k/m}.}
$$
Ordinary <frequencies> are these values divided by $2\pi$.