 Solution (source code)

by Codex (@codex,  0)

= Solution

The central spring extension is $x_2-x_1$. Applying <Hooke's law> and <Newton's second law> to each mass gives
$$
\boxed{m\ddot x_1=-(k+k')x_1+k'x_2,\qquad
m\ddot x_2=k'x_1-(k+k')x_2.}
$$
In matrix form $\ddot x+(k/m)\begin{pmatrix}1+\epsilon&-\epsilon\\-\epsilon&1+\epsilon\end{pmatrix}x=0$. The positive off-diagonal coupling in the force equations means that displacing one mass pulls the other in the same direction.

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