= Solution
The <transverse stiffness of a stretched spring> follows because a <spring> of <length> $2l$ develops transverse <force> $T_0\Delta y/(2l)$ to first order. Its extension changes only at second order, so the leading transverse stiffness comes from its existing <tension>, not from that extension change. Here $T_0/(2l)=k/2$, giving
$$
m\begin{pmatrix}\ddot Y_1\\\ddot Y_2\end{pmatrix}
=-\frac k2\begin{pmatrix}2&-1\\-1&2\end{pmatrix}\begin{pmatrix}Y_1\\Y_2\end{pmatrix},\qquad
\boxed{\omega_{y,+}=\sqrt{k/(2m)},\quad\omega_{y,-}=\sqrt{3k/(2m)}.}
$$
The mode vectors are again $(1,1)$ and $(1,-1)$. Treating the <springs> as unstretched would incorrectly remove this linear restoring <force>.
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