= Solution
The independent <normal modes> span the two displacement coordinates. With arbitrary real constants $A,B,C,D$, the general real solution is
$$
\boxed{\begin{pmatrix}y_1\\y_2\end{pmatrix}
=(A\cos\omega_-t+B\sin\omega_-t)\begin{pmatrix}1\\1\end{pmatrix}
+(C\cos\omega_+t+D\sin\omega_+t)\begin{pmatrix}2\\-1\end{pmatrix}.}
$$
Four constants are required for two second-order <ordinary differential equations>, corresponding to two initial displacements and two initial <velocities>.
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