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$$

Y =

\begin{pmatrix}

y_1 \\ y_2

\end{pmatrix}

$$

can be written as a linear combination of the eigenvectors

$$

Y = \alpha_1(t) v_1 + \alpha_2(t) v_2

$$

where the expansion coefficients generally depend on time. Inserting this into the differential equation gives you separated differential equations for the ##\alpha##, i.e., the differential equation for ##\alpha_1## does not depend on ##\alpha_2## and vice versa.