= Solution
Put $y^{(0)}=y$ and $y^{(1)}=\dot y$. Then
$$
\boxed{\frac d{dt}\begin{pmatrix}y^{(0)}\\y^{(1)}\end{pmatrix}
=\underbrace{\begin{pmatrix}0&1\\-k^2&-2\gamma\end{pmatrix}}_{M}
\begin{pmatrix}y^{(0)}\\y^{(1)}\end{pmatrix}.}
$$
The second row is the original <damped harmonic oscillator> equation and the first row defines the auxiliary <velocity>. This converts a second-order <ordinary differential equation> into two first-order equations without changing the required two initial values.
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