Definition:

A system of first order differential equations is defined as such: $D_t x_i = \sum\limits_{j=1}^n (P_{ij} (t) x_j) + g_i (t)$ for $1 \le i \le n$ $\Leftrightarrow$ $D_t \vec{x} = \begin{bmatrix}P_{ij} (t) \end{bmatrix}_{ij} \vec{x} = \begin{bmatrix}g_1(t) \\ \vdots \\ g_n(t) \end{bmatrix}$