연립 1계 선형 미분방정식
$$\begin{cases} &\frac{d}{dt} x_1(t) = a_{11}(t)x_1(t)+a_{12}(t)x_2(t) +.... + a_{1N}(t) x_N(t) + b_1(t) \\ & \frac{d}{dt} x_2(t) = a_{21}(t)x_1(t)+a_{22}(t)x_2(t) +.... + a_{2N}(t) x_N(t)+ b_2(t)\\ & \frac{d}{dt} x_3(t) = a_{31}(t)x_1(t)+a_{32}(t)x_2(t) +.... + a_{3N}(t) x_N(t) +b_3(t) \\ &\hspace{10em}\vdots \\ & \frac{d}{dt} x_N(t) = a_{N1}(t)x_1(t)+a_{N2}(t)x_2(t) +.... + a_{NN}(t) x_N(t) + b_N(t)\end{cases}$$
이것을 행렬로 나타낼 수 있다. $\mathbf{A}(t) =\left[ a_{ij}(t)\right]_{i=1,2,...,N, j=1,2,...,N}, \mathbf{x}(t) = \left(x_1(t),x_2(t), ... , x_N(t) \right)^T, \mathbf{b}(t) = \left( b_1(t),...,b_N(t) \right)$ 라고 한다면
$$\begin{align} \frac{d}{dt}\mathbf{x}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{b}(t) \end{align} $$