In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial differential equations.

A first-order linear system of ODEs is a system in which every equation is first order and depends on the unknown functions linearly. Here we consider systems with an equal number of unknown functions and equations. These may be written as

${\displaystyle {\frac {dx_{j}}{dt}}=a_{j1}(t)x_{1}+\ldots +a_{jn}(t)x_{n}+g_{j}(t),\qquad j=1,\ldots ,n}$

where ${\displaystyle n}$ is a positive integer, and ${\displaystyle a_{ji}(t),g_{j}(t)}$ are arbitrary functions of the independent variable t.

A first-order linear system of ODEs may be written in matrix form:

${\displaystyle {\frac {d}{dt}}{\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}a_{11}&\ldots &a_{1n}\\a_{21}&\ldots &a_{2n}\\\vdots &\ldots &\vdots \\a_{n1}&&a_{nn}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}+{\begin{bmatrix}g_{1}\\g_{2}\\\vdots \\g_{n}\end{bmatrix}},}$

or simply

${\displaystyle \mathbf {\dot {x}} (t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {g} (t)}$.