---
tag: 0A6QZ367GX
kind: definition
name: System of linear differential equations
aliases: [linear-system-of-odes]
msc: [34A30]
defines: 037ZNY7NAG
uses: [0P97V20RW7, 0PB7K8M7FG]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_differential_equation
    locator: System of linear differential equations
  - url: https://en.wikipedia.org/wiki/Matrix_differential_equation
    locator: lead section
---

## Definition

Let $n \ge 1$, let $I$ be an open interval, let $A : I \to \mathbb{R}^{n \times n}$ be a continuous matrix-valued function and let $b : I \to \mathbb{R}^n$ be continuous.
The system of linear differential equations with coefficient matrix $A$ and right side $b$ is the equation
$$x'(t) = A(t)\, x(t) + b(t)$$
for an unknown differentiable function $x : I \to \mathbb{R}^n$.
The system is homogeneous if $b = 0$ on $I$.

## Remarks

In coordinates, it is the system $x_i' = \sum_{j=1}^{n} a_{ij}(t)\, x_j + b_i(t)$ for $i = 1, \dots, n$.
For each $t_0 \in I$ and $x_0 \in \mathbb{R}^n$, there is exactly one solution on all of $I$ with $x(t_0) = x_0$.
The solutions of the homogeneous system form a vector space of dimension $n$. A basis gives a [[fundamental matrix|0MQ24RBG93]].
A [[linear equation of order n|0PB7K8M7FG]] becomes such a system by [[reduction to a first-order system|0TH9CPW3VS]].

## Explanations

### L1 human

Several unknown functions change at the same time, and each rate is a linear combination of all the unknowns.
In matrix form, the system looks like one linear first-order equation, with a matrix in place of a number.