Ordinary differential equations Linear equations and systems Definition

System of linear differential equations

Builds on Ordinary differential equation Linear differential equation of order n

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. A linear equation of order n becomes such a system by reduction to a first-order system.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Order of a differential equation definition
Linear differential equation of order n definition
System of linear differential equations definition

Tag	`0A6QZ367GX`
URI	`m2://0A6QZ367GX`
Aliases	`linear-system-of-odes`
Kind	definition
Writing	W0 stub
MSC	34A30
Defines	System of linear differential equations
Uses	Ordinary differential equation, Linear differential equation of order n

System of linear differential equations

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