System of linear differential equations

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.

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.