Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations

Let $a<b$ be real numbers and $k\ge1$ a natural number. Let $A$ assign to each $t\in[a,b]$ a real $k\times k$ matrix $A(t)$, and $g$ assign to each $t\in[a,b]$ a vector $g(t)\in\mathbb{R}^{k}$ (Euclidean space), all entries and components being continuous functions of $t$ on $[a,b]$; continuity of a real-valued function on an interval is understood as continuity of a map of metric spaces, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric. Integrals of matrix- or vector-valued functions are taken entrywise as Riemann integrals of continuous functions (existing by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval; degenerate intervals by the convention of Mean-Square Riemann Integral of a Family of Random Variables); products are the matrix product and matrix-vector product, and $I_k$ is the identity matrix.

1. (Fundamental solution) There is exactly one assignment $\Phi$ of a real $k\times k$ matrix $\Phi(t)$ to each $t\in[a,b]$, with continuous entries, such that

$\Phi$ is called the fundamental solution of $A$ on $[a,b]$.

2. (Invertibility) For every $t\in[a,b]$ the matrix $\Phi(t)$ is invertible, and the assignment $\Psi(t)=\Phi(t)^{-1}$ has continuous entries and is the unique continuous solution of

3. (Variation of constants) For every $\xi\in\mathbb{R}^{k}$, the function

has continuous components and is the unique such function satisfying