The Initial Value Problem for a First-Order System of Ordinary Differential Equations

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $a,b,c\in\mathbb{R}$ with $a<c\le b$. Derivatives of maps on $[a,c]$ are those of Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (i) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii), taken relative to the interval $[a,c]$, which contains the two distinct points $a$ and $c$ by Basic Facts about Intervals of the Real Line (v).

(i) Let $n\in\mathbb{N}$, let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a map and let $\xi\in\mathbb{R}^{n}$. The initial value problem on $[a,c]$ with right-hand side $F$ and initial value $\xi$ is the system

$$x'(t)=F\bigl(t,x(t)\bigr)\quad\bigl(t\in[a,c]\bigr),\qquad x(a)=\xi,$$

for an unknown map $x:[a,c]\to\mathbb{R}^{n}$; $F(t,x(t))$ is defined since $t\in[a,c]\subseteq[a,b]$ and $x(t)\in\mathbb{R}^{n}$.

(ii) Let $p,q\in\mathbb{N}$, let $F:[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$ be a map and let $\Xi\in\mathcal{M}_{p\times q}(\mathbb{R})$. The initial value problem on $[a,c]$ with right-hand side $F$ and initial value $\Xi$ is the system

$$X'(t)=F\bigl(t,X(t)\bigr)\quad\bigl(t\in[a,c]\bigr),\qquad X(a)=\Xi,$$

for an unknown map $X:[a,c]\to\mathcal{M}_{p\times q}(\mathbb{R})$; $F(t,X(t))$ is defined since $t\in[a,c]\subseteq[a,b]$ and $X(t)\in\mathcal{M}_{p\times q}(\mathbb{R})$.