We assume the reader is already familiar with first- and second-order linear ordinary differential equations (ODEs) with constant coefficients, either homogeneous or not. Here, we review results on the solutions to systems of *N* first-order linear ODEs.

We first consider the existence and uniqueness of a *C*^{1} solution *X*: [*a, b*] → to the problem

where is the given *initial datum*, **Q** is a given real *N* × *N* matrix and *F* : [*a, b*] → is a given continuous function.

**Lemma C.1 (Grönwall)** *Let W* *C*^{1}(]*a, b*[, ) *satisfy the inequality*

*for some α* ≥ 0 *and β >* 0. *Then*

*Proof.* Let *∊ >* 0. The function is strictly ...

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