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A First Course in Probability and Markov Chains by Laura Poggiolini, Giuseppe Modica

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Appendix C

Systems of linear ordinary differential equations

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.

C.1  Cauchy problem

We first consider the existence and uniqueness of a C1 solution X: [a, b] → image to the problem

image

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

C.1.1  Uniqueness

Lemma C.1 (Grönwall) Let W image C1(]a, b[, image) satisfy the inequality

image

for some α ≥ 0 and β > 0. Then

image

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

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