This section records for later reference some basic facts about linear systems of ordinary differential equations.

DEFINITION. Let X(.) : R → M^n×n be the unique solution of the matrix ODE

We call X(.) a fundamental solution, and sometimes write

the last formula being the definition of the exponential e^tM. Observe that

X⁻¹ (t) = X(−t).

THEOREM 1.1 (SOLVING LINEAR SYSTEMS OF ODE).

(i) The unique solution of the homogeneous system of ODE

(ii) The unique solution of the nonhomogeneous system

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