Existence and Uniqueness Theorem for 1st Order IVPs

Theorem
If the $$n \times n$$ matrix function $$A(t)$$ and vector function $$b(t)$$ are continuous on an interval $$I$$, then the Initial Value Problem
 * $$ x' = A(t)x + b(t), \, x(t_0) = x_0 $$

(where $$t_0 \in I$$) has a unique solution that exists on all of $$I$$.