It is a theorem (I think) that the equation:
$$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$
Has a unique global solution for any matrix $A(t)$ and vector $\mathbf{b}(t)$ whose entries are continuous functions of $t$.
My question is: Is there a relatively simple proof of existence and uniqueness? (By relatively simple, I mean not applying Picard-Lindelof).
In the case that $A$ is a constant matrix, one has a fairly easy proof using the matrix exponential. In the general case, perhaps one could use a time-ordered matrix exponential? Or perhaps there is a nice inductive proof based on the size of $A$?