Let $A\in C(\mathbb{R},\mathbb{R^{n\times n}}),B\in C(\mathbb{R},\mathbb{R^{n}})$. Prove that there exists a unique solution of the IVP $$ \dot{x}(t)=A(t)x+B(t),\quad x(0)=x_0 $$
An exercise in my tectbook is like this. I wonder whether it means to prove the existence of a unique global solution. If so, how can I solve it. I just learned the Picard Theorem, and it is about local condition, so I am confused.
Appreciate any help!
From the way the question is worded, $A(t)$ is a matrix $\mathbb{R}^n\to\mathbb{R}^n$.
To apply Picard's theorem, the rhs needs to satisfy a Lipshitz condition: $$|A(t)x_1+B(t)-A(t)x_2-B(t)|=|A(t)(x_1-x_2)|\le\|A(t)\||x_1-x_2|\le c|x_1-x_2|$$ since $A(t)$ (and its coefficients) are continuous in $t$, so $$\|A(t)\|^2\le\sum_{i,j}|A_{ij}(t)|^2\le c^2$$ on some neighborhood of the initial $t=0$.