Originally Posted by

**bkarpuz** Dear friends, I need some help with the existence and/or uniqueness of **global** solutions to first-order linear differential equations.

For instance, let $\displaystyle x_{0},t_{0}\in\mathbb{R}$ and $\displaystyle A,B\in C([t_{0},\infty),\mathbb{R})$ and consider the following differential equation

$\displaystyle

\begin{cases}

x^{\prime}(t)=A(t)x(t)+B(t),& t\geq t_{0}\\

x(t_{0})=x_{0}.&

\end{cases}

$

Which theorem ensures existence of global solutions to this initial value problem?

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