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Dear

Suppose that $p,f$ are locally essentially bounded Lebesgue measurable functions and consider the differential equation

$x'(t)=p(t)x(t)+f(t)$ almost for all $t\geq t_{0}$, and $x(t_{0})=x_{0}$.

By a solution of this equation, we mean a function $x$,

which is absolutely continuous in $[t_{0},t_{1}]$ for all $t_{1}\geq t_{0}$,

and satisfies the differential equation almost for all $t\geq t_{0}$ and $x(t_{0})=x_{0}$.

How can I prove existence and uniqueness in the sense of almost everywhere of solutions to this problem?

Thanks.

**MHB**members,Suppose that $p,f$ are locally essentially bounded Lebesgue measurable functions and consider the differential equation

$x'(t)=p(t)x(t)+f(t)$ almost for all $t\geq t_{0}$, and $x(t_{0})=x_{0}$.

By a solution of this equation, we mean a function $x$,

which is absolutely continuous in $[t_{0},t_{1}]$ for all $t_{1}\geq t_{0}$,

and satisfies the differential equation almost for all $t\geq t_{0}$ and $x(t_{0})=x_{0}$.

How can I prove existence and uniqueness in the sense of almost everywhere of solutions to this problem?

Thanks.

**bkarpuz**
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