i have searched in many books but i did not find a proof for the statement in the title. I know its linked with Cauchy's theorem, but i need a full and reasoned proof. Thanks.
2026-03-29 18:32:17.1774809137
prove that every solution of analytic ODE x˙=f(x,t) is analytic
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The Picard iteration in the proof of the Picard-Lindelöf theorem gives you a sequence $(\varphi_i)_{i\geq 0}$ converging in the sup-norm to $x$. However, all the $\varphi_i$ are analytic (prove this by induction, using that $f$ is analytic) and therefore $x$ is analytic as well.
Added much latter: A more complete answer about the real-analytic Picard-Lindelöf theorem can be found here Why does rational dependence of $f'$ on $f$ imply that $f$ is real-analytic?