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  "body": "\n## Summary\n\nPicard iteration is the construction of a solution of $x' = f(t, x)$, $x(t_0) = x_0$ as a limit of successive approximations.\nOne starts with the constant curve $\\varphi_0(t) = x_0$ and sets $\\varphi_{k+1} = P \\varphi_k$, where $P$ is the [[Picard operator|00WAVB68F9]]: $\\varphi_{k+1}(t) = x_0 + \\int_{t_0}^{t} f(s, \\varphi_k(s))\\, ds$.\nUnder the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]], the curves $\\varphi_k$ converge uniformly on the time interval to the unique solution.\n\n## Remarks\n\nExample: for $x' = x$, $x(0) = 1$, the iterates are $\\varphi_k(t) = \\sum_{j=0}^{k} t^j / j!$. They converge to $e^t$.\nLet $K$ be the Lipschitz constant and $T$ the length of the time interval. The error after $k$ steps is at most $C (K T)^{k} / k!$ for a constant $C$. See [[an iterate of the Picard operator is a contraction|08XDXP86XE]].\n\n## Explanations\n\n### L1 human\n\nStart with a rough guess, the constant $x_0$.\nPut the guess into the right side of the integral equation to get a better guess, and repeat.\nEach step improves the approximation, and the guesses converge to the solution.\n",
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