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  "body": "\n## Summary\n\nThe Euler method is a numerical method to approximate the solution of an [[initial value problem|075VBY4K4X]] $y' = f(t, y)$, $y(t_0) = y_0$.\nWith a step size $h \u003e 0$, it sets $t_{k+1} = t_k + h$ and $y_{k+1} = y_k + h\\, f(t_k, y_k)$ for $k = 0, 1, 2, \\ldots$. Then $y_k$ approximates $y(t_k)$.\nIt follows the tangent line of the solution for one step at a time.\n\n## Remarks\n\nThe error of one step, the local truncation error, is $\\frac{h^2}{2} y''(\\tau)$ for some $\\tau$ between $t_k$ and $t_{k+1}$, when $y$ is twice continuously differentiable.\nLet $f$ be $L$-[[Lipschitz|055VVSN00V]] in $y$ with $L \u003e 0$, and let $|y''| \\le M$ on the interval. Then the global error satisfies $|y(t_k) - y_k| \\le \\frac{h M}{2 L} \\left( e^{L (t_k - t_0)} - 1 \\right)$. The proof uses a [[discrete Grönwall inequality|0NTM3J5MFE]].\nSo the method is of first order: halving the step size roughly halves the error. It is the simplest Runge–Kutta method.\nLeonhard Euler proposed it in his book Institutionum calculi integralis.\n\n## Explanations\n\n### L1 human\n\nAt each step, we compute the slope at the current point and walk a short distance $h$ along it.\nThe result is a broken line that stays close to the true solution when $h$ is small.\nFor $y' = y$, $y(0) = 1$, the method gives $y_k = (1 + h)^k$, and with $h = 1/k$ this tends to $e$ at time $1$.\n",
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