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  "body": "\n## Definition\n\nLet $F\\left(x, y, y', \\dots, y^{(n)}\\right) = 0$ be an [[ordinary differential equation|0P97V20RW7]] of order $n$, with $F$ defined on a set $U \\subseteq \\mathbb{R}^{n+2}$.\nLet $I$ be an open [[interval|0Q1D85B8G6]] of $\\mathbb{R}$.\nA solution of the equation on $I$ is a function $u : I \\to \\mathbb{R}$ with these two properties.\nThe function $u$ is $n$ times differentiable on $I$, and $\\left(x, u(x), u'(x), \\dots, u^{(n)}(x)\\right) \\in U$ for each $x \\in I$.\nFor each $x \\in I$, $F\\left(x, u(x), u'(x), \\dots, u^{(n)}(x)\\right) = 0$.\nFor a system, $u$ takes values in $\\mathbb{R}^m$ and the same two conditions hold.\n\n## Remarks\n\nConsider the first-order system $x' = f(x)$, with a vector field $f$ on an open set of $\\mathbb{R}^m$. A solution is a curve whose velocity at each time is the value of $f$ at its position. Such a curve is an integral curve of $f$.\nThe function $u(x) = e^{2x}$ is a solution of $y' = 2y$ on $\\mathbb{R}$.\nA solution on $I$ restricts to a solution on each open subinterval of $I$. A [[maximal solution|01J7FZ65J0]] is a solution that has no extension to a larger interval.\n\n## Explanations\n\n### L1 human\n\nTo check that a function is a solution, put it and its derivatives into the equation.\nIf both sides are equal at each point of the interval, the function is a solution there.\nFor example, $u(x) = \\sin x$ gives $u'' + u = -\\sin x + \\sin x = 0$, so it solves $y'' + y = 0$.\n",
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