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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $I$ be an open interval, and let $a_0, \\dots, a_{n-1} : I \\to \\mathbb{R}$ be continuous.\nLet $S$ be the set of solutions $y : I \\to \\mathbb{R}$ of $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \\dots + a_0(x) y = 0$.\nThen $S$ is a real vector space of dimension $n$.\nMore precisely, for each $x_0 \\in I$, the map $S \\to \\mathbb{R}^n$, $y \\mapsto \\left(y(x_0), y'(x_0), \\dots, y^{(n-1)}(x_0)\\right)$ is a linear bijection.\n\n## Remarks\n\nThe [[superposition principle|0F2VBPXNP9]] gives that $S$ is a vector space. The map is injective by the uniqueness of solutions of the initial value problem, and surjective by their existence on all of $I$.\nA basis of $S$ is a [[fundamental system of solutions|0H4KNNCHPY]].\nExample: the solutions of $y'' + y = 0$ on $\\mathbb{R}$ are $C_1 \\cos x + C_2 \\sin x$, a space of dimension $2$.\n\n## Explanations\n\n### L1 human\n\nA solution is fixed by its starting data: the value and the first $n - 1$ derivatives at one point.\nThese are $n$ numbers that you can choose freely.\nSo there are exactly $n$ independent solutions, and each solution is a combination of them.\n",
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