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  "body": "\n## Definition\n\nLet $I$ be an open interval and let $a_0, \\dots, a_{n-1} : I \\to \\mathbb{R}$ be continuous. Let $S$ be the vector space of solutions on $I$ of the [[homogeneous linear equation|0EQZSAB9EC]] $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \\dots + a_0(x) y = 0$.\nA fundamental system of solutions of the equation is a basis $(y_1, \\dots, y_n)$ of $S$.\n\n## Remarks\n\nSince $S$ has [[dimension n|0GK68PZ6KH]], a family of $n$ solutions is a fundamental system if and only if it is linearly independent. The [[Wronskian test|0MHE4B1QCQ]] checks this.\nEach solution is then $c_1 y_1 + \\dots + c_n y_n$ for unique constants $c_1, \\dots, c_n$.\nExample: $(\\cos x, \\sin x)$ is a fundamental system of $y'' + y = 0$ on $\\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nA fundamental system is a set of building blocks.\nEach solution of the homogeneous equation is a unique combination of these $n$ solutions.\n",
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