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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $I$ be an open [[interval|0Q1D85B8G6]], and let $a_0, \\dots, a_{n-1}, b : I \\to \\mathbb{R}$ be continuous functions.\nThe linear differential equation of order $n$ with these coefficients and right side $b$ is\n$$y^{(n)} + a_{n-1}(x)\\, y^{(n-1)} + \\dots + a_1(x)\\, y' + a_0(x)\\, y = b(x)$$\nfor an unknown $n$ times differentiable function $y : I \\to \\mathbb{R}$.\nWith the operator $L y = y^{(n)} + a_{n-1} y^{(n-1)} + \\dots + a_0 y$, the equation is $L y = b$.\n\n## Remarks\n\nThe map $L$ is linear: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2$ for constants $c_1, c_2$.\nAn equation $a_n(x) y^{(n)} + \\dots + a_0(x) y = b(x)$ with continuous $a_n$ and no zero of $a_n$ on $I$ takes this form after division by $a_n$.\nThe equation is [[homogeneous|0EQZSAB9EC]] if $b = 0$. The case $n = 1$ is the [[linear first-order equation|0Z3YKRQZ48]].\nFor each $x_0 \\in I$ and each $(c_0, \\dots, c_{n-1}) \\in \\mathbb{R}^n$, there is exactly one solution on all of $I$ with $y^{(k)}(x_0) = c_k$ for $k = 0, \\dots, n-1$.\n\n## Explanations\n\n### L1 human\n\nIn a linear equation, the unknown $y$ and its derivatives appear only to the first power, each multiplied by a function of $x$.\nFor example, $y'' + x y' + y = \\cos x$ is linear, but $y'' + y^2 = 0$ is not.\nLinear equations have a rich theory, because sums and multiples of solutions behave well.\n",
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