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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $a_0, \\dots, a_{n-1}$ be real numbers, let $I$ be an open interval and let $b : I \\to \\mathbb{R}$ be continuous.\nA linear equation with constant coefficients is a [[linear differential equation|0PB7K8M7FG]] of the form\n$$y^{(n)} + a_{n-1}\\, y^{(n-1)} + \\dots + a_1\\, y' + a_0\\, y = b(x).$$\nIts homogeneous equation is the case $b = 0$, on the interval $I = \\mathbb{R}$.\n\n## Remarks\n\nThe function $e^{r x}$ solves the homogeneous equation if and only if $r$ is a root of the [[characteristic equation|0QDXMG46QW]] $r^n + a_{n-1} r^{n-1} + \\dots + a_0 = 0$.\nThe roots of this polynomial give a [[fundamental system|0H4KNNCHPY]]. See the cases of [[distinct real roots|0HJ0HDA6W7]], [[repeated roots|0HRVY67CNV]] and [[complex conjugate roots|0BEZX8F2XW]].\nThe [[harmonic oscillator|0YG0ZJPTWG]] $x'' + \\omega^2 x = 0$ is an example of order $2$.\n\n## Explanations\n\n### L1 human\n\nWhen the coefficients are constants, the derivative of $e^{r x}$ is $r e^{r x}$, again a multiple of the same function.\nSo the equation becomes a polynomial equation in $r$.\nThis changes calculus into algebra.\n",
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