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  "body": "\n## Definition\n\nLet $k \\ge 1$ and $a \u003c b$.\nLet $L \\colon [a, b] \\times \\R^{k+1} \\to \\R$, written $L(x, z_0, z_1, \\dots, z_k)$, be a function of class $C^{k+1}$, and write $L_{z_j}$ for its [[partial derivative|09EGC6F8PT]] in $z_j$.\nConsider the functional\n$$J(y) = \\int_a^b L\\big(x, y(x), y'(x), \\dots, y^{(k)}(x)\\big) \\, dx.$$\nThe *Euler–Poisson equation* of $J$ is the equation\n$$\\sum_{j=0}^{k} (-1)^j \\frac{d^j}{dx^j} \\Big( L_{z_j}\\big(x, y(x), y'(x), \\dots, y^{(k)}(x)\\big) \\Big) = 0, \\qquad a \\le x \\le b,$$\nfor an unknown $y \\in C^{2k}([a, b])$.\n\n## Remarks\n\nFor $k = 1$ it is the [[Euler–Lagrange equation|PS0VV149ET]] $L_{z_0} - \\frac{d}{dx} L_{z_1} = 0$ of an [[integral functional|JRQZ803Q5Y]].\nA minimiser $y \\in C^{2k}([a, b])$ of $J$ among functions with prescribed $y^{(j)}(a)$ and $y^{(j)}(b)$ for $0 \\le j \\le k - 1$ solves it.\nTo see this, take variations $\\eta$ with $\\eta^{(j)}(a) = \\eta^{(j)}(b) = 0$ for $j \u003c k$, integrate the term with $\\eta^{(j)}$ by parts $j$ times, and apply the fundamental lemma of the calculus of variations.\n\nExample: for $k = 2$ and $L = \\frac{1}{2} z_2^2$, one has $L_{z_0} = L_{z_1} = 0$ and $L_{z_2} = z_2$, so the equation is $y'''' = 0$.\nEvery cubic polynomial solves it.\nNon-example: $y(x) = x^4$ does not solve this equation, since $y'''' = 24$.\n",
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