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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^d$ be a bounded domain, that is, a connected open set, with boundary $\\partial \\Omega$ of class $C^1$ and outward unit [[normal vector|0WQJJCCTC8]] $\\nu$.\nLet $c_1, c_2$ be nonzero real constants and let $g \\colon \\partial \\Omega \\to \\R$ be a given function.\nFor an equation for $u$ on $\\Omega$, the *Robin boundary condition* is\n$$c_1\\, u(x) + c_2\\, \\frac{\\partial u}{\\partial \\nu}(x) = g(x) \\quad \\text{for all } x \\in \\partial \\Omega.$$\nFor a second-order ordinary differential equation on $[a, b]$, where the outward normal derivatives are $-y'(a)$ and $y'(b)$, it reads\n$$c_1\\, y(a) - c_2\\, y'(a) = g_a, \\qquad c_1\\, y(b) + c_2\\, y'(b) = g_b.$$\n\n## Remarks\n\nIf $c_2 = 0$, the condition becomes a [[Dirichlet condition|EETVA6QYX4]]; if $c_1 = 0$, it becomes a [[Neumann condition|GS3KT2H645]].\nSome authors allow these cases and also let $c_1, c_2$ be functions on the boundary.\n\nExample: $y'' = 0$ on $[0, 1]$ with $y(0) - y'(0) = 0$ and $y(1) + y'(1) = 3$, so $c_1 = c_2 = 1$.\nThe function $y = 1 + x$ gives $y(0) - y'(0) = 1 - 1 = 0$ and $y(1) + y'(1) = 2 + 1 = 3$, so it solves this [[boundary value problem|TM8VQJWGHZ]].\n\nNon-example: the conditions $y(0) = 0$ and $y(1) = 0$ are not of Robin type, because no derivative appears; they are Dirichlet conditions.\n\nA Sturm–Liouville problem usually has separated conditions $\\alpha_1 y(a) + \\alpha_2 y'(a) = 0$ and $\\beta_1 y(b) + \\beta_2 y'(b) = 0$, which include the Robin case.\n",
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