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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $p_0, \\ldots, p_{n-1}$ be [[continuous functions|0X3WGX95FE]] on $[a, b]$, with $n \\ge 1$.\nLet $L y = y^{(n)} + p_{n-1} y^{(n-1)} + \\cdots + p_0 y$ be the [[linear differential operator|0QQRP17Z7S]] on $[a, b]$.\nLet $B_1, \\ldots, B_n$ be linear forms of the shape $B_i y = \\sum_{j=0}^{n-1} \\left( \\alpha_{ij}\\, y^{(j)}(a) + \\beta_{ij}\\, y^{(j)}(b) \\right)$ with real constants $\\alpha_{ij}, \\beta_{ij}$.\nAssume that the [[boundary value problem|TM8VQJWGHZ]] $L y = 0$, $B_1 y = \\cdots = B_n y = 0$ has only the solution $y = 0$.\nA *Green's function* of this problem is a function $G \\colon [a, b] \\times [a, b] \\to \\R$ with the following property.\nFor every continuous $f$ on $[a, b]$, the function\n$$y(x) = \\int_a^b G(x, s)\\, f(s)\\, ds$$\nis a solution of $L y = f$, $B_1 y = \\cdots = B_n y = 0$.\n\n## Remarks\n\nUnder the assumption, the Green's function exists, and the solution above is the only one.\nFor fixed $s$, the function $x \\mapsto G(x, s)$ solves $L y = 0$ on $[a, s)$ and on $(s, b]$ and satisfies the boundary conditions.\nIts derivatives of order up to $n - 2$ are continuous at $x = s$, and its derivative of order $n - 1$ jumps by $1$ there.\nFormally, $L_x G(x, s) = \\delta(x - s)$.\n\nExample: for $L y = y''$ on $[0, 1]$ with $y(0) = y(1) = 0$, take $G(x, s) = x (s - 1)$ for $x \\le s$ and $G(x, s) = s (x - 1)$ for $x \\ge s$.\nIt vanishes at $x = 0$ and $x = 1$, it is linear in $x$ on each side of $s$, and $\\partial_x G$ jumps from $s - 1$ to $s$, a jump of $1$.\nFor $f = 1$ it gives $y(x) = \\frac{x (x - 1)}{2}$, and indeed $y'' = 1$ and $y(0) = y(1) = 0$.\n\nNon-example: for $L y = y''$ on $[0, 1]$ with $y'(0) = y'(1) = 0$, the constants solve the homogeneous problem, so the assumption fails.\nThere is no Green's function: $y'' = 1$ has no solution with these conditions, since $y'(1) - y'(0) = \\int_0^1 y'' = 1$.\n",
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