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  "body": "\n## Summary\n\nA Green's function of a linear differential operator $L$ with boundary conditions is a kernel $G(x, s)$ that turns the source term into the solution.\nThe function $y(x) = \\int G(x, s) f(s)\\, ds$ solves $L y = f$.\nFormally it is the response of the system to a unit point source at $s$, that is, $L G(\\cdot, s) = \\delta_s$ with the [[Dirac delta|0KPHENZ5S7]].\nIt turns a differential equation into an integral formula, and it is basic in boundary value problems, potential theory and physics.\nThe [[Green's function for the Laplacian|M36SD56MWQ]] is the main case for partial differential equations.\nSee the [[definition|ZH4V0WHSXT]] for ordinary differential equations.\n\n## Explanations\n\n### L1 human\n\nThink of a taut string fixed at both ends.\nPushing it at one point $s$ with a unit force bends it into a shape $G(x, s)$.\nA general load $f$ is a sum of many point pushes, so the string takes the shape $\\int G(x, s) f(s)\\, ds$.\nFor $y'' = f$ on $[0, 1]$ with $y(0) = y(1) = 0$, the Green's function is $G(x, s) = x (s - 1)$ for $x \\le s$ and $G(x, s) = s (x - 1)$ for $x \\ge s$.\n",
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