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  "body": "\n## Summary\n\nAn integral equation is an equation in which the unknown function appears under an integral sign.\nThe typical linear example is $\\varphi(x) = f(x) + \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds$, where $f$ and the [[kernel|GSMBXT7FPK]] $K$ are given and $\\varphi$ is unknown.\nMany problems for differential equations, such as [[initial value problems|0FXDW992DS]] and [[boundary value problems|TM8VQJWGHZ]], can be rewritten as integral equations.\nThe integral form often needs less smoothness and suits fixed point and operator methods.\n\n## Explanations\n\n### L1 human\n\nIn an ordinary equation the unknown is a number; here the unknown is a whole function.\nThe function appears once on its own and once inside an integral, which mixes its values over a range of points.\nA picture: the value of $\\varphi$ at $x$ is a known value $f(x)$ plus a weighted sum, an integral, of the values of $\\varphi$ over a range.\nFor example, $\\varphi(x) = 1 + \\int_0^x \\varphi(s) \\, ds$ has the solution $\\varphi(x) = e^x$, since $1 + \\int_0^x e^s \\, ds = e^x$.\n",
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