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  "body": "\n## Summary\n\nA linear integral equation is an [[integral equation|R3SG1SN7JQ]] in which the unknown function enters linearly, both outside and inside the integral.\nIts general form is $a(x) \\varphi(x) = f(x) + \\lambda \\int K(x, s) \\varphi(s) \\, ds$ with a given [[kernel|GSMBXT7FPK]] $K$.\nThe solutions form an affine space, so linear algebra and operator theory apply.\nFredholm and Volterra equations are the main classes.\n\n## Explanations\n\n### L1 human\n\nLinear means that sums and multiples of solutions of the equation with $f = 0$ are again solutions.\nThe integral acts on the unknown like a matrix acts on a vector.\nSo a linear integral equation is a continuous version of a system of linear equations.\nFor example, $\\varphi(x) = x + \\int_0^1 x s \\, \\varphi(s) \\, ds$ is linear, while an equation with $\\varphi(s)^2$ under the integral is not.\n",
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