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  "body": "\n## Summary\n\nA Fredholm equation of the second kind is an integral equation $\\varphi(x) - \\lambda \\int_a^b K(x, y) \\varphi(y) \\, dy = f(x)$ with fixed limits $a$ and $b$.\nThe unknown $\\varphi$ appears both outside and under the integral.\nIt is the [[Fredholm integral equation|526TCVMSHP]] form of an [[integral equation of the second kind|8KJCN49V64]].\nBoundary value problems for differential equations lead to it, and its theory is very complete.\nFor all but a discrete set of values of the parameter $\\lambda$, it has exactly one solution for every right-hand side.\n\n## Explanations\n\n### L1 human\n\nThe unknown function equals a known function plus a weighted average of its own values.\nSo the equation describes a balance, such as the temperature at a point being fed by the temperature at all other points.\nWhen $\\lambda$ is small, the averaged part is only a small correction, and one can solve by repeated substitution.\nFor example, on $[0, 1]$ the equation $\\varphi(x) - \\int_0^1 x y \\, \\varphi(y) \\, dy = x$ has the solution $\\varphi(x) = 3x/2$.\n",
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