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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $\\varphi \\colon [a, b] \\to \\C$ be an unknown [[function|0EPAT51DXQ]].\nFor each $x \\in [a, b]$, let $I_x \\subseteq [a, b]$ be an interval of integration, either the fixed interval $[a, b]$ or the interval $[a, x]$.\nAn *integral equation* for $\\varphi$ is an equation of the form\n$$F\\left(x, \\varphi(x), \\int_{I_x} H\\left(x, s, \\varphi(s)\\right) ds\\right) = 0 \\quad \\text{for all } x \\in [a, b],$$\nwhere $F$ and $H$ are given functions.\nA *solution* is a function $\\varphi$ for which the [[integrals|0W29TBTY12]] exist and the equation holds for every $x \\in [a, b]$.\n\n## Remarks\n\nExample: $\\varphi(x) = x + \\int_0^1 x s \\, \\varphi(s) \\, ds$ on $[0, 1]$ is an integral equation with $I_x = [0, 1]$.\nTry $\\varphi(x) = c x$: the integral is $c x / 3$, so $c = 1 + c/3$ and $c = 3/2$.\nSo $\\varphi(x) = 3x/2$ is a solution.\n\nNon-example: $\\varphi'(x) = \\varphi(x)$ is a differential equation, not an integral equation, since $\\varphi$ does not appear under an integral.\nIts integral form $\\varphi(x) = \\varphi(0) + \\int_0^x \\varphi(s) \\, ds$ is an integral equation.\n\nThe same definition is used with a region of $\\R^m$ in place of $[a, b]$, and with vector-valued unknowns.\nWhen $I_x = [a, b]$ the equation is of Fredholm type; when $I_x = [a, x]$ it is of Volterra type.\n",
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