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  "body": "\n## Definition\n\nLet $a \u003c b$, let $a_0, f \\colon [a, b] \\to \\C$ be given functions, let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a given [[kernel|GSMBXT7FPK]], and let $\\lambda \\in \\C$.\nFor each $x \\in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$.\nA *linear integral equation* for the unknown function $\\varphi \\colon [a, b] \\to \\C$ is an [[integral equation|R3SG1SN7JQ]] of the form\n$$a_0(x) \\varphi(x) = f(x) + \\lambda \\int_{I_x} K(x, s) \\varphi(s) \\, ds \\quad \\text{for all } x \\in [a, b].$$\n\n## Remarks\n\nExample: $\\varphi(x) = \\sin x + \\int_0^{\\pi} \\cos(x - s) \\varphi(s) \\, ds$ is linear, with $a_0 = 1$, $f = \\sin$, $K(x, s) = \\cos(x - s)$ and $\\lambda = 1$.\n\nNon-example: $\\varphi(x) = 1 + \\int_0^1 x s \\, \\varphi(s)^2 \\, ds$ is not linear, because $\\varphi$ enters through $\\varphi(s)^2$.\nIf $\\varphi$ solves it, then $2\\varphi$ does not solve the equation with $f$ replaced by $2$.\n\nThe equation is of the first kind when $a_0 = 0$ and of the second kind when $a_0 = 1$.\nIt is of the third kind when $a_0$ is a [[continuous function|0X3WGX95FE]] that vanishes somewhere but not everywhere.\nIf $a_0$ has no zeros, dividing by $a_0$ gives an equation of the second kind.\nIn operator notation the equation reads $a_0 \\varphi = f + \\lambda K \\varphi$, where $K$ also denotes the integral operator.\n",
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