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  "body": "\n## Definition\n\nLet $a \u003c b$, let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[kernel|GSMBXT7FPK]], let $f \\colon [a, b] \\to \\C$ be given, and let $\\lambda \\in \\C$.\nFor each $x \\in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$.\nAn *integral equation of the second kind* is a [[linear integral equation|AYZCKG34QK]] of the form\n$$\\varphi(x) = f(x) + \\lambda \\int_{I_x} K(x, s) \\varphi(s) \\, ds \\quad \\text{for all } x \\in [a, b],$$\nfor the unknown function $\\varphi \\colon [a, b] \\to \\C$.\n\n## Remarks\n\nExample: $\\varphi(x) = 1 + \\lambda \\int_0^1 \\varphi(s) \\, ds$ on $[0, 1]$, with $K = 1$.\nAny solution is a constant $c$ with $c = 1 + \\lambda c$, so for $\\lambda \\ne 1$ the unique solution is $\\varphi = 1/(1 - \\lambda)$.\nFor $\\lambda = 1$ there is no solution.\n\nNon-example: $\\int_0^1 \\varphi(s) \\, ds = 1$ is not of the second kind, since $\\varphi$ does not appear outside the integral; it is of the first kind.\n\nIn operator notation, $(I - \\lambda K) \\varphi = f$, where $I$ is the identity and $K$ is the integral operator.\nSome authors write $\\varphi(x) - \\lambda \\int K \\varphi = f(x)$ or put $\\lambda$ into the kernel.\n",
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