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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]], and let $f \\colon [a, b] \\to \\C$ be given.\nFor each $x \\in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$.\nAn *integral equation of the first kind* is a [[linear integral equation|AYZCKG34QK]] of the form\n$$\\int_{I_x} K(x, s) \\varphi(s) \\, ds = f(x) \\quad \\text{for all } x \\in [a, b],$$\nfor the unknown function $\\varphi \\colon [a, b] \\to \\C$.\n\n## Remarks\n\nExample: $\\int_0^1 (x + s) \\varphi(s) \\, ds = x$ on $[0, 1]$ is of the first kind.\nWith $\\alpha = \\int_0^1 \\varphi$ and $\\beta = \\int_0^1 s \\varphi(s) \\, ds$ it says $\\alpha x + \\beta = x$, so $\\alpha = 1$ and $\\beta = 0$.\nThe function $\\varphi(s) = 4 - 6s$ satisfies both, since $\\int_0^1 (4 - 6s) \\, ds = 1$ and $\\int_0^1 (4s - 6s^2) \\, ds = 0$; so do infinitely many others.\n\nNon-example: $\\varphi(x) = x + \\int_0^1 (x + s) \\varphi(s) \\, ds$ is not of the first kind, because $\\varphi$ also appears outside the integral.\n\nWith $I_x = [a, b]$ it is a Fredholm equation of the first kind; with $I_x = [a, x]$ it is a Volterra equation of the first kind.\nThe example shows that the solution need not be unique.\nThe right side $f(x) = x^2$ shows that it need not exist, since the left side is always affine in $x$.\n",
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