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  "body": "\n## Definition\n\nLet $a \u003c b$, let $\\Delta = \\set{(t, s) : a \\le s \\le t \\le b}$, let $K \\colon \\Delta \\to \\C$ be a [[kernel|GSMBXT7FPK]], and let $f \\colon [a, b] \\to \\C$ be a given function.\nA *linear Volterra equation of the first kind* is the equation\n$$\\int_a^t K(t, s) \\, x(s) \\, ds = f(t), \\quad t \\in [a, b],$$\nfor an unknown function $x$ on $[a, b]$.\nThe unknown appears only inside the [[integral|0W29TBTY12]], and the upper limit of integration is the variable $t$.\n\n## Remarks\n\nExample: $\\int_0^t (t - s) \\, x(s) \\, ds = t^3$ on $[0, 1]$ is of the first kind with $K(t, s) = t - s$.\nDifferentiating twice gives $x(t) = 6t$, and indeed $\\int_0^t (t - s) \\, 6s \\, ds = 3t^3 - 2t^3 = t^3$.\nNon-example: $x(t) - \\int_0^t x(s) \\, ds = 1$ is not of the first kind, since $x$ also appears outside the integral; it is a [[Volterra equation of the second kind|ATPSAHYG7C]].\nA necessary condition for a continuous solution when $K$ is continuous is $f(a) = 0$, since the integral vanishes at $t = a$.\nOften $K$ is required to be continuous on $\\Delta$, but weakly singular kernels such as $(t - s)^{-1/2}$ are also allowed, as in [[Abel's integral equation|G4GWBYTW7B]].\n",
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