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  "body": "\n## Summary\n\nA Volterra equation of the first kind is an equation $f(t) = \\int_a^t K(t, s) \\, x(s) \\, ds$ in which the unknown function $x$ appears only under the integral.\nIt is the [[Volterra integral equation|DND8CBEVXT]] analogue of an [[integral equation of the first kind|ZDS77Q0F06]].\nSolving it is like inverting an integration, so it behaves like differentiation and is in general ill-posed.\nWhen the kernel is smooth and does not vanish on the diagonal, it can be reduced to an equation of the second kind.\n\n## Explanations\n\n### L1 human\n\nThe data $f(t)$ is an accumulated quantity: a weighted total of the unknown $x$ over all past times $s \\le t$.\nThe task is to recover the rate $x$ from the totals.\nWith $K = 1$ and $a = 0$, the equation $f(t) = \\int_0^t x(s) \\, ds$ has the solution $x = f'$, provided $f$ is differentiable and $f(0) = 0$.\nFor instance, $f(t) = t^2$ gives $x(t) = 2t$.\nA small wiggle in $f$ can produce a large change in $f'$, which is why such equations are delicate.\n",
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