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  "body": "\n## Summary\n\nA Volterra equation of the second kind is an equation $x(t) = f(t) + \\lambda \\int_a^t K(t, s) \\, x(s) \\, ds$, in which the unknown appears both outside and inside the integral.\nIt is the [[Volterra integral equation|DND8CBEVXT]] case of an [[integral equation of the second kind|8KJCN49V64]].\nUnlike the Fredholm case, it has exactly one continuous solution for every value of $\\lambda$ when the kernel and the function $f$ are continuous.\nInitial value problems for linear ordinary differential equations lead to equations of this kind.\n\n## Explanations\n\n### L1 human\n\nThe value of the unknown at time $t$ is a known input plus a weighted memory of its own past values.\nSo one can compute the solution forward in time, starting from $x(a) = f(a)$.\nFor example, $x(t) = 1 + \\int_0^t x(s) \\, ds$ means $x' = x$ with $x(0) = 1$, so $x(t) = e^t$.\nBecause only the past enters, there is no resonance, and the equation can always be solved.\n",
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