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  "body": "\n## Summary\n\nA Volterra integral equation is an integral equation in which the integral runs from a fixed point up to the variable $t$.\nThe typical form of the second kind is $x(t) = f(t) + \\int_a^t K(t, s) x(s) \\, ds$.\nEvery [[initial value problem|0FXDW992DS]] for an ordinary differential equation can be written as a Volterra equation of the second kind, in general a nonlinear one.\nUnder mild conditions, the linear equation of the second kind has a unique solution, given by a convergent series of iterated kernels.\n\n## Explanations\n\n### L1 human\n\nIn a Volterra equation, the value at time $t$ is built from a known part and a weighted sum over all earlier times.\nSo the solution depends only on the past, as in a process that unfolds in time.\nThe equation $x(t) = 1 + \\int_0^t x(s) \\, ds$ is the integral form of $x' = x$, $x(0) = 1$.\nIts solution is $x(t) = e^t$.\n",
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