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  "body": "\n## Summary\n\nA Volterra integro-differential equation is an [[integro-differential equation|YRKW2KX2CW]] in which the integral runs only over the past.\nThe integral goes from a fixed initial time up to the current time.\nSo the rate of change at time $t$ depends on the present state and on the history up to $t$, not on the future.\nTogether with an initial condition it is equivalent to a [[Volterra integral equation|DND8CBEVXT]], and under Lipschitz conditions it has a unique solution.\nSuch equations model materials with memory, such as viscoelastic bodies, and populations whose growth depends on past sizes.\n\n## Explanations\n\n### L1 human\n\nImagine a spring made of a material that remembers how it was stretched before.\nIts force now depends on its current stretch and on a weighted sum of all past stretches.\nNewton's law then gives an equation with both a derivative and an integral over the past: a Volterra integro-differential equation.\nA small example is $x'(t) = \\int_0^t x(s) \\, ds$ with $x(0) = 1$, whose solution is $x(t) = \\cosh t$.\n",
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