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  "body": "\n## Summary\n\nAn integro-differential equation involves both derivatives and integrals of the unknown function.\nA typical form is $x'(t) = f(t, x(t)) + \\int_{t_0}^{t} K(t, s, x(s)) \\, ds$, in which the integral describes memory of the past.\nSuch equations appear in viscoelasticity, population dynamics with age structure, circuit theory and epidemiology.\nThey are studied with the Laplace transform, by turning them into integral equations, or as functional differential equations.\n\n## Explanations\n\n### L1 human\n\nIn an ordinary differential equation, the rate of change depends on the present state.\nIn an integro-differential equation, it can also depend on an accumulated total over the past.\nIn an electric circuit with a resistor, an inductor and a capacitor, the voltage across the capacitor is the integral of the current.\nSo the circuit law mixes the current, its derivative and its integral.\nThe equation $x'(t) = -\\int_0^t x(s) \\, ds$ with $x(0) = 1$ has the solution $x(t) = \\cos t$.\n",
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