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  "body": "\n## Summary\n\nA Volterra equation of convolution type is a [[Volterra integral equation|DND8CBEVXT]] whose kernel depends only on the difference $t - s$.\nIts integral term is the convolution $\\int_0^t k(t - s) \\, x(s) \\, ds$ on the half-line.\nSuch equations describe systems whose rules do not change in time, and the [[Laplace transform|0V49JEZW06]] turns them into algebraic equations.\nRenewal equations, Abel's equation and many models with memory are of this type.\n\n## Explanations\n\n### L1 human\n\nThe weight given to a past value $x(s)$ depends only on how long ago it was, the lag $t - s$, and not on the date $s$ itself.\nThis is the integral version of a linear differential equation with constant coefficients.\nFor example, $x(t) = 1 + \\int_0^t e^{-(t - s)} x(s) \\, ds$ has memory that fades exponentially with the lag.\nIts solution is $x(t) = 1 + t$, as one can check by computing the integral.\n",
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