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  "body": "\n## Definition\n\nLet $t_0 \\in \\R$, $T \u003e 0$, $I = [t_0, t_0 + T]$ and $n \\ge 1$.\nLet $f \\colon I \\times \\R^n \\to \\R^n$ and $K \\colon I \\times I \\times \\R^n \\to \\R^n$ be continuous, and let $x_0 \\in \\R^n$.\nA *Volterra integro-differential equation* with initial value $x_0$ is the problem of finding a continuously differentiable $x \\colon I \\to \\R^n$ with\n$$x'(t) = f\\big(t, x(t)\\big) + \\int_{t_0}^{t} K\\big(t, s, x(s)\\big) \\, ds \\quad \\text{for } t \\in I, \\qquad x(t_0) = x_0.$$\nIt is *linear* when $f(t, x) = A(t) x + b(t)$ and $K(t, s, x) = B(t, s) x$ for continuous matrix functions $A$, $B$ and a continuous $b$.\nEquations with higher derivatives of $x$, or with derivatives of $x$ inside the integral, are included in the same name.\n\n## Remarks\n\nExample: $x'(t) = \\int_0^t x(s) \\, ds$, $x(0) = 1$, with $f = 0$ and $K(t, s, x) = x$.\nDifferentiating gives $x'' = x$ with $x(0) = 1$ and $x'(0) = 0$, so $x(t) = \\cosh t$.\nCheck: $\\int_0^t \\cosh s \\, ds = \\sinh t = x'(t)$.\n\nNon-example: $x'(t) = \\int_0^1 x(s) \\, ds$ for $t \\in [0, 1]$ has an integral over the fixed interval $[0, 1]$, including future times, so it is an integro-differential equation of Fredholm type, not of Volterra type.\n\nIntegrating from $t_0$ to $t$ turns the problem into the [[Volterra integral equation|DND8CBEVXT]]\n$$x(t) = x_0 + \\int_{t_0}^{t} \\left[f\\big(s, x(s)\\big) + \\int_{t_0}^{s} K\\big(s, r, x(r)\\big) \\, dr\\right] ds.$$\nIf $f$ and $K$ are Lipschitz in their last variable, uniformly in the others, successive approximations converge on all of $I$ and give a unique solution.\nThis generalizes the [[initial value problem|0FXDW992DS]] for ordinary differential equations, which is the case $K = 0$.\n",
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