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  "body": "\n## Definition\n\nLet $\\mathbb{T}$ be a time scale, that is, a nonempty closed subset of $\\R$.\nLet $\\sigma$ be its forward jump operator, and let $\\mathbb{T}^\\kappa$ be as in the [[definition of the delta derivative|C4B1FA1H0J]].\nFor a function $x \\colon \\mathbb{T} \\to \\R^n$, write $x^\\Delta$ for its delta derivative, taken componentwise.\nLet $f \\colon \\mathbb{T} \\times \\R^n \\to \\R^n$ be a function.\nThe *first-order dynamic equation* on $\\mathbb{T}$ with right side $f$ is\n$$x^\\Delta(t) = f(t, x(t)), \\qquad t \\in \\mathbb{T}^\\kappa.$$\nA *solution* is a function $x \\colon \\mathbb{T} \\to \\R^n$ that is delta differentiable at every $t \\in \\mathbb{T}^\\kappa$ and satisfies the equation there.\nDynamic equations of higher order are defined in the same way with iterated delta derivatives.\n\n## Remarks\n\nFor $\\mathbb{T} = \\R$, the delta derivative is the ordinary derivative and the equation is the [[ordinary differential equation|0NT64JV96A]] $x' = f(t, x)$.\nFor $\\mathbb{T} = \\Z$, the equation reads $x(t + 1) = x(t) + f(t, x(t))$, a first-order [[recurrence relation|0FYX44GP92]].\nExample: on $\\mathbb{T} = h\\Z$ with $h \u003e 0$, the function $x(t) = (1 + h p)^{t/h}$ solves $x^\\Delta = p x$, since $\\frac{x(t + h) - x(t)}{h} = \\frac{(1 + hp) - 1}{h} x(t) = p x(t)$.\nNon-example: on $\\mathbb{T} = \\Z$, the function $x(t) = e^t$ does not solve $x^\\Delta = x$, because $x(t + 1) - x(t) = (e - 1) e^t \\ne e^t$.\nAt a point $t$ with $\\sigma(t) \u003e t$, the equation fixes $x(\\sigma(t))$ from $x(t)$, so solutions of initial value problems advance across gaps by an explicit step.\n",
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