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  "body": "\n## Definition\n\nLet $X$ be a set, let $k \\ge 1$ be an integer, and let $f \\colon \\N \\times X^k \\to X$ be a function.\nLet $c_0, \\dots, c_{k-1} \\in X$.\nThe *initial value problem* for the [[recurrence relation|07TZV87Y0T]] of [[order|SNRN446BHC]] $k$ given by $f$, with initial values $c_0, \\dots, c_{k-1}$, is the system\n$$u_{n+k} = f(n, u_n, \\dots, u_{n+k-1}) \\quad (n \\ge 0), \\qquad u_0 = c_0, \\ \\dots, \\ u_{k-1} = c_{k-1}.$$\nA *solution* is a sequence $(u_n)_{n \\ge 0}$ in $X$ that satisfies all these equations.\n\n## Remarks\n\nEvery such problem has exactly one solution.\nThe terms $u_k, u_{k+1}, \\dots$ are computed one after the other from the recurrence, and induction on $n$ shows that no other choice is possible.\nExample: $u_{n+2} = u_{n+1} + u_n$ with $u_0 = 0$ and $u_1 = 1$ has the solution $0, 1, 1, 2, 3, 5, \\dots$, the Fibonacci numbers.\nNon-example: giving only $u_0 = 0$ for this order $2$ recurrence is not an initial value problem in this sense.\nBoth $0, 0, 0, \\dots$ and $0, 1, 1, 2, \\dots$ satisfy the recurrence with $u_0 = 0$, so one initial value is too few.\nInitial conditions may also be placed at another starting index $n_0$ instead of $0$.\n",
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