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  "body": "\n## Statement\n\nLet $U$ be an open subset of $\\mathbb{R}^n$, let $f : U \\to \\mathbb{R}^n$ be a function. Let $\\gamma : I \\to U$ be a solution of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ on an interval $I$.\nLet $c \\in \\mathbb{R}$ and let $I - c = \\{t - c : t \\in I\\}$.\nThen the function $\\delta : I - c \\to U$ given by $\\delta(t) = \\gamma(t + c)$ is also a solution of $x' = f(x)$.\n\n## Remarks\n\nProof: by the chain rule, $\\delta'(t) = \\gamma'(t + c) = f(\\gamma(t + c)) = f(\\delta(t))$.\nSo it is enough to solve an initial value problem with initial time $0$. The solution with $x(t_0) = x_0$ is $t \\mapsto \\varphi(t - t_0)$, where $\\varphi$ solves $x(0) = x_0$.\nThe result fails for non-autonomous equations. For $x' = t$, the solution $t^2/2$ is a solution, but $(t+1)^2/2$ is not.\n\n## Explanations\n\n### L1 human\n\nIf the law of motion does not change with time, a motion that starts later is the same motion, only delayed.\nSo shifting a solution in time gives another solution.\n",
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