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  "body": "\n## Statement\n\nLet $I$ be an open interval, let $A : I \\to \\mathbb{R}^{n \\times n}$ and $b : I \\to \\mathbb{R}^n$ be continuous, and let $\\Phi$ be a [[fundamental matrix|0MQ24RBG93]] of $x' = A(t) x$ on $I$.\nLet $t_0 \\in I$ and $x_0 \\in \\mathbb{R}^n$.\nThen the unique solution on $I$ of $x' = A(t) x + b(t)$, $x(t_0) = x_0$ is\n$$x(t) = \\Phi(t)\\, \\Phi(t_0)^{-1} x_0 + \\Phi(t) \\int_{t_0}^{t} \\Phi(s)^{-1}\\, b(s)\\, ds, \\qquad t \\in I.$$\n\n## Remarks\n\nFor a constant matrix $A$, take $\\Phi(t) = e^{tA}$. The formula becomes $x(t) = e^{(t - t_0) A} x_0 + \\int_{t_0}^{t} e^{(t - s) A}\\, b(s)\\, ds$.\nThe formula comes from the trial solution $x(t) = \\Phi(t)\\, c(t)$: the equation gives $\\Phi(t)\\, c'(t) = b(t)$. This is [[variation of parameters|0Y6TH5PQVS]] for systems.\nFor $n = 1$, it is the [[solution formula for linear first-order equations|06B9W45XV0]].\n\n## Explanations\n\n### L1 human\n\n\nThe first term is the free motion from the initial state.\nThe integral adds up the effect of the force $b(s)$ at each earlier time $s$, carried forward to time $t$ by the homogeneous system.\n",
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        "url": "https://en.wikipedia.org/wiki/Matrix_exponential"
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