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  "body": "\n## Statement\n\nLet $I$ be an open interval, and let $A \\colon I \\to \\R^{n \\times n}$ and $f \\colon I \\to \\R^n$ be continuous.\nLet $\\Phi$ be a [[fundamental matrix|0H1GN6P9R5]] of the homogeneous [[linear system|037ZNY7NAG]] $x' = A(t) x$ on $I$, and put $\\Phi(t, s) = \\Phi(t) \\Phi(s)^{-1}$ for $t, s \\in I$.\nLet $t_0 \\in I$ and $x_0 \\in \\R^n$.\nThen the [[initial value problem|0FXDW992DS]]\n$$x' = A(t) x + f(t), \\qquad x(t_0) = x_0$$\nhas exactly one solution on $I$, and it is\n$$x(t) = \\Phi(t, t_0)\\, x_0 + \\int_{t_0}^{t} \\Phi(t, s)\\, f(s)\\, ds.$$\nIf $A$ is constant, then $\\Phi(t, s) = e^{(t - s) A}$, the [[matrix exponential|020M04FN1H]].\n\n## Remarks\n\nDifferentiating under the integral sign gives $x'(t) = A(t) \\Phi(t, t_0) x_0 + \\Phi(t, t) f(t) + \\int_{t_0}^t A(t) \\Phi(t, s) f(s)\\, ds = A(t) x(t) + f(t)$, since $\\Phi(t, t)$ is the identity.\nUniqueness holds because the difference of two solutions solves the homogeneous system with initial value $0$, so it is zero.\nThe formula is the matrix form of [[variation of parameters|0Y6TH5PQVS]].\n\nExample: for $x' = -x + 1$ and $x(0) = 0$, one has $\\Phi(t, s) = e^{-(t - s)}$.\nThe formula gives $x(t) = \\int_0^t e^{-(t - s)}\\, ds = 1 - e^{-t}$, and indeed $x' = e^{-t} = -x + 1$.\n\n## Explanations\n\n### L1 human\n\nThe principle says how to solve a linear equation with an outside force once one can solve it without the force.\nAt each instant $s$, the force gives the system a small kick $f(s)\\, ds$.\nThat kick then evolves by the unforced equation from time $s$ to time $t$.\nThe solution is the free evolution of the initial state plus the sum of all these evolved kicks.\n\n### L2 human\n\nThe function $\\Phi(t, s)$ is often called the state transition matrix or propagator.\nThe same idea works for evolution equations such as the heat equation and the wave equation, where $\\Phi(t, s)$ becomes a solution operator.\nIn the theory of $C_0$-semigroups, the formula defines the mild solution of an inhomogeneous abstract Cauchy problem.\nFor a boundary value problem, the analogous object is the [[Green's function|SH6JY9RN2E]].\n",
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