Variation of constants formula for linear systems

Statement

Let $I$ 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 of $x' = A(t) x$ on $I$. Let $t_0 \in I$ and $x_0 \in \mathbb{R}^n$. Then the unique solution on $I$ of $x' = A(t) x + b(t)$, $x(t_0) = x_0$ is $$x(t) = \Phi(t)\, \Phi(t_0)^{-1} x_0 + \Phi(t) \int_{t_0}^{t} \Phi(s)^{-1}\, b(s)\, ds, \qquad t \in I.$$

Remarks

For a constant matrix $A$, take $\Phi(t) = e^{tA}$. The formula becomes $x(t) = e^{(t - t_0) A} x_0 + \int_{t_0}^{t} e^{(t - s) A}\, b(s)\, ds$. The 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 for systems. For $n = 1$, it is the solution formula for linear first-order equations.

Explanations

L1 human

The first term is the free motion from the initial state. The integral adds up the effect of the force $b(s)$ at each earlier time $s$, carried forward to time $t$ by the homogeneous system.