---
tag: 0S1KG0XE40
kind: statement
role: theorem
name: Variation of constants formula for linear systems
aliases: [duhamel-formula]
msc: [34A30]
uses: [0A6QZ367GX, 0MQ24RBG93, 0YC8JT513J, 0Y6TH5PQVS]
sources:
  - url: https://en.wikipedia.org/wiki/Matrix_exponential
    locator: 'Inhomogeneous case generalization: variation of parameters'
  - url: https://en.wikipedia.org/wiki/Variation_of_parameters
    locator: Description of method
---

## 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|0MQ24RBG93]] 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|0Y6TH5PQVS]] for systems.
For $n = 1$, it is the [[solution formula for linear first-order equations|06B9W45XV0]].

## 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.