Proposition 3

Tag 08CX, permanent. Cite it as [Bourbaki, Tag 08CX].

Functions of a Real Variable, chapter IV, § 2. LINEAR DIFFERENTIAL EQUATIONS

With the notation of th. 2 (IV, p. 179), for every point $(t_0, \mathbf{x}_0)$ of $J \times \mathbf{E}$ the integral of the linear equation (2) defined on $J$ and equal to $\mathbf{x}_0$ at $t_0$ is given by the formula

\mathbf{u}(t) = C(t, t_0).\mathbf{x}_0 + \int_{t_0}^t C(t, s).\mathbf{b}(s)\,ds.

The method which leads to formula (10), which consists of taking the function $\mathbf{z}$ as a new unknown function, is often called the "method of variation of parameters".

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

IV, p. 179