Remark 1

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

Functions of a Real Variable, chapter IV, § 1. EXISTENCE THEOREMS

There can be infinitely many integrals of a differential equation (1) taking the same value at a given point. For example, the scalar differential equation $x' = 2\sqrt{|x|}$ has all the functions defined by

\begin{array}{ll} u(t) = 0 & \text{for } -\beta \leq t \leq \alpha \\ u(t) = -(t + \beta)^2 & \text{for } t \leq -\beta \\ u(t) = (t - \alpha)^2 & \text{for } t \geq \alpha \end{array}

as integrals taking the value 0 at the point $t = 0$ , for any positive numbers $\alpha$ and $\beta$ .

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

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