Exercise 2, § 1

Given two numbers $b > 0, M > 0$ and an arbitrary number $\varepsilon > 0$, give an example of a scalar differential equation $x' = g(x)$ such that $|g(x)| \leq M$ for $|x| \leq b$ and which admits an integral $x = u(t)$ continuous on the interval $] - b/M - \varepsilon, b/M + \varepsilon[$, but which does not have a finite limit at the point $x = \frac{b}{M} + \varepsilon$ (define $g$ so that the integral considered has a continuous derivative on $] - b/M - \varepsilon, b/M + \varepsilon[$, this derivative being equal to the constant $M$ on $] - b/M, b/M[$).