Lemma 1

Let $a(x), b(x)$ be two continuous real functions that do not change sign on an interval $[x_0, +\infty[$. If, on this interval, the function $y(x)$ is continuous and differentiable, and satisfies the identity

(1)

then there exists an interval $[x_1, +\infty[$ on which $y$ does not change sign.

Indeed, let us put $z(x) = y(x) \exp \left( - \int_{x_0}^x a(t) \, dt \right)$ (cf. IV, p. 183); then, by (1), $z'(x) = b(x) \exp \left( - \int_{x_0}^x a(t) \, dt \right)$. If $b(x) \geqslant 0$ for $x \geqslant x_0$ then $z$ is increasing on this interval, so either is $< 0$ on all this interval, or is zero on an interval $[x_1, +\infty[$, or else is $> 0$ on an interval $[x_1, +\infty[$; since $y$ has the same sign as $z$ the proposition is proved in this case. The argument is similar if $b(x) \leqslant 0$ for $x \geqslant x_0$.