Grönwall's inequality

Statement

Let $a < b$ and let $I$ be an interval of the form $[a, b]$, $[a, b)$ or $[a, \infty)$. Let $I^\circ$ be the interior of $I$. Differential form: let $\beta, u : I \to \mathbb{R}$ be continuous, let $u$ be differentiable on $I^\circ$, and assume $u'(t) \le \beta(t)\, u(t)$ for each $t \in I^\circ$. Then $$u(t) \le u(a) \exp\left( \int_a^t \beta(s)\, ds \right) \quad \text{for each } t \in I.$$ Integral form: let $\beta, u : I \to \mathbb{R}$ be continuous with $\beta \ge 0$, let $\alpha : I \to \mathbb{R}$ be non-decreasing, and assume $$u(t) \le \alpha(t) + \int_a^t \beta(s)\, u(s)\, ds \quad \text{for each } t \in I.$$ Then $$u(t) \le \alpha(t) \exp\left( \int_a^t \beta(s)\, ds \right) \quad \text{for each } t \in I.$$