i have the following Osgood Lemma: let $f(x,y)$ a function such as $|f(x,y_1)-f(x,y_2)| \leq h(|y_2 - y_1|)$ for all $(x,y_1)$ and $(x,y_2)$ in an opena $\Omega \subset \mathbb{R}^2$. We suppose that function $h$ is continuous on each point $u \in ]0,\alpha]$, stricly positive and $\lim_{\epsilon \to 0^+} \displaystyle\int_{\epsilon}^{\alpha} \dfrac{d u}{h(u)}=+\infty$, with $\alpha > 0$. Then for all $(x_0,y_0)$ on $\Omega$, the problem $y'=f(x,y)$ admits at last one solution.
My question is: please can you give me an example to show how we can apply this theorem to prouve existence of solution of Cauchy problem $y'=f(x,y), y(x_0)=y_0$.
Thank you in advance.