# Existance and unicity of Cauchy problem

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 $$y'=f(x,y), y(x_0)=y_0$$.