Q:

guarantee existence but not uniqueness

How can I use the theorem of existence and uniqueness for 1st order differential equations to guarantee existence but not uniqueness

Theorem: Let $R$ be a rectangular region in the $xy$-plane defined by $$a\leq x\leq b, ~~c\leq y\leq d$$ that contains the point $(x_0,y_0)$ in its interior. If $f(x,y)$ and $\partial f/\partial y$ are continuous on $R$ , then there exists an interval $I$ centered at $x_0$ and a unique function $y(x)$ defined on $I$ satisfying the initial-value problem $$y'=f(x,y),~~y(x_0)=y_0$$

For example for
$$\frac{dy}{dx}=y^\frac{1}{3}\:\:\:\:\:\:\:\:\:\:\:\: y(0)=0$$
Since x=0 in the initial conditions the theorem would say that there's not a unique solution, but how to use it to prove the existence of solutions?

A:

The existence and uniqueness theorem actually has two parts.  The existence part (sometimes called the Peano existence theorem) requires only continuity of $f$, the uniqueness part (the Picard–Lindelöf theorem) has stronger requirements (actually the requirement is that $f$ is locally Lipschitz in the $y$ variable).

