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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?

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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).

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    $\begingroup$ I swear to god you explained this theorem to me in that line better than any other site I've found on google. Thanks. $\endgroup$ – Overcode Sep 3 '15 at 0:48

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