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?


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).

  • 1
    $\begingroup$ 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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