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consider the following equation: $$ y'=y^{\frac{1}{3}}, \,y(0)=0 $$

My question is how can I prove the existence and uniqueness of solutions of this initial value problem without solving the equation?

Can anyone help me with this? Thanks!

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You can prove existence by writing down a solution. But uniqueness does not hold. –  Hans Engler Apr 18 at 19:53
I got it, thanks a lot. –  user133140 Apr 18 at 19:58
@user133140, the trivial solution comes to mind. –  Nameless Apr 18 at 19:59
@HansEngler, oh you are right, it is not if and only if. –  Nameless Apr 18 at 20:02
The theorem you used just now is a little bit different from the theorem I learned, and in my text book, f(x,y) should be linear. Maybe I need to look up other text books. @Nameless –  user133140 Apr 18 at 20:09

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