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## Initial value problem $dy/dx=y^{1/3}$, $y(0)=0$ has one of the following solution

I came across the following problem which says:

The initial value problem $y'=y^{1/3}$, $y(0)=0$ has:

- $\begingroup$ The answer depends on your definition of $y^{1/3}$ when $y<0$. $\endgroup$ – Christian Blatter Nov 28, 2012 at 18:57
- 2 $\begingroup$ @ChristianBlatter No, it doesn't. The answer is "infinitely many solutions" regardless of what happens for $y<0$. Moreover, the only sensible interpretation in this context would be $y^{1/3}=-|y|^{1/3}$ for $y<0$. $\endgroup$ – Federico Dec 5, 2018 at 16:37

## 2 Answers 2

The answer is "none of the above". There are infinitely many solutions.

- 2 $\begingroup$ I see why this happened. This is because the function is ill defined on the negative axis. So hope the question asker takes note. $\endgroup$ – Gautam Shenoy Nov 29, 2012 at 6:26
- 1 $\begingroup$ i know the uniqueness criteria for first order differential equations and from that criteria i know that the above problem doesn't have a unique solution. so can we say either unique solution or infinitely many solutions? $\endgroup$ – PAMG Feb 16, 2016 at 4:03
- 3 $\begingroup$ @GautamShenoy I don't know what do you mean by "the function is ill defined on the negative axis", but it has nothing to do with the problem. The answer is "infinitely many solutions" regardless of what happens for $y<0$. Moreover, the only sensible interpretation in this context would be $y^{1/3}=-|y|^{1/3}$ for $y<0$. $\endgroup$ – Federico Dec 5, 2018 at 16:39
- 1 $\begingroup$ $y=0$ looks like a solution, corresponding to $\alpha=+\infty$ $\endgroup$ – Henry Oct 1, 2022 at 15:50

2.y=0 for $x \le 0$ and $y=(2/3x)^{(3/2)}$ (positive squ root)

3.similarly consider neg squ root.

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