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I was asked this as a hw problem. However Im confused what the solution in the back of the book means.

determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point $(X_0 , Y_0 )$ in the region.

$$\frac{dy}{dx} = y^{\frac{2}{3}}$$

Any value plugged into y should give it a unique solution right? So a unique solution should span (- infinity, infinity)

However in the back of the book it says the answer is :

half-planes define by either y<0 or y <0

What does that answer mean?

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    To start can you find the function $y$ which solve the differential equation. – 2017-01-28
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    @marshalcraft the anti-derivative (3y^(5/3)) / 5 . Then using both the derivative and the function y ,would I determine what values of y won't give me a unique solution. – 2017-01-28
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    When they say unique solution, they mean only one funtion satisfies or is a solution to the d.e. You get constants from integration. With out specifying the value of those constants there isn't a unique solution or it is not enough constraints. Apparently $X_0, Y_0$ for some values, the solutions to the d.e. aren't unique meaning there are multiple functions which satisfy the initial condition and the d.e. – 2017-01-28
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    Also that isn't a solution to the above d.e. – 2017-01-28
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    To put it another way. You have a function $f(x)$ and the derivative $f'(x)$ is the cube root of the square of $f$. Also $f(X_0)=Y_0$ for some $X_0$ and $Y_0$, additionally there must only be one such function which satisfies this for the $X_0$ and $Y_0$. Apparently for half of the possible range of $X_0$ this isn't the case. – 2017-01-28

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