MHB Does the Existence and Uniqueness Theorem Guarantee Solutions for dy/dx = 2xy²?

find_the_fun
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Given $$\frac{dy}{dx} =2xy^2$$ and the point $$y(x_0)=y_0$$ what does the existence and uniqueness theorem (the basic one) say about the solutions?

1) $$2xy^2$$ is continuous everywhere. Therefore a solution exists everywhere
2) $$\frac{\partial }{\partial y} (2xy^2) = 4xy$$ which is continuous everywhere. Therefore the solution is unique everywhere.

Is this all? What does the point $$y(x_0)=y_0$$ have to do with it? I actually couldn't find any fully worked examples of the existence and uniqueness theorem. Is there a way I should be writing the answers that is more mathy?
 
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find_the_fun said:
Given $$\frac{dy}{dx} =2xy^2$$ and the point $$y(x_0)=y_0$$ what does the existence and uniqueness theorem (the basic one) say about the solutions?

1) $$2xy^2$$ is continuous everywhere. Therefore a solution exists everywhere.

Actually not. $f(x,y)$ must be Lipschitz continuous in $y$ and continuous in $x$ in order to guarantee existence and uniqueness. $f(x,y)=2xy^2$ is continuous in $x$, but it is not Lipschitz continuous in $y$. Now, it is locally Lipschitz continuous in $y$; if you wanted to argue that on any finite interval containing $x_0$ there exists a unique solution, you'd be on solid ground.
 
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