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Take for exemple: for $y'= 2\sqrt{y}$ the 0 function and the functions :$ (x-c)^2$.

How can somebody say there infinite solutions for the initial value $y(0)=0$ ?

And how to find out that are there not infinite solutions for $y(1)=1$ ?

I read many time: use Picard Lindelöf to show the existence and the singularity of a solution, can you teach me on this exemple how to do it?

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Solution of for $y′=2\sqrt y$ is $y=(ax+b)^2$ becomes $y=(ax)^2$ for $y(0)=0$; a is arbitrary constant; hence infinite solutions.

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