ordinary differential equations – Does Picard’s Theorem hold?

We have the following Cauchy problems and we want to check if Picard’s Theorem holds to get existence and uniqueness.

  • $y'(x)=cos (xy^2)+y^{1+a} , y(x_0)=2, ain (0,1)$

  • $y'(x)=e^{xy}+y^a, y(x_0)=10, ain (0,1)$

So we have to check if the function of the right side is Lipschitz continuous, right?

I think the first one is not Lipscitz continuous in $y$ because of the term $cos (xy^2)$.

Ad also the second one… The exponential function isnot bounded and so it is not Lipschitz continuous.

So I think none of them satisfy Picard’s theorem. Is that correct ?