So I am supposed to have a closer look at the initial value problem $$y' = y^2 + e^{-x^2}, y(0) = 0$$ The first point to show is whether the DEQ has a solution for $|x| \leq 1/2$.For this I used the Picard-Lindelöf theorem, which gives me the following inequality: $$x \leq min(A, B/{(B^2 + 1)}) $$ with $|x| \leq A$ and $|y| \leq B$. So I can just set $A,B$ accordingly to get $|x| \leq 1/2$. This however would imply to me that I can just make $A$ very big and $B$ as close to $0$ as needed to extend the solution interval for $x$ to the whole real numbers. (This seems strange and wrong to me but I can't find the mistake). Lastly I am supposed to give a reason why this DEQ causes problems for the Picard iteration. After looking at a few iterations I can say that the integral blows up heavily but this shouldn't be a sufficient argument against the iteratin, so if someone has an idea where the problem really is: I would really appreciate a hint...
Thanks in advance
The second term in the minimum, $\frac{B}{1+B^2}$, goes to zero as $B$ goes to infinity, the denominator grows faster than the numerator. It takes indeed its largest value $\frac12$ at $B=1$.
You can get a slightly larger interval of existence by computing an upper bound, as $e^{-x^2}<1$ the solution has as upper bound the solution of $u'=u^2+1$, giving $x\le \frac\pi2$.