I am not having good knowledge of ordinary differential equation so it may not be good question but i am unable to solve it.
I have to find maximal interval of unique solution using Picard existence and uniqueness theorem for the ODE $$y'=\sqrt{x^2+y^2},\quad y(0)=0$$ I tried it by taking taking rectangular region $|x|\leq a, |y|\leq b$ as in existence and uniqueness theorem. Solution exists and unique in interval $(-h,h)$ where $h=\min(a,\frac{b}{M})=\min({a, \frac{b}{\sqrt{a^2+b^2}}})$ and $M=\sup f(x)$ over rectangular region $|x|\leq a, |y |\leq b$ as usual. Now I put $a=\frac{b}{\sqrt{a^2+b^2}}$ and find $a^4+a^2b^2=b^2$ which implies $|a|\leq 1$. So according to me $h=1$ but answer is given $h=\frac{1}{\sqrt{2}}$.
Where is my mistake?