Question $1(10$ points$)$

Theorem $(Existence$ and Uniqueness$)$: Let $Rbe$ a rectangular region $in$ the $xy-$ plane

defined $byaxb,cyd{x}_0,y_{0}f(x,y)$ and $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}$ are continuous $onR,$ then there exists some interval

$I_0$:$(x_0-h,x_0+h),h>0$ contained $ina,b,$ and a unique function $y(x_0)=y_0$

Determine whether the theorem guarantees a unique solution for:

$\frac{dy}{dx}=x^2+y^2,y(0)=1$