Chapter 2.4, Problem 8P

In each of Problem $7$ through $12$, state where in $ty$- plane the hypothesis of Theorem $\mathrm{2.4.2}$ are satisfied.

$y\text{'}={\left(1-t^2-y^2\right)}^{1/2}$.

Theorem $\mathrm{2.4.2}$:

Let the functions $f$ and $\partial f/\partial y$ be continuous in some rectangle $\alpha <t<\beta ,\gamma <t<\delta$ containing the point $\left(t_0,y_0\right)$. Then, in some interval $t_0-h<t<t_0+h$ contained in $\alpha <t<\beta$, there is unique solution $y=\varphi \left(t\right)$ of the initial value problem

$y\text{'}=f\left(t,y\right),y\left(t_0\right)=y_0$.