Determine the contants and the largest interval such that the solution of the IVP exist and unique on rectangular area. $i-)\; x^{\prime}=x^2, \; x(0)=1,\;R=\{(t,x):|t|\leq2,\;|x-1|\leq2\}$.
Let $f(t,x)=x^2\;$ and $$\;|f(t,x)|=|x^2|\leq9=L$$ The function satisfying the Lipschitz Condition on R, since;
$$\left|\dfrac{\partial f}{\partial x}\right|=|2x|\leq6$$ So,
$$h=\min\left(a,\frac{b}{L}\right)=\min\left(2,\frac{2}{9}\right)=\frac{2}{9}$$
Hence, the largest interval is;
$$\left|t-0\right|\leq h=\frac{2}{9} \rightarrow -\frac{2}{9}\leq t \leq \frac{2}{9}$$
Such that the solution of IVP exist and unique on R
Is it true? Thanks in advance!