For each initial$-$value problem, determine whether Theorem $1$ applies to show the

existence of a unique solution in an open interval I containing $x=0.$

$($a$)\frac{dy}{dx}=x{y}^{\frac{4}{3}},y(0)=0$

$($b$)\frac{dy}{dx}=x{y}^{\frac{1}{3}},y(0)=0$

$($c$)\frac{dy}{dx}=x{y}^{\frac{1}{3}},y(0)=1$