$(a)$ Solve the differential equation $y^{\text{'}}=\frac{1}{2}x\sqrt[\phantom{2}]{1-16y^2}.$

$($ for $-\frac{}{2}{x}^2+C\frac{}{2})$

$(b)$ Solve the initial$-$value problem $y^{\text{'}}=\frac{1}{2}x\sqrt[\phantom{2}]{1-16y^2},y(0)=0.$

$($ for $-\sqrt[\phantom{2}]{\frac{}{2}}x\sqrt[\phantom{2}]{\frac{}{2}})$