Solve the given initial$-$value problem in which the input function $g(x)$ is discontinuous. $[$Pint: Solve the problem en two intervals, and then find a solution so that $y$ and $y^{\text{'}}$ are continuous at $x=\frac{}{2}.]$

$y^{2}4y=g(x),y(0)=1,y^{\text{'}}(0)=1,$ where

$g(x)=\{\begin{array}{cc}sin(x),& 0x\frac{}{2}\end{array}$

$0,x>\frac{}{2}$

$y(x)=\{\begin{array}{cc},& 0x\frac{}{2}\\ ,& x>\frac{}{2}\end{array}$