can someone take a look at this

$\displaystyle y^{'}(t)+y(t)=sint$

$\displaystyle y(0)=0$

$\displaystyle y^{'}e^x+ye^x=e^xsint$

$\displaystyle \int y^{'}e^x=\int e^xsintdx$

$\displaystyle ye^x=\frac{e^x}{2}(sinx-cosx)+C$

$\displaystyle y=\frac{1}{2}(sinx-cosx)+Ce^{-x}$

$\displaystyle 0=\frac{1}{2}(0-1)+C$

$\displaystyle C=\frac{1}{2}$

$\displaystyle y=\frac{1}{2}(sinx-cosx)+\frac{1}{2}e^{-x}$

oh yes it was a stupid mistake I have corrected it thank you