1.3.3.1 \(y^{\prime }+y=\sin \left ( x\right ) \) with one IC \(y\left ( 0\right ) =0\)

The general solution is \(y=-\frac {\cos x}{2}+\frac {\sin \left ( x\right ) }{2}+c_{1}e^{-x}\) and since one IC is given, then we set up one equation. This equation is

\begin{align*} 0 & =-\frac {\cos \left ( 0\right ) }{2}+\frac {\sin \left ( 0\right ) }{2}+c_{1}e^{-\left ( 0\right ) }\\ 0 & =-\frac {1}{2}+c_{1}\end{align*}

And now we solve for \(c_{1}\).