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

Since we have second order ode with only one IC, we expect that there will remain one constant in the final solution that can not be solved for. The general solution is

\[ y=c_{1}\cos x+c_{2}\sin x+\frac {\sin x}{2}-x\frac {\cos x}{2}\]

One equation is generated for each IC. This gives

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

Hence \(c_{1}=0\) and the particular solution is

\[ y=c_{2}\sin x+\frac {\sin x}{2}-x\frac {\cos x}{2}\]

We see \(c_{2}\) remains unresolved.