Chapter 4.6, Problem 20E

Use the method of variation of parameters to show that

$y\left(t\right)=c_1\cos t+c_2\sin t+{\displaystyle {\int }_0^{t}f\left(s\right)\sin \left(t-s\right)ds}$

is a general solution to the differential equation

$y^{\prime \prime }+y=f\left(t\right)$

where $f\left(t\right)$ is a continuous function on $\left(-\infty ,\infty \right)$.[Hint: Use the trigonometric identity $\sin \left(t-s\right)=\sin t\cos s-\sin s\cos t$.]