Chapter 6.3, Problem 37E

To determine

To prove:

If $a_n{y}^{\left(n\right)}\left(x\right)+a_{n-1}{y}^{\left(n-1\right)}\left(x\right)+\mathrm{...}+a_{0}y\left(x\right)=f\left(x\right)$ and $f\left(x\right)=a\cos \left(\beta x\right)+b\sin \left(\beta x\right)$, then the particular solution is of the form $y_p\left(x\right)=x^s\left(A\cos \left(\beta x\right)+B\sin \left(\alpha x\right)\right)$, where $s$ is chosen to be the smallest nonnegative integer such that $x^s\cos \left(\beta x\right)$ and $x^s\sin \left(\beta x\right)$ are not solutions to the corresponding homogeneous equation.