Chapter 6.3, Problem 36E

Use the annihilator method to show that if $f\left(x\right)$ in $\left(4\right)$ has the form $f\left(x\right)=B{e}^{ax}$, then equation $\left(4\right)$ has a particular solution of the form $y_p\left(x\right)=x^{s}B{e}^{ax}$, where $s$ is chosen to be the smallest nonnegative integer such that $x^s{e}^{ax}$ is not a solution to the corresponding homogeneous equation.

$a_n{y}^{\left(n\right)}\left(x\right)+a_{n-1}{y}^{\left(n-1\right)}\left(x\right)+\cdots +a_{0}y\left(x\right)=f\left(x\right)$, $\left(4\right)$