Chapter 8.1, Problem 12E

To determine

To proof:

The polynomial $\varphi \left(x\right)$ has derivatives of all orders on $\left(a,b\right)$ if $y=\varphi \left(x\right)$ is a solution to the differential equation $y^{″}+p\left(x\right)y^{\prime }+q\left(x\right)y=g\left(x\right)$ on the interval $\left(a,b\right)$, here $p$, $q$, and $g$ possess derivatives of all orders.