Chapter 8.5, Problem 18E

To determine

To show:

The second linearly independent solution of the equation $a{y}^{″}+b{y}^{\prime }+cy=0$, where $a$, $b$, and $c$ are constants with $r_0$ as a repeated root of the auxiliary equation $a{r}^2+br+c=0$ is $y_2\left(t\right)=t{e}^{r_{0}t}$.