The differential operator $D^2+12 D +72$ has the form $D^2 - 2\alpha D +\alpha^2+\beta^2$ where $\alpha =$ and $\beta =$ .

Therefore $D^2+12 D +72$ should annihilate the function $f=e^{-6x}\cos\!\left(6x\right)$ and $g = e^{-6x}\sin\!\left(6x\right)$. We will check that for $e^{-6x}\cos\!\left(6x\right)$. Note that when we compute the derivative and second derivative we will get terms that have $e^{-6x}\sin\!\left(6x\right)$ in them, so we will have to account for them below.

Compute $D^2\left(e^{-6x}\cos\!\left(6x\right)\right)$, $12 D\left(e^{-6x}\cos\!\left(6x\right)\right)$, and $72 e^{-6x}\cos\!\left(6x\right)$. Place the coefficients from the terms $e^{-6 x}\cos(6 x)$ and $e^{-6 x}\sin(6 x)$ in the table. The columns of the table should add to zero.