Question

. find a differential operator that annihilates the given function.

$x^4-x^2+11$

Short answer

$D^5$is the differential operator that annihilates the given function.

Step-by-step solution

Differentiate the function continuously

Let the function be$f\left(x\right)=x^4-x^2+11$

Then

$$\begin{gathered} f\text{'}\left(x\right)=4x^3-2x \\ f\text{'}\text{'}\left(x\right)=12x^2-2 \\ f\text{'}\text{'}\text{'}\left(x\right)=24x \\ {f}^{\left(4\right)}\left(x\right)=24 \\ {f}^{\left(5\right)}\left(x\right)=0 \end{gathered}$$

Hence$D^5\left[f\right]=0$

Then $D^5$is the differential operator that annihilates the given function.