Question

find a differential operator that annihilates the given function.

$e^{5x}$

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$f\left(x\right)=e^{5x}$

Then

$$\begin{gathered} f\text{'}\left(x\right)=5e^{5x}=5f\left(x\right) \\ \Rightarrow f\text{'}\left(x\right)-5f\left(x\right)=0 \end{gathered}$$

Hence$(D-5)\left[f\right]=0$

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