Question

In Problems 15–26 find a linear differential operator that annihilates

the given function.

cos 2x

Short answer

(D²+ 4)

Step-by-step solution

Annihilator Operator

If L is a linear differential operator with constant coefficients and f is a sufficiently differentiable function such that

$L\left(f\right(x\left)\right)=0$

then L is said to be an annihilator of the function.

Using the Differential operators

We have the function,

cos 2x

And we have to annihilate it using a linear differential operator as the following technique:

$\cos \beta \chi$or $\sin \beta \chi$is being annihilated by the linear differential operator $(D^2+{\beta }^2)$.

Then we can annihilates the given function as

$$\begin{gathered} \left(D^2+2^2\right)(\cos 2x)=0 \\ (D^2+4)\left(\cos 2x\right)=0 \end{gathered}$$

Hence, the used differential operator is (D²+ 4)