Question

In Problems 11–14 verify that the given differential operator annihilates the indicated functions.

${\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{64}\mathbf{;}\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{cos}\mathbf{8}\mathbf{x}\mathbf{-}\mathbf{5}\mathbf{sin}\mathbf{8}\mathbf{x}$

Short answer

Proved

Step-by-step solution

Annihilator Operator

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

$\mathrm{L}\left(\mathrm{f}\right(\mathrm{x}\left)\right)=0$

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

Using the Differential operators

To verify that the differential operator$({\mathrm{D}}^2+64)$annihilates the function $\mathrm{y}=2\cos 8\mathrm{x}-5\sin 8\mathrm{x}$, we have to verify that$({\mathrm{D}}^2+64)\mathrm{y}=0$as the form$\mathrm{L}\left(\mathrm{y}\right)=0$by applying the differential operator to the both sides of function and differentiate as the following:

$$\begin{gathered} ({\mathrm{D}}^2+64)\mathrm{y}=({\mathrm{D}}^2+64)(2\cos 8\mathrm{x}-5\mathrm{sim}8\mathrm{x}) \\ ={\mathrm{D}}^2(2\cos 8\mathrm{x}-5\sin 8\mathrm{x})+64(2\cos 8\mathrm{x}-5\sin 8\mathrm{x}) \\ ={{\mathrm{D}}^2}^{}\left(2\cos 8\mathrm{x}\right)-{\mathrm{D}}^2\left(5\sin 8\mathrm{x}\right)+128\cos 8\mathrm{x}-320\sin 8\mathrm{x} \\ =\mathrm{D}(-16\sin 8\mathrm{x})-\mathrm{D}\left(40\cos 8\mathrm{x}\right)+128\cos 8\mathrm{x}-320\sin 8\mathrm{x} \\ =-128\cos 8\mathrm{x}+320\sin 8\mathrm{x}+128\cos 8\mathrm{x}-320\sin 8\mathrm{x} \\ =0+0 \\ =0 \end{gathered}$$

Hence the differential operator annihilates the indicated function.