Question

In Problems 11–14 verify that the given differential operator annihilates the indicated functions.${\mathit{D}}^{\mathbf{4}}\mathit{;}\mathit{y}\mathit{=}\mathit{10}{\mathit{x}}^{\mathbf{3}}\mathit{-}\mathit{2}\mathit{x}$

Short answer

Proved

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(f(x))=0

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

Using the Differential operators

$y=10x^3-2$To verify that the differential operator D⁴ annihilates the function $\mathbf{y}\mathbf{=}\mathbf{10}{\mathbf{x}}^{\mathbf{3}}\mathbf{-}\mathbf{2}\mathbf{x}$, we have to verify that D⁴y=0 as the form L(y)=0 by applying the differential operator to the both sides of function and differentiate as the following:

localid="1667825587332" $$\begin{gathered} {\mathrm{D}}^4\mathrm{y}={\mathrm{D}}^4(10{\mathrm{x}}^3-2\mathrm{x}) \\ ={\mathrm{D}}^4\left(10{\mathrm{x}}^3\right)-{\mathrm{D}}^4\left(2\mathrm{x}\right) \\ ={\mathrm{D}}^3\left(30{\mathrm{x}}^2\right)-{\mathrm{D}}^3\left(2\right) \\ ={\mathrm{D}}^2\left(60\mathrm{x}\right)-0 \\ =\mathrm{D}\left(60\right) \\ =0 \end{gathered}$$

Hence the differential operator annihilates the indicated function.