Question

(a): As in problem 12,
${\mathit{D}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{D}\mathbf{+}\mathbf{1}$linear?

(b): Is${\mathit{x}}^{\mathbf{2}}{\mathit{D}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathit{D}\mathbf{+}\mathbf{7}$ a linear operator?

Short answer

1. Differentiating the operator${\mathrm{D}}^2+2\mathrm{D}+1$with the objects being operated on are the function f (x) by differentiating it with respect to x representing a linear operator.

Differentiating the operator${\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7$with the objects being operated on are the function f (x) by differentiating it with respect to x representing a linear operator

Step-by-step solution

 Step 1: Definition of the linear operator

An operator is said to be a linear operator if the following relations are satisfied if

$\mathbf{O}\left(A+B\right)\mathbf{=}\mathbf{O}\left(A\right)\mathbf{+}\mathbf{O}\left(B\right)\mathbf{}\mathbf{and}\mathbf{,}\mathbf{}\mathbf{O}\left(\mathrm{kA}\right)\mathbf{=}\mathbf{kO}\left(A\right)$

where, is a number, and and, are numbers, functions, vectors, etc.

 Step 2: Parameters

Given the following Differentiating operator${\mathrm{D}}^2+2\mathrm{D}+1$.

It is to show that the differentiating operator ${\mathrm{D}}^2+2\mathrm{D}+1$,which operates on the function f (x) by differentiating it with respect to x representing a linear operator.

Operation on differentiating operators ddx for O(A+B)=O(A)+O(B) and O(kA)=kO(A)

Find $\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right].$

$$\begin{gathered} \frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)\right]+\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{g}\left(\mathrm{x}\right)\right] \\ \frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\mathrm{k}\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)\right] \\ \mathrm{Where},\mathrm{k}\mathrm{is}\mathrm{a}\mathrm{constant}. \end{gathered}$$

Operation on differentiating operators d2dx2for O(A+B)=O(A)+O(B) and O(kA)=kO(A) 

$\mathrm{Find}\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right].$

$$\begin{gathered} {\mathrm{D}}^2\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \\ =\frac{\mathrm{d}}{\mathrm{dx}}\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \\ =\frac{\mathrm{d}}{\mathrm{dx}}\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)\right]+\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{g}\left(\mathrm{x}\right)\right]\right\} \\ =\frac{\mathrm{d}}{\mathrm{dx}}\frac{\mathrm{df}\left(\mathrm{x}\right)}{\mathrm{dx}}+\frac{\mathrm{d}}{\mathrm{dx}}\frac{\mathrm{dg}\left(\mathrm{x}\right)}{\mathrm{dx}} \end{gathered}$$

Solve further.

localid="1658984281233" $$\begin{gathered} {\mathrm{D}}^2\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{f}\left(\mathrm{x}\right)+\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{g}\left(\mathrm{x}\right) \\ ={\mathrm{D}}^2\left[\mathrm{f}\left(\mathrm{x}\right)\right]+{\mathrm{D}}^2\left[\mathrm{g}\left(\mathrm{x}\right)\right] \\ \mathrm{Find}={\mathrm{D}}^2\left[\mathrm{kf}\left(\mathrm{x}\right)\right] \\ {\mathrm{D}}^2\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{kf}\left(\mathrm{x}\right)\right] \\ =\mathrm{k}\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{f}\left(\mathrm{x}\right)\right] \\ ={\mathrm{kD}}^2\left[\mathrm{f}\left(\mathrm{x}\right)\right] \\ \mathrm{Where},\mathrm{k}\mathrm{is}\mathrm{a}\mathrm{constant}. \end{gathered}$$

Operation on differentiating operators d2dx2+2ddx+1 for O(A+B)=O(A)+O(B) and O(kA)=kO(A)

Find $\left[{\mathrm{D}}^2+2\mathrm{D}+1\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]$.

$$\begin{gathered} \left[{\mathrm{D}}^2+2\mathrm{D}+1\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\left[\frac{d^2}{d{x}^2}+2\frac{d}{dx}+1\right]\left[f\left(x\right)+g\left(x\right)\right] \\ =\frac{d^2}{d{x}^2}\left[f\left(x\right)+g\left(x\right)\right]+2\left[f\left(x\right)+g\left(x\right)\right]+\left[f\left(x\right)+g\left(x\right)\right] \end{gathered}$$

Solve further.

$$\begin{gathered} \frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{f}\left(\mathrm{x}\right)+\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{g}\left(\mathrm{x}\right) \\ \mathrm{Also},\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)\right]+\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{g}\left(\mathrm{x}\right)\right]. \end{gathered}$$

Substitute the parameters in the above equation.

role="math" localid="1658986125885" $$\begin{gathered} \left[{\mathrm{D}}^2+2\mathrm{D}+1\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\left[\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}+2\frac{\mathrm{d}}{\mathrm{dx}}+1\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \\ =\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]+2\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]+\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \\ =\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{f}\left(\mathrm{x}\right)+\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{g}\left(\mathrm{x}\right)+2\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{g}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right) \\ =\left[\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{f}\left(\mathrm{x}\right)+2\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)\right]+\left[\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{g}\left(\mathrm{x}\right)+2\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{g}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \end{gathered}$$

Solve further.

$$\begin{gathered} \left[{\mathrm{D}}^2+2\mathrm{D}+1\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\left[\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}+2\frac{\mathrm{d}}{\mathrm{dx}}+1\right]f\left(x\right)+\left[\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}+2\frac{\mathrm{d}}{\mathrm{dx}}+1\right]g\left(x\right) \\ =\left[{\mathrm{D}}^2+2\mathrm{D}+1\right]f\left(x\right)+\left[{\mathrm{D}}^2+2\mathrm{D}+1\right]g\left(x\right) \\ Also, \\ \left[{\mathrm{D}}^2+2\mathrm{D}+1\right]\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\left[\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}+2\frac{\mathrm{d}}{\mathrm{dx}}+1\right]\left[kf\left(x\right)\right] \\ =\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[kf\left(x\right)\right]+2\frac{d}{dx}\left[kf\left(x\right)\right]+\left[kf\left(x\right)\right] \end{gathered}$$

Also, differential operators D and $D^2$are linear operators.

$$\begin{gathered} \frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\mathrm{k}\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{f}\left(\mathrm{x}\right)\right] \\ ={\mathrm{kD}}^2\left[\mathrm{f}\left(\mathrm{x}\right)\right] \\ \mathrm{And} \\ \frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\mathrm{k}\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)\right] \\ \mathrm{Which}\mathrm{implies}\mathrm{that} \end{gathered}$$

$$\begin{gathered} \left[{\mathrm{D}}^2+2\mathrm{D}+1\right]\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\left[\frac{d^2}{d{x}^2}+2\frac{d}{dx}+1\right]\left[kf\left(x\right)\right] \\ =\frac{d^2}{d{x}^2}\left[kf\left(x\right)\right]+2\frac{d}{dx}\left[kf\left(x\right)\right]+\left(1\right)\left[kf\left(x\right)\right] \\ =k\frac{d^2}{d{x}^2}\left[f\left(x\right)\right]+2k\frac{d}{dx}\left[f\left(x\right)\right]+kf\left(x\right) \\ =k\left[\frac{d^2}{d{x}^2}+2\frac{d}{dx}+1\right]f\left(x\right) \end{gathered}$$

And further

$\left[D^2+2D+1\right]\left[kf\left(x\right)\right]=k\left[D^2+2D+1\right]f\left(x\right)$

Where k is a constant

Therefore, it has been shown that Differentiating the operator $D^2+2D+1$ with the objects being operated on is the function f (x) by differentiating it with respect to representing a linear operator.

(b)

Parameters

Given the following Differentiating operator ${\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7$.

It is to show that the differentiating operator ${\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7$,which operates on the function f(x) by differentiating it with respect to x representing a linear operator.

Operation on differentiating operators    ddx for O(A+B)=O(A)+O(B) and O(kA)=kO(A)

$$\begin{gathered} Find\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right] \\ \frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]+\frac{d}{dx}\left[f\left(x\right)\right] \\ \frac{d}{dx}\left[kf\left(x\right)\right]=k\frac{d}{dx}\left[f\left(x\right)\right] \end{gathered}$$

Where k is a constant

Operation on differentiating operators d2dx2for O(A+B)=O(A)+O(B) and O(kA)=kO(A)

$\mathrm{Find}{\mathrm{D}}^2\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right].$

role="math" localid="1658988852066" $$\begin{gathered} {\mathrm{D}}^2\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \\ =\frac{\mathrm{d}}{\mathrm{dx}}\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \\ =\frac{\mathrm{d}}{\mathrm{dx}}\left\{\frac{\mathrm{df}\left(\mathrm{x}\right)}{\mathrm{dx}}+\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{g}\left(\mathrm{x}\right)\right]\right\} \\ =\frac{\mathrm{d}}{\mathrm{dx}}\frac{\mathrm{df}\left(\mathrm{x}\right)}{\mathrm{dx}}+\frac{\mathrm{d}}{\mathrm{dx}}\frac{\mathrm{dg}\left(\mathrm{x}\right)}{\mathrm{dx}} \end{gathered}$$

Solve further.

$$\begin{gathered} {\mathrm{D}}^2\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{f}\left(\mathrm{x}\right)+\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{g}\left(\mathrm{x}\right) \\ ={\mathrm{D}}^2\left[\mathrm{f}\left(\mathrm{x}\right)\right]+{\mathrm{D}}^2\left[\mathrm{g}\left(\mathrm{x}\right)\right] \end{gathered}$$

$$\begin{gathered} Find{D}^2\left[kf\left(x\right)\right] \\ {D}^2\left[kf\left(x\right)\right]=\frac{d^2}{d{x}^2}\left[kf\left(x\right)\right] \\ =k\frac{d^2}{d{x}^2}\left[kf\left(x\right)\right] \\ =k{D}^2\left[f\left(x\right)\right] \end{gathered}$$

Where k is a constant

Operation on differentiating operators x2d2dx2-2xddx+7 for O(A+B)=O(A)+O(B) and O(kA)=kO(A)

$$\begin{gathered} Find\left[x^2{D}^2-2xD+7\right]\left[f\left(x\right)+g\left(x\right)\right] \\ \left[x^2{D}^2-2xD+7\right]\left[f\left(x\right)+g\left(x\right)\right]=\left[x^2\frac{d^2}{d{x}^2}-2x\frac{d}{dx}+7\right]\left[f\left(x\right)+g\left(x\right)\right] \\ =x^2\frac{d^2}{d{x}^2}\left[f\left(x\right)+g\left(x\right)\right]-2x\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]+7\left[f\left(x\right)+g\left(x\right)\right] \end{gathered}$$

Solve further

$$\begin{gathered} \frac{d^2}{d{x}^2}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d^2}{d{x}^2}f\left(x\right)+\frac{d^2}{d{x}^2}g\left(x\right) \\ Also,\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]+\frac{d}{dx}\left[g\left(x\right)\right] \end{gathered}$$

Substitute the parameters in the above equation.

$\left[x^2{D}^2-2xD+7\right]\left[f\left(x\right)+g\left(x\right)\right]=\left[x^2\frac{d^2}{d{x}^2}-2x\frac{d}{dx}+7\right]\left[f\left(x\right)+g\left(x\right)\right]$

= $x^2\frac{d^2}{d{x}^2}\left[f\left(x\right)+g\left(x\right)\right]-2x\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]+\left(7\right)\left[f\left(x\right)+g\left(x\right)\right]$

Solve further

$$\begin{gathered} \frac{d^2}{d{x}^2}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d^2}{d{x}^2}f\left(x\right)+\frac{d^2}{d{x}^2}g\left(x\right) \\ Also,\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[g\left(x\right)\right] \end{gathered}$$

Substitute the parameters in the above equationlocalid="1658994082917" $$\begin{gathered} \left[{\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}+7\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \\ ={\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]+\left(7\right)\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right] \\ =\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{f}\left(\mathrm{x}\right)-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}\left(\mathrm{x}\right)+7\mathrm{f}\left(\mathrm{x}\right)\right]+\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\mathrm{g}\left(\mathrm{x}\right)-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}\left(\mathrm{x}\right)+7\mathrm{g}\left(\mathrm{x}\right)\right] \end{gathered}$$

Solve Further

$$\begin{gathered} \left[{\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}+7\right]f\left(x\right)+\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}+7\right]g\left(x\right) \\ =\left[x^2{D}^2-2xD+7\right]f\left(x\right)+\left[x^2{D}^2-2xD+7\right]g\left(x\right) \end{gathered}$$

Also

localid="1658993473906" $$\begin{gathered} \left[{\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7\right]\left[\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\right]=\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}+7\right]f\left(x\right)+\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}+7\right]g\left(x\right) \\ =\left[x^2{D}^2-2xD+7\right]f\left(x\right)+\left[x^2{D}^2-2xD+7\right]g\left(x\right) \\ \left[{\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7\right]\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}+7\right]\left[\mathrm{kf}\left(\mathrm{x}\right)\right] \\ =x^2\frac{d^2}{d{x}^2}\left[kf\left(x\right)\right]-2x\frac{d}{dx}\left[kf\left(x\right)\right]+\left(7\right)\left[kf\left(x\right)\right] \end{gathered}$$

Also, differential operators $\mathrm{D}\mathrm{and}{\mathrm{D}}^2$are linear operators.

$$\begin{gathered} \frac{d^2}{d{x}^2}\left[kf\left(x\right)\right]=k\frac{d^2}{d{x}^2}\left[f\left(x\right)\right] \\ =k{D}^2\left[f\left(x\right)\right] \\ And \\ \frac{d}{dx}\left[kf\left(x\right)\right]=k\frac{d}{dx}\left[f\left(x\right)\right] \end{gathered}$$

Which implies that

localid="1658994089991" $$\begin{gathered} \left[{\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7\right]\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}+7\right]\left[\mathrm{kf}\left(\mathrm{x}\right)\right] \\ ={\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{kf}\left(\mathrm{x}\right)\right]-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{kf}\left(\mathrm{x}\right)\right]+\left(7\right)\left[\mathrm{kf}\left(\mathrm{x}\right)\right] \\ =\left(\mathrm{k}\right)\left({\mathrm{x}}^2\right)\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}\left[\mathrm{f}\left(\mathrm{x}\right)\right]-\left(2\mathrm{x}\right)\left(\mathrm{k}\right)\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{f}\left(\mathrm{x}\right)\right]+\left(7\right)\mathrm{kf}\left(\mathrm{x}\right) \\ =\mathrm{k}\left[{\mathrm{x}}^2\frac{{\mathrm{d}}^2}{{\mathrm{dx}}^2}-2\mathrm{x}\frac{\mathrm{d}}{\mathrm{dx}}+7\right]\mathrm{f}\left(\mathrm{x}\right) \end{gathered}$$

And further

$\left[{\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7\right]\left[\mathrm{kf}\left(\mathrm{x}\right)\right]=\mathrm{k}\left[{\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7\right]\mathrm{f}\left(\mathrm{x}\right)$

Therefore, it has been shown that Differentiating the operator ${\mathrm{x}}^2{\mathrm{D}}^2-2\mathrm{xD}+7$ with the objects being operated on is the function f(x) by differentiating it with respect to x representing a linear operator.