91Ó°ÊÓ

Given information are

$\begin{array}{l}D=\frac{d}{dx}\\ Dy=\frac{dy}{dx}\\ D^{2}y=\frac{d}{dx}\left(\frac{dy}{dx}\right)\end{array}$

It is given that

$\begin{array}{c}(D−a)(D−b)=(D−b)(D−a)\\ =D^2−(b+a)D+ab\end{array}$,here.$D=\frac{d}{dx}$

Use the algebraic equation:

$\mathbf{(}{\mathit{a}}^{\mathbf{3}}\mathbf{+}{\mathit{b}}^{\mathbf{3}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}\mathbf{(}{\mathbf{a}}^{\mathbf{2}}\mathbf{−}\mathbf{a}\mathbf{b}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{)}$

a)

Let us consider:

$\begin{array}{c}(D−a)(D−b)\\ =D(D−b)−a(D−b)\\ =\frac{d}{dx}(\frac{dy}{dx}−by)−a(\frac{d}{dx}−by)\\ =\frac{d^{2}y}{d{x}^2}−b\frac{dy}{dx}−a\frac{dy}{dx}−aby\end{array}$

Further solve the equation

$\begin{array}{c}=\frac{d^{2}y}{d{x}^2}−(b+a)\frac{dy}{dx}−aby\\ =D^2−(b+a)D+ab\end{array}$

Similarly,

$\begin{array}{c}(D−b)(D−a)\\ =D(D−a)−b(D−a)\\ =\frac{d}{dx}(\frac{dy}{dx}−ay)−b(\frac{d}{dx}−ay)\\ =\frac{d^{2}y}{d{x}^2}−a\frac{dy}{dx}−b\frac{dy}{dx}−aby\end{array}$

Solve further

=$D^2−(b+a)D+ab$

Hence, it is proved

(b)

Let us consider:

$\begin{array}{c}(D^3+1)=(D+1)(D^2−D+1)\\ =D(D^2−D+1)+1(D^2−D+1)\\ =\frac{d}{dx}(\frac{d^{2}y}{d{x}^2}−\frac{dy}{dx}+y)+1(\frac{d^{2}y}{d{x}^2}−\frac{dy}{dx}+y)\\ =\frac{d^{3}y}{d{x}^3}−\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+\frac{d^{2}y}{d{x}^2}−\frac{dy}{dx}+y\end{array}$

Further solve the equation

$\begin{array}{c}=\frac{d^{3}y}{d{x}^3}+0+0+y\\ =\frac{d^{3}y}{d{x}^3}+y\\ =D^3+1\end{array}$

(c)

Let us consider:

$\begin{array}{c}Dx=\frac{d}{dx}\left(xy\right)\\ =x\frac{dy}{dx}+\frac{dx}{dx}y\\ =x\frac{dy}{dx}+\left(1\right)y\end{array}$

Solve further,

$\begin{array}{c}=x\frac{dy}{dx}+y\\ =x\frac{dy}{dx}+1\end{array}$

(d)

Let us consider:

$\begin{array}{c}(D−x)(D+x)=D(D+x)−x(D+x)\\ =\frac{d}{dx}(\frac{dy}{dx}+xy)+x(\frac{dy}{dx}+xy)\\ =\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y−x\frac{dy}{dx}−x^{2}y\end{array}$

Solving further,

$\begin{array}{c}=\frac{d^{2}y}{d{x}^2}+y−x^{2}y\\ =\frac{d^{2}y}{d{x}^2}+1−x^2\\ =D^2+1−x^2\\ =D^2−x^2+1\end{array}$