Chapter 8.5, Problem 31P

Let $D$ stand for $d/dx,$ that is, $Dy=dy/dx;$ then $D^{2}y=D(Dy)=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d^{2}y}{d{x}^2},D^{3}y=\frac{d^{3}y}{d{x}^3},\text{ etc}\text{. }$ $D$ (or an expression involving $D$ ) is called a differential operator. Two operators are equal if they give the same results when they operate on y. For example, $D(D+x)y=\frac{d}{dx}\left(\frac{dy}{dx}+xy\right)=\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=\left(D^2+xD+1\right)y$ so we say that $D(D+x)=D^2+xD+1$.

In a similar way show that:

(a) $(D-a)(D-b)=(D-b)(D-a)=D^2-(b+a)D+ab$ for constant $a$ and $b.$

(b) $D^3+1=(D+1)\left(D^2-D+1\right).$

(c) $Dx=xD+1.$ (Note that $D$ and $x$ do not commute, that is, $Dx\ne xD.$ )

(d) $(D-x)(D+x)=D^2-x^2+1,$ but $(D+x)(D-x)=D^2-x^2-1$.

Comment: The operator equations in (c) and (d) are useful in quantum mechanics; see Chapter 12, Section 22.