## Wednesday, October 12, 2022

### Introducing Basic Differential Operators

Differential Operators are useful, and often shorthand ways to approach writing differential equations - including systems of equations. They can be written in various forms, but basically:

D x u  =   ux   e.g.

du/ dx =   ux

Are first degree ordinary differential equations.

But we begin by using basic forms, say to obtain derivatives.   In general:

(D x y) n  =   dn y / dxn       n=  1, 2, 3….

Then:  (D x y) 2  =   d2 y/ dx2

(D x y) 3  =   d3 y/ dx3

And so forth:

Examples:

a)  Find:  (D x y) 2   (4x3  +  3x

Then:  d2 y/ dx2   (4x3  +  3x

=  12 x +   6x

b)  (D x y)  (4/ x)

Then:  y/ dx   (4/x) =  -4/  x

c)  Find:  (D x y) 2   (sin 4x)

Then:  d2 y/ dx2   (sin 4x)

Take 1st derivative first:

dy/ dx (sin 4x)  = 4 cos (4x)

Then take derivative again, i.e.

dy/ dx (4 cos (4x))

=  - 16 sin (4x)

Of special interest is the inverse differential operator, viz.

(D-1  x n )   =    ( xn+1 )  /  (n + 1)

Similarly for higher order::

(D-2  x n )   =  (D-1 ) ( xn+1 )  /  (n + 1) =

xn+2  /  (n  +1 )  (n  +  2)

And for higher order (k) in general:

(D-k  x n )   =    (xn+k )   (n+ 1) (n + 2)...(n  +   k)

Thus,  1/D stands for the integral, e.g.     F(x)dx  but with denominator corrected for as given by inverse formulae above.

And is the inverse differential operator

And 1/ D n  stands for successive (n)  integration.

So we see the inverse operator has the same effect as integration, but disregarding the constants of integration.

Suggested Problems:

Evaluate each of the following:

1) (D x y)  (ln x / 1+  x)

2) (D x y)  ½ ( x   -  -x)

3) D-1  ( 11 x)

4)  (D x y)    ( e x   ln x)

5)  D-1  (sin x  cos x)

6)  D-1  (tanh x )

7)  (D x y)  (sech 2 x)

8)   D-2  (3 x )

9)  D-1  ( cosh 2  3x)

10)   D-2  (1/  x

11) D-1  ( 3x2 )

12) (D  - a) ax

13) (D  - a)-1   ax

14) (D- 1)

15(D  - 1)-1   x4