Monday, August 28, 2023

Revisiting Differential Operators.

 Differential Operators are basically shorthand ways of writing derivatives and by extension writing more concise differential equations.  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.


For example:  d4y/ dt4

Can be rewritten: (D t y) 4 

And:   2d3y/ dt3  

Can be rewritten:  2(D t y)3 

Then rewrite the differential equation:

d4y/ dt4 – 2d3y/ dt3  –7 d2y/ dt2  + 20  dy/dt  –12 y = 0

Using differential operators. 

Using the preceding hints for the notation, we have:

(D t y) 4 -  2(D t y)- 7 (D t y) 2  + 20 (D t y) - 12y = 0

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


Suggested Problems:


1)  Rewrite the DE below with differential operators:


   5 d5y/ dt5 - 10 dy/dt –25y = 0


2) Rewrite  the DE below in standard derivative form and solve:


(D x y)   =   3x2   -  1  


3)  Does  (D x y) 2  =   D 2 x y  ?  Explain.


4) Evaluate each of the following:


i) (D x y)  (ln x / 1+  x)


ii) (D x y)  ½ ( x   -  -x)   


iii) D-1  ( 11 x)  


 iv)  (D x y)    ( x   ln x)


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