Differential Equations. Single Operator.

Question

Write as a single operator action on y

(D+cotx)(D-cotx)y

 

 

The answer is :

(D^(2) +1)y

 

Just need steps to get there. 

Answer 1

John,

 

If you treat the D-operator as an algebraic expression and simplify it, here's what happens:

 

(D + cot x)(D - cot x) = (D² - cot² x).

 

If you were to apply this to a function y, you would NOT get the same result as applying (D² + 1)!

 

Here is what happens in each case.

 

(D² - cot² x)y = d²y/dx² - (cot² x)y

 

while

 

(D² + 1)y = d²y/dx² + y.

 

The only way these could be equal is if 1 = -cot² x, which is not true for any connected interval of real numbers.

 

However you got your answer, it is not correct.