1. Consider the square of the derivative operator D^2.
A. Show that D^2 is a linear function.
B. Find the eigen-functions and corresponding eigen-values of D^2.
C. Give an example of an eigen-function of D^2 that is not an eigen-function of D.
Solution
I will try to attempt only the first , part
A)
if D is the derivative , then it is a linear operator
by defination of derivative rules , D(*x+*y) = *D(x)+*D(y)
now D^2 = D(D ) or D.D
D^2(*x+*y) = D(D(*x+*y)) = D (*D(x)+*D(y))
= *D.D(x)+*D.D(y) [again using the same rules of derivatives)
= *D^2(y) +*D^2(y)
hence D^2 is also a linear operator