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# Eigenfunctons and eigenvalues of d^2/dx^2

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Consider the square of the derivative operator D^2

(a) Show that D^2 is a linear operator
(b) Find the eigenfunctions and corresponding eigenvalues of D^2.
(c) Give an example of an eigenfunction of D^2 which is not an eigenfunction of D.

https://brainmass.com/physics/properties/eigenfunctions-eigenvalues-143521

#### Solution Preview

D is a linear operator. If f and g are two functions and a and b two complex numbers, then:

D[a f + b g] = a D[f] + b D[g] (1)

Which is the defining property of a linear operator (or equivalent to other set of defining properties)

The operator D^2 is defined as:

D^2f = ...

#### Solution Summary

A detailed solution is given. The expert considers the square of the derivative operators and linear operators.

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