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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.

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    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.