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Eigenfunctions of a linear operator

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Suppose that O1 and O2 are eigenfunctions of a linear operator A with eigenvalues a1 and a2, respectively, and that we construct a linear combination Y=C1O1 + C2O2. Under what conditions does Y became an eigenfunction of A? Show reasoning.

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In any relationship of the type AF = aF, where F is a function, A is an operator and a is a number, F is said to be an eigenfunction of the operator A with eigenvalue equal ...

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The expert analyzes the eigenfunctions of a linear operator. Linear combinations of functions and their conditions are given.

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Eigenfunctions and eigenvalues of d^2/dx^2

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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