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Eigen function and eigenvalues

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I have attached a problem involving eigenfunctions and eigenvalues. I would sure appreciate a complete explanation with work.

Consider a linear operator ... with two sets of eigenfunctions and eigenvalues:
...
with ... Now consider a linear combination of eigenfunctions, ... . Show whether or not ... is an eigenfunction of ...

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The expert examines Eigen functions and eigenvalues.

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Hello and thank you for posting your question to Brainmass.
The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most ...

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