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That's why A and B were related, by complex conjugation.
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Here, if I take the complex conjugate of x,
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e^(α+·t), if it's real,
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it remains itself.
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So, A just goes into A*, B goes into B*,
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and you want the answer to be unchanged.
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That requires A equal to A* and B equal to B*.
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Thanks for pointing out that here.
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We do want A equal to A*
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and B equal to B*.
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So, what I'm telling you is go take the solution,
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conjugate it and equate it to itself,