The problem is stated as follows:

"Show that with the initial conditions has an infinite sequence of eigenfunctions with distinct eigenvalues. Identify the eigenvalues explicitly."

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seems to yield the trivial solution, so . The general solution is then . The first initial condition gives , and then the second gives

I've tried to solve for by writing the LHS as a single sine function, and separately by dividing with , but neither approach seems to give a good way of giving an explicit formula for .

(The best I've been able to do with the single sine function is to write as . Consequently, . But then , so I wouldn't call this explicit.)

Thoughts?