Your eigenvalues are given by the intersection of this curves.
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."
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.)