That equation is the same as , an "Euler type" or "equipotential" equation. The change of variable t= ln(x) changes it to an equation with constant coefficients. You should be able to find the general solution. Obviously, y(x)= 0, for all x, satifies the equation as well as y(1)= y(2)= 0. The eigenvalues are the values of u for which there are other, non-trivial, solutions. And the eigenfunctions are the non-trivial solutions.