So the first part of this question was already asked but the answer was found using a method not mentioned in the question. The substitution of x = e^t is important. Here is the question:
By substituting $x = e^t$, find the normalized eigenfunctions $y_n(x)$ and the eigen-values $λ_n$ of the operator $L$ defined by $$Ly=x^2y′′+ 2xy′+\dfrac{1}{4}y$$ over $$1≤x≤e$$ with $y(1)=y(e)=0$. Assume that the weight function is unity. Find, as a series $\sum_n a_ny_n(x)$, the solution of $Ly=x^{−1/2}$ subject to the same boundary conditions.