Nontrivial solution of a fourth-order DE with variable coefficients$\frac{d^2}{dx^2}\left(f(x)\frac{d^2y(x)}{dx^2}\right)-ag(x)y(x)=0$

Question

How to find the general closed-form solution of the following eigenvalue problem? \begin{equation} \frac{d^2}{dx^2}\left(f(x)\frac{d^2y(x)}{dx^2}\right)-ag(x)y(x)=0 \\ \text{where} \ 0< f(x),g(x)\leq 1,\ 0\leq x \leq 1,f(x) \text{ and } g(x) \text{ are continuous in }[0,1], \\ a>0 \text{ is the eigenvalue, and} \ y(x) \text{ is the mode shape to be solved in terms of } f(x) \text{ and } g(x). \\ \text{Boundary conditions: } y(0)=0,{y}'(0)=0, y(1)=0,{y}'(1)=0 \end{equation} It is a beam vibration problem. Tried many methods, failed by far. Are there any references to this problem? Thanks in advance!