I am trying to solve a 4th order boundary value problem with variable coefficients, namely the problem of a rotating cantilever beam:

$u'''' - \frac{((1-x^2)u')'}{2\eta} - \frac{\alpha}{\eta}((1-x)u')' - \frac{\mu}{\eta} u = 0$

with boundary conditions

$u(0) = 0, u'(0) = 0, u''(1) = 0, u'''(1) = 0$

where

$\eta, \alpha,\mu = \text{const}_x$, $u = u(x)$

I am not really interested in the form of solution itself. What I want to know is the natural frequencies $\mu$ values corresponding to solutions.

I know that it can be solved using Finite Elements Method but I am not familiar with this one. Is this the only option for me?

Thank you very much in advance!