boundary conditions eigenvalue problem

Question

Consider the following eigenvalue problem \begin{equation} \mathcal {L} x(s) = \lambda x(s), \end{equation} where \begin{equation} \mathcal {L} = \alpha \partial^4_s + (s^2-1)\partial^2_s + s \partial_s + 1, \end{equation} with $\alpha = \rm{const}$ and $s\in[0,1]$. In order to find the eigenvalues and the eigenfunctions, the operator $\mathcal L$ has to be translated into a matrix form. What is the matrix that satisfies the boundary conditions $u(0) = u_s(0)=0$ and $u_{ss}(1)=u_{sss}(1)=0$?