I have the following linear ordinary differential problem
\begin{equation} \begin{cases} EI y'''' - \left[M_z(z) x'\right]'' - \left[ T(z)y'\right]' = -\alpha y \\ EI x'''' - \left[M_z(z) y'\right]'' - \left[ T(z)x'\right]' = -\alpha x \end{cases} \end{equation}
where
$$ M_z(z) = M_0 = const \quad \quad \text{and} \quad \quad T(z) = -T_0\left(1-\tfrac{z}{Z}\right) $$
The $'$ denotes differentiation with respect to $z$. This system has to be solved for $z \in \left[0, Z\right] $ with the following conditions:
$$ \begin{cases} x = y = x'' = y'' = 0 \quad \text{at}\,\, z=0 \\ x = y = x' = y' = 0 \quad \text{at}\,\, z=Z \end{cases} $$
Then, in matricial form one can rewrite it as
\begin{equation} \begin{cases} Y' = MY \\ f(Y(0)) = 0 \\ g(Y(Z)) = 0 \end{cases} \end{equation}
where $Y = \left( y, y', y'', y''', x, x', x'', x''' \right)$ and
$$ f(\mathbf{v}) = \begin{pmatrix} v_1 \\ v_3 \\ v_5 \\ v_7 \end{pmatrix} \quad \quad g(\mathbf{v}) = \begin{pmatrix} v_1 \\ v_2 \\ v_5 \\ v_6 \end{pmatrix} $$
Of course, there is always the trivial solution $Y=0$, but there should be non trivial solutions that at the moment I am not able to catch (by simply using a method like the bvp4 provided by MATLAB/python). Any suggestion?