Is there a way to predict in advance whether the solution to a sixth-order nonlinear ODE will exhibit symmetry about half of its domain?
The ODE is expressed as $f(x, y', y'', y''', y'''', y''''', y'''''') = 0$, with boundary conditions at $x=0$ and $x=1$ as: $y(0) = y(1) = 0$ and $[b_i(x, y, y', y'', y''', y'''')]_{x \,=\, 0\, \text{and}\, 1} = 0$ for $i$ in ${1,2}$. The ODE is relevant to a mechanics problem. While numerically solving the ODE in Auto-07p, I'm observing symmetric solutions for $y'$ around $x=1/2$. However, I'm unsure if the symmetry of $y'$ is a coincidence in the solutions I got or is the only possibility. Is there a method to pre-check whether the system will inherently yield a symmetric $y'$ around the midpoint $x=1/2$?"