Given the equation $ u''=f(u)$ with symmetric boundary conditions $u(-a)=u(a)=u_0$, is there any proof that possibly relies on dynamical systems techniques to show that the solutions are symmetric (i.e., even)? I've been trying to find a reference to this result for a while now, but I couldn't find anything related to this simple scenario.
EDIT: The function $f$ has a definite sign. For additional reference: What I am trying to find is the proof of something like Problem 3 in Section 2.7.1 of the book "Introduction to Partial Differential Equations and Hilbert Space Methods" of Karl E. Gustafson.