I am unable to prove the following statement (which is numerically motivated). For a function $F(t_1,t_2)$ fulfilling the following ODE $$\partial_{t_1}^2F(t_1,t_2) + E(t_1)^2F(t_1,t_2)=0$$ for $(t_1,t_2)\in[t_0,t_f]^2$ with boundary conditions (upper sign for symmetric BC lower for antisymmetric BC) $$F(t_0,t)=\pm F(t,t_0)=f(t)$$ $$\partial_x F(x,t)|_{x=t_0}=\pm\partial_x F(t,x)|_{x=t_0}$$
the solution is symmetric/antisymmetric, i.e. $$F(t_1,t_2)=\pm F(t_2,t_1).$$
I do not have any information about what ODE $F$ fulfills in its second argument. $E$ can be any smooth function. It is not dependent on $t_2$.
Not sure if Physics or Maths SE is the right place for this.
