Free-endpoint normal mode of a coupled variational functional
= Free-endpoint normal mode of a coupled variational functional
For
$$
I[y,z]=\int_0^{x_0}(y'^2+z'^2+2yz)\,dx,
\qquad y(0)=z(0)=0,
$$
with free terminal values, the Euler-Lagrange equations and natural conditions give a nonzero stationary family exactly when $x_0=(k+\tfrac12)\pi$. It is
$$
y=C\sin x,
\qquad z=-C\sin x.
$$