Solution
= Solution
The Lagrangian is
$$
F(X,Y,\nabla X,\nabla Y)=X^{-2}(|\nabla X|^2+|\nabla Y|^2).
$$
The <Euler-Lagrange equation> for $X$ is
$$
-2X^{-3}(|\nabla X|^2+|\nabla Y|^2)
-\operatorname{div}(2X^{-2}\nabla X)=0.
$$
Expanding the divergence and simplifying gives
$$
\boxed{\Delta X=\frac{|\nabla X|^2-|\nabla Y|^2}{X}}.
$$
The equation for $Y$ is
$$
\boxed{\operatorname{div}\left(\frac{\nabla Y}{X^2}\right)=0}.
$$