已知函数 $f(u, v)$ 具有 2 阶连续偏导数, 且函数 $g(x, y)=f(2 x+y, 3 x-y)$ 满足 $\frac{\partial^2 g}{\partial x^2}+\frac{\partial^2 g}{\partial x \partial y}-6 \frac{\partial^2 g}{\partial y^2}=1$.
(1) 求 $\frac{\partial^2 f}{\partial u \partial v}$;
(2) 若 $\frac{\partial f(u, 0)}{\partial u}=u e^{-u}, f(0, v)=\frac{1}{50} v^2-1$, 求 $f(u, v)$ 的表达式.