方法一
$\begin{aligned} &\frac{y}{y-x}-\frac{4x}{2x+y}\\ &=\frac{y}{y-x}-\frac{4x+2y-2y}{2x+y}\\ &=\frac{y}{y-x}+\frac{2y}{2x+y}-2\\ &=y\left(\frac{2}{2y-2x}+\frac{2}{2x+y}\right)-2\\ &=\frac13\left((2y-2x)+(2x+y)\right)\left(\frac{2}{2y-2x}+\frac{2}{2x+y}\right)-2\\ &=\frac23\left(2+\frac{2x+y}{2y-2x}+\frac{2y-2x}{2x+y}\right)-2\\ &\geq\frac23\left(2+2\sqrt{\frac{2x+y}{2y-2x}\cdot\frac{2y-2x}{2x+y}}\right)-2\\ &=\frac83-2=\boxed{\frac23} \end{aligned}$