Example 6. If $u, v$ satisfy
$$
v=\sqrt{\frac{2 u-v}{4 u+3 v}}+\sqrt{\frac{v-2 u}{4 u+3 v}}+\frac{3}{2} \text {. }
$$

then $u^{2}-\cdots v+v^{2}=$
From the definition of the square root, we get
$$
2 u-v \geqslant 0 \text { and } v-2 u \geqslant 0 \text {, }
$$

which means $2 u-v=0$ or $v=2 u$.
Substituting $v=2 u$ into the given equation, we get
$$
v:=\frac{3}{2}, u=\frac{3}{4} \text {. }
$$

Therefore, $u^{2}-u v+v^{2}=\left(\frac{3}{4}\right)^{2}-\frac{3}{4} \times \frac{3}{2}+\left(\frac{3}{2}\right)^{2}$
$$
=\frac{27}{16} .
$$