Level curve and PDE

Question

Suppose you have the following PDE sistem: $$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial^2 u}{\partial y^2}$$ $$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$ If we set $\overline{x}=ax$ and $\overline{y}=by$, performing substitutions we notice that for $a=b^2$ and $\overline{v}=bv$ we obtein the original PDE sistem. In particular we notice that $u(b^2x,by)=u(x,y)$ $\forall b>0$. So the question is: how can we proof that $u$ is costant on the curve of equation: $$Y=\frac{y}{\sqrt{x}}\sqrt{X}$$ ? Thank you for the answers.