Consider the equation

$$2 \frac{\partial^{2} u}{\partial x^{2}}+3 \frac{\partial^{2} u}{\partial y^{2}}-7 \frac{\partial^{2} u}{\partial x \partial y}=0$$

for the function $u(x, y)$, where $x$ and $y$ are real variables. By using the change of variables

$$\xi=x+\alpha y, \quad \eta=\beta x+y$$

where $\alpha$ and $\beta$ are appropriately chosen integers, transform $(*)$ into the equation

$$\frac{\partial^{2} u}{\partial \xi \partial \eta}=0$$

Hence, solve equation $(*)$ supplemented with the boundary conditions

$$u(0, y)=4 y^{2}, \quad u(-2 y, y)=0, \quad \text { for all } y$$