Find the most general solution of the equation

$$6 \frac{\partial^{2} u}{\partial x^{2}}-5 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=1$$

by making the change of variables

$$\xi=x+2 y, \quad \eta=x+3 y .$$

Find the solution that satisfies $u=0$ and $\partial u / \partial y=x$ when $y=0$.