= Solution
The <chain rule> gives $\partial_x=\partial_\xi+\beta\partial_\eta$ and $\partial_y=\alpha\partial_\xi+\partial_\eta$. The coefficients of $u_{\xi\xi}$ and $u_{\eta\eta}$ are $2+3\alpha^2-7\alpha$ and $2\beta^2+3-7\beta$. Their integer roots are \b[$\alpha=2$ and $\beta=3$]. Thus choose <characteristic coordinates> $\xi=x+2y$, $\eta=3x+y$, whose Jacobian determinant is $-5\ne0$. The coefficient of $u_{\xi\eta}$ is $4\beta+6\alpha-7(1+\alpha\beta)=-25$, so the equation becomes $-25u_{\xi\eta}=0$, equivalently $u_{\xi\eta}=0$.
Integrating the transformed <hyperbolic partial differential equation> gives $u=F(\xi)+G(\eta)$. Along $x=-2y$ the coordinates are $(0,-5y)$, so the zero <boundary condition> gives $G(\eta)=-F(0)$ for every real $\eta$. Along $x=0$ they are $(2y,y)$, so $F(2y)-F(0)=4y^2$, hence $F(\xi)=\xi^2+F(0)$. The constants cancel and the unique resulting solution is
$$
\boxed{u(x,y)=(x+2y)^2.}
$$
It directly satisfies both <boundary conditions> and $2u_{xx}+3u_{yy}-7u_{xy}=4+24-28=0$.
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