 Solution

Eliminate the two pure second derivatives by taking $\alpha ,\beta$ to be the roots of $z^2+2Bz+C=0$. They are distinct real numbers exactly whenThus the required function is $H(B)=B^2$. The mixed coefficient becomes $4(C-B^2)\ne 0$, so the transformed equation is $y_{\xi \eta }=0$. Integrating once in each coordinate giveswith arbitrary sufficiently smooth functions $F,G$. This is the characteristic form of a strictly hyperbolic partial differential equation.