 Solution

Since ${\partial }_x=\alpha {\partial }_{\xi }+\beta {\partial }_{\eta }$ and ${\partial }_y={\partial }_{\xi }+{\partial }_{\eta }$,If $\alpha ,\beta$ are the distinct real roots of $a{s}^2+bs+c$, then $A=C=0$ and $B=(4ac-b^2)/a\ne 0$. Thus $v_{\xi \eta }=0$. For $s^2+3s+2=(s+1)(s+2)$, take $\xi =y-x$, $\eta =y-2x$, obtainingfor arbitrary twice differentiable functions $f,g$.