Solution
= Solution
{c}
The <multivariable chain rule> gives $\partial_x=\partial_\xi+\partial_\eta$ and $\partial_t=\alpha\partial_\xi+\beta\partial_\eta$. Substitution into the <differential equation> yields
$$
\boxed{(\alpha^2+2B\alpha+C)y_{\xi\xi}
+2\{\alpha\beta+B(\alpha+\beta)+C\}y_{\xi\eta}
+(\beta^2+2B\beta+C)y_{\eta\eta}=0.}
$$
The <coordinate transformation> is invertible exactly when $\alpha\ne\beta$.