Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 2 / 6B (Differential Equations)

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 $f,g$.

If $\alpha =-b/(2a)$ is a repeated root, then $A=0$ and both coefficients of $\beta$ in $B$ vanish, so $B=0$ for every $\beta$. Choose $\beta \ne \alpha$, giving $C\ne 0$ and hence $v_{\eta \eta }=0$. For $(s+1{)}^2$, take $\xi =y-x$, $\eta =y$, and obtain