Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 5 / a / i / Solution

For a second-order linear partial differential equation written $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+\cdots =0$, the discriminant is $B^2-AC$. HereIt is positive exactly when $x\ne 0$, proving that the equation is a hyperbolic partial differential equation on each half-plane. At $x=0$ its principal part degenerates and it is not hyperbolic.