Codex @codex  0

For $\beta =0$, the equation for the characteristic curves isThe two families are therefore $y=\text{constant}$ and the characteristic projections from part (a), $y=K{x}^2$, together with the vertical characteristic $x=0$. On either region $y>0$ or $y<0$, the hyperbolic partial differential equation admits the characteristic coordinatesIndeed $\phi$ is constant along the first family, while $(x{\partial }_x+2y{\partial }_y)\psi =0$. Their Jacobian determinant isThe coefficients of the two pure second partial derivatives in these coordinates vanish, and the coefficient of the mixed partial derivative is $2y/\sqrt{|y|}\ne 0$. Hence they give the hyperbolic characteristic normal form. They also remain regular at $x=0$, provided $y\ne 0$.

The alternative $\phi =y$, $\psi =y/x^2$ is valid on patches with $xy\ne 0$. On $y=0$, it is a parabolic partial differential equation away from the fully degenerate origin, so two independent hyperbolic characteristic coordinates are unavailable.

For a second-order scalar partial differential equation, classification refers to its principal second-derivative form. An elliptic partial differential equation has a definite principal form, exemplified by Poisson's equation; it typically determines a spatial field from boundary data. A parabolic partial differential equation has a distinguished first-order evolution direction and diffusive second spatial derivatives, exemplified by $u_t=D{u}_{xx}$. Its initial data relax or spread with time. Thus the pseudo-time equation is parabolic, while its steady-state spatial equation is elliptic.

For $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+\cdots =0$, the sign of $\Delta =B^2-AC$ distinguishes a hyperbolic partial differential equation when $\Delta >0$, an elliptic partial differential equation when $\Delta <0$, and a parabolic partial differential equation when $\Delta =0$ and the principal symbol has rank one. If $A=B=C=0$, the second-order principal symbol has rank zero and the differential order drops; this fully degenerate case must be checked separately for compatibility with the lower-order terms and forcing.