= Mixed-type partial differential equation
{title2=$B^2-AC\text{ changes sign}$}
A second-order <partial differential equation> is of mixed type when the type of its principal part changes across its domain. For $A u_{xx}+2B u_{xy}+C u_{yy}=0$, the sign of $B^2-AC$ distinguishes a <hyperbolic partial differential equation>, <parabolic type> and an <elliptic partial differential equation>. For example, $u_{xx}+\operatorname{sign}(y)u_{yy}=0$ is elliptic above $y=0$, hyperbolic below it, and degenerate on the interface. On its hyperbolic side, the change $\xi=x+y$, $\eta=x-y$ reduces it to $u_{\xi\eta}=0$; above it is the <Laplace equation>. Type classification alone supplies no matching rule for the normal derivative.
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