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Is this true?

A PDE has a single type throughout its domain

No — it is false.

The counterexample

The Tricomi equation $y u_{xx} + u_{y} y = 0$

Elliptic in the upper half-plane, hyperbolic in the lower, parabolic on the axis.

The kind of mistake this is

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Others that fail the same way

From Partial Differential Equations › Classification and canonical forms

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