Is the following statement true? A PDE has a single type throughout its domain

No. The Tricomi equation y uₓₓ + u_yy = 0 Elliptic in the upper half-plane, hyperbolic in the lower, parabolic on the axis.

Counterexample

Is this true?

A PDE has a single type throughout its domain

No — it is false.

The counterexample

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

Why it fails

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

Keep practicing

Boundary and endpoint

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

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.