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Part BCSIR NET December 2023

Classification and canonical forms

The partial differential equation $x^{2} u_{xx} - 2$ xy $u_{x}$ ᵧ $- 3 y^{2} u_{y} y + u_{x} - u_{y} = 0$ is

A.elliptic in {(x, y) : y > 0}
B.parabolic in {(x, y) : x > 0, y > 0}
C.hyperbolic in {(x, y) : xy ≠ 0}✓
D.parabolic in {(x, y) : xy ≠ 0}

Solution

$B^{2} - 4$ AC $= 4 x^{2} y^{2} + 12 x^{2} y^{2} = 16 x^{2} y^{2} > 0$ wherever xy ≠ 0: hyperbolic.

Related counterexample: A PDE has a single type throughout its domain

From Partial Differential Equations › Classification and canonical forms

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