The bookUnit 3 · Partial Differential Equations59 / 83

Classification and canonical forms

Why this is asked: B² − 4AC decides the type pointwise; the sign can change with the point (Tricomi). Reduce to canonical form by solving the characteristic ODE dy/dx = (B ± √(B²−4AC))/2A.

Classification of second-order PDE

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See it move

Classifying a second-order PDE, and the type that changesinteractivefree

Step 1 / 4The discriminant

For $A u_{xx} + B u_{x y} + C u_{y y} + \dots = 0$ , compute $B^{2} - 4 A C$ :

$< 0$ elliptic (Laplace)
$= 0$ parabolic (heat)
$> 0$ hyperbolic (wave)

The trap here

“A PDE has a single type throughout its domain” — false

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

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