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

Partial Differential EquationsODE, PDE & Applied Mathematicsfree

One discriminant decides everything — including equations that are elliptic in one region and hyperbolic in another.

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

Step 1 / 4The discriminant

For Auxx+Buxy+Cuyy+⋯=0Au_{xx} + Bu_{xy} + Cu_{yy} + \cdots = 0, compute B2−4ACB^2-4AC:

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