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Classify a second order PDE

Short answer

For u_xx + x² u_yy = 0, discriminant B²−4AC = −4x². Hence elliptic where x≠0 (negative), and parabolic on the line x=0 (zero).

Write down the classification rules for a general second order partial differential equation and use them to classify the equation u_xx + x²u_yy = 0.

A general second order PDE in two variables is A u_xx + B u_xy + C u_yy + (lower order terms) = 0. Its type depends on the sign of the discriminant B² − 4AC at the point considered:

ConditionTypeExample
B² − 4AC < 0EllipticLaplace equation
B² − 4AC = 0ParabolicHeat equation
B² − 4AC > 0HyperbolicWave equation

Given: u_xx + x² u_yy = 0. Here A = 1, B = 0, C = x².

B² − 4AC = 0 − 4x² = −4x²

  • For x ≠ 0, B² − 4AC < 0, so the equation is elliptic.
  • On the line x = 0, B² − 4AC = 0, so it is parabolic there (the equation degenerates to u_xx = 0).

Answer: elliptic everywhere except on the y-axis (x = 0), where it is parabolic.

✓ Answer checked by Claude Opus 5.5 (high effort)

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