TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05

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Constant-coefficient second-order equations in two variables have canonical principal forms

Statement

Let

$A u_{x x} + 2 B u_{x y} + C u_{y y} + lower-order terms = 0$

have constant real coefficients in its principal part.

If $B^{2} - A C < 0$ , an invertible linear change of variables and multiplication by a nonzero scalar reduce the principal part to $u_{ξ ξ} + u_{η η}$ .

If $B^{2} - A C > 0$ , such a change reduces it to $u_{ξ ξ} - u_{η η}$ .

If $B^{2} - A C = 0$ but $(A, B, C) \neq (0, 0, 0)$ , such a change reduces it to $u_{ξ ξ}$ .

Facts & Assumptions

Given: The constant symmetric matrix $M = (\begin{array}{cc} A & B \\ B & C \end{array})$ of the principal quadratic form.

[L1]

A symmetric second-order principal part has a signature normal form whose signature is coordinate invariant (A symmetric second-order principal part has a coordinate-invariant signature normal form).

[L2]

In two variables the sign of $B^{2} - A C$ is coordinate invariant (In two variables, type and characteristic directions are coordinate invariant).

[L3]

The discriminant is $B^{2} - A C$ (The discriminant for a second-order equation in two variables).

Proof

technique · direct

1.1

By [L1], an invertible linear change of variables diagonalizes the constant principal matrix to $diag (1, 1)$ , $diag (1, - 1)$ , $diag (1, 0)$ , or the negative of one of these matrices, and multiplying the equation by a nonzero scalar removes the global sign.

2.1

By [L3], these three normal forms have discriminants $- 1$ , $1$ , and $0$ , and [L2] says that the sign of the discriminant is the invariant datum distinguishing them; therefore the principal part reduces to the Laplace, wave, or rank-one parabolic form listed in the statement.

L2L3step 1.1∎

Depends on

Used by

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Sources

Victor Ivrii, Partial Differential Equations (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (standard reference, not scraped)