Partial differential equation

Mathematical statement

This slice fixes bivariate quadratic polynomial candidates u and linear constant-coefficient operators and computes first- or second-order residuals exactly. The second-order principal part is a*u_xx+b*u_xy+c*u_yy, with discriminant b²−4*a*c. This convention must be explicit.

Representative formula (exact TeX)

L(u)=g

Assumptions

• Exercises handle only symbolic polynomial derivatives, point residuals, principal-part classification and polynomial identities.
• A zero point residual proves only that point check. Only an identically zero symbolic residual polynomial proves the specified polynomial satisfies the operator equation globally.

Symbol scope

• L — linear PDE operator
• u — candidate unknown function

Intuition, definition, and misconceptions

Operator coefficients and the mixed-term convention determine the classification formula.

A point residual is not a proof of a global solution.

A successful candidate check does not automatically supply a general solution, existence, uniqueness or well-posedness.

Do not infer a global solution from zero sampled residuals or mix the discriminants for b*u_xy and 2*b*u_xy conventions.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Partial derivative — PDE operators use first or higher partial derivatives.
• Equation — A PDE is an equation involving functions and partials.
• Differentiating polynomial sums, products, and compositions — Polynomial partials follow from derivative rules.

Verification boundary

Only the specified linear constant-coefficient polynomial slice is checked; general solution theory is not handled.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add boundary-value problems, weak solutions and well-posedness.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.