Ordinary differential equation

Mathematical statement

On a specified interval, an ordinary differential equation relates an unknown single-variable function y and its finite-order derivatives. Its order is the highest derivative order occurring in the equation. A zero substitution residual checks only whether a given candidate satisfies the relation over the specified test range.

Representative formula (exact TeX)

F(y,D(y))=0

Assumptions

• The unknown is a real-valued function of one real variable; the derivatives and coefficients used are explicit.
• Exercises are restricted to integer-coefficient polynomial candidates and exact residuals at specified points. Existence, uniqueness and global extension are not inferred from the equation alone.

Symbol scope

• F — differential relation
• y — unknown function

Intuition, definition, and misconceptions

ODE order comes from the highest derivative actually present, not the polynomial candidate’s degree or the number of equation terms.

Recompute y, y′, y″ and the stated equation when checking a candidate. A zero residual at one point is generally insufficient for a solution on an interval.

Initial data can narrow candidate solutions, but existence and uniqueness require appropriate additional hypotheses, such as relevant continuity or Lipschitz conditions, and their corresponding theorems.

Do not turn success at one point into an interval-wide solution, or infer a unique solution merely from an ODE and initial data.

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

• Real derivatives at a point and finite difference quotients — Derivatives express change in differential equations.
• Function mapping — The unknown is a function on its domain.
• Equation — A solution must satisfy the differential relation.

Verification boundary

This slice checks derivative-order annotations, point residuals and initial conditions for polynomial candidates; it proves no existence/uniqueness theorem.

not_formalized

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

Next obligation

Develop initial-value problems, solution intervals, existence/uniqueness hypotheses and standard first-order equation classes.

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.