# Initial-value problem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

An initial-value problem adds function or derivative values at a specified point to a differential equation; candidate checks are distinct from existence and uniqueness.

mko-initial-value-problem · definition · v0.10.0



## Mathematical statement

An initial-value problem consists of an ordinary differential equation and initial data for the same unknown function at a specified point. A first-order problem commonly gives y(x0)=y0; a second-order problem typically needs y(x0)=y0 and y′(x0)=v0. A candidate solution must satisfy both the equation and the initial data.





## Representative formula (exact TeX)

y(a)=b





## Assumptions


- Exercises are restricted to integer-coefficient polynomial candidates, specified finite residual grids and explicit initial values.
- Finite-grid and initial-data agreement only audits a candidate; it does not establish existence, uniqueness or a general solution.





## Symbol scope


- y — unknown function
- a — initial point
- b — initial function value





## Intuition, definition, and misconceptions

Initial data are additional constraints. Writing them down alone guarantees neither a solution nor uniqueness.

Audit a polynomial candidate’s ODE residuals and initial data separately. Failure of either obligation prevents it from satisfying the specified initial-value problem.

Zero residuals at finitely many points are still not a functional identity on an entire interval.

Do not check initial values without the equation or treat zero sampled residuals as an interval-wide solution.





## Adopted derivation guide

[Initial-value problem: auditing four polynomial candidates](https://ocme.evemisslab.com/en/derivations/derivation-mko-initial-value-problem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Ordinary differential equation](https://ocme.evemisslab.com/en/mko/mko-ordinary-differential-equation/) — An existing ODE supplies the equation part of the initial-value problem.
- [Domain, codomain, and range](https://ocme.evemisslab.com/en/mko/mko-domain-codomain-range/) — The initial point must lie in the common domain of the candidate solution and equation.





## Verification boundary

This slice checks candidates and finite grids only; existence, uniqueness and general solution methods remain unproved.

not_formalized

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





## Next obligation

Establish existence/uniqueness conditions, maximal solution intervals and standard first-order solution methods.





## 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/initial-value-problem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-initial-value-problem/index.md)