DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Extensions, maximal solutions, maximal intervals, and global solutions

Definition

Let $x : I \to R^{n}$ and $y : J \to R^{n}$ solve the same IVP. The solution $y$ is an extension of $x$ when $I \subseteq J$ and $y ∣_{I} = x$ . It is a proper extension when $I ⊊ J$ .

A solution is maximal when it has no proper extension. Its domain is its maximal interval of existence. A solution is global when its domain is the entire time projection allowed by the ODE domain; for a vector field on $R \times R^{n}$ , this means domain $R$ .

Depends on

Used by

Every Picard–Lindelöf initial value problem has one maximal solution on an open interval Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Ch. 2 (standard reference, not scraped)
Jiri Lebl, Basic Analysis I, Section 6.3 (standard reference, not scraped)