False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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A maximal ODE solution need not have a closed interval domain

Statement

False claim: a maximal solution of an ODE has a closed interval as its domain of definition.

Facts & Assumptions

Given: The solution $x (t) = 1 / (1 - t)$ of $x^{'} = x^{2}$ with $x (0) = 1$ .

[L1]

Every IVP has a unique maximal solution on an open interval (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval).

[F1]

The maximal solution domain is open in the time-state variables (The maximal solution domain is open).

Refutation

technique · direct

1.1

The explicit solution $x (t) = 1 / (1 - t)$ solves $x^{'} = x^{2}$ for all $t < 1$ and [given] cannot be extended through $t = 1$ , so its maximal interval is $(- \infty, 1)$ .

given

2.1

This domain is open and not closed. That matches [L1] and [F1], and it [F1, L1, step 1.1] directly refutes the false claim.

F1L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Nigel Hitchin, Differentiable Manifolds, Appendix §10.2 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.2 (standard reference, not scraped)