Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-29
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  • 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.
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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′=x2 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
1.1given

The explicit solution x(t)=1/(1−t) solves x′=x2 for all t<1 and [given] cannot be extended through t=1, so its maximal interval is (−∞,1).

2.1F1L1step 1.1∎

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

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