ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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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.

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Smooth dependence in an ODE with a parameter

Example

For the parameter-dependent ODE

$x^{'} (t) = λ x (t), x (0) = x_{0},$

the solution is

$x (t; λ, x_{0}) = e^{λ t} x_{0} .$

It depends smoothly on both the initial value $x_{0}$ and the parameter $λ$ .

Facts & Assumptions

Given: The parameter-dependent scalar ODE $x^{'} = λ x$ , $x (0) = x_{0}$ .

[L1]

Smooth parameter-dependent ODEs depend smoothly on the parameter (Smooth dependence of ODE solutions on parameters).

[L2]

Smooth autonomous ODEs depend smoothly on initial data (Smooth dependence of solutions on initial data).

Verification

technique · direct

1.1given

The curve $x (t) = e^{λ t} x_{0}$ satisfies $x (0) = x_{0}$ and [given] $x^{'} (t) = λ e^{λ t} x_{0} = λ x (t)$ , so it solves the ODE.

2.1L1L2step 1.1

Differentiating the explicit formula gives [L1, L2, step 1.1] $\partial_{x_{0}} x (t; λ, x_{0}) = e^{λ t}$ and $\partial_{λ} x (t; λ, x_{0}) = t e^{λ t} x_{0}$ , and higher derivatives are again polynomial multiples of $e^{λ t}$ . Thus the solution depends smoothly on both data variables, exactly as [L1] and [L2] predict.

3.1step 2.1∎

So this ODE is a concrete instance of smooth dependence on initial data and [step 2.1] parameters.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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 §10.3 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.4 (standard reference, not scraped)