TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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The fundamental theorem for nonautonomous smooth ODEs

Statement

Let $D \subseteq R \times R^{n}$ be open and let $F : D \to R^{n}$ be smooth. For every base point $(t_{0}, x_{0}) \in D$ there are $h > 0$ and an open neighbourhood $W$ of $(t_{0}, x_{0})$ such that each initial pair $(s, y) \in W$ has a unique solution on $[s - h, s + h]$ , and the resulting local solution map is smooth in the initial time and initial state.

Facts & Assumptions

Given: A smooth nonautonomous field $F (t, x)$ and a base point $(t_{0}, x_{0}) \in D$ .

[L1]

Autonomous smooth ODEs have unique local smooth flows (The fundamental theorem for autonomous smooth ODEs).

[L2]

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

Proof

technique · direct

1.1

Introduce an extra variable $s$ and consider the autonomous system below on $R \times R^{n}$ .

$s^{'} (t) = 1, x^{'} (t) = F (s (t), x (t)) .$

Its right-hand side is smooth. By [L1], this augmented autonomous system has a unique local smooth flow.

2.1

The first equation forces $s (t) = s_{0} + t$ when $s (0) = s_{0}$ , so the second equation becomes exactly the original nonautonomous system with initial time $s_{0}$ . Reading the initial value $(s_{0}, x_{0})$ as a parameter, [L2] makes the resulting solution map smooth in $(s_{0}, x_{0})$ . This is precisely the claimed local theorem for nonautonomous smooth ODEs.

L2step 1.1∎

Depends on

Used by

A nonautonomous equation made autonomous by adjoining time Example

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Sources

Nigel Hitchin, Differentiable Manifolds, Appendix §10.3 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.3 (standard reference, not scraped)