3 Maximal Solutions, Comparison and Dependence on Data

The Picard–Lindelöf theorem is local: it produces a solution on a short interval whose length depends on bounds for \(f\) near the initial point. Example 2.3.2 showed how conservative that interval can be. This section asks how far solutions really extend, shows that they can fail to extend only by leaving every compact set, typically by blowing up, gives conditions that rule this out, develops comparison theorems that bound solutions without solving for them, and shows that solutions depend differentiably on their initial values.

Throughout, \(f\) is continuous on an open set \(D \subseteq \mathbb R \times \mathbb R^n\) and locally Lipschitz in \(x\), so that the Picard–Lindelöf theorem applies near every point of \(D\).

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