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Is this true?

A locally unique solution exists for all time

No — it is false.

The counterexample

This one comes with the Notes pack.

See pricing

85 counterexamples are free — including most of the ones Part C leans on hardest.

The kind of mistake this is

Existence vs uniqueness

A theorem giving one was read as giving both.

Drill statements like this
ODE

Others that fail the same way

From Ordinary Differential Equations › Existence–uniqueness, Picard, Lipschitz

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