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Part CCSIR NET June 2023bounded-rhs-global-existence

Bounded rhs global existence

Let be bounded. Consider the initial-value problem (P): x′(t) = f(x(t)), t > 0, x(0) = 0. Which of the following statements are true?

  1. A.(P) has solution(s) defined for all t > 0.✓
  2. B.(P) has a unique solution.✓
  3. C.(P) has infinitely many solutions.
  4. D.The solution(s) of (P) is/are Lipschitz.✓

Solution

locally Lipschitz ⇒ unique local solution (Picard). |x′| ≤ sup|f| the solution cannot blow up, so it is global, and it is Lipschitz with constant sup|f|.

The trap it tests

Existence vs uniqueness

A theorem giving one was read as giving both.

Drill statements like this

Related counterexample: Every IVP has a unique solution

More on this topic

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

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