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Part BCSIR NET December 2023dalembert-limit

Dalembert limit

Consider the Cauchy problem for the wave equation utt−4uxx=0u_tt - 4u_{xx} = 0 on R×(0,∞)\mathbb{R} \times (0, \infty), with u(x, 0) = e−1/x2e^{-1/x^{2}} for x ≠ 0 (and 0 at x = 0), and u_t(x, 0) = x e−x2e^{-x^{2}}. Which one of the following is true?

  1. A.limt→∞lim_{t\to\infty} u(5, t) = 1✓ Correct
  2. B.limt→∞lim_{t\to\infty} u(5, t) = 2
  3. C.limt→∞lim_{t\to\infty} u(5, t) = 1/2
  4. D.limt→∞lim_{t\to\infty} u(5, t) = 0

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Drill statements like this

Related counterexample: All the classical equations satisfy a maximum principle

More on this topic

The chapter behind this: The three classical equations — free to read

From Partial Differential Equations › Laplace, heat and wave equations: separation of variables

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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